oneill_2003_phd

Full-waveform Reflectivity for

Modelling, Inversion and Appraisal of

Seismic Surface Wave Dispersion in

Shallow Site Investigations

Adam O’Neill

This thesis is presented for the degree of

Doctor of Philosophy

of The University of Western Australia

School of Earth and Geographical Sciences.

December, 2003

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Abstract

Surface wave inversion has become very popular in recent years for shallow nonde-

structive testing of both layered natural (soil and rock) and artificial (concrete and road

base) structures, both onshore and offshore. This research investigates and successfully

overcomes the issues of dispersion repeatability and dominant higher modes associated

with the inversion of surface wave (‘ground-roll’) dispersion measured from multichannel

shot gathers in shallow (<30 m) investigations. The contribution of this research is the

rapid inversion of the full surface wavefield with realistic dispersion errors, to a layered

shear velocity structure, in both synthetic and field data. Combined with objective model

appraisal methods, the limitations of surface wave inversion for engineering, environmental

and exploration applications are quantitatively revealed.

The kernel of this research is a forward dispersion modelling method based on full-

waveform P -SV reflectivity synthetic seismograms. This method accounts for all wave-

fields, in both the near- and far-fields, and field acquisition parameters are incorporated,

such as geophone offsets, source types and wavefield component. Through standard plane-

wave transform of the synthetic shot gathers, the process for calculating the theoretical

dispersion mimics that of the field dispersion. This removes any systematic errors with

mode identification which is common when plane-wave modelling methods are applied in

the inversion. The algorithm is verified by comparison with publications concerned with

dominant higher modes in dispersion curves, and, attempts to filter the data to suppress

higher modes are ill-advised.

Three aspects of surface wave inversion are addressed:

1. Dispersion repeatability;

2. Full surface-wavefield inversion; and

3. Model appraisal;

each synthetically tested and practised on field data. The synthetic models comprise nor-

mally and irregularly dispersive structures, the latter including both low and high velocity

layers (LVL/HVL). These models generate dominant higher modes, due to large elastic

constant contrasts and reversals, and extend to 14 m depth. The field sites investigated

were generally within this depth range.

Dispersion repeatability evaluation tests reveal that the maximum measurable wave-

length is 0.4 that of the spread length, so called ‘near-field’ effects only manifest to about

0.1 of surface wavelength and that the frequency-slowness transform is preferable to the

frequency-wavenumber transform for multichannel dispersion observation, except in in-

versely dispersive sites. The most influential sources of noise in measured dispersion are

additive Gaussian white noise exceeding 2.5%, trace-to-trace static shifts exceeding 1 ms

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and geophone positioning errors. Geophone coupling and tilt errors induced dispersion

errors an order of magnitude less. Source depth is a factor when explosive sources are

used, but only due to the preferential generation of dominant higher modes with small

increments in depth, even in normally dispersive cases. On average, when error sources

are considered individually, at high frequency the dispersion repeats to about 1% and is

Gaussian distributed. However, at low frequency, uncertainty increases nonlinearly and

mimics the theoretical slowness resolution with frequency, which is dependent on spread

length, with a Lorentzian distribution of measured phase velocities. When all error sources

are combined, along with near offset, channel density and spread length variations, the

dispersion scatter sums as a Lorentzian distribution for all frequencies, thus, theoretical

Gaussian error propagation is not valid.

Field repeatability tests confirm the numerical error tests with a large number of re-

dundant data. All common shot, common midpoint and walkaway gathers show large

errors in dispersion at low frequency, which increases nonlinearly with a Lorentzian distri-

bution. However, a Gaussian distribution and approximate 1% error threshold at higher

frequencies is observed. Artificially increasing channel numbers by shot walkaway is ac-

ceptable with dispersion scatter retaining a Gaussian distribution. Dispersion from CMP

gathers is similar to that of common shot gathers over coincident geophone positions. In

all cases, synthetic and field, the dominant factor controlling accurate measurement of low

frequency dispersion is longer spread length, with larger uncertainty around the frequen-

cies of modal transitions. These are the basis for a nonlinear noise model for dispersion of

real data.

Inversion of dominant higher modes employs the full-waveform reflectivity forward

modelling kernel with a damped linearised optimisation (Occam’s inversion) solving for

the layer shear velocity only. This removes the need for modal identification and allowing

analytic partial derivatives of the ‘effective’ phase velocity to be employed for efficiency.

The use of plane-wave modelling methods results in a LVL not being detected and HVL

being interpreted as deeper and thicker than true. In general, the assumed layering has

the largest influence on the RMS accuracy of the final shear velocity structure where thin

layers generate model artifacts. Poisson’s ratio is required to be estimated to within about

50% but density has no influence. The starting model shear velocity is best overestimated

in irregularly dispersive cases with realistic dispersion errors to avoid convergence to local

minima. Realistic dispersion errors also cause misinterpretation of deeper layers. A LVL

is better resolved if directly under caprock, while soft layers under a buried HVL have

very poor resolution, as does the homogenous half-space shear velocity.

Field data inversion was conducted for geotechnical (Telfer gold mine), civil (Perth

Convention Centre site and highway road cutting) and earthquake (Hyden fault scarp)

engineering applications, in highly irregularly dispersive layering. At the Telfer site, the

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observed dispersion curve exhibited strong dominant higher modes, very similar to syn-

thetic LVL and HVL cases, the dispersion discontinuities properly modelled with the

reflectivity method. Shallow trenching to remove overburden and plant geophones on the

horizon of interest causes scattering and non-interpretable dispersion. Normally dispersive

artificial backfill is best surveyed with short spreads to prevent reflections from containing

walls. At the Perth Convention Centre site, the target 5 m thick sandy layer at 20 m

depth (known from borehole methods) is not interpreted from the walkaway surface wave

dispersion. Synthetic modelling of a borehole model shows this is the case with both 55

m and 69 m spread lengths. However, the shallow zones correlated well with the seismic

cone penetrometer trend. At the road cutting site, the thickness and strength of a surficial

laterite caprock for rippability potential correlated well with geological expectation, where

direct arrivals and refractions were not recognised. The inline component better records

direct arrivals for more accurate Poisson’s ratio evaluation. At the Hyden site, a thick,

stiff layer at shallow depth is interpreted, which correlates with Ground Penetrating Radar

profiles. The rollalong, 48-fold seismic data proved useful to stack dispersion curves from

several near offsets and average surface wavefield scattering at frequencies above 50 Hz.

Model appraisal by deterministic and Monte Carlo unconstrained data likelihood, in-

corporating dominant higher modes in a linearised inversion framework, shows that the

shear velocity and depth of a LVL under caprock is well resolved, with a well-defined

global minimum. The shear velocity of a buried HVL shows multiple local minima, but

the solution at the global minimum is well resolved. With more realistic dispersion errors,

both LVL and HVL thicknesses are poorly constrained, least so a LVL under a shallow

HVL. The distribution of inverted shear wave velocities is generally asymmetrical and/or

bimodal, with best fitting models showing lower shear velocity than the true values at

depth. However, a LVL under a shallow HVL shows preference to higher shear velocity

and smoothing between the homogenous half-space and overlying layer shear velocities oc-

curs in all cases. A stack of constant thickness layers generally offers the best compromise

between true and assumed structures. Field models all show non-Gaussian error propaga-

tion. Shear velocity of basement under thick overburden shows the largest shear velocity

nonuniqueness, but a shallow HVL is well resolved in depth and shear velocity. A 5 m

thick, stiff layer at 20 m depth cannot be detected by surface wave dispersion inversion.

Surface waveform phase matching is excellent in homogenous compacted fill at all offsets,

in irregularly dispersive sites best at near offsets and in inversely dispersive caprock best

at far offsets. Group velocity envelope matching is good in all cases.

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Contents

Abstract iii

List of Tables xv

List of Figures xvii

Acknowledgements xxix

Synopsis xxxiii

1 Introduction 1

1.1 Historical, application and methodology overview . . . . . . . . . . . . . . . 1

1.1.1 Earthquake seismology . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Engineering applications . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Exploration geophysics . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.4 Application overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.5 General procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Surface wave properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Guided wave overview . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.2 Elastic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.3 Rayleigh wave theory . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.4 Phase and group velocity . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.5 Wave dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.6 Higher modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.7 Spherical spreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.8 Intrinsic attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Dispersion observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.1 Large-scale seismology . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.2 Small-scale seismology . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Dispersion modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.1 Matrix methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.2 Propagator theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4.3 Numerical considerations . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4.4 Published computer codes . . . . . . . . . . . . . . . . . . . . . . . . 22

1.5 Inversion to Earth models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.5.1 Dispersion inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.5.2 Inverse optimisation methods . . . . . . . . . . . . . . . . . . . . . . 23

1.5.3 Alternative inversion methods . . . . . . . . . . . . . . . . . . . . . . 25

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2 Outline and methods of thesis 29

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Synthetic methods, models and verification . . . . . . . . . . . . . . . . . . 31

2.2.1 Synthetic seismogram generation . . . . . . . . . . . . . . . . . . . . 31

2.2.2 Source functions and numerical aspects . . . . . . . . . . . . . . . . 32

2.2.3 Dispersion observation . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.4 Layered models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.5 Numerical procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.6 Algorithm verification . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2.7 Effective phase velocity . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.3 Dominant higher modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.3.1 Experimental observations . . . . . . . . . . . . . . . . . . . . . . . . 51

2.3.2 Theoretical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.4 Low-frequency discrepancies . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.4.1 Near-field effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.4.2 Spread and pixel resolution . . . . . . . . . . . . . . . . . . . . . . . 61

2.4.3 Higher-mode contamination . . . . . . . . . . . . . . . . . . . . . . . 62

2.4.4 ‘Low-frequency effects’ . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.5 Data resolution and accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.5.1 Dispersion processing . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.5.2 Source and receiver layouts . . . . . . . . . . . . . . . . . . . . . . . 68

2.5.3 Individual acquisition errors . . . . . . . . . . . . . . . . . . . . . . . 69

2.5.4 Dispersion repeatability and robustness . . . . . . . . . . . . . . . . 72

2.6 Dominant higher mode inversion . . . . . . . . . . . . . . . . . . . . . . . . 74

2.6.1 Existing inversion methods . . . . . . . . . . . . . . . . . . . . . . . 74

2.6.2 A new inversion method . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.7 Model resolution and accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.7.1 Physical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.7.2 Intrusive tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.7.3 Alternative geophysical datasets . . . . . . . . . . . . . . . . . . . . 82

2.7.4 Inference methods for model appraisal . . . . . . . . . . . . . . . . . 83

3 Dispersion resolution and accuracy: Synthetic testing 87

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.2 Dispersion processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.2.1 Spread layouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.2.2 Spread length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.2.3 Trace padding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.2.4 Spatial aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

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3.2.5 Discussion and comparisons with previous work . . . . . . . . . . . . 99

3.3 Spread layouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.3.1 Homogenous half-space . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.3.2 Synthetic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.3.3 Case 1 near offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.3.4 Case 2 near offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.3.5 Case 3 near offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.3.6 Near-offset error distributions . . . . . . . . . . . . . . . . . . . . . . 114


3.4 Individual acquisition errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.4.1 Geophone positioning . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.4.2 Geophone tilt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.4.3 Geophone coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.4.4 Source parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.4.5 Equipment errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3.4.6 Additive noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3.4.7 Methods for acquisition error testing . . . . . . . . . . . . . . . . . . 121

3.4.8 Case 1 errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3.4.9 Source depth influence . . . . . . . . . . . . . . . . . . . . . . . . . . 130

3.5 Repeatability and model resolution . . . . . . . . . . . . . . . . . . . . . . . 133

3.5.1 Acquisition errors and processing parameters . . . . . . . . . . . . . 133

3.5.2 Case 1 repeatability . . . . . . . . . . . . . . . . . . . . . . . . . . . 133



3.5.5 Repeatability statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 141

3.5.6 Effective dispersion power . . . . . . . . . . . . . . . . . . . . . . . . 146

3.5.7 Partial derivative analysis . . . . . . . . . . . . . . . . . . . . . . . . 147


3.6 Conclusions for this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

3.6.1 Fundamental limitations . . . . . . . . . . . . . . . . . . . . . . . . . 154

3.6.2 Near offset and spread layout . . . . . . . . . . . . . . . . . . . . . . 154

3.6.3 Individual acquisition errors . . . . . . . . . . . . . . . . . . . . . . . 154

3.6.4 Repeatability and sensitivity . . . . . . . . . . . . . . . . . . . . . . 155

3.6.5 Noise model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

3.6.6 SASW perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

4 Dispersion repeatability: Field tests 159

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

4.2 Hyden fault scarp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

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4.2.1 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

4.2.2 Test data and source analysis . . . . . . . . . . . . . . . . . . . . . . 162

4.2.3 Fixed-offset variations . . . . . . . . . . . . . . . . . . . . . . . . . . 165

4.2.4 Near-offset variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.2.5 Common-shot gather versus common-midpoint gather . . . . . . . . 172

4.3 Perth Convention Centre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.3.1 Acquisition equipment . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.3.2 Walkaway data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.3.3 Observed dispersion curves . . . . . . . . . . . . . . . . . . . . . . . 174

4.3.4 Numerical error estimation . . . . . . . . . . . . . . . . . . . . . . . 181

4.3.5 Partial derivative analysis . . . . . . . . . . . . . . . . . . . . . . . . 182


4.4.1 Hyden fault scarp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

4.4.2 Perth Convention Centre . . . . . . . . . . . . . . . . . . . . . . . . 183

5 Dispersion inversion: Methods and synthetic tests 185

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

5.2 Inverse theory review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

5.2.1 Ill-posed problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

5.2.2 Linear problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

5.2.3 Linearised problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

5.2.4 Regularisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

5.3 Proposed inversion procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 190

5.3.1 Data and noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

5.3.2 Regularised optimisation algorithm . . . . . . . . . . . . . . . . . . . 191

5.3.3 Model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

5.3.4 Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

5.3.5 Statistical analysis measures . . . . . . . . . . . . . . . . . . . . . . 196

5.3.6 Convergence misfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

5.4 Numerical summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

5.4.1 Layer parameterisation . . . . . . . . . . . . . . . . . . . . . . . . . 200

5.4.2 Damping parameter range . . . . . . . . . . . . . . . . . . . . . . . . 203

5.4.3 Experimental dispersion curves . . . . . . . . . . . . . . . . . . . . . 203

5.4.4 Inversion parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

5.4.5 Some concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

5.4.6 Issues to be tested . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

5.5 Case 1 inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

5.5.1 Effects of inversion kernel and dispersion uncertainty . . . . . . . . . 208

5.5.2 Effects of assumed model parameters and layer interfaces . . . . . . 213

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5.6 Case 2 inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223



5.7 Case 3 inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235



5.8 Other layer thickness settings . . . . . . . . . . . . . . . . . . . . . . . . . . 246


5.9.1 Modelling kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

5.9.2 Dispersion errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

5.9.3 Poisson’s ratio (σ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

5.9.4 Starting model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

5.9.5 Layer interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

6 Dispersion inversion: Field applications and pitfalls 253

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

6.2 Site locations and significance . . . . . . . . . . . . . . . . . . . . . . . . . . 254

6.2.1 Telfer gold mine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

6.2.2 Perth Convention Centre . . . . . . . . . . . . . . . . . . . . . . . . 256

6.2.3 Road cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

6.2.4 Hyden fault scarp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

6.3 Telfer gold mine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260


6.3.2 Data quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260


6.3.4 Inversion models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272





6.4.4 Resolution limitations . . . . . . . . . . . . . . . . . . . . . . . . . . 291

6.5 Road cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293








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6.6.4 Seismic reflection profiles . . . . . . . . . . . . . . . . . . . . . . . . 313

6.7 Pitfalls in dominant higher mode inversion . . . . . . . . . . . . . . . . . . . 316

6.7.1 ‘Apparent’ guided waves . . . . . . . . . . . . . . . . . . . . . . . . . 316

6.7.2 Lamb waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

6.7.3 Oscillating mode transitions . . . . . . . . . . . . . . . . . . . . . . . 324


6.8.1 Successful inversions . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

6.8.2 Inversion pitfalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

7 Model parameter resolution and appraisal 329

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

7.2 Numerical methods for model inference . . . . . . . . . . . . . . . . . . . . 330

7.2.1 Monte Carlo model generation . . . . . . . . . . . . . . . . . . . . . 330

7.3 Synthetic model resolution and appraisal . . . . . . . . . . . . . . . . . . . . 334

7.3.1 Deterministic parameter resolution . . . . . . . . . . . . . . . . . . . 334

7.3.2 Monte Carlo model resolution . . . . . . . . . . . . . . . . . . . . . . 342

7.3.3 Monte Carlo parameter statistics . . . . . . . . . . . . . . . . . . . . 347

7.4 Telfer gold mine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

7.4.1 High velocity layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

7.4.2 Low velocity layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

7.4.3 Waveform matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 351


7.5.1 Waveform matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

7.5.2 Hard layer resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

7.6 Road cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

7.6.1 Granitic overburden . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

7.6.2 Laterite caprock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365


7.7.1 Common shot data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

7.7.2 Rollalong data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375


7.8.1 Synthetic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

7.8.2 Field models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

8 Conclusions and recommendations 379

8.1 Conclusions from this work . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

8.1.1 Dominant higher mode simulation and inversion . . . . . . . . . . . 379

8.1.2 Realistic dispersion curve uncertainty . . . . . . . . . . . . . . . . . 381

8.1.3 Model resolution and accuracy appraisal . . . . . . . . . . . . . . . . 382

xiii

8.2 Recommendations for further work . . . . . . . . . . . . . . . . . . . . . . . 384

8.2.1 Lateral discontinuity effects . . . . . . . . . . . . . . . . . . . . . . . 384

8.2.2 Model appraisal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

8.2.3 Higher mode and coupled inversion . . . . . . . . . . . . . . . . . . . 385

8.2.4 Alternative optimisations . . . . . . . . . . . . . . . . . . . . . . . . 385

8.2.5 Waveform inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

8.2.6 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

8.3 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

A Glossary 389

A.1 Selected terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

A.2 Mathematical symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

Bibliography 391

xiv

xv

List of Tables

1.1 Range of surface wave applications . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Comparison of seismic and invasive tests . . . . . . . . . . . . . . . . . . . . 5

1.3 Limitations of seismic surface wave surveying . . . . . . . . . . . . . . . . . 5

1.4 Guided wave types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Elastic parameter definitions at small strain . . . . . . . . . . . . . . . . . . 9

1.6 Earthquake dispersion methods . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 Typical 1D synthetic models . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2 Tokimatsu model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3 Higher mode literature examples . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4 Surface wave field tests correlated to borehole logs . . . . . . . . . . . . . . 79



3.1 Dispersion accuracy RMS values with fixed trace spacing . . . . . . . . . . 92

3.2 Dispersion accuracy RMS values with fixed spread length . . . . . . . . . . 92

3.3 Trace padding effects on RMS accuracy values . . . . . . . . . . . . . . . . 97

3.4 Acquisition parameter error ranges for synthetic seismograms . . . . . . . . 134

3.5 Seismogram and dispersion processing parameters . . . . . . . . . . . . . . 135

5.1 Recent full-simulation inversion methods . . . . . . . . . . . . . . . . . . . . 190

6.1 Field sites investigated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

6.2 Summary of borehole logs near the Perth Convention Centre surface wave

survey spread centre. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

6.3 Summary of surface wave spreads locations at Telfer gold mine . . . . . . . 261

6.4 Summary of surface wave recording and processing parameters at Telfer

gold mine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

6.5 Model parameters for Bietigheim . . . . . . . . . . . . . . . . . . . . . . . . 318

xvi

xvii

List of Figures

1.1 Typical land seismic wavefields . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Rayleigh wave motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Phase and group velocity of superposed waves . . . . . . . . . . . . . . . . . 11

1.4 Rayleigh wave dispersion curves . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 CSMP geometry in SASW testing . . . . . . . . . . . . . . . . . . . . . . . 17

1.6 CRMP geometry in SASW testing . . . . . . . . . . . . . . . . . . . . . . . 17

1.7 Geometry in MASW testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1 Analytic source functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2 Shear velocity of the Tokimatsu et. al. models . . . . . . . . . . . . . . . . . 37

2.3 Synthetic shot gathers of the Tokimatsu et. al. models . . . . . . . . . . . . 37

2.4 Frequency-slowness transforms of the Tokimatsu et. al. model shot gathers 38

2.5 Frequency-wavenumber transforms of the Tokimatsu et. al. shot gathers . . 38

2.6 Phase velocity dispersion of the Tokimatsu et. al. plane-wave transforms . 39

2.7 Approximate inversion of the Tokimatsu et. al. dispersion . . . . . . . . . . 39

2.8 Poisson’s ratio effect on half-space phase velocity . . . . . . . . . . . . . . . 41

2.9 Models for forward dispersion calculations . . . . . . . . . . . . . . . . . . . 42

2.10 Models for dispersion inversion methods . . . . . . . . . . . . . . . . . . . . 42

2.11 Cases 2 and 3 dispersion of Tokimatsu et. al. . . . . . . . . . . . . . . . . . 44

2.12 Site B and C dispersion of Roesset et. al. . . . . . . . . . . . . . . . . . . . 44

2.13 Case 3 and 4 dispersion of Gucunski and Woods . . . . . . . . . . . . . . . 44

2.14 Case 4 and 5 dispersion of Ganji et. al. . . . . . . . . . . . . . . . . . . . . . 45

2.15 Case 2 and 3 dispersion of Lai and Rix . . . . . . . . . . . . . . . . . . . . . 45

2.16 Berkheim and Bietigheim site dispersion of Forbriger . . . . . . . . . . . . . 45

2.17 Two theoretical dispersion cases of Sezawa and Kanai . . . . . . . . . . . . 46

2.18 Comparison with Al-Hunaidi dispersion . . . . . . . . . . . . . . . . . . . . 46

2.19 Plane-wave filtering Case 2 of Tokimatsu et. al. . . . . . . . . . . . . . . . . 49

2.20 Filtered dispersion of Cases 2 and 3 of Tokimatsu et. al. . . . . . . . . . . . 49

2.21 Spurious SASW phase lags associated with dominant higher modes . . . . . 52

2.22 Field example of a dominant higher mode observed by MASW . . . . . . . 54

2.23 Theoretical dispersion curve branches for flexural and Rayleigh modes . . . 57

2.24 Modelled dominant higher modes observed by MASW . . . . . . . . . . . . 58

2.25 Homogenous half-space dispersion curves . . . . . . . . . . . . . . . . . . . . 63

2.26 Homogenous half-space dispersion at near and far shot offsets . . . . . . . . 63

2.27 Vertical and horizontal dispersion of Tokimatsu et. al.Case 1 . . . . . . . . 64

3.1 100 m/s half-space dispersion accuracy with frequency . . . . . . . . . . . . 90

xviii

3.2 500 m/s half-space dispersion accuracy with frequency . . . . . . . . . . . . 90

3.3 500-100 m/s normal dispersion accuracy with frequency . . . . . . . . . . . 91

3.4 100-500 m/s inverse dispersion accuracy with frequency . . . . . . . . . . . 91

3.5 Resolution limitations with spread length . . . . . . . . . . . . . . . . . . . 94

3.6 Trace padding effects on dispersion accuracy for a 100 m/s half-space . . . 95

3.7 Trace padding effects on dispersion accuracy for a 500 m/s half-space . . . 95

3.8 Trace padding effects on accuracy for a normal dispersion curve . . . . . . . 96

3.9 Trace padding effects on accuracy for an inverse dispersion curve . . . . . . 96

3.10 Projection of a normal dispersion curve onto the f − k transform . . . . . . 97

3.11 Low-frequency accuracy of normal dispersion curves . . . . . . . . . . . . . 98

3.12 Low-frequency accuracy of normal dispersion curves at higher padding . . . 98

3.13 Wavelength resolution limitations with spread length . . . . . . . . . . . . . 102

3.14 Homogenous half-space f − p dispersion curve variation with near offset . . 105

3.15 Homogenous half-space f − p dispersion value with near offset . . . . . . . . 106

3.16 Homogenous half-space f − p dispersion error analysis . . . . . . . . . . . . 106

3.17 Case 1 f − k dispersion variation with near offset . . . . . . . . . . . . . . . 108

3.18 Case 1 f − p dispersion variation with near offset . . . . . . . . . . . . . . . 108

3.19 Case 1, 96-channel dispersion at low frequency . . . . . . . . . . . . . . . . 109

3.20 Case 1 near-offset dispersion error analysis . . . . . . . . . . . . . . . . . . . 109

3.21 Case 2 f − k dispersion variation with near offset . . . . . . . . . . . . . . . 110








3.29 Homogenous half-space, near offset error distribution . . . . . . . . . . . . . 115

3.30 Case 1, near offset error distribution . . . . . . . . . . . . . . . . . . . . . . 115



3.33 Representative responses due to variable geophone coupling . . . . . . . . . 120

3.34 Case 1 geophone positioning and tilt error tests . . . . . . . . . . . . . . . . 124

3.35 Case 1 geophone positioning error distribution . . . . . . . . . . . . . . . . 124

3.36 Case 1 geophone coupling and static error tests . . . . . . . . . . . . . . . . 125

3.37 Case 1 source parameter and depth effects . . . . . . . . . . . . . . . . . . . 125

3.38 Case 1 source depth range dispersion . . . . . . . . . . . . . . . . . . . . . . 126

3.39 Case 1 additive noise tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

xix

3.40 Case 1 additive noise error analysis . . . . . . . . . . . . . . . . . . . . . . . 127

3.41 Ranking of error influences for Case 1 . . . . . . . . . . . . . . . . . . . . . 127

3.42 Case 2 source type and depth range dispersion . . . . . . . . . . . . . . . . 128

3.43 Case 3 source type and depth range dispersion . . . . . . . . . . . . . . . . 128

3.44 Synthetic shot gathers of the Tokimatsu et. al. models with 1.4 m deep

explosive sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.45 Frequency-slowness transforms of the Tokimatsu et. al. shot gathers from

1.4 m deep explosive sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.46 Random spread layouts and positional errors . . . . . . . . . . . . . . . . . 135

3.47 Case 1 repeatability by f − k . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3.48 Case 1 repeatability by f − p . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3.49 Case 1 thresholded repeatability by f − p . . . . . . . . . . . . . . . . . . . 137

3.50 Case 2 repeatability by f − k . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3.51 Case 2 thresholded repeatability by f − k . . . . . . . . . . . . . . . . . . . 138


3.53 Case 3 repeatability by f − p . . . . . . . . . . . . . . . . . . . . . . . . . . 139



3.56 Case 1 repeatability distributions by f − p . . . . . . . . . . . . . . . . . . . 142

3.57 Case 1 thresholded repeatability distributions by f − p . . . . . . . . . . . . 142



3.60 Reduced χ2 fit of Gaussian distribution by f − p . . . . . . . . . . . . . . . 144

3.61 Dispersion power curves from synthetic dispersion error analysis by f − k . 145

3.62 Dispersion power curves from synthetic dispersion error analysis by τ − p . 145

3.63 Analytic partial derivative analysis of Case 1 . . . . . . . . . . . . . . . . . 148

3.64 Standardised partial derivative analysis of Case 1 . . . . . . . . . . . . . . . 148





3.69 Dispersion noise model compared to actual errors . . . . . . . . . . . . . . . 157

4.1 Photographs of source types used at the Hyden fault scarp . . . . . . . . . . 161

4.2 Vertical component common shot gathers from Hyden fault scarp . . . . . . 163

4.3 Radial component common shot gathers from Hyden fault scarp . . . . . . 163

4.4 Transverse component common shot gathers from Hyden fault scarp . . . . 164

4.5 Source type analysis at Hyden fault scarp . . . . . . . . . . . . . . . . . . . 164

4.6 Full spread dispersion at Hyden fault scarp by f − k . . . . . . . . . . . . . 166

xx

4.7 Full spread dispersion at Hyden fault scarp by f − p . . . . . . . . . . . . . 166

4.8 Full spread dispersion error analysis at Hyden fault scarp by f − p . . . . . 167

4.9 Near-field dispersion at Hyden fault scarp by f − p with 28 Hz geophones . 167

4.10 Far-field dispersion at Hyden fault scarp by f − k with 8 Hz geophones . . 168

4.11 All power curves of the dispersion at Hyden fault scarp by f − k . . . . . . 168

4.12 Dispersion variation with near offset at Hyden fault scarp by f − p . . . . . 170

4.13 Dispersion error with near offset analysis at Hyden fault scarp by f − p . . 170

4.14 Vertical component CMP gathers from Hyden fault scarp . . . . . . . . . . 171

4.15 CSG and CMP dispersion at Hyden fault scarp by f − p . . . . . . . . . . . 171

4.16 Schematic source and receiver locations for walkaway shooting at the Perth

Convention Centre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

4.17 Representative shot walkaway gathers from the Perth Convention Centre . . 175

4.18 Near offset dispersion analysis of 3 m geophone spacing gathers at Perth

Convention Centre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

4.19 Walkaway dispersion analysis with 1 m geophone spacing at the Perth Con-

vention Centre by τ − p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

4.20 Walkaway dispersion analysis with 3 m geophone spacing at Perth Conven-

tion Centre by τ − p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

4.21 Maximum power curves of the dispersion at the Perth Convention Centre

by τ − p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

4.22 Walkaway dispersion error analysis with 1 m geophone spacing at the Perth

Convention Centre by τ − p . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

4.23 Borehole logs at the Perth Convention Centre . . . . . . . . . . . . . . . . . 178

4.24 Combined dispersion errors of layered model based on SC2 model at the

Perth Convention Centre by τ − p with threshold . . . . . . . . . . . . . . . 179

4.25 Standardised partial derivative analysis of the SC2 model . . . . . . . . . . 179

5.1 Synthetic model layer shear velocity solution subspaces . . . . . . . . . . . . 198

5.2 Forward PSV dispersion flowchart . . . . . . . . . . . . . . . . . . . . . . . . 201

5.3 Occam’s inversion flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

5.4 Layering thickness parameterisations . . . . . . . . . . . . . . . . . . . . . . 202

5.5 Optimal damping parameter range . . . . . . . . . . . . . . . . . . . . . . . 203

5.6 Case 1 PSV inversion with 3% dispersion errors and true parameters . . . . 209

5.7 Case 1 FSW inversion with 3% dispersion errors and true parameters . . . 209

5.8 Case 1 PSV inversion with 3% dispersion errors and true parameters ne-

glecting low frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

5.9 Case 1 FSW inversion with 3% dispersion errors and true parameters ne-

glecting low frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

5.10 Case 1 PSV inversion with realistic dispersion errors and true parameters . 211

xxi

5.11 Case 1 FSW inversion with realistic dispersion errors and true parameters . 211

5.12 Case 1 PSV inversion with realistic dispersion errors and true parameters

neglecting low frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

5.13 Case 1 FSW inversion with realistic dispersion errors and true parameters

neglecting low frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

5.14 Case 1 PSV inversion with 3% dispersion errors and relative Poisson’s ratio

errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

5.15 Case 1 PSV inversion with realistic dispersion errors and relative Poisson’s

ratio errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

5.16 Case 1 PSV inversion with 3% dispersion errors and constant Poisson’s

ratio underestimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215


ratio overestimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

5.18 Case 1 PSV inversion with realistic dispersion errors and constant Poisson’s




5.20 Case 1 PSV inversion with 3% dispersion errors and relative density errors 217

5.21 Case 1 PSV inversion with 3% dispersion errors and various starting shear

velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

5.22 Case 1 PSV inversion with realistic dispersion errors and various starting

shear velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

5.23 Case 1 PSV inversion with 3% dispersion errors and various layer subdivisions219

5.24 Case 1 PSV inversion with realistic dispersion errors and various layer sub-

divisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

5.25 Case 1 PSV inversion with 3% dispersion errors and various constant thick-

ness layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

5.26 Case 1 PSV inversion with realistic dispersion errors and various constant

thickness layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220


ness layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

5.28 Case 1 PSV inversion with realistic dispersion errors and various constant

thickness layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221






xxii











velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230



5.41 Case 2 PSV inversion with realistic dispersion errors and 200 m/s, 16 layer

starting model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

5.42 Case 2 PSV inversion with realistic dispersion errors and automatic, 16

layer starting model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231


ness layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232


ness layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

5.45 Case 2 PSV inversion with realistic dispersion errors and automatic, thick

layer starting models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

5.46 Case 2 PSV inversion with realistic dispersion errors and automatic, thin

layer starting models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233













xxiii




velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241



5.59 Case 3 PSV inversion with 3% dispersion errors and various thin layer stacks242

5.60 Case 3 PSV inversion with 3% dispersion errors and various thick layer stacks242

5.61 Case 3 PSV inversion with realistic dispersion errors and various thin layer

stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

5.62 Case 3 PSV inversion with realistic dispersion errors and various thick layer

stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

5.63 Case 2 PSV inversion with geometrically increasing layer thickness . . . . . 247

5.64 Case 2 FSW inversion with geometrically increasing layer thickness . . . . . 247

5.65 Case 3 PSV inversion with geometrically increasing layer thickness . . . . . 248

5.66 Case 3 FSW inversion with geometrically increasing layer thickness . . . . . 248

6.1 Map of Western Australia with surface wave sites . . . . . . . . . . . . . . . 255

6.2 Borehole logs near the Perth Convention Centre surface wave survey spread

centre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

6.3 Walkaway shooting procedure at Telfer gold mine . . . . . . . . . . . . . . . 262

6.4 Shot gathers from Line 1 at Telfer gold mine . . . . . . . . . . . . . . . . . 263










6.14 Frequency-slowness transforms from Telfer gold mine tunnel pit 1 . . . . . . 269

6.15 Phase velocity dispersion from Telfer gold mine tunnel pit 1 . . . . . . . . . 269

6.16 Early time shot gathers from Telfer gold mine tunnel pit 1 . . . . . . . . . . 270

6.17 Frequency-slowness transforms from Telfer gold mine mill and waste dump

sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

6.18 Processed frequency-slowness transforms from Telfer gold mine mill and

waste dump sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

6.19 Phase velocity dispersion from Telfer gold mine mill and waste dump sites . 271

xxiv

6.20 Telfer gold mine tunnel pit Line 1 forward shot PSV inversion . . . . . . . 273

6.21 Telfer gold mine tunnel pit Line 1 reverse shot PSV inversion . . . . . . . . 273

6.22 Telfer gold mine tunnel pit Line 1 forward shot FSW inversion . . . . . . . 274

6.23 Telfer gold mine tunnel pit Line 1 forward shot PSV shallow inversion . . . 274

6.24 Telfer gold mine tunnel pit Line 2 forward shot PSV inversion . . . . . . . 275

6.25 Telfer gold mine mill site Line 9 forward shot PSV inversion . . . . . . . . 275

6.26 Telfer gold mine mill site Line 9 forward shot FSW inversion . . . . . . . . 276

6.27 Telfer gold mine mill site Line 9 reverse shot PSV inversion . . . . . . . . . 276

6.28 Telfer gold mine mill site Line 9 forward shot PSV shallow inversion . . . . 277

6.29 Telfer gold mine mill site Line 9 forward shot PSV deep inversion . . . . . 277

6.30 Telfer gold mine waste dump site Line 11 forward shot PSV inversion . . . 278

6.31 Telfer gold mine waste dump site Line 11 forward shot FSW inversion . . . 278

6.32 Telfer gold mine waste dump site Line 11 forward shot PSV shallow inversion279

6.33 Telfer gold mine waste dump site Line 11 forward shot PSV deep inversion 279

6.34 Telfer gold mine waste dump site Line 11 reverse shot PSV inversion . . . . 280

6.35 AGC early time data from Telfer gold mine waste dump Line 11 . . . . . . 280

6.36 Frequency-slowness transforms from Perth Convention Centre . . . . . . . . 285

6.37 Frequency-slowness transforms from Perth Convention Centre with reduced

upper velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

6.38 Walkaway dispersion analysis with 1 m geophone spacing at the Perth Con-

vention Centre by f − k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

6.39 Walkaway dispersion analysis with 3 m geophone spacing at Perth Conven-

tion Centre by f − k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

6.40 Perth Convention Centre 1 m geophone spacing PSV borehole inversion . . 287

6.41 Perth Convention Centre 3 m geophone spacing PSV borehole inversion . . 287

6.42 Perth Convention Centre 3 m geophone spacing FSW borehole inversion . . 288

6.43 Perth Convention Centre 1 m geophone spacing PSV blind inversion . . . . 288

6.44 Perth Convention Centre 3 m geophone spacing PSV blind inversion . . . . 289

6.45 Perth Convention Centre synthetic 1 m geophone spacing PSV borehole

inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

6.46 Perth Convention Centre synthetic 3 m geophone spacing PSV borehole

inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

6.47 Shot gathers from the road cutting site . . . . . . . . . . . . . . . . . . . . . 294

6.48 Frequency-slowness transforms from the road cutting site . . . . . . . . . . 294

6.49 Phase velocity dispersion from the road cutting site . . . . . . . . . . . . . . 295

6.50 Road cutting site Line 1 (BH9) PSV borehole inversion . . . . . . . . . . . 295

6.51 Road cutting site Line 1 (BH9) PSV subdivided borehole inversion . . . . . 296

6.52 Road cutting site Line 1 (BH9) PSV blind inversion . . . . . . . . . . . . . 296

xxv

6.53 Road cutting site Line 5 (SAB23) PSV borehole inversion . . . . . . . . . . 297

6.54 Road cutting site Line 5 (SAB23) PSV subdivided borehole inversion . . . 297

6.55 Road cutting site Line 1 (BH9) PSV blind inversion . . . . . . . . . . . . . 298

6.56 Road cutting site cross section . . . . . . . . . . . . . . . . . . . . . . . . . 298

6.57 AGC early time data from the Hyden fault scarp test array . . . . . . . . . 303

6.58 Frequency-slowness transforms from the Hyden fault scarp test array . . . . 303

6.59 Processed frequency-slowness transforms from the Hyden fault scarp test

array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

6.60 Phase velocity dispersion from the Hyden fault scarp test array . . . . . . . 304

6.61 Frequency-wavenumber stacking of rollalong data at the Hyden fault scarp . 305

6.62 Hyden fault scarp test array raw vertical component PSV inversion . . . . 307

6.63 Hyden fault scarp test array raw horizontal component PSV inversion . . . 307

6.64 Hyden fault scarp test array processed vertical component PSV inversion . 308

6.65 Hyden fault scarp test array processed horizontal component PSV inversion 308

6.66 Hyden fault scarp stacked PSV inversion . . . . . . . . . . . . . . . . . . . 310

6.67 Hyden fault scarp stacked FSW inversion . . . . . . . . . . . . . . . . . . . 310

6.68 Hyden fault scarp GPR profiles . . . . . . . . . . . . . . . . . . . . . . . . . 311

6.69 Hyden fault scarp shear velocity profile . . . . . . . . . . . . . . . . . . . . . 311

6.70 Hyden fault scarp P -wave reflection profile . . . . . . . . . . . . . . . . . . 312

6.71 Reflectivity modelling with homogenous overburden . . . . . . . . . . . . . 315

6.72 Reflectivity modelling with shallow HVL . . . . . . . . . . . . . . . . . . . . 315

6.73 Data from Narrows Bridge foreshore showing strong surface waves . . . . . 317

6.74 Dispersion from Narrows Bridge foreshore showing dominant higher modes 317

6.75 AGC early time data from Narrows Bridge foreshore . . . . . . . . . . . . . 318

6.76 Synthetic data of the Bietigheim model . . . . . . . . . . . . . . . . . . . . 319

6.77 Synthetic dispersion of the Bietigheim model . . . . . . . . . . . . . . . . . 319

6.78 Bietigheim PSV inversion with modelled dispersion errors . . . . . . . . . . 320

6.79 Narrows Bridge PSV inversion with modelled dispersion errors . . . . . . . 320

6.80 Data from the road cutting site Line 4 (SAB18) with Lamb waves . . . . . 322

6.81 Dispersion from the road cutting site Line 4 (SAB18) showing Lamb waves 322

6.82 Road cutting site Line 4 (SAB18) PSV inversion with modelled dispersion

errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

6.83 Data from the railway tunnel excavation site showing strong overlapping

modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

6.84 Data from reciprocal shot at the railway tunnel excavation site . . . . . . . 325

6.85 Fundamental mode dispersion from the railway tunnel excavation site . . . 325

6.86 Railway tunnel excavation site FSW inversion with modelled dispersion

errors and ‘borehole’ interfaces . . . . . . . . . . . . . . . . . . . . . . . . . 326

xxvi

6.87 Railway tunnel excavation site FSW inversion with modelled dispersion

errors and ‘blind’ interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

7.1 Monte Carlo model generation . . . . . . . . . . . . . . . . . . . . . . . . . 332

7.2 Case 1, layer 4 resolution test with 3% data errors . . . . . . . . . . . . . . 336

7.3 Case 1, layer 4 resolution test with realistic data errors . . . . . . . . . . . . 336







7.10 Case 1 Monte Carlo shear velocity PDF . . . . . . . . . . . . . . . . . . . . 342

7.11 Case 1 Monte Carlo shear velocity PDF by the FSW method . . . . . . . . 343

7.12 Case 1 Monte Carlo shear velocity likelihood ranges . . . . . . . . . . . . . 343

7.13 Case 1 Monte Carlo shear velocity PDF with equally distributed perturbations344

7.14 Case 1 Monte Carlo shear velocity likelihood ranges with equally distributed

perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344





7.19 Case 1 Monte Carlo model parameter statistics with realistic dispersion errors348

7.20 Case 1 Monte Carlo model parameter statistics with 3% dispersion errors . 348



7.23 Telfer gold mine mill site Line 9 Monte Carlo shear velocity PDF . . . . . . 352

7.24 Telfer gold mine mill site Line 9 shear velocity likelihood ranges . . . . . . . 352

7.25 Telfer gold mine mill site Line 9 Monte Carlo shear velocity PDF with

random layer to layer perturbations . . . . . . . . . . . . . . . . . . . . . . 353

7.26 Telfer gold mine mill site Line 9 shear velocity likelihood ranges with random

layer to layer perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

7.27 Telfer gold mine waste dump Line 11 Monte Carlo shear velocity PDF . . . 354

7.28 Telfer gold mine waste dump Line 11 shear velocity likelihood ranges . . . . 354

7.29 Telfer gold mine waste dump Line 11 shallow layer Monte Carlo shear ve-

locity PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

7.30 Telfer gold mine waste dump Line 11 shallow layer shear velocity likelihood

ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

7.31 Telfer gold mine tunnel pit Line 1 observed and modelled shot gathers . . . 356

xxvii

7.32 Telfer gold mine tunnel pit Line 1 observed and modelled traces . . . . . . . 356

7.33 Telfer gold mine mill site Line 9 observed and modelled traces . . . . . . . . 357

7.34 Telfer gold mine waste dump Line 11 observed and modelled traces . . . . . 357

7.35 Perth Convention Centre observed and borehole modelled traces . . . . . . 360

7.36 Perth Convention Centre observed and borehole modelled traces . . . . . . 360

7.37 Perth Convention Centre hole SC2 simplified models . . . . . . . . . . . . . 361

7.38 Perth Convention Centre hard layer resolution planes . . . . . . . . . . . . . 361

7.39 Perth Convention Centre hard layer at 20 m depth resolution . . . . . . . . 362

7.40 Perth Convention Centre hard layer at 20 m depth resolution . . . . . . . . 362

7.41 Road cutting site Line 1 Monte Carlo shear velocity PDF . . . . . . . . . . 366

7.42 Road cutting site Line 1 shear velocity likelihood ranges . . . . . . . . . . . 366

7.43 Road cutting site Line 5 Monte Carlo shear velocity PDF . . . . . . . . . . 367

7.44 Road cutting site Line 5 shear velocity likelihood ranges . . . . . . . . . . . 367

7.45 Road cutting site Line 5 observed and modelled shot gathers . . . . . . . . 368

7.46 Road cutting site Line 5 observed and modelled traces . . . . . . . . . . . . 368

7.47 Hyden fault scarp vertical component Monte Carlo shear velocity PDF . . . 371

7.48 Hyden fault scarp vertical component shear velocity likelihood ranges . . . 371

7.49 Hyden fault scarp inline component Monte Carlo shear velocity PDF . . . . 372

7.50 Hyden fault scarp inline component shear velocity likelihood ranges . . . . . 372

7.51 Hyden fault scarp vertical component observed and modelled traces . . . . 373

7.52 Hyden fault scarp inline component observed and modelled traces . . . . . . 373

7.53 Hyden fault scarp stacked dispersion Monte Carlo shear velocity PDF . . . 374

7.54 Hyden fault scarp inline component shear velocity likelihood ranges . . . . . 374

7.55 Hyden fault scarp vertical component observed rollalong data and modelled

traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

xxviii

xxix

Acknowledgements

My supervisors Mike Dentith and Ron List were instrumental in managing this project

to finality. It was definitely beneficial having two supervisors, who’s specialised areas

of geophysics and mathematics respectively, combined to aid my understanding of both

the practical and theoretical sides of seismology. Mike is of the School of Earth and

Geographical Sciences (previously known as Department of Geology and Geophysics)1

and Ron is now retired from the School of Mathematics and Statistics2 of The University

of Western Australia (UWA).

In addition, the Postgraduate Research and Scholarships Office of UWA are acknowl-

edged for provision of a University Postgraduate Award (UPA) for this 3.5 year project3.

A number of other people who deserve special mention acknowledged below:

Sebastiano Foti shared a great deal of knowledge on surface wave inversion with me

from a civil engineer’s point of view. His geotechnical experience broadened my under-

standing on the applications and limitations of the method in site evaluation. While

Sebastiano was working under contract in the Centre for Offshore Foundation Systems4

during 2001-2002 to be able to discuss various issues directly. Moreover, we collected and

shared data together from local sites using the COFS-built seismograph. Sebastiano is

currently at the Politecnico di Torino, where he completed a PhD in 20005.

Thomas Forbriger’s research also employed the reflectivity method as a forward disper-

sion operator for surface wave inversion. Many important observations from his work are

present in this thesis, in particular the resolution limitation due to spread length, and we

communicated a great deal. Thomas is currently at the Black Forest Observatory6, previ-

ously the University of Frankfurt7, but completed his PhD at the University of Stuttgart

in 20018.

Klaus Holliger of the Swiss Federal Institute of Technology9 and Michael Roth, cur-

rently at NORSAR10, provided the Fortran reflectivity synthetic seismogram code which

forms the kernel of this research. It was originally developed by Gerhard Muller of the

University of Frankfurt11.

Glenn Rix of The Georgia Institute of Technology12 originally suggested the study of

1http://www.geol.uwa.edu.au/ mdentith/mdentith.html2https://www.maths.uwa.edu.au/People/list3http://www.research.uwa.edu.au/4http://www.cofs.uwa.edu.au/5http://www2.polito.it/research/soilmech/foti/6http://www-gpi.physik.uni-karlsruhe.de/pub/forbriger/7http://www.geophysik.uni-frankfurt.de/˜forbrig/8http://www.geophys.uni-stuttgart.de/thof/9http://www.aug.ig.erdw.ethz.ch/people/klaus/klaus.html

10http://www.norsar.no/NORSAR/Staff.html11http://www.geophysik.uni-frankfurt.de/˜gmueller/12http://www.ce.gatech.edu/˜grix/surface wave.html

xxx

error propagation in shallow surface wave inversion and was always readily approachable.

His MATLAB toolbox (co-authored by Carlo Lai) was used extensively, particularly the

analytic partial derivative calculation and Occam’s linear inversion code.

Other people I communicated with and provided invaluable information and ideas were

(listed alphabetically):

David Boore of the US Geological Survey13;

Leo Brown of ConocoPhillips, USA14;

Vaclav Bucha of Charles University, Czechoslovakia15;

Kristen Buchanan (nee Beaty) of The University of Alberta, Canada16;

Nenad Gucunski of Rutgers University, USA17;

Koichi Hayashi of OYO Corporation, Japan18;

Yoshiaki Hisada of Kogakuin University, Japan19;

Barbara Luke of The University of Nevada, USA20;

Paul Michaels of Boise State University, USA21;

Nils Ryden of Lund University of Technology, Sweden22; and

Mauricio Sacchi of The University of Alberta, Candada23.

Field data collection and site information with the assistance of the following people, all

currently based in Perth, Western Australia:

Telfer gold mine–Eric Wedepohl and Greg Turner of Geoforce Pty. Ltd.24, data courtesy

of Newcrest Mining Limited;

Perth Convention Centre–COFS of UWA and Arup Australasia25;

13http://quake.wr.usgs.gov/˜boore/14http://conocophillips.com/15http://sw3d.mff.cuni.cz/staff/bucha.htm16http://laser.phys.ualberta.ca/˜kbuchan/17http://www.cait.rutgers.edu/faculty/gucunski/18http://www.oyo.co.jp/english/instru e/soft/seisimager/19http://kouzou.cc.kogakuin.ac.jp/Member/Boss/hisada index .ehtml20http://www.ce.unlv.edu/˜bluke/21http://cgiss.boisestate.edu/˜pm/home.html22http://www.tg.lth.se/misc/gt.htm#PeopleGT23http://rubble.phys.ualberta.ca/˜sacchi/SEISMIC LAB/24http://www.geoforce.com.au/25http://www.arup.com.au/

xxxi

Road cutting and railway excavation–Nathan Narendranathan and staff of A.L. Tech-

nologies Pty. Ltd.26; and

Hyden fault scarp–Brian Evans, Domenic Howman, Milovan Urosevic and Miroslav Bra-

janovski of Curtin University of Technology27.

Last, but never least, my wife Maki and daughter Riuna are thanked for their support

as always.

26http://www.altechnologies.com.au/27http://www.geophysics.curtin.edu.au/

xxxii

xxxiii

Synopsis

Chapter 1 is a literature review of the theory and methods of surface wave inversion.

It is based around the conventional acquisition-processing-inversion process and includes

some applications of shallow surface wave surveying. More complicated phenomena to be

addressed in this thesis are only mentioned briefly.

Chapter 2 introduces the full-waveform P -SV reflectivity forward dispersion modelling

algorithm, which is the basis of dispersion calculation in this thesis. Applied to synthetic

models with large elastic parameter reversals and contrasts, the phenomena of dominant

higher modes and near field effects is introduced and why these problems require a new

inversion procedure.

Chapter 3 is a rigorous numerical evaluation of the factors influencing the experimen-

tal resolution and accuracy of multichannel surface wave dispersion. The field test is

simulated incorporating typical data acquisition and processing errors of engineering site

investigation applications. The measured dispersion uncertainty is used to develop a new

noise model for observed dispersion in shallow engineering structures and how these affect

dispersion sensitivity.

Chapter 4 shows repeated field tests employing the error influences of the synthetic

dispersion repeatability. A variety of sources, geophones and offsets are incorporated,

along with additive noise. The uncertainty envelopes observed in the field verify those

of the numerical tests and errors in dispersion measured from CMP and walkaway shot

gathers are quantified and the distribution modelled.

Chapter 5 outlines a new surface wave dispersion inversion algorithm based on reflec-

tivity synthetic seismograms. A linearised optimisation scheme is applied (Occam’s inver-

sion), taking into account dominant higher modes, low frequency effects and the acquisition

error envelopes observed in the synthetic tests. Comparisons with traditional source-free

plane-wave methods are also made and the effects of some common assumptions (layer

thicknesses, Poisson’s ratio, starting shear velocity etc.), incorporating regularisation, are

illustrated.

Chapter 6 illustrates some shallow field applications of the new inversion method,

particularly for cases where dominant higher modes are generated and traditional plane-

wave methods are unsuitable. Dispersion uncertainty is also be incorporated, either based

on repeated tests or a noise model. From comparisons to borehole data some limitations

of the method become evident.

Chapter 7 uses both grid search and Monte Carlo techniques to better define the reso-

lution and accuracy of inverted shear wave velocity models. Both synthetic and field data

models are appraised using these procedures. This method of practical inference theory

is made complete by the incorporation of previously defined noise envelopes. Waveform

matching of observed and modelled seismograms is illustrated as a qualitative model ap-

xxxiv

praisal method.

Chapter 8 summarises the main conclusions and recommendations of this work, namely,

the improved inversion of dispersion with dominant higher modes observed at many engi-

neering sites. Realistic data uncertainty allows analysis of error propagation into the final

models and further research avenues in shallow surface wave inversion are suggested.

1

CHAPTER 1

Introduction

1.1 Historical, application and methodology overview

Surface waves are strongly dispersive and this property is employed in the inversion

for several reasons. Firstly, it is a robust property that can be quite easily observed and

not contaminated by other wavefields. Secondly, forward modelling techniques exist to

generate this data rapidly and accurately for layered Earth models. Finally, compared

to the inversion of waveforms, the nonlinearity of the inverse problem is greatly reduced,

allowing linearised optimisation procedures to be more efficiently applied. The efficiency

is aided by having the partial derivatives of phase velocity with respect to model param-

eters (essentially layer shear velocity) available in analytic form. For all these reasons,

surface wave dispersion will generally be a favourable dataset over the fundamental data

(waveforms) or some other secondary observable, for imaging the Earth in a variety of

applications.

1.1.1 Earthquake seismology Earthquake seismology involves seismograms mea-

sured over great distances from earthquake or nuclear explosion sources. The earliest use

of body waves was in the 1850’s but is was not until the 1920’s that surface wave dispersion

began to play an important role in global seismology, due to both improvements in instru-

mentation and mathematical developments. The early reviews of [12, 153, 156, 157, 228]

are excellent overviews of surface wave observation, inversion and applications in earth-

quake seismology. Chronological histories of earthquake seismology are in [5, 25] and the

history of developments of surface wave theory are summarised in [35, 147].

Modelling of a layered Earth from surface waves propagated over a great circle path

from a single seismogram were first studied in the 1920’s. At that time, theoretical foun-

dations restricted layered Earth models derived from surface wave data to only two or

three layers over a half space and the results mainly supported oceanic and continen-

tal crustal thicknesses estimated using body waves. In the 1950’s and 1960’s the growing

global network of seismographs, combined with improved theory, magnetic tape recording,

digital acquisition and the recording of atomic test explosions, provided additional data

over more source-receiver paths. With more detailed modelling methods, discoveries were

made of the low velocity zone in the upper mantle, as well as its anisotropy and lateral

heterogeneity, and surface wave inversion became an interpretational tool in its own right

[328].

Tomographic inversions of surface wave [216, 294] and body wave [338] properties for

crust and upper mantle structure are regularly performed. While there are theoretical

limits to the resolution of Earth models, the accuracy is becoming progressively better

with time. Source problems (fault mechanisms, natural/explosive discrimination etc.)

are also studied with surface waves and are an important part of nuclear test-ban treaty

2 Chapter 1. Introduction

monitoring [312].

1.1.2 Engineering applications Early civil and geotechnical engineering applica-

tions of surface wave dispersion were apparently motivated by observations and theoretical

developments by earthquake seismologists. The early work is summarised well in [344] and

important historical notes are made in [55, 80, 135, 154, 268]. While the method was first

applied in the 1930’s for shallow asphalt and concrete road slabs and subgrades, the fre-

quencies measured (tens of Hz) actually correlated with deeper soil properties, the zone

which was conventionally tested with body wave measurements [74].

While theoretical dispersion of guided waves were well known, only simple layered

cases were studied and even the propagating wavefield was debated among Rayleigh, Love

or Lamb (plate) waves. The methods were usually called ‘dynamic testing’ or ‘wave

propagation / vibration methods’ or simply ‘seismic or surface wave methods’. They are

now generally referred to as ‘continuous surface wave’ (CSW) methods and these single

channel methods are mostly obsolete. The most readily accessible reports from the mid

1950’s to 1960’s are [132, 135, 133, 134] and many commercial pavement testing systems

were available by the late 1960’s [344]. However, interpretation methods still mostly relied

on simple wavelength-depth approximations of propagating waves.

Research progressed slowly through the late 1960’s and 1970’s, mostly for asphalt and

road base testing applications, unlike the unlike rapid advancement of theory and methods

in earthquake seismology. In the 1980’s the 2-channel method of ‘spectral analysis of

surface waves’ (SASW) was introduced [67, 112, 222, 219, 220, 221, 264, 282, 299], and

many geotechnical applications followed [1, 4, 323, 325]. However, automated, iterative

inversion was not applied until the later 1980’s to early 1990’s, which are the norm in the

most recent work [34]. Field methods by civil engineers are now dominated by multichannel

measurements, both linear [259] and spatial [373] arrays. Soil damping estimation with

surface waves is another recent application [261, 84], traditionally accomplished with body

wave measurements [365, 130].

More recently, techniques for investigating lake and ocean bed sediments have been

researched, again traditionally using seismic refraction and reflection [75, 184, 242]. These

are focused on offshore engineering problems such as oil production platform and pipeline

construction. Other novel engineering applications are buried object detection [90], ar-

chaeological investigation [181], downhole logging [138] and materials science [321].

Earthquake engineering is another field which requires accurate estimates of in situ shear

wave velocity. In earthquake-prone areas, it is important to know the resonant frequencies

of the shallow (upper tens of metres) of a site, so structures can be designed to avoid these

harmonics and minimise damage during extended vibration. Average shear wave velocity

is interpreted for liquefaction studies and estimates of economic loss [310, 13]. It is ironic

that surface waves, which are often the most destructive arrivals from an earthquake, can

1.1. Historical, application and methodology overview 3

be applied (on a much smaller scale) for earthquake hazard studies [34, 179, 125].

Environmental applications to date focus on depth to basement and stratigraphic

mapping to determine probable contaminant flow paths [207, 205] or ‘time-lapse’ seasonal

monitoring of the near surface mechanical properties [21, 347].

1.1.3 Exploration geophysics While the earliest seismic reflection surveys for

petroleum exploration were made around the 1920’s, reports of surface waves in an ex-

ploration context appeared in the 1950’s [60, 61], based on theory developed from obser-

vations of explosions during WWII and of subsequent atomic bomb tests, studied. While

the early underwater work was academic and the on-land studies primarily intended to

better understand surface waves in order to combat them in reflection surveying, this

quote from [59] in 1951 is particularly insightful,

...dispersion curves could constitute an effective tool for studying the superficial

layers near the surface of the ground, particularly as regards the distribution

of shear speeds.

considering that exploration geophysicists of that time were apparently unaware of the

earlier advances made by civil and geotechnical engineers. The later 1950’s and 1960’s are

noticeably void of exploration geophysics applications of dispersive surface waves. This

is possibly due to the development in the mid-1950’s of common midpoint (CMP) stack-

ing and Vibroseis for improved reflection seismic profiling [297]. In addition, theoretical

advancements were not fully appreciated, due to lack of digital computing. Later, in

the 1970’s, studies of source generated noise employed the dispersive characteristics of

ground-roll for static corrections of land S-wave seismic data [73, 131, 318].

By the 1980’s a definite interest was re-emerging and rigorous multichannel dispersion

analysis being performed [10, 50, 89, 192, 345]. Around this time, applications of surface

wave propagation theory allowed for dispersive noise removal in on-land [116], coal seam

[33] and floating ice [27] seismic surveys. In the late 1990’s, the acronym ‘multichannel

analysis of surface waves’ (MASW) was introduced [236, 239]. Like its 2-channel equivalent

(SASW), the basic dispersion observation and modelling techniques are unchanged from

the developments made much earlier in earthquake seismology, and the focus in recent

work almost entirely on engineering applications [354], with none to date in a mineral

exploration context.

1.1.4 Application overlap The primary difference between applications is depth

scale, which is intrinsically linked to the frequency of measurements and/or the recording

array aperture or wavefield propagation distance. These are summarised in Table 1.1.

In general, earthquake applications measure waves generated by earthquakes or nuclear

test explosions and engineering applications usually employ artificial sources like impact

(hammer or weight drop), explosion (gun or dynamite) or vibratory. However, the ‘basin’


Table 1.1: Range of surface wave applications showing approximate frequencies and spatial

ranges involved. ‘Spatial range’ implies either depth of investigation, recording array

aperture or wavefield propagation path

Application Main frequency Spatial range

(a) Earthquake

Mantle mHz 100’s km

Crust 0.01’s Hz 10’s km

Basin 0.1’s Hz 100’s m ∼ km’s

(b) Engineering

Deep Hz 10’s ∼ 100’s m

Shallow 10’s Hz metres

Road kHz centimetres

Materials MHz millimetres

and ‘deep engineering’ investigations can be equally accomplished with passive or ac-

tive measurements. Passive sources are those which are not artificially initiated, usually

called microseismic events. They may be either natural (atmospheric/oceanic disturbance

or small volcano quakes) or cultural (usually traffic or construction equipment). Active

sources are artificially initiated and for long propagation distances are usually explosions,

such as large downhole detonations or mining blasts.

In engineering applications, comparisons with other site investigation methods and

some limitations are in Tables 1.2 and 1.3.

1.1.5 General procedure As with many geophysical techniques, any surface wave

application involves the standard four step procedure of acquisition, processing, inversion

and interpretation, comprising:

1 Acquire time series data with a strong surface wave component;

2 Process the data to accurately obtain a dispersion curve;

3 Invert the dispersion curve into a layered shear velocity model ; and

4 Interpret the model into a geological framework.

The inversion (Step 3) actually comprises two stages:

3a Estimate a model employing the theory of surface wave propagation and mathematical

optimisation; and

3b Appraise the model for its accuracy, either deterministically or statistically.

These issues (Steps 3a and 3b) are the focus of the research of this thesis.

1.1. Historical, application and methodology overview 5

Table 1.2: Comparison of seismic and invasive tests for shallow site investigation.

Aspect Seismic Seismic Borehole/

surface waves refraction penetrometer

Non-invasive / Yes Yes, except No, also

non-destructive shot hole environmental

Detects velocity Yes, under caprock Not nonuniquely Yes, within

reversals or buried horizons log sampling

Supply shear Yes, with vertical Yes, with shear wave Yes, with shear source

wave velocity geophones and hammer sources and detectors and 3C detectors

Survey among traffic Yes, due to large Not usually, also Yes, except for

or industrial noise signal strength in rain or wind public obstruction

Survey over asphalt Yes, towed arrays Problematic due to Yes, with

roads and concrete also possible to grazing waves coring bits

Investigate rockfill Yes, for bulk Yes, but large Not with

and solid waste properties degree of scattering penetrometer tests

Safe hammer Yes, 67% of energy Often explosives N/A

sources suitable to surface waves are required

Model vertical Yes, with Yes, with more Yes, but not

velocity gradations confidence nonuniqueness from zero depth

In-field Yes, rapid Not usually Not usually

processing and conclusive performed performed

Table 1.3: Limitations of surface wave surveying, relative to body wave and invasive

methods.Aspect Seismic Seismic Borehole/

surface waves refraction penetrometer

Detect lateral Yes, but Yes, Yes, but

lateral changes approximate rigorously empirically

Detect very No, unless Yes, within Yes, within

thin layers quite shallow resolution log sampling

Valid in No Yes, within Yes, within

steep topography several degrees logistics


1.2 Surface wave properties

Lord Rayleigh first quantified the mechanics of a wave travelling along a solid half-

space / free-space interface (simplified Earth’s surface) in 1885 [254]. However, Rayleigh

did not comment on the probable source mechanisms and dispersive nature of these waves,

to be later suggested by Lamb in 1904 [255]. However, rigorous derivation of Rayleigh (R)

waves dispersion (velocity variation with with wavelength in the presence of a vertical

variation in shear modulus) was by Love in 1911 [61, 332]. Love is better known for the

theoretical foundation of the L-waves, named after him, which comprised the transverse

motion seen in early seismograms and only exist in a low speed overlying layer [156]. Love

waves arrive slightly before Rayleigh waves and have a shallower penetration of particle

motion.

The typical wavefields observed in a land shot gather are shown in Figure 1.1 [363].

A shallow explosive or surface impact source generates fundamental mode ’ground-roll’ in

any environment but the conditions here were conducive for both higher mode and guided

wave generation, which would be a relatively thin, low velocity overburden with high

velocity basement. These types of source generated wavefields and their characteristics

are now discussed.

1.2.1 Guided wave overview A waveguide can comprise one or two interfaces

and there are several wave types which require these conditions for propagation. Rayleigh

waves are one of the group of guided waves, those whose amplitude decreases away from

the interface and are dispersive. These are summarised in Table 1.4, where ‘air’ indicates

a free-space and ‘solid’ indicates an elastic half-space. Note that Love waves actually

require a waveguide with two boundaries for propagation, thus are not strictly ‘surface’

waves [327].

The waves of Table 1.4 are more correctly called boundary [327] or interface [5] waves.

Other names include ‘trapped’, ‘normal’, ‘locked’, ‘propagating’ or ‘channel’ modes. To

practitioners, guided waves usually imply the shingled arrivals just after the direct and

refracted wavefields on shot gathers due to multiply reflected acoustic (P ) waves in a two-

interface waveguide. They are more problematic in marine [166] than land [272] seismic

reflection. In marine settings they can be generated in the water column and/or low speed

sediments, either at the seafloor or at depth. They were first observed and theoretically

explained from observations made during WWII [39, 60]. While early marine work treated

seafloor sediments as liquid layers [251], a solid seafloor required an extension of the theory

[252]. Similar wave guides exist on land, but are less common, usually in low velocity

sediments where the lower interface is the shallow water table [265]. Low velocity coal

seams also generate this type of wave, where effort is directed towards their compression for

better fracture detection [26, 33, 169]. In older literature, these are often called ‘damped’

or ‘leaky’ modes. This is because normal mode theory is used to define the dispersion of

1.2. Surface wave properties 7

Figure 1.1: Typical wavefields observed in a land seismic shot gather with shallow explosive

source, including fundamental and higher mode surface waves, guided and refracted waves,

the air wave (∼340 m/s) and the optimum reflection window (ORW). Direct wave arrivals

are probably at similar velocity to the higher mode surface waves. [363].


Table 1.4: Various guided wave types.

Wave name Guide type Wave class Derived

Rayleigh air-solid interface surface 1885

Love air-solid layer surface 1911

Lamb solid plate in air flexural 1916

Stoneley solid-solid interface interface 1924

Scholte liquid-solid interface interface 1942

guided waves, where in a liquid waveguide the secular function is explicit and analytic but

in solids is an implicit determinant [94].

Physically, leaky modes are due to multiple reflections in a wave guide at less than the

critical angle, thus energy can ‘leak’ out. Propagating modes can be similarly visualised,

but the angle of incidence is greater than the critical [131]. Mathematically, leaky modes

arise from branch cuts on complex frequency planes which do not satisfy the radiation

condition (decay exponentially and no upgoing waves from the half-space). Propagating

modes are of roots which do satisfy the radiation condition [5, p253] which are on the real

axis within certain density and shear moduli limits [94].

All dispersive wave modes comprise the Green’s function of a layered Earth [5, p586].

Rayleigh, Stoneley and Scholte waves are dispersive when there are variations in elastic

properties away from the interface. However, even in a homogenous half-space, gravity

induces dispersion of a few percent [327]. Stoneley waves can be used for fracture (per-

meability) mapping between boreholes and Scholte waves for imaging seafloor sediments.

Scholte waves show very similar dispersion to free-space Rayleigh waves. Dispersive Love

waves are only generated where there is an increase in shear modulus with depth [135]

and Lamb (flexural) waves are always dispersive in a free plate. Lamb wave dispersion is

also observed in a stiff asphalt layer [243] and floating sea ice [27] at high frequency.

1.2.2 Elastic parameters Seismic wave propagation depends on the elastic pa-

rameters of the propagation medium, outlined in Table 1.5 [297, 298]. The two Lame pa-

rameters (µ,λ) describe the linear stress-strain relation in an isotropic solid, µ (sometimes

denoted G) being the shear modulus (resistance to shearing) and λ having no physical

explanation [297]. The amount λ+ 2µ is sometimes called the constrained modulus (M)

and is equal to ρα2. Bulk modulus relates to volumetric change under hydrostatic pressure

(resistance to compression) and Young’s modulus (sometimes denoted Y ) relates to length

change under elongation.

Poisson’s ratio (σ, sometimes denoted ν) is a useful factor relating the shear (β) and

compressional (α) wave velocities by

β

α=

[

0.5 − σ

1 − σ

]1/2

(1.1)


Table 1.5: Common elastic parameter definitions at small strain, relative to Lame’s con-

stants µ and λ.

Parameter Definition

Young’s modulus E = (3λ+2µ)µλ+µ

Bulk modulus k = λ+ 23µ

Poisson’s ratio σ = λ2(λ+µ)

Compressional velocity α =(

λ+2µρ

)1/2

Shear velocity β =(

µρ

)1/2

While the theoretical range of σ is from -1 to 0.5, most physical materials are within 0 and

0.5. The upper limit represents a fluid (β = µ = 0) and the lower limit indicates an incom-

pressible solid. Over this range in a homogenous half-space, the Rayleigh wave to shear

wave velocity ratio (c/β) varies between 0.862 and 0.955 [3, 76], a good approximation

being [3]cRβ

=0.862 + 1.14σ

1 + σ(1.2)

A Poisson solid is a perfectly elastic body where λ = µ, σ = 0.25, αβ =

√3 and

c/β = 0.9194 [297]. Most hard rocks and consolidated sediments have a Poisson’s ra-

tio around 0.3. In saturated sediments it may reach 0.45 or above [1] (low β, high α) and

in unconsolidated sands may be 0.2 or less (β and α both low). Sediments saturated in

excess of 99% still support shear wave propagation, albeit with very low β, and a nominal

α of 1500 m/s is attained only at 100% saturation. In most practical applications, elastic

constants are assumed frequency independent and isotropic. However, the shear strain

has a marked influence on the effective shear modulus. The strain induced by small-scale

seismic waves is in the order of 10−6 or 0.0001%, where deformation is elastic and moduli

are maximum [161, 202].

1.2.3 Rayleigh wave theory At the free surface of a homogenous half-space, the

constructive interference of compressional (P ) and vertical shear (SV ) waves will produce

a propagating wave in the x − z plane with retrograde elliptical particle motion at the

surface. The primary properties of free Rayleigh waves are [76]:

1. Vanishing of stress (τ) at the free surface;


Figure 1.2: Rayleigh wave particle motion in a homogenous, isotropic half space is retro-

grade at the surface, passing through purely vertical at about λ/5 then becoming prograde

at depth [55].

2. Amplitude decreases exponentially with depth; and

3. Displacement (u,w) is independent of y.

If the displacement is a combination of compressional (φ) and rotational (ψ) potential

solutions of the form

φ = A exp i(kx− ωt) (1.3)

ψ = B exp i(kx− ωt) (1.4)

then applying the free surface boundary conditions (τ31 = τ33 = 0) and solving for A and

B reveals Rayleigh’s equation [254]:

(

2 − c2

β2

)2

= 4

(

1 − c2

α2

)

1

2(

1 − c2

β2

)

1

2

(1.5)

There is always a real root for Equation 1.5 if

0 < c < β < α (1.6)

and it is under these conditions that R-waves exist [76]. Rayleigh’s wave equations are

derived in [5, 38, 76, 297, 327, 332].

The particle motion of a propagating Rayleigh wave in a homogenous half space is

illustrated in Figure 1.2, showing the transition from retrograde to prograde elliptical with

depth. Since the Rayleigh wave solutions comprise both a compressional and rotational

component, the eccentricity of the ellipse depends on Poisson’s ratio, defined in Section

1.2.2. The phase velocity of homogenous R-waves is a nonlinear function of Poisson’s ratio


Figure 1.3: A modulated, propagating wave showing phase and group velocities [332].

[327] and must be less than than shear wave velocity (c < β) else energy would be radiated

into body waves. R-waves are best generated from sources at the surface or shallow depth

and, along with L-waves, comprise the largest motions seen on shot records and are the

most destructive components of earthquake wavetrains [297].

1.2.4 Phase and group velocity In a homogenous half space, the various com-

ponents of a polychromatic wave propagate at the same horizontal velocity. The phase

velocity (c) is that of the individual peaks and troughs on a seismogram and the group

velocity (U) is that of the trace envelope or the wave packets with a Gaussian frequency

spectrum [249]. In special cases it can be shown that U is the velocity of energy prop-

agation and for acoustic waves c is the velocity along the ray path. The case of two

interfering waves where a high frequency vibration is modulated by a lower frequency

envelope is shown in Figure 1.3.

For a harmonic propagating wave of the form

u(x, t) = A exp i(kx− ωt) (1.7)


the phase velocity is given by

c =ω

k(1.8)

and the group velocity is

U =dω

dk= c+ k

dc

dk= c

(

1 − kdc

dω

)

−1

(1.9)

The factor kx − ωt in Equation 5.14 is the phase. In the case of surface waves, the

phase velocity is an average of the propagation over the receiver array, which is the case

for SASW and MASW. The group velocity reflects average propagation over the source-

receiver path [314] and is always slower than the phase velocity. Where the group velocity

dispersion curve reaches a local extreme, an Airy phase occurs, which is energy from a

range of frequencies arriving almost simultaneously. Single earthquake stations measure

group velocity, but requires knowing the epicentral distance and source time.

1.2.5 Wave dispersion All propagating guided elastic waves show dispersion in a

layered Earth [88], which is the variation of velocity with the frequency (or wavelength) of

the oscillations [311]. For P and S body waves in a wave guide in elastic media it is hardly

negligible, but the surface waves exhibit this to a high degree [190]. In a density stratified

solid the source wavelet decomposes into a spectrum of surface waves all travelling at

different velocities. Because of this, it thus changes shape as it propagates in x and t [327]

and disperses.

Theoretically, the implicit frequency-phase velocity relationship emerged from the very

early mathematics [292]. Physically, surface wave dispersion occurs due to the distribution

of particle motion with depth [264]. Significant particle motion in high frequency (short

wavelength) surface waves is confined to the shallow Earth and conversely low frequency

(long wavelength) components induce particle motion at greater depths. In a layered Earth

the variation of material properties with depth implies that particle motion thus surface

wave velocity is a function of this distribution. In a homogenous half-space Rayleigh waves

are non-dispersive and Love waves do not exist.

In raw seismograms, dispersion can be recognised as the change in pulse shape with

propagation distance. A monochromatic source pulse will disperse to arrive at an observa-

tion station as a disturbance of longer duration. A monochromatic source function (such

as from a fixed frequency vibrator) does not change shape. However, the useful representa-

tion of dispersion is the dispersion curve, which shows the frequency-velocity relationship

which can be exploited to interpret the Earth structure. Note that some authors write

that the waves are dispersive [349] while others imply the medium is dispersive [327].

1.2.6 Higher modes Like a vibrating string, surface wave modes are distinct eigen-

functions of the wavefield. Higher modes can be physically described as constructive in-

terference between the curved ray paths of multiply reflected wavefronts in a layer of finite


Figure 1.4: Rayleigh wave dispersion curves showing the fundamental mode (M11) and

first three higher modes (symmetric M12 and antisymmetric M21 and M22) for a single

elastic layer over a half space [332].

thickness [3, 177]. They travel at higher phase velocities than the fundamental mode and

it is the superposition of these propagating normal modes which comprise the surface wave

motion seen on seismograms.

At a given frequency, surface waves only propagate with certain wavenumbers or vice-

versa. Figure 1.4 shows the phase velocities permitted at certain wavenumbers (which

is interchangeable with frequency) which are the eigenvalues of the propagating wave

solutions. The dispersion relation between ω and k [349] indicates there will be a number

of eigenfunction solutions to

ω = f(k) (1.10)

the different functions indicating different modes. Only a finite number of modes is possible

[5, 76] and there is a cutoff frequency for modes above the fundamental, below which the

wave will not propagate [76, 5]. The displacement eigenfunctions of higher mode Rayleigh

waves at any given frequency show higher amplitudes at depth, indicating that they provide

more information for deeper layers than the fundamental mode alone [145].

1.2.7 Spherical spreading Unlike body waves which attenuate due to spherical

spreading at a rate of r−2 along the free surface, surface waves attenuate much slower at

r−1/2, where r is distance from the source. Guided waves near a modal cutoff frequency


(i.e. body waves at the surface) attenuate at r−1 due to constructive interference [76].

In most experimental work and mathematical derivations, the plane wave assumption

is maintained, whereby waves are assumed to propagate homogeneously from a distant

source, thus wavefronts are neither spreading, converging nor decaying (evanescent).

1.2.8 Intrinsic attenuation Aside from geometric spreading and scattering at dis-

continuities, seismic wave amplitude is degraded by anelastic losses during wave propa-

gation, known as intrinsic or material attenuation [373]. In seismology, the dimensionless

quality factor (Q) is defined as [5, 297, 332]

1

Q(ω)=

∆E

2πE= 2D (1.11)

where ∆E is the energy loss per cycle at a given frequency and E is the peak strain energy.

D is the damping ratio and is preferred in engineering applications. There will be different

Q values for both P - and S-waves, denoted Qα and Qβ respectively, and in practise are

usually assumed frequency independent [261].

1.3. Dispersion observation 15

1.3 Dispersion observation

There are various methods for measuring phase and group velocity dispersion from

time series data and usually depend on the application. Broadly, the methods are divided

between those mainly used in large-scale earthquake seismology and those for small-scale

engineering and exploration seismology.

1.3.1 Large-scale seismology Methods for measuring dispersion from seismograms

of earthquake or nuclear explosions at permanent or mobile recording stations are sum-

marised with references in Table 1.6. By far the most widely used method is the multiple

filter technique (MFT) which is a time-frequency decomposition for group velocity. It

is also applied on smaller scale, such as analysing quarry blasts [186]. The two-station

‘phase difference method’ correlates with SASW and linear array ‘frequency-wavenumber’

is equivalent to MASW.

Table 1.6: Methods for analysis of dispersion from earthquake seismograms.

Method Group Phase

velocity velocity

Single- Moving window analysis (MWA) [19] ‘Peak and trough’ [19, 72, 249]

station Multiple filter technique (MFT) [71, 72, 188]

Wavelet transform [178, 361]

Two- Deconvolution/cross-correlation [19, 72, 249] Phase difference [19, 72, 249]

station

Linear N/A Frequency-wavenumber [224]

array

Areal N/A Triangular array [19, 72, 249]

array Frequency-wavenumber [44]

1.3.2 Small-scale seismology In site investigations, both the source parameters

(place, time and usually frequency) are accurately known and arrays placed to favourably

record propagating waves. The earliest methods employed single receivers, later two-

channel (SASW) and multi-channel (MASW). Only MASW methods are used in this

thesis.

Single-channel: CSW Usually called the steady state method [255] or continuous sur-

face wave (CSW) method [1] and been employed in the studies by [135, 323]. This method

involves moving a single geophone away from an oscillating, harmonic source and iden-

tifying successive positions where the received signal is in phase with previous positions.

The distance between these positions is then equal to one wavelength λ, assuming no cycle

skips. Since the vibrator frequency (f) is known, the phase velocity dispersion curve is


simply [311]

c(f) = fλ (1.12)

Points that are π/2 out of phase are also measured, corresponding to a half-wavelength.

Usually several geophone positions are deployed for a least squares regression of phase

velocity. More rapid field techniques employ two or more signals simultaneously measured

from geophones a distance d apart, thus the phase difference φ can be directly seen at all

offsets. Assuming d < λ the wavelength will be λ = 2πd/φ and knowing the vibrator fre-

quency the phase velocity calculated from Equation 1.12. Multichannel CSW is described

in [195] and is a precursor to the impact source method, SASW.

Single-channel: transfer function This is a more recent method than SASW, intro-

duced for geotechnical applications [373, 260]. It involves fixing an accelerometer to the

source and measuring the empirical Green’s function by deconvolving it from a single re-

ceiver. Alternatively, one geophone signal in a linear array can be deconvolved from one or

more of the others for a linear average. This method is most suited for coupled dispersion

and attenuation measurements [84].

Two-channel: SASW This method became popular with the advent of small, digital

receivers in the 1970’s for civil engineering and geotechnical applications. The procedures

are summarized in [55, 102, 282, 217, 311]. Impact, random or swept frequency oscillators

can all be used. Figures 1.6 and 1.5 show the two commonly adopted field layouts:

• Common Source Midpoint (CSMP); and

• Common Receiver Midpoint (CRMP); and

It is similar to the two-channel CSW method except that each of the recorded signals

is digitised as A(t) and B(t) and a Fast Fourier Transform (FFT) applied to calculate

the phase function φ(f) for each signal. From this a phase difference function 4φAB(f)

between the two signals can be calculated, thus the frequency dependent travel time

function between the two receivers is

t(f) =4φ(f)

2πf(1.13)

Knowing the distance between the receivers (d) the phase velocity function becomes

c(f) =d

t(f)(1.14)

Most engineering publications present the dispersion in terms of surface wavelength using

the relation λ = c/f . Since the phase velocity depends on the intergeophone phase dif-

ference, the only corrupting factor is static error between the two signals. CRMP is more

commonly reported in applications where a variety of distances for d are used, with x1 kept

equal to d, and incrementing in factors of two in both forward and reverse source positions.


A(t) B(t)........................................................................................

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x2.................................................................................................................................................

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x1.................................................................................................................................................

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d....................................................................................................................................................................................

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................

Figure 1.5: Common source mid point (CSMP) geometry in SASW testing [102, 161]. S

are the source locations and A and B are the receivers.

A(t) B(t)........................................................................................

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x2.................................................................................................................................................

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..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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x1.................................................................................................................................................

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d....................................................................................................................................................................................

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..................................................................................................................................................................................................

................

Figure 1.6: Common receiver mid point (CRMP) geometry in SASW testing [102, 161].

S are the source locations and A and B are the receivers.


Accelerometers are generally used for high frequencies / close spacings and geophones for

deeper investigations, however, displacement transducers can be equally employed.

The dispersion curve calculated from these methods is often noisy and segments those

for different d spacings generally do not coincide. More objective noise analyses can

be made using cross-correlation of the two input signals A(t) and B(t). The coherance

function is given as [55, 217]

γ2(f) =|GAB(f)|2GA(f)GB(f)

(1.15)

where GA(f) and GB(f) are the linear (auto-power) spectra of A(t) and B(t) that is

GA(f) = A(f)A(f) (1.16)

GB(f) = B(f)B(f) (1.17)

and A(f) is the complex conjugate of A(f). It is usually averaged over n records (receiver

spacings) as

GA(f) =1

n

n∑

i=1

Ai(f)Ai(f) (1.18)

The cross-power spectrum is

GAB(f) = A(f)B(f) (1.19)

or usually averaged as

GAB(f) =1

n

n∑

i=1

Ai(f)Bi(f) (1.20)

Phase velocities calculated using Equations 1.13 and 1.14 are retained at frequencies where

γ2 is 0.9 or greater. Note that the coherance function is analogous to the square of the

cross-correlation coefficient and as such ranges between 0 and 1. Modifications to the

method include a multiple filter/crosscorrelation technique [9] or use of the harmonic

wavelet transform for the phase decomposition [244]. Employing Hankel functions as the

transform kernels overcome the plane wave assumption of the Fourier transform for more

accuracy in the near field [65]. SASW has been shown equivalent to f − k transform with

infinite trace padding [82].

Multichannel frequency-wavenumber: MASW f − k In this method, a multi-channel

shot record is transformed from the t − xdomain into f − k domain by a 2D FFT. The

raw data can be filtered or windowed to enhance ground roll but ideally requires equi-

spaced receivers, which is shown in Figure 1.7. The surface wave dispersion is picked

from the dominant ridge in f − k space, the phase velocity relationship being f/k. Since

surface waves are usually the strongest arrivals they also dominate the power of the f −k spectrum to assist wavefield identification. However, a large number of traces are usually

necessary to provide sufficient wavenumber resolution without over-interpolation and avoid

spatial aliasing. Higher modes can usually be more accurately extracted with sufficient k

resolution [89, 96]. Methods to improve spectral resolution with limited 12- or 24-channel


R1(t) . . . R12(t)........................................................................................

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x.................................................................................................................................................

..............

................

..................................................................................................................................................................................................

................

d.................

......

........

..............................

........

Figure 1.7: Multi-channel field layout showing a linear spread of 12 geophones.

data includes stacking f − k spectra of individual common shot gathers (CSG) [85, 370]

or common midpoint gathers (CMP) [108, 110].

Multichannel frequency-slowness: MASW f − p The f − p method was originally

proposed in [198] whereby a time-distance (t−x) shot record is plane-wave transformed into

a intercept-slowness (τ−p) wavefield by slant-stack [331, 363], the slowness limits correctly

chosen to encompass the surface wavefield. The τ − p image is then FFT transformed in

the intercept (τ) direction to form a frequency-slowness (f − p) image and the ridge of

this spectrum is the dispersion curve, phase velocity being the inverse of slowness. Similar

to the f − k method, many traces at small interval are ideally required to avoid spatial

aliasing [89], but are not required to be at equal spacing. Like the f−k method, it assumes

plane wavefronts thus is strictly invalid in the near-field, where cylindrical wavefronts are

diverging. However, it has been shown to be accurate in the near field [79] but with less

resolution than other techniques for higher mode detection [237, 240].

Vibroseis Similar to the steady state method, a vibratory source oscillates over a

programmed sweep and linear moveout events are picked off the records. This method

has been utilised in [236, 239] and the swept frequency allows detection of higher modes

which take appreciable energy [240].

Two-dimensional arrays Using short period microseisms, comprising natural (atmo-

spheric, volcanic etc.) or cultural (traffic, machinery etc.) sources, surface wave dispersion

can be calculated from time series data over two dimensional sensor arrays by f −k meth-

ods or spatial autocorrelation analysis [324, 322]. With nonlinear arrays, both active or

passive sources can be equally employed [373, 374].


1.4 Dispersion modelling

Prior to 1950, methods used to calculate surface wave dispersion involved manually

solving large systems of simultaneous equations. As such, only simple two layer cases

were ever considered [12, 147]. The Thomson-Haskell scheme of 1950/1953 [320, 107]

allowed many layers to be modelled and was a springboard for the development of many

parallel matrix methods. The matrix methods solve the eigenvalue problem of the system

of differential equations which defines surface wave propagation in a layered half-space.

Most construct a dispersion equation, which is an implicit function of frequency, phase

velocity (wavenumber) and the thicknesses, elastic parameters and damping of the layers.

The dispersion curves are then the roots (eigenvalues) for possible modes of propagation

at any particular frequency. Other names for the dispersion equation are the ‘period’ or

‘secular’ equation/function.

1.4.1 Matrix methods Analytic matrix methods for construction of the dispersion

equation include [35, 82, 91, 161]:

1. Transfer matrix - Most commonly used method, especially in earthquake and explo-

ration seismology, after [320, 107];

2. Stiffness matrix - Complementary method, often favoured by engineers, after [141];

and

3. Reflection-transmission matrix - Generalised R/T coefficients for the entire stack of

layers by [143];

All these methods essentially allow the propagation of the stress-displacement field through

the layer stack from a known value at a reference depth. They are analytically exact and all

equivalent [35], but still employ piecewise-constant approximations within the layers [51]

for propagating the stress-displacement field through the layer stack. The fundamental

differences are how the matrix determinant, which defines the implicit dispersion equa-

tion, is calculated. They are collectively known as propagator matrix methods which will

distinguish them from the approximate linearised methods, which employ finite element

techniques.

Approximations based on numerical finite element methods, usually referred to as

linearised stiffness matrix methods [91, 344], are also used to solve the surface wave eigen-

problem. By subdividing the layers, the trancendental functions (complex exponentials)

can be represented as closed form in wavenumber. This is usually for the benefit of full

simulation (Section 2.3.2) by removing the need for costly wavenumber integration, but is

not rigorous and allow error propagation. Other methods used for the dispersion equation

which do not involve analytic propagator matrices include:

1. Finite element methods [185];

1.4. Dispersion modelling 21

2. Finite difference methods [30]; and

3. Direct numerical integration [317].

Numerical methods solve an initial-boundary value problem while matrix methods are the

solution to a two-point boundary value problem. The advantages of these over propa-

gator matrix methods, such as the modelling of lateral heterogeneities with finite differ-

ence/elements and continuous velocity-depth functions with numerical integration, how-

ever are rarely used in practice. Similarly, analytic dispersion solutions such as for a

Gibson half space [353] which are very rapid are rarely used due to the unrealistic model.

1.4.2 Propagator theory All matrix methods assume a half-space of homoge-

neous, isotropic and linearly elastic layers. The transmission (or radiation) boundary

conditions for traction are strictly:

1. Vanishing of stresses at the free surface (z = 0);

2. Continuity of displacements and normal and tangential stress components (i.e. force)

between layers; and

3. Displacement amplitude decays exponentially with depth, to vanish at z = ∞.

The last condition implies that there is no upcoming wave in the homogenous half space,

often referred to as a radiation condition, and as such only propagates homogeneously in

the x direction in this region.

The term propagator matrix actually comes from the work of Gilbert and Backus in

1966 [95], and the Thomson-Haskell method is a special case of their generalisation. The

eigenvalue problem derivation of the propagator matrices and how they transfer the stress-

displacement field, rather than seismic wave amplitudes, through a system of homogenous

layers are derived in [12, 5, 145, 332], and also as part of the experimental reports in

[219, 282, 325, 366].

1.4.3 Numerical considerations The numerical implementation of plane-wave,

propagator matrix methods comprises:

1. Dispersion equation calculation; and

2. Root bracketing for eigenvalues;

which are subject to several numerical considerations.

High frequencies Even though the determinant for the dispersion equation usually

remains finite or identically zero, a problem can arises when individual products in the

determinant used to construct the dispersion function. The high frequency problem is well

known for Rayleigh wave dispersion and has been researched by several authors [2, 69,

151, 291]. Most applications will not encounter this problem at practical frequencies and

more recent methods are not subject to the overflow [2, 49].


Stiff-over-soft layers Under some conditions the determinant used to create the sec-

ular function will be complex, thus root computation becomes more difficult [161]. This

indicates there are no Rayleigh waves (whose amplitude decreases with depth) propagat-

ing beyond a critical frequency [282]. In the case of a single stiff layer overlying a softer

half space the roots will be real only for wavelengths much greater than the upper layer

thickness. For shorter wavelengths only the real part of the determinant is generally taken

[311]. Similar to the high frequency problem, this case is common in pavement evaluation

testing.

Underwater Another problem occurs in sites overlain by water, where new wave types

are generated. If the R-wave velocity of the bottom sediments is less than the P -wave

velocity of the overlying water, so-called Scholte waves are generated. Conversely they are

called generalised Rayleigh waves and for both types the velocity is slightly less than VR

at the Earth-air interface [184, 311, 242].

Root finding In calculating the roots (zeros) of the dispersion equation most authors

(such as [290]) choose to fix the frequency (ω) then iterate over the desired range of phase

velocities c until the function changes sign. The precise value of c for a particular ω can

then be found by narrowing down by any number of methods, such linear or quadratic

interpolation, Hessian interpolation etc. Kennett [145] explains the advantages of fixing

the velocity and varying frequency, primarily for the reflection-transmission (R/T ) matrix

representation [35]. Since the dispersion equation is highly oscillatory, there will often be

a tradeoff between accuracy and efficiency in solving for the roots [219].

Multimodal solutions Not all the roots of the secular equation correspond to true modal

solutions of surface waves. Real roots occur when c < VS < VP and correspond to the

normal propagation modes. If c becomes greater than VS or VP complex roots will arise

and these are the modes of propagation of leaking modes. They give rise to complex phase

velocities, the imaginary part describing the intrinsic damping of the system which causes

attenuation with distance. Real solutions for these guided waves must be considered using

P -wave layer matrices, since these modes comprise multiply reflected acoustic waves in

the near surface.

1.4.4 Published computer codes While many codes have been developed as part

of research theses, they are generally not published. The published ‘Fast Surface Wave’

Fortran code of [290] is based on the transfer matrix method and is widely used. Two more

codes, both employing transfer and linearised stiffness matrix methods were published in

[344]. The transfer matrix methods are also in [117], which also provides modal summation

synthetic seismograms. Similar transfer matrix methods are part of the Matlab code of

[161], based on the R/T methods of [49, 122], the original Fortran code of [122] also

available. Matlab code for isotropic dispersion is listed in [47].

1.5. Inversion to Earth models 23

1.5 Inversion to Earth models

1.5.1 Dispersion inversion By far the most widely used forward modelling kernels

in the inversion of surface wave dispersion are the matrix methods [35]. The geophysical

model comprises a stack of flat, homogenous layers, so there is an immediate systematic

error in the inversion of real Earth data where structure is almost certainly not. However,

the approximation is accurate enough, and these methods were first applied in earthquake

seismology [64], much later in shallow applications.

Dispersion is primarily sensitive to subsurface shear wave velocity [59, 352], thus VS is

usually the only parameter which is sought. The earliest dispersion inversions employed

numerical partial derivatives, [64] but development of analytical partial derivatives [316]

allowed more rapid inversions and are more commonly used. Some, especially nonlinear or

global optimisation, incorporate layer thicknesses. Density is invariably excluded due to

its negligible effects, but will depend on the amount of expected contrast and estimation

uncertainty in the parameter [40], which for near surface materials, are quite low. Nev-

ertheless, calculation of elastic parameters such as the shear and constrained moduli are

only affected to first order from estimation errors in density or Poisson’s ratio.

Like inversion of other geophysical data, the procedure suffers from the ill-posed nature

of inverse problems [223], worsened with noise in the data. That is, even with infinite,

noise-free data and a perfectly parameterised model, the solution will not be unique.

For this reason, ‘fine’ Earth structure is especially doubtful from conventional dispersion

inversion [152]. This problem is even more so for group velocity inversion due to the

derivative relationship with phase velocity [153]. Surface wave inversion does not suffer the

same degree of nonuniqueness as body wave inversion, however resolution of a low velocity

layer is generally poor with a realistic set of surface wave data [335, 362]. The surface

wavetrain is both a function of the source mechanism and the Earth structure traversed,

thus the inversion actually comprises two problems, not completely independent [144].

1.5.2 Inverse optimisation methods Inversion techniques aim to minimise an

objective function, which comprises the RMS error between the observed and forward

modelled data. Historically, inverse problems in geophysics are linearised by appropriate

transforms and optimisation techniques are local least-squares methods [170]. The early

publications on the subject are still valuable references today [16, 127, 350, 128], and some

of these used surface wave inversion for demonstration.

However, the objective functions in almost all geophysical inversion are nonlinear and

local search methods can be made to account for this by iteratively ‘jumping’ or ‘creeping’

through model space [284]. Global search methods do not rely on a smoothly varying

objective function and are sometimes referred to as nonlinear optimisation. Linear and

nonlinear actually refer to the (assumed) relative change in objective function with per-

turbation in model parameters. Local and global refer to the way in which an objective


function is scanned for an ‘optimal’ solution. An early overview of surface wave inversion

methods is in [144].

It is often the ‘forward’ or ‘direct’ problem of calculating values on an exactly known

structure which which provides the most difficulty in an inverse problem [245]. The effi-

ciency of the forward calculation is of vital importance if an inversion algorithm is to be

computationally feasible. Linear minimisation is a computationally more efficient method

since it requires less direct calculation, however, it is prone to poor accuracy and instability.

In the case of surface waves, weakly contrasting and/or gradational boundaries represent

a linear problem, however sharp contrasts causes the dispersion curve to become a more

nonlinear function of model parameters, more readily addressed with global optimisation.

Local inversion Inversion is performed using a plane-wave matrix method to model

the phase and/or group velocity dispersion for a layered Earth model [144]. The modelled

dispersion is compared to the observed dispersion and model parameters adjusted by the

assumption of local linearity to match the modelled and observed curves to within a

user-defined RMS error. Model parameters are usually confined to the layer shear wave

velocities, both since sensitivities are largest and partial derivatives are available in analytic

form. Nonlinear optimisation in a Gauss-Newton framework can also be applied.

Least-squares optimisation were applied in the early work of [64] and have been the

mainstay method of most earthquake and engineering surface wave inversions as well as

that of current commercially available systems. Higher mode incorporation can assist the

convergence in some cases [89, 357].

Global inversion Global optimisation methods are less widely publicised but have been

employed, while still assuming 1D models and using the matrix modelling methods. They

do not assume any linearity, and, being derivative free, other model parameters such as

layer thicknesses can be introduced. By scanning a much broader model space, the global

minimum is more likely to be found. The drawback is computational inefficiency, since

many hundreds or thousands of iterations are required.

Genetic algorithms have been applied to upper crustal investigations from both Love

[362] and Rayleigh [186] wave data as well as for pavement evaluation by SASW [124].

Simulated annealing is a related method, which has been successfully used in multi-mode

MASW inversions [20, 22] as well as in crust-mantle investigations [194]. Other Monte

Carlo methods include clustering algorithms [163]. Neural networks have also been used

at both crustal [58] and site scales (by SASW) [199, 351] for SASW dispersion inversion,

as yet unreported from a MASW perspective.

Coupled inversion Simultaneously inverting surface wave dispersion and another dataset

which describes a parameter other than shear wave velocity is a mathematically better

posed problem. Attenuation and dispersion are intrinsically related and employed in

coupled inversions in both earthquake [165] and engineering [260] investigations for simul-


taneous VS and Q models. Employing amplitude ratios and particle orbits along with

dispersion can also reduce ambiguities [323].

Coupled receiver function and surface wave dispersion inversions have been employed in

crust and upper mantle investigations [137, 230]. In shallow engineering, coupled electrical-

surface wave inversions are described in [53, 114, 211]. In all cases, the two parameters

are required to be confined within the same layer interfaces.

1.5.3 Alternative inversion methods This chapter will conclude with a men-

tion of some methods which do not employ the matrix methods to invert a traditionally

observed dispersion curve, including:

1. Algebraic;

2. Waveforms;

3. Plane wave spectra;

4. Feature vector;

5. Tomography; and

6. Numerical methods.

Algebraic methods These methods are a direct inversion of the dispersion curve based

algebraic approximations. While they are very rapid, essentially instantaneous, they are

rarely applied nowadays due to the unrealistic models which they produce.

One method employed in the very early engineering surface wave surveying involves

simple scaling of the observed dispersion curve. Measured wavelength (λ) is reduced to

approximate depth by:

z′ =λ

k(1.21)

where k is the reduction factor. Next, c at each wavelength is converted to approximate

β′, usually by a simple multiplication factor of 1.1, that is β′ = 1.1c.

The λ scaling in early work simply assumed a depth of λ/2 [135]. Later, attempts were

made to optimise the parameters for the transform. One suggestion was to use λ/2 in more

homogenous sites and λ/4 where larger velocity contrasts exist, with λ/3 as a reasonable

compromise [202]. Similarly, λ/3.3 was recommended by [341, 342]. Later analytic studies

of normally dispersive trends also conclude that λ/3.3 is an appropriate reduction but for

more mild vertical heterogeneity the factor is closer to λ/2 [92]. The reduction factor has

also been shown to be offset dependent [342].

The c scaling is based on the theoretical ratio of Rayleigh wave velocity to homogenous

half-space shear velocity over the possible physical range of Poisson’s ratio (σ). The

fraction c/β ranges from 0.862 to 0.955 over the σ range of 0 to 0.5. Usual assumption


of perfect elasticity (σ=0.25) gives c/β=0.914 but a more correct estimate can be made

from Equation 1.2 [3]. In early CSW and SASW work, this was the only method for

backcalculation of moduli [1]. Nowadays, the β′ − z′ function is often referred to as

approximate inversion [82] and employed to generate an initial model for the more complex

inversion methods. The approximate inversion method is also summarised in [4, 82, 195,

202].

There do exist analytic solutions to simple 2-layer cases but have limited use [22, 96].

Other algebraic methods include a linear c−z model [1], nonlinear regressions [92, 250, 341]

and Gibson half-space [336, 337]. Lamb wave dispersion can be calculated analytically and

is most useful for asphalt and road base interpretation [133], but can also be applied in

geological cases with a stiff surface layer [243]. Only the ‘approximate inversion’ and

‘Gibson half-space’ methods have been reported in field studies by practitioners, such as

landfill surveys [41] and Vibroseis MASW dispersion analysis [353]. These cases correlated

well with borehole depths and discrete layered models respectively.

However, all these algebraic methods are strictly invalid for rigorous interpretation of

dispersion in irregularly layered sites [258]. Nevertheless, they can be used for a rough first

impression of the site characteristics and as a basis for automatic generation of a starting

model for linearisaed inversion, as will be demonstrated in this thesis.

Waveform inversion Inversion of body waves (acoustic or reflected waves) has been

employed in exploration geophysics for quite a number of years [174], usually performed

in the spectral domain [266]. In surface wave inversion, it was earthquake seismology

which first realised the shortcomings of dispersion inversion and utilised observed wave-

forms for estimating deep structure [225]. Many employ a ‘secondary observable’, such

as time-frequency transformation [45] or differential [246] of the data prior to inversion.

The use of receiver functions, which are the vertical component deconvolved out of the

radial component, eliminates source effects [300] but valid for body wave phases only.

Forward surface waveform calculations usually employ modal summation synthetics, since

travel paths are long and body waves not required. For the inversion, global optimisation

including genetic algorithms [279], neural networks [164] and neighbourhood algorithms

[364] would be suitable.

In shallow surface wave inversion, synthetic seismograms have been used qualitatively

for model appraisal [50], assisting mode recognition [188] and appraising attenuation mod-

els [212]. They have also been employed for modelling seafloor seismic data [75]. A

waveform inversion using modal summation synthetics in [6] was used to create unique

dispersion curves over laterally heterogenous structure, however, a true waveform inversion

has not been reported in shallow surface wave surveying. In earthquake seismology, mod-

elling and interpreting lateral discontinuities with synthetic seismograms is a consistent

research topic [68, 231].


1D plane-wave spectra Synthetic seismograms in the Fourier domain are a slightly

better posed problem than in the time domain. An approach in [314, 315] inverted the

mode coefficients of observed seismograms by multimodal seismograms. The ellipticity

from 2-component seismograms was preferred due to a stronger sensitivity of near surface

layers. A similar approach in [187] inverted the observed spectra by a conjugate gradient

optimisation with numerical partial derivatives. The starting model in this case was was

ironically determined from the multichannel phase and group velocity dispersion.

Feature vector An approach in [203] utilised an inner product between the spectral

envelopes of two synthetic seismic traces to create a feature vector. By developing a

set of training curves and assuming the feature vector is unique to a particular layered

earth, observed data could be inverted by a neural network. This method required some a

priori knowledge (eg. borehole data) in order to create a training set at a particular site

and performance in complex layering was not reported.

Tomography More recently tomographic imaging in deep [216] and shallow [176] appli-

cations has been used to map 3D variations in shear velocity and Q, but only qualitatively.

Generally, group velocity is used, recorded over multiple source-receiver paths with an areal

array, to provide the necessary quantity of travel times.

Numerical methods These incorporate finite difference (FD) and finite element (FE)

procedures. They have been employed for a direct inversion of physical modelling data

[57, 174] or material damping estimation [368]. While 2D or 3D features can only be

readily modelled with finite difference methods, they are currently not practical for efficient

inversion.


29

CHAPTER 2

Outline and methods of thesis

2.1 Introduction

The primary objective of this thesis is to develop an alternative to current 1D multi-

channel surface wave inversion procedures and provide a better indication of the limitations

of the method for use in engineering site investigation. The motivation for these aims is

based on shortcomings revealed in both published literature and field experiences with

surface wave inversion surveying. Although Chapter 1 showed that surface wave inver-

sion is quite accurate in most situations, there are some engineering scenarios which the

method fails to correctly interpret. It is thought that these are due to the following:

1. Lateral discontinuities;

2. Higher mode contribution; and

3. Resolution and accuracy of: (a) Data and (b) Model.

The first problem (lateral discontinuities) is probably the biggest offender in misinterpret-

ing a layered earth structure from field dispersion data. It arises from large 2D and 3D

structures in the vicinity of the source and receiver array(s). To date, all analytic surface

wave modelling schemes assume flat layering, both between the source and receiver(s) or

along the receiver array. Numerical (finite difference) methods can model lateral discon-

tinuities, within a resolution limit, but at the expense of time and accuracy. However, it

has been shown numerically and experimentally that certain discontinuities such as dip-

ping strata and fault steps can be interpreted using 1D methods with minor errors [313].

The lateral discontinuity problem is highly intractable and, as such, this thesis remains

confined to 1D layering.

The second problem (higher mode contribution) occurs due to the complex wavefield

generated in some shallow engineering scenarios. In an ’irregularly dispersive’ site, when

layer shear velocity contrasts and reversals become high, the normally continuous disper-

sion curves (fundamental and higher modes) are represented by an ‘effective’ dispersion,

which is a combination of two or more modes, often exhibiting ‘mode jumping’. When

traditional modelling methods are used, the combination of: (a) Incorrect mode identifi-

cation and; (b) Inaccurate forward simulation causes a systematic error in inverted model

parameters. This problem requires a new forward modelling kernel to replace the tra-

ditional plane-wave propagator matrix methods, by both correctly modelling ‘dominant

higher modes’ and not requiring mode identification.

The third problem (resolution and accuracy) involves two issues for the observed data

and inverted models respectively: (a) The relatively poor understanding of uncertainty

in measured dispersion and; (b) How data errors propagate through the optimisation

30 Chapter 2. Outline and methods of thesis

procedure into the final model. The data resolution and accuracy issue requires a thorough

investigation of how field layouts, acquisition errors and dispersion processing impart on

the measured dispersion curve, which is the data used as input for inversion. Model

resolution and accuracy must then consider how these errors propagate through to the

final model parameters in addition to the inherent nonuniqueness in the inversion for a

rigorous appraisal of the surface wave inversion results.

The problems 2 and 3 (listed above) are addressed in the following order in this thesis:

1. Data resolution and accuracy (Problem 3a);

2. New inversion procedure (Problem 2);

3. Quantitative model appraisal (Problem 3b).

Note that the resolution and accuracy problem is divided into two parts: (a) The acqui-

sition/processing stage (data) and; (b) Inversion appraisal stage (model), to be addressed

in separate chapters. Starting with synthetic models, the methods are then applied to

field data. For each item above to be addressed, this chapter outlines the issues involved

and a review of related and previous research. However, we will begin by describing the

methods and models to be employed in this research, where dominant higher modes and

low-frequency effects are the important features.

2.2. Synthetic methods, models and verification 31

2.2 Synthetic methods, models and verification

Two procedures to forward calculate dispersion will be used in this work. The tradi-

tional plane-wave propagator matrix method, based on the ‘Fast Surface Wave’ code of

[290] will be called the FSW method. The new procedure for calculating surface wave

dispersion, based on full-waveform P -SV reflectivity code for synthetic seismograms [214],

will be called the PSV method. This section describes the new forward dispersion cal-

culation procedure, basically plane-wave transform of a synthetic shot gather, illustrated

with the 1D models to be employed throughout this thesis.

2.2.1 Synthetic seismogram generation Synthetic seismogram generation em-

ploys methods to solve the wave equation in a medium from a realistic source. This can

be achieved both by analytic or numerical (finite difference) methods. Numerical meth-

ods can solve for all wavefields and model lateral discontinuities, but are approximate

and moreover computationally expensive [167, 339]. Analytic methods are restricted to

plane, horizontal layers and the sub-class of ray theory methods only incorporates body

wave contributions. Those which incorporate surface waves are the integration/summation

procedures of [214]:

1. Wavenumber or slowness integration;

2. Wavenumber summation;

3. Modal summation.

The reflectivity method is a wavenumber or slowness integration method, so called be-

cause of the recursive use of reflection-transmission coefficients for the calculation of the

integrand [343]. While it was initially derived for body waves only [87], the relationship

between the layer matrices and R/T properties in [143, 146] allowed surface wave contri-

butions to be incorporated [147]. This is usually called the extended reflectivity method

[214], where all internal and surface multiples are incorporated. A history of developments

in the reflectivity method are summarised in [191].

In essence, the procedure involves [346]:

1. Source potential decomposed into plane-wave components;

2. Plane-wave response of the layered medium evaluated; and

3. Summation over all plane-wave components.

Step 2, the elastic wavefield or medium response (Green’s function) calculated in the

f − k or f − p domain, is the most expensive part of the computation [191]. Herein is the

overlap with with matrix methods for the calculation of the plane-wave response. While

the transfer [106] or stiffness matrix [180] methods can be used, the recursive calculation


of reflectivities and transmissivities [214, 343] are numerically stable for all frequencies

and slownesses. Nevertheless, in a weakly irregularly dispersive model, the transfer and

R/T methods have been shown to be identical [35]. The R/T matrices are of size 2×2 for

P -SV waves and SH scalar contributions are not incorporated. Intrinsic damping is incor-

porated by use of complex elastic constants [214], usually assumed frequency independent

at the frequency ranges observed in practice.

Step 3 is the integration or summation along horizontal wavenumber or slowness fol-

lowed by inverse Fourier transformed to recover the time domain response. This is repre-

sented as [191]

u(x, ω) =

∫ inf

0ω2pu(ω, p)J0(ωpx)dp (2.1)

where (ωpx) are frequency, slowness and offset respectively, J0 is the Bessel function of

order zero and u(ω, p) is one component of the reflectivity function. Cylindrical spread-

ing is incorporated by the use of Bessel functions in the source function decomposition.

Summation of near and far field responses provides vertical and inline seismograms (in the

P -SV case) at any source offset.

One restriction is that only flat-layered models can be modelled and no intermediate

solutions are possible, prohibiting ‘snapshots’ such as in finite-difference methods. While

spherical Earth and dipping layer approximations do exist [346], horizontal layers are

the traditional models used in surface wave inversion. Another weaknesses is that many

slownesses are required to be summed to truly represent the full wavefield. Fortunately

in these tests, the dominant desired component are surface waves. Thus, a relatively

narrow slowness band can be used while still recovering the bulk of the normal body

waves (direct, refracted, guided and most hyperbolic reflections). This means that the

number of slownesses over which to integrate can be minimised without risking aliasing.

Aliasing due to insufficient slownesses is evident from strong sinusoidal ringing down the

traces.

2.2.2 Source functions and numerical aspects While dispersion can be ob-

served from the Green’s functions [281], the reflectivity method requires a source spectrum

for realistic seismograms. The source band limitations, as opposed to impulse response,

provide more realistic seismograms. Figure 2.1. The Berlage wavelet is causal and provides

a broad spectrum with a similarity to recorded impact [78, 270] and explosive [155, 372]

source time histories, thus it was adopted as the ‘standard’ wavelet in this work. Only

vertical impact point sources at the surface are employed. A damping factor (α) of 8f ,

where f is the centre frequency was applied.

While the PSV method is not constrained by issues with the plane-wave propaga-

tor matrix methods (Section 1.4.3), there are other factors such as aliasing (in time or

distance), windowing and attenuation, outlined in [191]. However, for modelling shallow


−1

0

1K

uppe

rSource function (normalised)

−40−20

0Power spectrum (dB)

−1

0

1

Ric

ker

−40−20

0

−1

0

1

Gab

or

−40−20

0

−1

0

1

Ber

lage

−40−20

0

−1

0

1

Ray

leig

h

−40−20

0

−0.02 −0.01 0 0.01 0.02−1

0

1

Time (s)

Dira

c

0 50 100 150 200 250−40−20

0

Frequency (Hz)

Figure 2.1: Displacement (solid line) and velocity (dashed line) functions in the time (left)

and frequency (right) domains for five analytic source wavelets of 40 Hz centre frequency.

References for each are: Kupper [158]; Ricker [256]; Gabor [123]; Berlage [11]; and Rayleigh

(only real part shown) [123].

surface waves some important points of note are:

1. Source spectrum bandwidth: Low-frequency cosine rolloff of 0-fG Hz and fN/2-FN

high frequency rolloff, where fG is the geophone resonant frequency (Hz) and fN the

sampling Nyquist frequency (Hz);

2. Minimum number of slownesses: Limited by farthest receiver, usually 100 per 48 m

of far-offset, however 200 or more when the top layer velocity is very slow;

3. Phase velocity and integration limits for calculation of R/T matrices: 0.4βmin to

1.5αmax, where βmin and αmax are the minimum and maximum shear and compres-

sional velocities in the model; and

4. Phase velocity tapers to damp numerical arrivals at the phase velocity limits: 0.8βmin

to αmax.

Usually a 4.5 Hz lower corner frequency is a good compromise between the 1 Hz or 2 Hz


‘preferred’ surface wave geophone to the ‘standard’ 8 Hz refraction geophone. No over- or

under-damping is simulated, nor spurious geophone frequencies.

2.2.3 Dispersion observation To extract the multichannel dispersion of time-

distance (t−x) shot gathers, either frequency-slowness (f−p) [198] or frequency-wavenumber

(f − k) [89] plane-wave transforms are used, introduced in Chapter 1. Some practical as-

pects of the transforms are outlined in [179], which include normalising the power at each

frequency for display purposes. The f − p method comprises a slant-stack transformation

of the shot gather into the τ − p domain, followed by FFT down the τ axis to create a

f − p plane. The f − k transform comprises a cascaded FFT, first across the traces (k),

then down (f).

The Nyquist frequency for either method is where slownesses or wavenumbers exceed

the Nyquist values. In the case of f − p, this is nonlinear with frequency as:

pN =1

f∆x(2.2)

and by f − k it is a constant

kN =0.5

∆x(2.3)

where f is the frequency (Hz) and ∆x is the trace spacing (m). Aliased slownesses or

wavenumbers can be recovered by reflecting the negative portions of each transform and

concatenating with the positive slowness or wavenumber portion.

For f − p, the least squares slant-stack described in [363] and implemented in Matlab

[276] was used, followed by FFT in the τ direction. The τ − p alias can be avoided by

using a fine sampling of slowness defined by [331] as

∆p < 1/xrfmax (2.4)

where xr is the spread length (m) and fmax is the highest frequency (Hz) desired to be

reconstructed. In any case, since surface waves are generally confined to a narrow slowness

range combined with the fact that full wavefield reconstruction is not required means that

adequate ∆p can be achieved with a modest number of slownesses.

As for slowness or wavenumber resolution, this can be improved by interpolation,

since the number of traces is usually much smaller than samples in the time dimension.

Slowness (p) interpolation is accomplished by setting a slowness vector for the slant-

stack with finer sampling than required to simply overcome aliasing. A limit exists in

τ − p transformations of real data using least-squares due to memory constraints, since

a large matrix inverse is computed. With 512 time samples, the maximum number of

slownesses allowed was 256. This is a second reason for preferring a narrow slowness band,

which also accomplishes the bulk of the undesired wavefield suppression. Wavenumber (k)

interpolation is accomplished by merely zero padding the gather to the desired number

of traces. This can be arbitrarily large, but not usually larger than the padding used in


Table 2.1: Typical 1D synthetic models with real units taken from past publications.

The depth ranges indicate a depth to the top of the half space. The model descriptors

in parenthesis describe both the average absolute in, and contrasts of, shear velocities.

Absolute averages are: ‘slow’ (<250 m/s); ‘medium’ (250-400 m/s); and ‘fast’ (>400m/s).

Average contrasts between layers are: ‘low’ (<1.5); ‘moderate’ (1.5-2.5); and ‘high’ (>2.5).

‘Shallow’ ‘Intermediate’ ‘Deep’

(<5m) (5-15m) (>15m)

Tuomi and Hiltunen [330] Tokimatsu et al [325] Szelwis and Behle [315]

(slow/moderate) (slow/moderate) (medium/moderate)

Ganji et al [91] Roth and Holliger [273] Gucunski and Woods [101]

(slow/moderate) (fast/moderate) (medium/moderate)

Park and Kim [244] Foti [82] Lai and Rix [161]

(medium/low) (fast/low) (fast/low)

Table 2.2: Parameters of the 1D models from Tokimatsu et al [325], where h are the

thicknesses (m), ρ the densities (g/cc), VP and VS are the P - and S-wave velocities (m/s)

and σ the layer Poisson’s ratios.

Case 1 Case 2 Case 3

h ρ VP VS σ VS σ VS σ

2 1.8 300 80 0.46 180 0.22 80 0.46

4 1.8 1000 120 0.49 120 0.49 180 0.48

8 1.8 1400 180 0.49 180 0.49 120 0.49

∞ 1.8 1400 360 0.46 360 0.46 360 0.46

the time dimension. Frequency resolution is usually not a concern, as the number of time

points and sample rate can be easily chosen within aliasing limits.

From the plane-wave transforms, the peak of the dominant lobe is picked to return

a set of (f, p) or (f, k) points which are converted to phase velocity at each frequency

by either 1/p or f/k respectively. The phase velocity dispersion curve represents a path

average over the spread and the ‘sounding point’ is conventionally assumed at the spread

centre. More suitable plane-wave transforms accommodate cylindrical spreading, such as

the is the Fourier-Bessel f − p method of [78].

2.2.4 Layered models There is no standard testing model for surface wave inver-

sion testing. However, over the years many models have emerged, including synthetic

(used for numerical testing), inverted (results of field tests) and borehole models (based

on downhole or penetrometer logs). Table 2.1 shows some typical irregularly dispersive

models from the literature. Other cases with velocity reversals and large contrasts are

[43, 99, 100, 182, 271] and the dimensionless cases in [293, 281, 269]. Some normally dis-


persive synthetic models include [114, 86, 356] and those based based on borehole data

include Treasure Island [161] and ISC’98 [82]. Other interesting models from surface

wave inversion confirmed with borehole/geological data are [22, 78]. Asphalt and road

base structures inferred from surface wave testing over large scale physical models are in

[9, 133, 268, 351].

The synthetic models to be considered in this work are the cases from [325, 322],

hereafter referred to as Case 1, 2 and 3. Case 1 is normally dispersive while Cases 2

and 3 are irregularly dispersive. In Case 2, the second layer is a low velocity layer (LVL)

and in Case 3 it is a high velocity layer (HVL). These are relatively soft (low VS) layered

models, probably based on the reclaimed soils of many sites in Japan [322] chosen as

a basis for the synthetic tests of this thesis because of the nature of the dispersion they

generate - dominant higher modes due to large velocity reversals and/or contrasts between

the layers - over depth ranges required for typical shallow, engineering site investigations.

The physical parameters are summarised in Table 2.2 and, while not given in the original

publication, attenuation has been assumed as Qα=100 and Qβ=45 for all layers.

2.2.5 Numerical procedure The numerical procedure for generating a dispersion

curve by full waveform P -SV reflectivity will now be demonstrated using the shear velocity

models of [325], shown in Figure 2.2. The synthetic shot gathers with a surface impact

source for these models are shown in Figure 2.3. An spread of 48 geophones at 1 m spacings

with 5 m near offset have been used. Sampling rate is 2 ms for 512 samples. These are

trace normalised to the RMS mean of each trace, usual for displaying surface waves.

Plane-wave transform of these shot gathers to f − p and f − k are shown in Figures

2.4 and 2.5. In both cases, the data were zero padded in time to 1024 samples and

slowness/wavenumber interpolated to 256 points. The upper slant-stack phase velocity

limit is 350 m/s and the lower limit 71 m/s (Case 1 and 3) and 110 m/s (Case 2). While

not as obvious, a velocity fan using these limits is also applied to the f − k images.

The double Nyquist has been chosen, which can be seen as the curved black line at high

f − p (Figure 2.4) or at 1 m−1 in f − k (Figure 2.5 and the picked dispersion is shown as

the white line which follows the dominant ridge in f − p or f − k space.

The black curves in Figure 2.6 are the f − p dispersion curves and the grey bands

show the theoretical FSW dispersion for each model, the edges of which are the modal

dispersion curves. In Figure 2.6(a), the PSV generated dispersion for Case 1 matches the

conventional FSW dispersion well, following the edge of the first grey band (fundamental

mode). However, in Cases 2 and 3, the PSV dispersion transitions (‘jumps’) up to higher

phase velocities (higher modes). These are also evident from the discontinuities in the

f − p or f − k planes of Figures 2.4 and 2.5. While a propagating surface wavetrain is

a superposition of ‘normal modes’, when higher modes are more energetic, they become

known as ‘dominant higher modes’. In Case 2 these occur at about 20 Hz intervals, starting


0 100 200 300 400

0

2

4

6

8

10

12

14

16

18

20

Shear velocity (m/s)

Dep

th (

m)

0.46

0.49

0.49

0.46

(a)

0 100 200 300 400

0

2

4

6

8

10

12

14

16

18

20


Dep

th (

m)

0.22

0.49

0.49

0.46

(b)

0 100 200 300 400

0

2

4

6

8

10

12

14

16

18

20


Dep

th (

m)

0.46

0.48

0.50

0.46

(c)

Figure 2.2: Shear velocity of the 1D models of Tokimatsu et. al. [325]: (a) Case 1; (b)

Case 2 and; (c) Case 3. The values within each layer are the Poissons ratios.

10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Offset (m)

Tim

e (s

)

(a)

10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Offset (m)

Tim

e (s

)

(b)

10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Offset (m)

Tim

e (s

)

(c)

Figure 2.3: Full-waveform P -SV synthetic common shot gathers of the 1D models of

Tokimatsu et. al. [325]: (a) Case 1; (b) Case 2 and; (c) Case 3.


Slowness (s/m x10−3)

Fre

quen

cy (

Hz)

(a)

4 6 8 10 12 14

10

20

30

40

50

60

70


Fre

quen

cy (

Hz)

(b)

4 6 8

10

20

30

40

50

60

70

80

90

100

110

Slowness (s/m x10−3)F

requ

ency

(H

z)

(c)

4 6 8 10 12 14

10

20

30

40

50

60

70

Figure 2.4: Frequency-slowness (f − p) transforms of the synthetic shot gathers of the 1D

models of Tokimatsu et. al. [325], showing the picked spectral maxima: (a) Case 1; (b)

Case 2 and; (c) Case 3.

Wavenumber (/m)

Fre

quen

cy (

Hz)

(a)

0.2 0.4 0.6 0.8

10

20

30

40

50

60

70

Wavenumber (/m)

Fre

quen

cy (

Hz)

(b)

0.2 0.4 0.6 0.8

10

20

30

40

50

60

70

80

90

100

110

Wavenumber (/m)

Fre

quen

cy (

Hz)

(c)

0.2 0.4 0.6 0.8

10

20

30

40

50

60

70

Figure 2.5: Frequency-wavenumber (f −k) transforms of the synthetic shot gathers of the

1D models of Tokimatsu et. al. [325], showing the picked spectral maxima: (a) Case 1; (b)

Case 2 and; (c) Case 3.


Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

(a)

20 40 6050

100

150

200

250

300

350

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

(b)

20 40 60 80 10050

100

150

200

250

300

350

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

(c)

20 40 6050

100

150

200

250

300

350

Figure 2.6: PSV frequency-phase velocity (f − c) dispersion from the f − p transforms of

the 1D models of Tokimatsu et. al. [325]. The plane-wave modal FSW dispersion images

are shown as grey bands: (a) Case 1; (b) Case 2 and; (c) Case 3.

0 100 200 300 400

0

2

4

6

8

10

12

14

16

18

20

Shear and phase velocity (m/s)

Dep

th a

nd w

avel

engt

h (m

)

(a)

0 100 200 300 400

0

2

4

6

8

10

12

14

16

18

20


Dep

th a

nd w

avel

engt

h (m

)

(b)

0 100 200 300 400

0

2

4

6

8

10

12

14

16

18

20


Dep

th a

nd w

avel

engt

h (m

)

(c)

λ/1λ/2λ/3λ/4

Figure 2.7: Phase velocity-reduced wavelength of the dispersion from the f −p transforms

of the 1D models of Tokimatsu et. al. [325]. Four wavelength reduction factors have been

applied and phase velocity is unscaled from the original f − c and the true shear velocity

models are shown in grey: (a) Case 1; (b) Case 2 and; (c) Case 3.


at about 25 Hz while in Case 3 only one dominant higher mode is evident over 10-15 Hz.

Although in Figure 2.6(b) and (c) the FSW method correlates with segments of the

PSV dispersion, the transition frequencies to dominant higher modes cannot be predicted,

thus it would be inadequate as an inversion kernel for the observed dispersion. More exam-

ples of these are outlined in Sections 2.2.6 and 2.3. In addition, there unusual behaviour

of the dispersion at low frequency which does not appear systematic. This is traditionally

called the ‘near-field effect’ and some aspects and new thoughts on this are discussed in

Section 2.4.

The f − c dispersion curve can be displayed as approximate depth-velocity by calcu-

lating a reduced wavelength by Equation 1.21 and multiplying c by 1.1. These are shown

with various reduction factors of the wavelength in Figure 2.7. In the normally dispersive

Case 1 (Figure 2.7(a)), the dispersion approximates the trend of the true shear velocity,

best for a wavelength reduction between 2 and 3. However, in the irregularly dispersive

cases (Figure 2.7(b) and (c)), the trend is less accurately reproduced. In Case 3, when a

HVL is present and dominant higher modes are generated, the correlation is invalid.

So, in summary, the forward modelling procedure comprises:

1. Set 1D model and synthetic acquisition parameters as used in field (offsets, sampling

rate, source type, geophone component etc.);

2. Generate time-offset synthetic shot gather (t− x) and return array in memory;

3. Process synthetic gather as per field data (gain, frequency/velocity window etc.);

4. Plane-wave transform as per field data (f −k or τ −p, at desired padding / slowness

limits etc.) and normalise power at each frequency; and

5. Automatically pick spectral peak at each frequency within defined f − k fan or

f − p band and return dispersion curve (f, c).

The processing stage, Step 3, is usually minimal, often simply a trace normalisation (to

RMS mean) and group velocity window with tapered edges. More complex pre-processing

such as [235] is not applied to ensure the surface wave modes retain their natural energy.

The transform stage (Step 4) often requires a few trials to obtain the best slowness limits

when using f−p. Starting with f−k with no velocity bounds is a good way for estimating

slowness limits for a τ − p transform. Picking dispersion from field data comprises Steps

3 to 5. The picking stage (Step 5) is usually done manually in this case, by encircling

the ridge in plane-wave space and can be quite subjective, since the surface wavefield is

often not immediately evident. Correct slowness limits and experience helps reduce the

nonuniqueness in picking dispersion. Similar synthetic cases with LVL’s and HVL’s will

now be used to verify this forward modelling procedure.


0 0.1 0.2 0.3 0.4 0.50.86

0.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96(a) 25 Hz

σ

c/β

0 0.1 0.2 0.3 0.4 0.50.86

0.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96(b) 70 Hz

σ

c/β

PSV 80 m/s PSV 120 m/s PSV 180 m/s PSV 360 m/s FSW Achenbach (1973)

Figure 2.8: Variation of half-space phase velocity to shear velocity ratio (c/β) with Pois-

son’s ratio (σ) by three methods: PSV dispersion (frequency dependent); FSW dispersion

(frequency independent); and an analytic approximation in [3]. Two frequencies are shown

to illustrate the low-frequency effects associated with the PSV method: (a) 25 Hz; and

(b) 70 Hz.

Table 2.3: Publications where dispersion curves with dominant higher modes have been

both: (a) Forward modelled from synthetic models by active source methods; and (b)

Inverted into a layered model.

Publication LVL model HVL model Forward kernel

(a) Forward dispersion

Tokimatsu et. al. [325] Case 2 Case 3 Transfer matrix

Roesset et. al. [101] Site B Site C Stiffness matrix

Gucunski and Woods [101] Case 3 Case 4 Stiffness matrix

(b) Dispersion inversion

Ganji et. al. [91] Case 4 Case 5 Linearised stiffness

Lai and Rix [161] Cases 2,3 N/A Transfer matrix

Forbriger [78]∗ Berkheim Bietigheim P -SV reflectivity

∗The dispersion was field data and the HVL case did not have an underlying LVL.


100 200 300 400

0

10

20 β (m/s)

Dep

th (

m)

0 0.1 0.2 0.3 0.4 0.5 σ

(a) Case 2 of Tokimatsu et al

100 200 300 400

0

10

20 β (m/s)

Dep

th (

m)

0 0.1 0.2 0.3 0.4 0.5 σ

(b) Case 3 of Tokimatsu et al

0 500 1000 1500

0

2

4 β (m/s)

Dep

th (

m)

0 0.1 0.2 0.3 0.4 0.5 σ

(c) Site B of Roesset et al

0 500 1000 1500 2000 2500

0

2

4 β (m/s)D

epth

(m

)

0 0.1 0.2 0.3 0.4 0.5 σ

(d) Site C of Roesset et al

150 200 250 300 350 400 450

0

20

40 β (m/s)

Dep

th (

m)

0 0.1 0.2 0.3 0.4 0.5 σ

(e) Case 3 of Gucunski and Woods

150 200 250 300 350 400 450

0

20

40 β (m/s)

Dep

th (

m)

0 0.1 0.2 0.3 0.4 0.5 σ

(f) Case 4 of Gucunski and Woods

Figure 2.9: Shear velocity (grey) and Poisson’s ratios (dotted) for the models of Ta-

ble 2.3(a).

150 200 250 300 350 400 450

0

10

20 β (m/s)

Dep

th (

m)

0 0.1 0.2 0.3 0.4 0.5 σ

(a) Case 4 of Ganji et al

150 200 250 300 350 400 450

0

10

20 β (m/s)

Dep

th (

m)

0 0.1 0.2 0.3 0.4 0.5 σ

(b) Case 5 of Ganji et al

250 300 350 400 450 500 550

0

20

40 β (m/s)

Dep

th (

m)

0 0.1 0.2 0.3 0.4 0.5 σ

(c) Case 2 of Lai and Rix

350 400 450 500 550 600 650

0

20

40 β (m/s)

Dep

th (

m)

0 0.1 0.2 0.3 0.4 0.5 σ

(d) Case 3 of Lai and Rix

0 500 1000 1500 2000

0

5

10 β (m/s)

Dep

th (

m)

0 0.1 0.2 0.3 0.4 0.5 σ

(e) Berkheim site of Forbriger

0 500 1000 1500 2000

0

10

20 β (m/s)

Dep

th (

m)

0 0.1 0.2 0.3 0.4 0.5 σ

(f) Bietigheim site of Forbriger

Figure 2.10: Shear velocity (grey) and Poisson’s ratios (dotted) for the models of Ta-

ble 2.3(b).


2.2.6 Algorithm verification An initial test is by modelling of a homogenous half-

space, as results can be corroborated with other analytic methods. This was done for four

shear velocities (VS) of 80, 120, 180 and 360 m/s at five different Poisson’s ratios (σ) of

0.01, 0.1, 0.2, 0.3, 0.4 and 0.49. Again, a 48-channel gather at 1 m spacings with 5 m near

offset is used, with an impulsive source and vertical component geophones. By picking the

phase velocity for each model, the variation of half-space phase velocity to shear velocity

ratio (c/β) with σ can be plotted. Dispersion generated with the FSW method was also

employed. In Figure 2.8, the solid curve is the analytic approximation in [3] and for for

each half-space shear velocity modelled, the curve is mimicked with both the PSV and

FSW dispersion values, at 25 and 70 Hz. The FSW dispersion is frequency independent

however at 25 Hz, the PSV dispersion produces poor estimates. At 75 Hz both methods

coincide, however at low σ the approximation is not recovered.

For more complex cases, comparison with the dispersion curves presented in publi-

cations by other workers is made. This also objectively compares different numerical

algorithms which have been developed to generate dominant higher modes. Similar logic

was applied in [8, 9], [91] and [271] who directly compared their methods with the results

in the previous publications of [133, 282], [268] and [99] respectively. Models with LVL

and HVL structures and dominant higher mode patterns similar to Case 2 and 3 of [325]

respectively are summarised in Table 2.3. They are shown in Figures 2.9 and 2.10.

In general, the model shear velocities were usually tabulated. However, of all the

models used in forward dispersion calculations (Table 2.3(a)), densities and Poisson’s

ratios were only supplied in [325] and and attenuation (quality factors) not given in any.

For the inversion models (Table 2.3(b)), all extra elastic parameters were provided, except

the damping in [91]. However, only shear wave attenuation (Qβ) was supplied, so Qα was

estimated by (9/4)Qβ . Where the model parameters were not tabulated ([78, 268]), they

had to be scaled off plots in the original publications. Each case of Table 2.3 was forward

modelled using the PSV method, where both the synthetic acquisition and processing

settings had to be defined based on sensible estimates. In [78], a ‘MASW’ dispersion

was observed and the source and and receiver positions were exactly known. However,

where the experimental parameters were unknown, these (along with the estimated elastic

parameters) will add systematic errors to the comparisons.

Comparisons of the modelled dispersion curves with figures from the original publica-

tions are illustrated in Figures 2.11 to 2.16. In general, the dominant higher modes are

modelled well, with major discrepancies only at low frequency, where the bandwidth of the

stiffness matrix methods extend to much lower frequencies than other methods. However,

as will be outlined in Section 2.4, the low frequency data is highly dependent on the spread

length. In Figure 2.14, the PSV method appears to also generate stronger modal transi-

tions than in the original publication of [268]. Similarly, in Figure 2.14, modal transitions


Figure 2.11: Comparison of the Case 2 and 3 dispersion from [325] with the PSV modelling

(thick grey line).

Figure 2.12: Comparison of the Site B and C dispersion from [268] with the PSV modelling

(thick grey line).


(thick grey line).



(thick grey line).


(thick grey line).

Figure 2.16: Comparison of two dispersion cases from [78] with the PSV modelling (thick

grey line).


Figure 2.17: Comparison of two cases of theoretical dispersion from [293] with the

PSV modelling (thick grey line).

Re<0,Im<0

Re<0,Im>0

Re>0,Im<0

Re>0,Im>0

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

100 200 300 400 500 600 7000

100

200

300

400

500

600

700

TheoreticalSASW MFT MASW

Figure 2.18: Dominant higher modes observed in the pavement model in [9], originally

conceived by [133]. The ‘Theoretical’ dispersion is originally from [133] and this, along

with the ‘SASW’ and ‘MFT’ points, are tabulated in [9]. The ‘MASW’ dispersion is from

the PSV method, observed by f − k transform. The grey bands are the FSW dispersion

function amplitudes, which are complex, thus require four shades.


in the original LVL dispersion data of [91] data are not evident and in the HVL case the

PSV dispersion has exceeded the scale of the original figure. However, that publication

employed the linearised stiffness matrix which is an approximate, non-rigorous solution.

The best comparison appears to be in Figure 2.16. This was to be expected, as both

employed the reflectivity method and plane-wave transform, where acquisition parameters

were nearly exactly duplicated.

In addition, two more cases with large contrast horizons are:

• Soft-over-stiff - theoretical dimensionless models of [293]; and

• Stiff-over-soft - the physical asphalt/road base model of [9].

For the dimensionless case of [293], sensible elastic parameters had to be assumed, which

were 1 m and 100 m/s for the thickness and shear velocity of the upper layer. The stiff

half-space shear velocity was then scaled appropriately based on the given shear moduli

ratios of 5:1 and 20:1 respectively. The model of [9] all elastic and (shear wave) damping

parameters supplied, in addition a table of dispersion data.

Figure 2.17 is particularly interesting. The original publication of [293] in 1935 was

focused on the discontinuity which occurs in a two-layer dispersion curve with increasing

rigidity of the underlying layer. At finite rigidity, the point of closest approach is called

an osculation point. The upper (dotted line) and lower (solid line) phase velocity curves

contact when the half-space shear modulus became infinitely rigid. The original work,

however, did not predict that dispersion would actually ‘jump’ up to the higher mode

branch. In Figure 2.17(b), this occurs at the osculation point when the ratio of upper to

lower rigidities reaches 1/20. The transition back to the fundamental mode was also not

predicted, shown by the grey dashed line, when the wavelength/layer thickness reaches 18.

It is prophetic that their manual calculations nearly 70 years ago correlate with modern

modelling and processing techniques.

Figure 2.18 models an asphalt and road base model originally devised in [133], later

applied in the studies of [8, 124, 91]. The ’Theoretical’ (normal mode solutions) of [133] are

plotted along with the dispersion measured from synthetic seismograms calculated by the

linearised stiffness matrix method [9] where two dispersion methods were implemented:

standard ‘SASW’ phase differences, and a multiple filter/crosscorrelation (‘MFT’) tech-

nique, based on waveforms calculated by a linearised stiffness technique. The ‘MASW’

dispersion is calculated from the PSV method by f−k transform, as opposed to the other

comparisons of Figures 2.11 to 2.16, where a τ − p transform was used. In addition, geo-

phone positions were from 0.1 m to 4.9 m (0.1 m spacing), with a 100 Hz source function.

At any frequency, the FSW dispersion function (grey bands) is complex and the real roots

of the normal mode dispersion is confined to very low frequencies up to about 120 Hz. It

is also singular at the P - and S-wave velocities of the homogenous half-space, being 549


m/s and 122 m/s respectively. Above 120 Hz, dominant higher modes occur, which the

standard SASW dispersion has not detected but the MFT method reproduces well.

If a forward modelling method which employed the roots of the propagator matrix

solution was to be employed in an inversion scheme, systematic errors would occur. In

addition, the standard SASW dispersion methods assume single mode propagation and

do not properly measure dominant higher modes. This dataset was extended to 500 kHz

in [124], where above 1000 Hz the dispersion is dominated by the flexural (Lamb wave)

mode. However, the broad bandwidth of over 4 orders of magnitude of frequency would

not normally be achieved in practise.

2.2.7 Effective phase velocity The ‘source-free’ versus ‘active-source’ theoretical

methods are often referred to as ‘plane-wave’/‘point load’ [282] or ‘two-dimensional’/‘three-

dimensional’ analyses [268]. ’Free-modes’ are often used by mathematicians for the plane-

wave case [35], which defines the modal phase velocity. In this thesis, the dispersion

measured in the field or by active-source simulation methods will be called the effective

phase velocity after [161], earlier called the apparent phase velocity [281, 322].

Invariably, effective dispersion will not correlate with plane-wave modal dispersion,

due to dominant higher modes and/or near-field effects. Plane-wave dispersion codes such

as the FSW method [290] have been employed by other researchers in theoretical and field

studies [352]. However, in practice, it relies crucially on modal identification and isolation,

which invariably can only be done for the fundamental mode. This can be sometimes

be accomplished in the time-distance domain with simple group velocity windows [21]

or frequency filtering of the shot gather [235]. However, these filtering methods require

the mode of interest to have appreciable energy over a broad frequency range and if

not, truncation and tapering of waveforms can introduce errors into the experimental

dispersion.

An attempt at isolating the fundamental mode in the Case 2 dispersion is shown

in Figure 2.19, where at high frequency it is expected at a phase velocity of about 125

m/s (slowness of 0.008 s/m). The ‘bow’ filtering method of [235] is implemented in the

f − p plane, as opposed to f − k, to facilitate ease of mode identification. Moreover, the

slowness limits of the slant-stack also accomplish a large portion of undesired wavefield sup-

pression. The original shot gather (Figure 2.6(b)) is transformed to f −p (Figure 2.19(b))

and a region outlined to reject the dominant higher modes, extending to minimum slow-

ness and maximum frequency. The inverse transform produces a shot gather with much

lower frequency content (Figure 2.19(c)) and the filtered f − p transform shows a new

spectral maxima (Figure 2.19(d)). However, it appears to be mostly contaminated by side

lobes and ringing effects, possibly due to poorly designed filter tapers. Nevertheless, when

the dispersion is picked from the filtered shot gather, it appears erroneous and does not

correlate with any FSW roots above 35 Hz, shown in Figure 2.20(a).


10 20 30 40 50

0

0.2

0.4

0.6

0.8

1

Offset (m)

Tim

e (s

)

(a)

Slowness (s/m)

Fre

quen

cy (

Hz)

(b)

5 6 7 8 9

x 10−3

0

50

100

150

200

250

10 20 30 40 50

0

0.2

0.4

0.6

0.8

1

Offset (m)

Tim

e (s

)

(c)

Slowness (s/m)

Fre

quen

cy (

Hz)

(d)

5 6 7 8 9

x 10−3

0

50

100

150

200

250

Figure 2.19: Plane-wave filtering Case 2 of Tokimatsu et. al.. (a) and (c) are the shot

gathers and (b) and (d) are the f −p transforms, raw and filtered respectively. The region

of rejection containing the dominant higher modes is outlined in (b).

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

(a)

20 40 60 80 10050

100

150

200

250

300

350

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

(b)

10 20 30 40 50 60 7050

100

150

200

250

300

350

Figure 2.20: Filtered dispersion of Cases 2 and 3 of Tokimatsu et. al.. The observed dis-

persion from shot gathers filtered to exclude higher modes are overlain over the FSW dis-

persion, shown as grey bands. (a) Case 2, and (b) Case 3.


The result of applying the same f − p filtering method to reject the dominant higher

mode at 10-15 Hz in Case 3 is shown in Figure 2.20(b). The fundamental mode is better

identified, but errors remain around the 8-16 Hz band. In these cases, and others shown

in this section, when significant energy is transferred to dominant higher modes, the

fundamental mode retains only negligible energy. This is often clear from the overlapping

nature of waveforms in t − x shot gathers and attempting to filter out the higher modes

leads to distortion of the wavefield and an erroneous observed dispersion.

In summary, the important points illustrated in this section are:

1. The PSV algorithm proposed for use in this thesis is sound in generating modal and

effective dispersion;

2. The nature of dominant higher modes in the dispersion curve which are generated

by LVL and HVL cases;

3. The behaviour of the dispersion at low frequency, traditionally called ‘near-field

effects’;

4. Filtering seismograms to extract the fundamental mode is not advised in cases with

dominant higher modes; and

5. Conventional plane-wave matrix methods are not suitable for inversion of dispersion

with dominant higher modes.

The last fact was stated by all the early forward modelling publications of Table 2.3(a) and

addressed in the inversion works of (Table 2.3(b). A brief literature review of dominant

higher mode and near-field effect observations and calculations will now follow.

2.3. Dominant higher modes 51

2.3 Dominant higher modes

In normally dispersive sites with low contrasts (and non-dispersive layering), the fun-

damental mode propagates with the most energy. Higher modes exist theoretically as

multiple solutions (eigenvalues) to the dispersion equation and are explained physically by

constructive interference between reflected and transmitted waves [82]. Inclusion of higher

modes, in addition to the fundamental, has been long known to aid the inversion of dis-

persion and waveforms in earthquake seismology, but most shallow surface wave methods

only observe and invert fundamental mode dispersion.

Fundamental mode dispersion is an inherent assumption is SASW [9]. In MASW,

both large near offsets [315] and long spreads of many channels [89, 126] are recommended

for improved modal isolation and resolution. Nevertheless, most works only employ the

fundamental mode [304, 352], also the case in passive microseismic measurements [179,

288]. More recent active source MASW works employ higher modes for improved depth

penetration and inverted model accuracy [22, 237, 238, 357, 356].

However, in these MASW cases, the dispersion was always multi-modal. That is, a

dominant fundamental mode with weaker higher modes, well separated in plane-wave

transform domain. Dominant higher modes are those where the higher modes of prop-

agating surface waves take more energy than lower modes. In addition, other modes of

vibration from flexural and guided wavefields can also become dominant, usually at much

higher frequencies and/or phase velocities of the surface wavefield.

2.3.1 Experimental observations In experimental data, dominant higher modes

manifest as discontinuous ‘jumps’ in the dispersion curve, between possible wavenumber

functions of surface wave propagation (modal dispersion). In SASW, these points can be

identified from fluctuations in the phase lag function [344] and in MASW as discontinuities

between lobes in plane-wave transform space. This is usually observed in shallow profiles

where stiffness does not increase monotonically with depth (irregularly dispersive) and/or

shear velocity or Poisson’s ratio contrasts are quite high. They are rarely observed in

earthquake seismology due to insufficient elastic contrasts [80]. However, in shallow surface

wave applications, the problem was first noted by CSW and SASW researchers in asphalt

investigations but many unconsolidated soil profiles can also generate dominant higher

modes.

Asphalt and road base Even in the early CSW measurements, large wavelength changes

between different frequencies caused discontinuities in the dispersion curves [118]. At

higher frequency (short wavelengths) the dispersion was noticed to be dominated by the

flexural vibrations of a free plate. These are known as Lamb waves and proved invaluable

for interpreting field data [133]. The phenomena was initially called the ‘frequency gap’ []

and the different dispersion branches were theoretically arranged in [133].


Figure 2.21: Spurious SASW phase lags associated with dominant higher modes, as ob-

served in [344, p102].

The term ‘mode jumping’ was later suggested in [344], where early SASW measure-

ments showed that these jumps occurred at frequencies where the phase lag between two

receivers showed an irregularity, an example shown in Figure 2.21. Mode jumps, mani-

fested in SASW from the departure of a monotonically increasing phase lag, were later

suspected to occur when the Rayleigh wavelength spanned two layers of markedly different

elastic properties [7]. This was shown from a CSW perspective in [8] where non-harmonic

surface oscillations are suggestive of multiple mode propagation. However, large discon-

tinuities in SASW dispersion were observed in the early reports over pavement systems

[112, 219]

Countermeasures to isolating a single mode in SASW testing are both by more careful

phase unwrapping [7] and from testing with various near-offsets and receiver spacings and

only retaining certain wavelengths. This was noted in [8], who both identified spurious

phase cycles and both higher mode and other wavefield contributions to cause the mode

jumping. Many SASW dispersion ‘filtering’ measures were intended, primarily for rejecting

data suffering from near-field effects, but also for creating a continuous dispersion curve by

excluding jumps which occurred at higher frequencies [91]. The more brute force method

is to simply reject dispersion points in the vicinity of spurious phase differences, which


coincide with poor coherance [217].

With SASW, a large ambiguity in the multiple dispersion curves measured over pave-

ment systems can occur [65], a good example is the large scatter in [218]. These fluctu-

ations have been attributed to reflected body waves [299]. It has also been noticed that

dispersion velocities from horizontal component geophones were somewhat higher than

those from the vertical component [112]. One explanation was a possible greater sensitiv-

ity of the horizontal component to P -waves. However, higher mode interference may have

also been in effect, as noted by [131].

Later MASW observations in showed sharp truncations of the fundamental mode and

branching to modes of flexural (Lamb) wavefields, where at higher frequencies, the trend

of dispersion is reversed and higher mode branches were more effectively observed with

accelerometers instead of geophones [233]. The theory of mode branches in a stiff plate

was conclusively field proven by higher frequency MASW analysis over asphalt in [275].

Soil and rock The early single channel dispersion curves of [154] show very similar

discontinuities to the theoretical curves of [293], in cases where a thin, soft layer overlies a

much stiffer half-space. Similar phenomena were noticed in [135, 119] in a clay overlying

gravel or moraine. While stepping the receiver away from the vibrator, it was noticed

that within a certain frequency band (30-50 Hz in [119] beyond a certain offset there

was an increasing distance between wavelengths, that is, larger velocity. The result was

two markedly different velocities being measured at frequencies within this band caus-

ing two branches of dispersion. The reasons at the time were stated as transverse wave

contamination [135].

Dominant higher modes are difficult to observe by SASW in unconsolidated soils,

usually associated with spurious phase lags and/or low coherance [7]. However, body or

reflected wave interference can also cause similar effects thus is not a conclusive indicator

[28]. In microseismic array measurements, clear discontinuities in the observed dispersion

are better observed in all components of [322]. Dispersion from transverse motion usually

correlated best with theoretical Love modes however, often matching theoretical Rayleigh

wave modes due to the omnidirectional nature of the wave propagation.

Branching of dispersion curves was also noted in some very early marine multichannel

studies [251]. While these were ‘leaky modes’, they followed normal mode dispersion theory

and occurred when a low velocity liquid layer was present. Sharp dispersion discontinuities

also occurred for air coupled surface waves, due to above ground explosions at sites where

the Rayleigh wave and sound velocities were similar [253]. Later studies also suggested the

presence of dominant higher modes. In [131], dissimilar phase velocities between vertical

and horizontal component records at 3 Hz were attributed to a stronger higher mode. The

dispersion curves showed clear a transition from the first to third mode at higher frequency

(6 Hz) the scatter at frequencies above 6 Hz due to an overlapping second mode. Leaky


Figure 2.22: Field example of a dominant higher mode observed by MASW [235]: (a) Shot

gather and; (b) Dispersion image, where a higher mode(s) dominate at frequencies above

35 Hz.

mode dispersion also showed discontinuities.

In the work of [259], active source, f − k dispersion at three sites showed large dis-

continuities. At one site, a LVL identified from a nearby VSSP borehole log generated

dominant higher modes at higher frequency (30-50 Hz). At the two sites with discontinu-

ities at lower frequencies (10-20 Hz), a HVL was identified with refraction/reflection tests.

While the body wave seismic inversions are not ultimately conclusive, the interpretations

of stiff layers and velocity reversals supports the the presence of dominant higher modes.

Similar large discontinuities were observed in the f − k dispersion in [329], but were ex-

cluded from the analysis which only incorporated the fundamental mode. Another in [21]

showed the sharp decay of the fundamental mode, where higher modes became dominant

at higher frequencies. These were strongly seasonal dependent and were inverted as con-

tinuous, multi-modal cases. Dominant higher modes are also observed in lower frequency

measurements in basins with thick sequences of low velocity sediments [296].

In recent MASW work, dominant higher modes were recognised in transformed vertical

impact source shot gathers in both soft [240, 242] and hard [241, 235, 234, 357] soils. Both

the f − p and a modified transform showed showed a transition from the fundamental to

the (assumed) first higher mode, within the 15-35 Hz range. Figure 2.22 shows a field

example from [235], where a higher mode dominates from about 35 Hz. In soft soils,

multiple mode transitions were not observed at higher frequencies but in the harder soils

[357] a further mode jump apparently occurred at about 40 Hz. The the new transform


described in [237, 235] showed apparently better resolution than the standard f−p method

of [198]. That work also showed the near-offset effects on higher modes, where they are

much stronger at larger source to near-receiver distances (17.1 m opposed to 2.4 m). The

reason was attributed to both better developed surface waves and less attenuation of higher

modes (deeper eigenfunctions) at larger offsets.

It is interesting to note that the conclusions in the results of [240, 242], less emphasis

was placed on on geological structure as a factor in generation of dominant higher modes

than acquisition layout effects, only suggested they are better developed in cases of low

shear velocity contrast. However, in [81], dominant higher modes are clearly observed in

two cases (shown in Figure 2.16), the character very similar to that observed in [240, 242,

235]. Full-wavefield inversion of the Bietigheim data in [80] could only be achieved by a

very steep gradient in shear velocity and Poisson’s ratio in the upper metre. While no

borehole data was available in the upper the upper 2 m in the sites studied in [240, 242],

it is likely that such a steep gradient existed, supported by the strong wavefield splitting

in the raw shot gathers. In [126] similar modal jumps occurred but were noticed to be

from the Rayleigh-wave higher modes to guided P -wave dispersion (normal versus leaky

modes). Jumps between normal modes were not evident due to the high velocity causing

almost overlapping surface and guided waves. Data with dominant higher modes was also

used in [241] to illustrate better modal identification with longer spreads and [235] focusing

on shot gather filtering to enhance the fundamental mode for inversion. In those cases,

the fundamental mode energy at high frequency usually permitted its isolation from the

higher energy, higher modes.

The pavement testing of [275] was extended to soil and rock cases in [243]. Branches

showing opposite dispersion trends (inverse at low phase velocity and normal at high phase

velocity) occurred in cases with velocity due to gas sands under saturated sand and a soft

formation underlying a well cemented limestone caprock which compared well to Lamb

wave (free plate) dispersion.

Other environments Love waves in coal seams are especially susceptible to dominant

higher mode generation [26, 33, 169] as are flexural waves in floating ice [27]. Some seafloor

sediments also show higher modes with more energy than the fundamental [149, 301].

2.3.2 Theoretical modelling The possible branches of a dispersion curve can be

theoretically modelled using the same matrix methods for plane surface wave propagation.

Either full-waveform (surface and body waves) or modal summation (surface waves only)

can be generated to analytically generate dominant higher modes. Only the active source

methods can also includes the effects of body waves and cylindrically spreading wavefronts,

which were initially intended for the study of near-field effects (Section 2.4).

The active source procedure as outlined in [269, 281] is:

1. Decompose load into a series of cylindrical (Bessel) functions, which is an expansion


in wavenumber (k);

2. Employ the active source wavenumbers for the system displacements and stresses

using the matrix methods (Green’s functions);

3. Integrate the convolution of the Green’s functions with the corresponding source

functions.

In other words, this procedure calculates the impulse response at specified points in the

medium due to a harmonic load at another point for displacement, velocity or acceleration,

that is, the Green’s functions of the medium, which are the response at one point due to a

(usually harmonic) loading at another points. It is essentially a procedure for calculating

synthetic seismograms as outlined in the reflectivity method of Section 2.2.1 but using a

stiffness matrix approach for step 2. Dispersion curves can then be calculated from these

time histories by the standard SASW method (phase of the cross spectrum). However, the

phase difference can be found directly from the complex Green’s functions, either between

two receivers (cross spectrum) or between the source and one receiver (transfer function)

[281]. It was also discovered that the response for small disc loads was almost exactly the

same as for point sources [269].

Note that this method for simulation is used because body waves are simulated, and

along with the standard SASW acquisition and processing effects. The solutions of dom-

inant higher Rayleigh modes can also be picked from the theoretical spectral maxima

in frequency-wavenumber space, without the need for calculating actual surface displace-

ments [351]. However, acquisition layout is usually a priority to properly invert the ob-

served wavefield dispersion [271]. Usually the stiffness and linearised stiffness are employed

in engineering applications [91], although the transfer or R/T matrix methods are equally

valid [271].

Asphalt and road base The dispersion of Lamb waves had been known since 1916 [344].

It is caused by the flexural vibrations of waves in an infinite solid plate bounded above

and below by an infinite fluid. While the early dispersion branches observed in [133]

were dominant higher modes, the theoretical modelling was based on two procedures: 1.

Flexural mode dispersion used to model the short wavelength data; and 2. Choosing real

roots of the standard plane-wave matrix methods which best match the field data at longer

wavelengths, an example shown in Figure 2.23. Flexural mode dispersion is quite accurate

in many situations [219] and these wavefields are also modelled well with numerical finite

difference methods [275].

The earlier full simulation SASW methods were generally aimed at investigating near-

field effects (Section 2.4) and dispersion discontinuities in pavement models not considered.

In [281, 269] only two layer cases were considered and dominant higher modes were not

explicitly stated. While small fluctuations occurred at mid-frequencies in [281], they were


Figure 2.23: Theoretical dispersion curve branches for flexural and Rayleigh modes over

a two-layer pavement system model [344, p81].

stated as being due to vertically propagating body wave reflections. Similar, but much

larger, discontinuities occurred in in [269] and appeared to be more likely from mode

transitions. However, in [282] a pavement model was tested and dispersion compared to

the plane-wave transfer matrix roots. It was clear that at successively higher frequencies,

the point load dispersion correlated with successively higher roots. While the dominance

of higher modes at higher frequency or mode jumping was not explicitly stated, this work

showed that it would occur with a stiff surface layer and was subsequently verified in [8].

Also in [9] it was shown that standard SASW cross spectra will not properly measure

the extreme jumps to higher modes observed in pavement systems. Thus, the SASW

method has a shortcoming in both modelling and observations when dominant higher

modes are present. Although SASW is formally equivalent to plane-wave transform with

infinite trace padding [85], multistation methods with long spreads are required to properly

resolve effective dispersion with dominant higher modes.

Soil and rock The early forward modelling SASW simulations for dominant higher

mode generation in soil models are listed in Table 2.3(a). Tokimatsu et. al. [325] employed

the transfer matrix formulation with a point source, Roesset et. al. [268] the linearised

stiffness matrix formulation and Gucunski and Woods [101] the full stiffness matrix method

for a circular footing. Again, these are all methods to calculate the Green’s functions

of a layered medium for synthetic seismograms, merely with different plane-wave matrix

methods, usually by wavenumber integration. One focus of those works were for evaluating

criteria for ‘filtering’ dispersion curves based on source to near receiver and receiver spacing


Figure 2.24: Dominant higher modes modelled by modal summation synthetic seismo-

grams and observed by MASW [235]: (a) and (c) are for fundamental mode only and; (b)

and (d) incorporate the fundamental and first higher mode, in time-offset and frequency-

phase velocity domains respectively.

to minimise undesirable effects.

In the literature, the case of a HVL is generally addressed less frequently than LVL

cases, only employed in the forward modelling cases of [99, 100, 161, 159, 271, 325] and

the inversion work of [91]. The earliest appearance of this HVL case is in [99], which was

also employed in [271], where stiff layer at 20 m depth generated a jump to the second

mode at around 10 Hz. The dispersion pattern is very similar to Case 3 of [325], except

in the latter a jump to the third mode also occurs. The models and dispersion of recent

work on HVL comprised of cemented layers in [43, 182] would also be expected to generate

dominant higher modes but only plane-wave methods were used in those works.

It is also interesting to note the dispersion of a LVL model employed in [351], also

used in [99, 100], which shows very clearly the transitions to dominant higher modes at

25 and 50 Hz in the wavenumber domain. These modal jumps occur are at very similar

frequencies to that in Case 2 of [325], albeit the markedly different layer thickness and

shear velocities. However, in terms of velocity to thickness ratios of each layer, in the

model of [351] they are 40 and 30 for layer 1 and 2 respectively and in [325] are 90 and 30.

These observations suggest that the scaling properties as suggested for modal dispersion

in [303] are valid for effective dispersion curves.

Theoretical MASW simulations in [82] employed the modal superposition code of [117].

Soil cases with a LVL were tested and both f − k and τ − p transforms showed clear tran-

sitions to higher modes at higher frequencies. The thicker the LVL the more transitions to

higher modes which occurred. That work also compared the conventional SASW proce-

dure and even though the method assumes a single propagating mode, similar transitions

occurred to dominant higher modes. Full-waveform wavenumber integration seismograms

were also generated, but only used for testing near-field effects. Modal summation syn-

thetic seismograms were used in [237] but higher modes were not generated. But in


[126, 235], use of the same code showed clear dominance of the first higher mode above

25 Hz, however, the soil model used was not stated. An example from [235] is shown in

Figure 2.24. While these do not include body wave contributions, there is similarity to

the observed modal structure in Figure 2.24

The MASW simulations of pavement systems [233] also employed the modal superpo-

sition code of [117]. While strong osculation points are evident, there is the constraint

that a stiff homogenous half-space must be employed. Wavenumber integration code gen-

erates full waveform synthetic seismograms without this constraint an also incorporate

body waves. Jumps to higher modes (branches) were generated very similar to observa-

tions. Later, numerical finite difference code was used to generate shot gathers in [275].

Recently, it has been shown that Lamb wave dispersion can apply in geological structures

with stiff caprock and shallow gas charged sand layers [243].

In [78, 80], use of P -SV reflectivity method allowed calculation of the full wavefield over

any range of offsets. In addition, he noted that the Fourier transform may be inaccurate

for longer wavelengths (near-field) while Bessel functions are cylindrical and do not assume

plane wavefronts. Thus the generation of a f − p-domain representation employed Bessel

functions in a Hankel transform instead of the sines and cosines of a Fourier transform.

This was suggested earlier from a SASW view in [65] and the images in [78] show dispersion

which is not asymptotic at low frequency and both correlates well with standard τ −p transforms and are almost perfectly recoverable.


2.4 Low-frequency discrepancies

At low frequency, any or all of the following effects may corrupt a dispersion curve:

1. Near-field effects (body and cylindrical wave spreading);

2. Spread length and plane-wave pixel resolution limit;

3. Dominant higher modes (shallow stiff layer).

When standard dispersion observation methods are used, almost all experimental and

full simulation numerical observations show a decrease in surface wave velocity at low

frequency in homogenous half-space or weakly normally dispersive structures. In exper-

imental data it is only regularly noticed in MASW observations [82, 50]. Experimental

SASW data with low coherance values at low frequencies are invariably filtered before

beng presented as dispersion curves [217]. However, numerical SASW studies clearly show

the drop in phase velocity at lower frequency in simple half-space cases, both in ana-

lytic [269, 281, 373] and numerical finite difference [90, 98] modelling. In the numerical

modelling, the discrepancy was attributed to standing waves [98].

Since in SASW, to adequately sample the short and long wavelengths, several spread

layouts are used at a site. As there is an overlap of frequencies/wavelengths from the

various source-receiver spacings, criteria for ‘filtering’ the dispersion curves by rejecting

certain wavelengths for a particular spread dimension were researched.

2.4.1 Near-field effects The seismic near-field is traditionally considered as the

vicinity from the source where body and surface wave amplitudes are similar [260, 283].

However, much earlier, it was noted that cylindrical spreading surface wavefield is an

important contributor to the measured wavelengths [133]. The cylindrical spreading effect

is also known as ‘model incompatibility’, that is, employing plane-wave transforms on a

cylindrical wavefield, [373].

A point source will generate cylindrical waves and successive wavelengths will occur

at distances proportional to the roots of a Bessel function [133]. Thus, in CSW dispersion

measurements when several receiver positions are used in walkaway fashion, the first wave-

length measured from the source would be about 3/4 of the equivalent plane wavelength.

However, after a distance of one wavelength, measurements are equivalent to plane waves

within experimental accuracy of the time.

In SASW or MASW, plane homogenous waves are assumed for the transform to fre-

quency domain. Since the near-field extends to greater radius for low frequency, anoma-

lous dispersion invariably appears at very low frequencies. The Hankel transform employs

cylindrical Bessel functions as the kernel and is suggested to be more suitable than the

Fourier transform at low frequency [65]. Bessel functions were later applied in multichan-

2.4. Low-frequency discrepancies 61

nel analyses by [78, 373] to combat the effects seen in dispersion curves at low frequency,

However, the τ − p transform is still quite accurate even in the near-field [79].

The other explanation was a coupling between Rayleigh and body waves near the

source [260, 269]. Extensive parametric studies using SASW simulations were made to

to describe the anomalous behaviour of the dispersion curve from the plane-wave trend,

usually at low frequencies, in [281], which were always explained as body wave effects

[282, 269], even though those methods included cylindrical surface wavefronts. It was

realised that the far offset to near offset ratio should be more than 2, at least 1.5, to best

match the plane-wave dispersion. In terms of reducing body wave contamination, MASW

is usually reported as being preferable to SASW [82, 322].

With SASW, although it did not always manifest at closer source-receiver spacings, the

experimental recommendation of [282] was to use the same receiver spacing as near offset.

Then, by rejecting wavelengths longer than half the receiver spacing would minimise body

wave interference. near-field effects are also model dependent. In irregular or inversely

dispersive soil profiles (where shear velocity does not increase with depth) body wave

interference may extend out to distances of two times the wavelength [260]. A review

of these types of recommended filtering criteria is summarised in [91], which were also

intended for producing a unique phase lag curve which was ‘contaminated’ with dominant

higher modes.

Another source of contamination noticed at long wavelength (low frequency) were

reflected body and surface waves [65, 299]. However, the dispersion discrepancies were not

as systematic as those of simple near offset variations. In [65] the recommendation was

to use accelerometers oriented inline, with source and receivers parallel to the reflecting

boundary. In [299], placing the source between the reflecting boundary and receivers with

the SASW line perpendicular to the boundary to minimise reflected waves. If the test

point is close to a boundary, like [65] the SASW line should be parallel to the boundary.

Unusual fluctuations in the mid-frequency ranges were also believed to be due to body

wave reflections and refractions. In other early numerical simulations [281], a strange

fluctuation also seemed to occur at a frequency of 3VS/4H, where VS is the upper (stiff)

layer velocity and H the layer thickness. This corresponds to the second natural frequency

of the layer, and was also noted in early SASW pavement tests as flexural and interfacial

dispersion [219].

2.4.2 Spread and pixel resolution The problem of poor resolution at low fre-

quency has been noted by practitioners. In [288], it was questioned which of (1) array

length, (2) high pass filtering of the Earth or (3) source bandwidth has the greatest influ-

ence of the usable frequency range, in particular the low-frequency uncertainty. In [301],

the maximum depth was roughly estimated as half the maximum wavelength, which in

normally dispersive sites occurs at the lowest detected frequency. This was said to be


limited by (1) source power, (2) detector frequency response and (3) source-receiver offset

range. Microtremor measurements in [173] also noted maximum wavelengths limited to

twice the array aperture.

As suggested in [78], the fundamental limitations of spread length of low-frequency

resolution are more influential than any other factor (such as near offset, geophone response

or source spectrum). Even a dispersion curve with strong signal to noise at low frequency

may not provide any accurate information at depth simply due to the lack of resolution.

In reality, however, it may not be desirable to increase spread length due to topographic

or geological discontinuities, or simply lack of space at the site. The important measure of

the phase velocity (or slowness) resolution dependence on spread length will be considered

in Chapter 3 and may play a dominant role in explaining ‘near-field effects’.

In terms of the pixel resolution of the plane-wave transform, this can be improved

by padding the input shot gather with null traces [85]. This is equivalent to wavenumber

interpolation (in the case of f−k), and while no new information is added, it assists visually

in clearer detection of lower-amplitude spectral lobes and dispersion at low-frequency. By

τ − p transform, a slowness vector with sufficient resolution achieves the equivalent in

f − p [331]. Nevertheless, the spread-length dependent resolution limitation remains.

2.4.3 Higher-mode contamination It is interesting to note that in the earlier

simulations of [281] the observed dispersion at low frequencies in severe soft-over-stiff

cases (large stiffness contrasts between the top layer and half-space) was higher than the

plane-wave dispersion. The reason at the time could not be explained, as the usual trend

was for phase velocity to be lower than the plane-wave case. It was attributed to ‘near-

field effects’, however, in light of later numerical work by [101, 268, 325], for cases with

a large contrast in stiffness at shallow depth, higher modes can become dominant. This

has also been shown experimentally in [322]. In the higher mode studies of [325], the

dominant higher mode associated with the HVL case is especially perturbed when the

near receiver is within one wavelength of the source. However, that aside, no other effects

due to near offset were evident. So, the model dependency element of near-field effects may

be simply more than differing degrees of body wave generation [260]. In the stiff-over-soft

cases of [281], low-frequency dispersion was also generally higher, but not indicative of a

discontinuity due to a dominant higher mode.

2.4.4 ‘Low-frequency effects’ While the unusual dispersion at low frequency may

due to all or any of the above effects, the poor phase velocity resolution at low frequency

due to a finite spread length expected to be the dominant factor. These kinds of issues must

be considered a priori before rejecting low-frequency dispersion points. Two examples

using the PSV method are now shown.

The first example employs the homogenous half-space dispersion curves used in the

half-space Poisson’s ratio tests of Section 2.2.6. The curves at a near offset of 5 m for


0 20 40 60 80 100 120 140 160 180 20050

100

150

200

250

300

350

400

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

PSVFSW

Figure 2.25: PSV dispersion of four homogenous half-spaces, of shear velocities 80, 120,

180 and 360 m/s. For each, five Poisson’s ratios were used, 0.01, 0.1, 0.2, 0.3, 0.4 and 0.49.

250

300

350

(a)

Pha

se v

eloc

ity (

m/s

)

0 10 20 30 40 50 60 70 80 90 100

250

300

350

(b)

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

PSVFSW

Figure 2.26: PSV dispersion of a 360 m/s homogenous half-spaces at five Poisson’s ratios,

0.01, 0.1, 0.2, 0.3, 0.4 and 0.49. (a) 5 m and (b) 197 m near offset.


Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

(a)

10 20 30 40 50 60 7050

100

150

200

250

300

350

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

(b)

10 20 30 40 50 60 7050

100

150

200

250

300

350

Figure 2.27: PSV dispersion of [325] Case 1 dispersion from both vertical and horizontal

inline synthetic shot gathers, over the FSW plane-wave dispersion bands.

the entire range of VS and σ combinations are shown in Figure 2.25. While the FSW dis-

persion is frequency independent as expected, the PSV dispersion is not. In particular,

the maximum measurable frequency is due to the alias frequency being reached. Other

features immediately evident are:

1. Higher Poisson’s ratio of a given half space causes larger measured phase velocity;

2. Larger phase velocities cause spurious dispersion at gradually higher frequencies,

due to longer surface wavelengths;

3. At low frequency, PSV dispersion is invariably lower than the frequency independent

FSW dispersion;

4. At very low frequency, PSV dispersion is invalid due to catastrophic loss of resolu-

tion.

When the near offset is shifted to 197 m and recorded with the same 48-channel spread,

the dispersion is markedly different, shown in Figure 2.26(b) for the 360 m/s half space.

There, the dispersion curve rises at low frequency, in contrast to dispersion at 5 m near

offset (Figure 2.26(a)), which decreases, in common with other published cases. These

effects will be further investigated in Chapter 3.

The second example is the PSV dispersion of the normally dispersive Case 1. The same

acquisition parameters as for Figure 2.25 are employed with both vertical and horizontal

inline geophones. In Figure 2.27, note how in both components, at about 4 Hz, the

PSV dispersion appears to asymptote and actually merges with the first higher mode.


This is actually a simple loss of resolution effect, not a true transition to a higher mode, and

is very similar to the asymptotic effect seen in the plane-wave transform of Bietigheim in

Figure 2.16. This highlights the chance that near-field effects and/or the loss of resolution

at low frequency can combine for a fundamental mode to jump up an apparently coincide

with a higher mode phase velocity.

However, the horizontal inline dispersion (Figure 2.27(b)) jumps to a dominant higher

mode at about 10 Hz, which appears to be a true dominant higher mode . The horizontal

component dispersion in [112] was also noticed to be higher than the vertical component,

attributed to greater sensitivity to body waves. While dominant higher modes at low

frequency are usually only generated in cases with a sharp and high contrast interface

at shallow depth, it appears that their strength depends on the wavefield component

measured.


2.5 Data resolution and accuracy

Inherent with any measurement is the resolution and accuracy. Resolution can be

thought of as the degree of fineness and accuracy as the degree of repeatability. Ideally,

one desires both to be high. In traditional surface wave inversion, the problem is com-

pounded in that the data, the dispersion curve, are actually a secondary observable of the

transformed fundamental datum, the seismogram. Any resolution limitations and inaccu-

racies propagate through the acquisition, processing and inversion processes to manifest

as limitations in the inverted model parameters.

In earthquake seismology, there was always known to be a systematic error in group

velocity curves, in part due to the poor resolution at low frequency in the time-frequency

analysis of single station recordings [77, 295]. In single station phase velocity dispersion,

errors in the source parameters contaminate the phase differences [215]. However, repeata-

bility of earthquake dispersion over a certain propagation path is difficult to measure since

exact source-receiver locations could rarely be duplicated. In engineering seismology, ex-

perimental repeatability can be measured easily. After the seismograms, the quantity in

error is the phase of the cross spectrum in SASW and frequency dependent wavenumber

or slowness coefficient maxima.

The objective is to implement a controlled synthetic resolution and accuracy anal-

ysis of the MASW test, in layered cases where dominant higher modes are generated,

incorporating:

1. Spread layouts, trace padding and dispersion observation method;

2. Individual acquisition errors, such as geophone placement/angle/coupling, source

types/frequency/depth, static errors, dead traces and external noise; and

3. Monte Carlo simulation of all acquisition and processing parameters and errors.

Each of the above is introduced in this section and appraisal of the propagation of disper-

sion repeatability errors to the inverted model parameters is introduced in Chapter 7.

2.5.1 Dispersion processing Before any numerical and field trials, it is vital to

understand how discrete sampling of the wavefield imposes a fundamental limitation on

the frequency dependent resolution of the dispersion curve. These limits may in fact be

greater than the envelopes due to acquisition and processing resolution and accuracy.

In large scale, P -wave reflection profiling, more channels allows a greater stacking

fold with less chance of timing and static errors. Current seismographs for this type of

surveying generally employ several hundred concurrent recording channels. In engineering

work this will generally be lower, at the expense of lower fold and/or increased field effort.

In MASW, naturally, we would like to measure a continuous wavenumber function. This

would require infinitely many channels spaced an infinitesimal distance apart with similar

2.5. Data resolution and accuracy 67

discretisation in time. As is well known, sampling and padding in the t direction is usually

of little concern, as modern seismographs can accomplish this for the frequency band of

interest, which for surface waves is the low. The low-frequency rolloff of detector response

will be ignored for the moment. However, in the x dimension, sampling and padding is of

great concern.

For measuring dispersion, more channels is not always better [241]. This is because

of the number of channels versus channel spacing tradeoff [85]. In discrete recording, a

larger number of channels with larger spacing is preferred to obtain maximum wavenumber

resolution. However, the problem of spatial aliasing and lateral discontinuity then becomes

prevalent. Analytical and experimental observations in [241] noted this. However, more

channels is an advantage to measure a highly dispersive wavefield over large offsets and

thus more accurately transform the gather.

A shot gather will comprise Nt time by Nx distance points. Considering an unpadded

f − k transform, Nf = Nt/2 frequency and Nk = Nx/2 wavenumbers will allow a total of

Nf × Nk possible dispersion points in the unaliased portion of the transform. However,

for surface wave modes, they are usually confined to a narrow fan and thus a somewhat

lower number of discrete velocities are available. By f − k transform, the maximum

number of dispersion points is strictly limited to the lower dimension, which is invariably

wavenumber. The wavenumber resolution is

∆k =kN

Nk=

1

Nx∆x(2.5)

where kN and associated parameters are defined in the Nyquist wavenumber (Equa-

tion 2.3). Note that zero and aliased wavenumbers are not included in this derivation

nor any x padding, thus Nk is Nx/2. As we will also compare f − k with f − p, it may

be easier to consider slowness, p which is the inverse of phase velocity, c = f/k and thus

defined as,

p =k

f(2.6)

Thus slowness resolution is directly proportional and can be written

∆p =∆k

f=

1

f

1

Nx∆x=

1

fL(2.7)

where f is the frequency (Hz) and L is the spread length (m) and is the same result as [78,

Eqn. III.22]. This is an absolute curve based on the physical array length, irrespective of

trace padding. Padding zero traces onto the end of the gather prior to 2D transformation

is equivalent to wavenumber interpolation in the spectral domain and in fact SASW is

essentially an f − k with infinite padding [85].

By f − p transform, the slowness vector is preset prior to slant stacking and just as in

f − k, resolution decreases with frequency, thus Equation 2.7 applies. This was derived

for a continuous frequency-slowness plane in [78, Eqn. III.22]. However, similar to the


phase of the cross spectrum in SASW, the wavenumber function observed by plane-wave

decomposition must monotonically increase with frequency for ideal representation of the

dispersion.

2.5.2 Source and receiver layouts From a SASW perspective, field layouts were

investigated numerically in [281, 282, 269]. In these studies, source and receiver spacings

were tested to try an minimise the so-called ‘near-field effects’ of Section 2.4. The basic

recommendation was to use equal near-offset and receiver spacing, expanded in multiples

of two. This was also investigated with dominant higher modes in mind by [101, 325],

where it became apparent that dispersion is offset dependent, especially in inversely dis-

persive earth. Moreover, vertical and horizontal components were affected differently.

An experimental evaluation of receiver layouts and frequency limitations for pavement

evaluations was made in [121]. The criteria for optimum receiver spacing and source-to-

near-receiver (near offset) arrangements to minimise undesired effects, usually manifested

at low frequency, are summarised in [91, 373]. They are extremely varied and as suggested

in [373] some will usually produce measured phase velocities which are too low due to

cylindrical wavefield spreading. The near offset limits are based on surface wavelengths,

which range anywhere from 0.25λ to 2.5λ. The receiver spacing is invariably kept equal

to the near offset (d1), however, smaller limits such as 0.5d1 are allowed. The rules are

more commonly used as ‘filtering’ criteria for rejecting certain wavelengths and ensure a

smoother, more unique dispersion curve. The long wavelength limit ranges anywhere from

0.5∆d to 16∆d, where ∆d is the SASW receiver spacing. The short wavelength limit is

less varied, from 0.25∆d to ∆d.

These types of rules have not been as rigorously researched from a MASW perspec-

tive. One rule (albeit only six-receiver tests), for an array length criteria of L > λ/3 was

suggested by [322] when a linear array is used. Dispersion resolution with various field

layouts was investigated with both synthetic and field data in [85]. Measuring surface

waves with several spreads of different geophones is the preferred procedure. However, for

effective dispersion, longer spread lengths are required. Higher mode resolution was also

tested experimentally with near offset in [238] and numerically with number of channels

in [237]. This was put into an analytic framework in [241]. However, the synthetic disper-

sion curves in these cases were observed from modal summation seismograms, thus while

dominant higher modes are generated, body waves are not included. In MASW, it was

observed that a near offset of 0.5λ is necessary for observing separated higher modes, but

a smaller near offset of 0.1λ is sufficient for the fundamental mode [238]. This was also

shown in [234], where from swept frequency records showed that wavelengths up to 60 m

appear as plane wavefronts from as little as 10 m near offset.

From the MASW tests, it was revealed that larger geophone spacings provide the

better wavenumber resolution necessary for phase velocity resolution at low frequency


(i.e. deep layer interpretation). In addition, more channels provides greater resolution,

but only if associated with a larger geophone spacing. In short, resolution is spread length

dependent and geophone spacing dictates the wavenumber alias and influence of lateral

discontinuity. With limited channels, f−k stacking of transformed gathers is recommended

over t − x stacking of raw shot gathers [85, 370], at the expense of field time. This can

also be accomplished by stacking dispersion from CMP gathers or individual rollalong

shot gathers with traces windowed to common offsets [108, 110]. These methods enhance

the modal lobes, especially those variable due to different shot offsets, and suppress other

coherant and scattered wavefields.

However, all these MASW tests were purely concerned with modal resolution, that is,

ability to detect a certain mode in plane-wave transform space. Shot offset was usually

recommended to be small (such as 5 m) to ensure near-field effects were only considered

in the numerical modelling with full-waveform wavenumber integration shot gathers in

[82]. A 24-channel gather at 1 m and 20 m near offset showed better agreement with the

theoretical effective phase velocity at the farther shot offset. This was supported by a

SASW analysis of the same modal and data, where filtering wavelengths up to twice the

near offset improved the dispersion quality. Aside from this work, a thorough, simulated

testing of multichannel surface wave offset and resolution has not been reported. Various

near offsets, number of channels, channel spacing and trace padding will be tested using

both f − k and f − p decomposition. By studying several models, both normally and

inversely dispersive, an overall conclusion will be achieved.

2.5.3 Individual acquisition errors The noise on surface wave dispersion curves

is generally estimated as a fraction of the phase velocity, either a percentage or statistical

measure such as standard deviation. Earthquake seismology has generally applied this

measure prior to inversion, generally citing a ‘red noise spectrum’ as surface waves are

generally normally dispersive. In shallow surface wave testing, repeated SASW field tests

on the same day [330], MASW tests on the same day [160] or MASW tests on several

different days [21, 23] confirm this assumption. SASW tests incorporating different source

types were made in [121].

Over several days the repeatability is more accurately examined, due to such errors

as geophone replanting and source locations. However the model may change over longer

time-lapses, such as the water table depth, thus differences not entirely due to acquisition

errors. To provide an accurate experimental error envelope, more rapid acquisition tech-

niques which do not degrade surface waves would be required. Surface wave dispersion

from rigidly connected geophone arrays with wooden [308], steel [307, 289] or grid [15]

frames or flexible land streamers [334] has not yet been reported.

Geophone positioning During acquisition, geophones are ideally inserted vertically

along a straight line and an equal distance apart. More likely, there will be a small


placement error in (x, y, z), reflected in the plane-wave decomposition. This transform

usually assumes equally spaced samples in time and space. In time, this will be true

to an almost negligible error. The τ − p transform (otherwise known as slant-stack or

radon) does not however require equidistant x-data points. Methods for the FFT with

non-uniformly spaced data do exist [17]. However, a multichannel shot gather will gen-

erally comprise a regular array of assumed equidistant data when a constant geophone

spacing has been chosen. The dispersion observed from a transformed shot gather will

then be contaminated by errors in the spectral image from which it was picked.

Geophone tilt In addition to position, there will also be an error in the required angle

of the receivers. Although Rayleigh wave dispersion should be the same for both vertical

and horizontal wavefields, it has been shown experimentally [112] and synthetically [325]

to be otherwise. A cluster test in [179], however, showed that sensors can be up to

20◦ off vertical with little effect. For geophone tilt in the SH-direction, ideally a Love

wave component should be vectorially added, further contaminating the observed Rayleigh

wavefield. However in these tests, particle motion will be confined to the x−z plane. Note

that in the theoretical derivation of the reflectivity matrix coefficients, both SH and SV

contributes to the other for point sources in the near-field [214]. These types of errors

would probably be mostly dependent on field conditions and crew diligence.

Geophone coupling This is often considered in reflection work [150, 66, 305] but not

for surface wave inversion, except for a comparison between spike and baseplate geophones

over asphalt [207, 209]. Indeed, surface waves comprise the largest amplitude and low-

est frequency portion of the seismogram over a narrow bandwidth mostly outside the

frequencies generally affected by coupling problems. However, very low frequencies and

high frequencies are often required in deep and shallow work respectively, so these types

of errors will be simulated. Geophone coupling is both near surface geology and source

parameter dependent, thus will be more difficult to accurately simulate.

Source parameters Another possible acquisition error is variable source function pa-

rameters of type, energy, depth, bandwidth and ghosting. In seismic reflection experiments

source effects have been thoroughly investigated [37, 63, 93, 111, 142, 197, 204, 206]. Source

repeatability was tested by [14].

The early multichannel surface wave studies of [59, 62] studied the effects of different

explosive source depths in boreholes and also above ground. The results mainly focused on

the lower amplitude Rayleigh waves from deeper shots, a fact well known from earthquake

seismology and the ‘inverse dispersion’ observed in [62] being due to the shot being below

the low velocity surface layer. This was the first, and apparently only, study of the effect

of source depth on observed dispersion. In [59, 62], particle motion was also observed at

various depths and were experimental proof of theoretical Rayleigh wave eigenfunctions.

The later work of [131] also showed the reduced surface wave energy with shot depth.


However, effects of different source depths on phase velocity were quantitatively measured

for a few low frequencies (3, 5 and 6.5 Hz) where dispersion was shown as independent

of source depth. That work also computed standard deviations from the phase-distance

plots (from which phase velocity was determined) as one or two percent for frequencies

where signal amplitude is strong.

In later SASW site investigation experiments, Variations in the observed dispersion

curve between different impact sources were observed in a pavement structure in [121].

Only the autopower spectra and coherance functions were compared and no mention of

dominant higher modes was made in this work, merely a recommendation to employ at

least two sources at a given site. The observation that heavy hammers best generated

low frequencies and light hammers high frequencies was also made in [322]. Impact,

explosion and shaker sources were compared by SASW and MASW in [82] with similar

observations. Impact and accelerated weight drop sources tested in [234] suggested that

the latter provided more energetic higher modes.

Trigger and timing delays should not be a concern for single shot gather processing

done in the frequency domain. However, two field methods for increasing channel density,

namely walkaway and differential shooting, can introduce phase errors. If the geophones

are left in place and the source refired at a new position, not only will scattering increase,

both source function and trigger delays between the spliced shot gathers can lead to the

plane-wave decomposition to be in error. As there has been no numerical source effect

investigation in a controlled MASW test, this work will be quite new. The field tests of

various source types (impact and explosive) made in this work will provide experimental

support.

Equipment errors These may include geophone frequency response (amplitude and

phase) and A/D timing and digitisation errors. Several geophones tested in [234] sug-

gested only that higher frequency geophones (40 Hz) may not properly allow dispersion

measurement less than about 10 Hz, limiting maximum depth of interpretation to 15 m.

However, others do not consider this a factor in low-frequency dispersion measurement,

where spread length is more influential [79].

Shot trigger errors can be simulated as a constant shift to all traces. However, these are

irrelevant for dispersion processing in the frequency domain. Only time domain stacking

will be affected, which is motivation for dispersion stacking of plane-wave spectra instead.

Channel to channel timing errors are more problematic but will be minimal for a modern

seismograph using cable transmission. With poor synchronisation of standalone receivers

or with radio delays these may be higher. The problem of dead traces and DC offsets,

however, is commonly observed in the field and can be simply reproduced, as well as

different digital resolution levels. Large DC shifts in traces, such as caused by the shot

trigger cable overlying takeout cables, can be simply removed but data are best to be


re-shot.

Additive noise The other types of errors include coherant (propagating) and stationary

wavefields not contributing to the surface wave dispersion. They may be source generated

such as direct, refracted, reflected, guided, air and scattered waves. Or other sources such

as cultural (vehicles, machinery, personnel) and natural (wind, rain, seismic). The effects

of spatially random noise on geophone arrays was investigated in [168]. While passive

microseismic measurements will not be considered, when those measurements are made

with a linear array there is also the problem of wavefront angle to the array [179].

2.5.4 Dispersion repeatability and robustness In shallow multichannel seismic

reflection, numerous investigations of the effects of variable acquisition and processing

parameters on stacked sections, particularly source types, have been published, such as

[63]. A simple spectral repeatability field test for time-lapse monitoring using reflection

was reported in [18]. The effect of random perturbations was investigated for seismic

refraction in [171, 227].

In 2-channel SASW field testing, an indication of the error envelope of the dispersion

curve can be inferred from the scatter in the overlapped segments from the different

receiver spacings [311]. A more comprehensive study of SASW repeatability was made

in [330]. Since not all sources of error associated with acquisition were quantified, it was

essentially a source repeatability test contaminated with ambient noise. It was discovered

that the average error in phase angle (thus phase velocity) were within 7% and are Gaussian

distributed at each frequency. The limitations of the study were that frequency range was

only 60-120 Hz and only two geophone re-plantings (1.8 and 2.4 m separation) were made.

However, since perfectly flat layers could not be guaranteed, different propagation paths

may have contributed. That work also incorporated an error propagation analysis through

a linear inversion to the final modal (Section 2.7.4).

Later, in [160], repeated shooting on the same day and MASW observation by f −k revealed a Gaussian scatter in observed dispersion curves. Since a fixed multichannel

array was used, lateral subsurface variability and geophone replanting were not factors.

Similar to [330], the largest errors were confined to narrow bands which were around

the resonant ambient noise frequencies. Repeated MASW field tests and observation by

f −p in [20, 23] gave an indication of typical dispersion envelopes over a larger time lapse.

While these errors incorporated source and receiver re-positioning uncertainties, due to

seasonal differences the Earth model was not constant for these tests. Their conclusion was

that surface wave surveys are repeatable, but without quantitative measures of dispersion

distribution.

These issues are investigated further in Chapter 3 for synthetic models by numerical

Monte-Carlo simulation of the MASW test combining all probable sources of error will be

run for an absolute repeatability envelopes, incorporating ‘near-field effects’ and dominant


higher modes. Chapter 4 shows the results of repeated field tests incorporating similar

error influences. However, lateral discontinuity will always introduce large systematic

errors in practice.


2.6 Dominant higher mode inversion

As outlined in Chapter 1, traditional surface wave modelling methods calculate the

‘modal dispersion’ or ‘normal modes’. These are the possible frequency-phase velocity

combinations for plane surface waves. As these are only possible modes, experimentally

it is invariably the fundamental mode (or ‘zeroth mode) with the lowest phase velocity

which is dominant and thus observed. In these cases, the site will be either normally

or weakly irregularly dispersive. Higher modes may also be identified, an example of

well-separated, broad-band modes is in [89]. When the experimental dispersion curves

are smooth with no discontinuities, usually only up to the the first or second higher

mode can be accurately identified and usually have much less power than the fundamental

and pervade over shorter frequency limits. Higher modes were studied by [263] from

measurements of particle motions down a borehole, with the conclusion that neglecting

them would not introduce significant error in the inversion. In SASW, this is an inherent

assumption anyway, since higher modes cannot be identified from the phase cross spectra

alone.

However, in many experimental surface wave dispersion curves, ‘dominant higher

modes’ are invariably present. These appear as discontinuities in the observed frequency-

phase velocity dispersion as outlined in Section 2.3. When a single dispersive mode cannot

be isolated over an appreciable frequency range, the traditional inversion will fail due to the

systematic error in the forward modelling procedure. This was noted in the publications

of dominant higher mode simulation [101, 268, 271, 325] but not based on the disper-

sion of field data. From experimental results, a quote in [34] summarises the requirement

succinctly:

At more complicated sites, surface-wave dispersion models that take into ac-

count receiver geometry, body-wave energy, and higher-modes of Rayleigh-wave

propagation may generally improve the solution.

2.6.1 Existing inversion methods The methods of Section 2.3.2 have been used

to simulate the full wavefield generated by an active-source, incorporating dominant higher

modes and other wavefields such as direct, refracted and reflected body waves. Either

transfer or stiffness matrix formulations have been employed, whereby the medium’s

Greens functions or seismograms are analysed by conventional techniques (SASW or

MASW) to model the dispersion.

SASW simulation The forward calculation of Ganji et al [91] simulated an active circu-

lar source at the surface in a linearised stiffness matrix formulation to reproduce the SASW

test. Both LVL and HVL cases were inverted with linear and nonlinear optimisations and

numerical partial derivatives. Later work by the same group employed similar modelling

with a global simulated annealing inversion [103] and [124] used the same kernel to invert

2.6. Dominant higher mode inversion 75

the ‘mode jumping’ associated with asphalt/road base cases with a genetic algorithm. To

fully exploit the active source simulations, however, better spectral analysis techniques

should be applied to ensure all modes are properly detected. This was demonstrated in [9]

for a synthetic pavement model where standard SASW did not recover correct transitions

to higher modes. Thus in addition to a systematic modelling error with plane-wave matrix

methods, there is a systematic observed error when SASW is used in severely irregularly

dispersive cases.

MASW simulation Forbriger [78] simulated a modified MASW test, where the P -SV

reflectivity method by surface impact and buried explosive sources was used to generate

a f − pdomain response as the kernel for the inversion of the transformed field data

by numerical partial derivatives. While the slowness limits are mainly concerned with

surface wave propagation, first P -wave arrivals were also coupled into the inversion for

better constraint on layer interfaces. Lai and Rix [161] employed the transfer matrix

formulation with a harmonic point source at the surface to measure effective dispersion

in the f − kdomain. Their work only addressed LVL cases [159], while HVL cases were

only synthetically modelled in [271]. However, they employed analytic partial derivatives

of effective phase velocity with respect to model parameters for an efficient inversion.

All these reports, SASW and MASW, accurately inverted cases of a soft layer under a

stiff surface (similar to Case 2). Only Ganji et al [91] addressed the buried stiff layer (Case

3). However, in their linearised formulation a sharp discontinuity in the dispersion curve

at low frequency was not generated. While stiff-layer inversion has been addressed with

fundamental mode dispersion [43], the inversion of dominant higher modes associated with

a HVL model such as Case 3 of Tokimatsu et. al. [325] has apparently not been reported

to date.

While dominant higher modes associated with a HVL have been identified in multichan-

nel field dispersion [259], only numerically generated dispersion curves were automatically

inverted in [91, 161]. The field dispersion in [271] did not show evidence of dominant

higher modes. Dispersion from a waste dump with compacted surface layer was inter-

preted in [55], employing the ‘3D’ solution of [268] but by manual iteration. To date, only

Forbriger [78] has both observed and automatically inverted the phenomena observed in

field data. In fact, dominant higher modes occurred in nearly three-quarters of the sites he

investigated. In all these schemes, errors in the data were either neglected or estimated,

thus the final models are not truly representative of the non-uniqueness of the method.

2.6.2 A new inversion method Both dominant higher modes and ‘low-frequency

effects’ are the motivation for developing a new surface wave inversion procedure. It is

proposed that synthetic seismograms calculated by the reflectivity method can be used for

the inversion of observed surface wave dispersion, based on the forward modelling scheme

of Section 2.2. The field test can be mimicked exactly and the dispersion of the modelled


seismograms will account for all the phenomena not properly modelled with traditional

methods. By still employing an observed dispersion curve, it can be robustly inverted to a

layered Earth by linear inversion methods, as analytic partial derivatives are still valid for

effective phase velocity [159]. Moreover, by incorporating realistic dispersion uncertainty,

a better indication of final model resolution and accuracy is inferred.

Chapter 5 shows the work on this hypothesis on synthetic models and Chapter 6 the

inversion of field data, which shows irregular dispersion and dominant higher modes, in

engineering applications.

2.7. Model resolution and accuracy 77

2.7 Model resolution and accuracy

There are at least four kinds of errors which limit the accuracy of an inverted model

[32]:

1. Raw and transformed data errors;

2. Inexact forward modelling;

3. Inappropriate parameterisation; and

4. Detection of the global minimum.

The search for a global minimum in error space has been extensively researched [280].

With any optimisation method, the usual assumption is that the global minimum has

been found and errors are due to the first three sources. However, it is very difficult to

discriminate between these, coupled with the inherent nonuniqueness of the geophysical

inverse problem. For all these reasons, the inversion solution is always considered as an

estimated model [286].

In many shallow surface wave application publications, effort is usually made towards

appraising the inverted shear velocity models, usually by correlating the results with the

‘true’ or expected structure. In evaluating synthetic algorithms, this can be done with

physical modelling datasets. Field data can be correlated with intrusive methods, such

as borehole logs, or other surface geophysical datasets. However, these do not comprise a

rigorous basis as are affected by their own inaccuracies. Where no method of correlation is

available, the inverted model can only be appraised statistically, which is part of the broad

field of inference theory. This section outlines methods for model inference or appraisal,

of which error propagation is a major part.

2.7.1 Physical modelling The optimum method for evaluating synthetic and field

results is with physical modelling datasets, since the structure being investigated is known

exactly. Linear elasticity can also be guaranteed and anisotropy also tested, as when the

real soil and rocks are usually replaced with plastic or metals. Small [319, 166] and large

[56] scale modelling for reflection shear wave and noise studies applications are common,

reports for surface wave studies are rare. The earliest trials were made by earthquake and

engineering seismologists in the 1950-60’s, mainly due to the lack of analytical methods

[344, p86]. Much later, frequencies in the kilohertz range were used by [184] for SASW

testing in a 0.4 m thick sandbox model, with both air and water above the receivers.

Sandbox model results were also used to confirm numerical finite element modelling in

[90]. Ultrasonic MASW tests by [109] were conducted over thin (5 and 10 mm) layered

models of acrylic and mortar and ultrasonic tests of asphalt and road base cores were used

to verify the early SASW results over pavement systems [8]. Larger-scale models have

been built to test 3D effects in SASW for road base investigations [299].


2.7.2 Intrusive tests As site investigations are a popular experimental base for

shallow surface wave inversion, the usual method for comparison is with borehole shear

velocity logs. These may be from downhole, crosshole or in-hole tests, summarised in

[4]. A summary comparison of surface and downhole methods for measuring shear wave

velocity is in [125].

Other destructive methods which do not provide shear velocity can be correlated em-

pirically, by relationships such as [229]:

VS = 97.0N0.314 (2.8)

where VS is the shear velocity (m/s) and N the standard penetrometer test (SPT) blow

count, based on the best-fit of multiple N -values with VS (m/s) from borehole logging.

Another SPT correlation rule is given in [75] as

G = 11.9N0.8 (2.9)

where G is the (small strain) shear modulus (Pa). Various sites were investigated for cone

penetrometer test (CPT) relationships by [196], where the log-log relationship for clays is

[104]

Gmax = 2.78qc1.335 (2.10)

where qc is the CPT cone resistance (kPa) and Gmax is the small strain shear modulus (in

kPa). Shear velocity can then be derived employing the relationship

Gmax = ρVS2 (2.11)

where ρ is the measured of assumed density (kg/m3) and VS is the shear velocity (m/s). In

these equations, Gmax is the tangential shear modulus, which is in the low strain or elastic

regime. It is usually MPa in amplitude. Other empirical correlations can be made with

generalised depth-shear velocity relationships for marine [105] and land [348] sediments.

Surface and intrusive test comparison limitations While surface wave inversion results

data are often compared to intrusive tests, the correlation may often be both invalid and/or

the borehole data inaccurate, for the reasons listed below:

1. Smaller volume sampled by borehole methods than surface seismic tests;

2. Lateral discontinuity and anisotropy between/around sampling locations and seismic

spread extends;

3. Accuracy of arrival time picking of downhole/crosshole seismic methods;

4. Higher strains of penetrative and deflectometer tests (over 1%) compared to surface

seismic wave propagation (<0.001%);


5. Modelling inaccuracy/nonuniqueness of downhole arrival times and/or empirical con-

version to layer velocities;

6. Frequency dependence of downhole measured velocities or ultrasonic laboratory tests

on cores;

7. Time lapse between downhole and surface seismic tests.

Often normalising the shear moduli provides better base for comparison, such as for liq-

uefaction studies [326] or compaction density evaluations [148]. Alternatively, comparing

slownesses is suggested as being more correct since they are linearly related [34].

Shallow surface wave and intrusive test comparisons In the early surface wave disper-

sion studies, correlations with borehole velocity [62] and geological [154] logs were made,

but only qualitatively. Only from the early 1980’s were quantitative correlations made

with inverted models and invasive methods. Table 2.4 outlines some of these, where the

abbreviations used are:

Site types: Soil = natural soil or rock; Fill = artificial fill material; Road = asphalt

surface; Marine = seafloor or riverbed;

Dispersion methods: CSW = 1-channel continuous surface wave method; SASW = 2-

channel spectral analysis of surface waves; MASW = multichannel analysis of surface

waves, where (Vib) indicates a Vibroseis source; MS = microseismic, where (linear)

employed a linear spread and (array) employed a 2D array; MFT = multiple filter

technique.

Intrusive methods: DH = downhole shear velocity (1- or 2-geophone, exact method un-

stated); VSSP = vertical seismic shear wave profile; CH = cross-hole shear velocity;

PS = in-hole suspension log; SCPT = seismic cone penetrometer test; CPT = cone

penetrometer test; SPT = standard penetrometer test; PIP = push in pressureme-

ter; FWD = falling weight deflectometer; Lab = undefined laboratory test on cores,

including RC (resonant column test), CT (cyclic torsional test) and TS (torsional

shear).

Table 2.4: Review of some published surface wave field tests correlated to invasive tests

of shear wave velocity.

Reference Site Dispersion Intrusive

type method method

[1]5 Soil CSW PIP

Continued on next page





type method method

[4] Soil SASW SCPT

[7] Road SASW CH

[13] Fill SASW SPT

[22] Soil MASW VSSP

[31]1 Soil SASW DH

[34]1 Soil SASW DH and PS

[42] Marine MFT CH

[43] Soil SASW CH

[55] Soil SASW CH

[67] Road SASW FWD

[70] Soil SASW DH

[75] Marine Waveforms SPT/BSMP2

[82, p173] Soil SASW/MASW CH

[83, p36] Soil SASW/MASW CH and SPT

[84] Soil MASW transfer CH, SCPT and Lab

[89] Soil MASW VS model

[97] Soil SASW Lab

[104] Soil SASW CPT and Lab

[112]5 Road SASW CH and FWD

[120]1 Soil SASW CH

[125] Soil SASW SCPT

MASW DH

[136] Soil MASW CH

[140]5 Concrete SASW Cut and visual

[139]3 Soil SASW RC

[148] Fill SASW RC (density)

[162] Fill SASW SCPT

[161, p218],[257] Fill SASW CH (for VS) and

and [261] CH/RC/TS (for Q)

[173]1 Fill MS (array) PS

[179] Soil MS (linear) PS and SCPT

[184, 183] Marine SASW SCPT

Continued on next page





type method method

[189]1 Soil MASW and MFT CH

[202]5 Soil CSW CPT

[222] Road SASW CH and FWD

[218] Road SASW CH

[219] Road SASW CH

[220] Road SASW FWD

[221] Road SASW FWD

[239, 352] Soil MASW (Vib) CH

[238, 240] Soil MASW DH

[242] Marine MASW Onshore DH

[258] Soil SASW CH

[259] Soil MASW and MS SCPT and VSP

[269] Road SASW CT and RC

[288] Soil MS BH

[311] Soil SASW CH

[322] Fill MS (array) DH

[323] Soil SASW DH and SPT

[324] Fill MS (array) DH and SPT

[329] Soil MASW CH and SPT

[353] Soil MASW (Vib) DH4

[358] Soil MASW DH (several)

[359] Soil MASW PS

[354]1 Soil MASW DH and PS

[366] Soil SASW CPT and CH

Notes:

1 Focused on surface/borehole comparisons;

2 SPT N -values give shear velocity from Equation 2.9 and

BSMP uses swell wave interaction;

3 Dispersion from a borehole SASW tool;

4 Surface wave model is a Gibson half-space; and

5 Approximate inversion.


When shear velocity data is presented, there is the first observation of poorer correla-

tion with depth. This is due to the obvious decrease of surface wave resolution with depth.

The mismatch is especially large among thin, large contrasting horizons at depth. Overall,

correlation is best in formations which comprise low velocity, low contrast horizons, i.e.

normally or weakly irregularly dispersive. Observations by [86, 34], showed that if Pois-

son’s ratios are assumed too low (such as 0.25), the inverted layer shear velocities will be

systematically high. This was the case in the initial studies of [31], where the water table,

thus rise in Poisson’s ratio, was not accounted for. As such, inverted shear velocities of

the deeper layers are often too high.

However, in very low contrast, normally dispersive sites, surface wave inversion ve-

locities are in general lower than the borehole log interval velocities, notably in marine

studies [242], albeit the good resolution of surface wave inversion in this range. A correla-

tion of shear velocities from surface wave inversion and PS-suspension logs is provided in

[229], where the correlation is good, except at velocities above 300 m/s where the surface

wave values are invariably too low. Note that surface wave inversion models are usually

extrapolated to the surface, whereas invasive tests generally start from a deeper position,

up to a few metres below the surface, thus the often poor correlation in shallow layers.

Damping, when measured, is also invariably lower than that from invasive tests, that is,

Q is overestimated. This is most likely due to the strong frequency dependence of this

parameter [161].

Where larger velocity contrasts near the surface are present, surface wave inversion

velocities are almost ubiquitously lower, especially the shallower layers [31]. This was

also noticed in marine sites [75]. The reason for underestimating the velocity around

a high velocity layer (HVL), such as a sandy or gravelly horizon in a geotechnical site

investigation, may be a result of ‘dominant higher modes’, either not being correctly

detected and/or modelled. As in [9], it can be seen that in inversely dispersive sites,

standard SASW may measure higher modes as the fundamental mode, which is of lower

phase velocity.

However, in testing over inversely dispersive asphalt and road base sites, the elastic

moduli by SASW compared to FWD tests were generally higher. This is summarised

well in [67]. The reason may be due to ‘superposed modes’ at high frequency being

smoothed and assumed as the fundamental mode at an erroneously high phase velocity.

Alternatively, the strong frequency dependence noted in laboratory test results in [269]

may be a factor.

2.7.3 Alternative geophysical datasets When no other data is available, sup-

porting information can be obtained from other geophysical datasets, either observed in the

field or generated from the surface wave models. One procedure is to generate waveforms

from the final models from dispersion inversion and compare to the observed waveforms


and has been used to appraise both velocity [81, 186] and attenuation [212] models. In [50]

a full elastic source was employed for a better appraisal of VS and Q models. But, unless

the waveforms are incorporated into the inversion, any correlations are purely qualitative.

Other geophysical datasets such as body-wave seismic, electrical, electromagnetic can

be used, but, these are mainly for providing better confidence on layer interfaces and

physical parameter correlations are only qualitative, except for shear wave refraction. In

shallow work, coupled inversions with refraction [78] or vertical electrical soundings (VES)

[53, 114, 211] are the most promising. In deeper work, coupling dispersion and body

wave receiver functions is one method to reduce ambiguity and since the model physical

parameters (velocities) are directly comparable [137, 230].

2.7.4 Inference methods for model appraisal The statistical appraisal of shear

velocity models combines the effects of random and systematic measurement and mod-

elling errors (acquisition errors, mode misidentification and inversion kernel deficiencies),

together with the inherent nonuniqueness of surface wave inversion. While Monte Carlo

inference analyses are treated in many reports of large scale earthquake seismology [280],

this has not been reported in shallow surface wave inversion, at least in a linearised optimi-

sation framework. Linear inversion produces a single model, but the statistical measures

of the final parameters are rigorously defined. In global optimisation methods, there is

no rigorous treatment for error propagation, but many hundreds or thousands of pseudo-

randomised iterations through a broad range of model space provide a statistical appraisal.

As for the nonuniqueness aspect alone, this too is well understood from studies at

earthquake scale [335], but in shallow work has been less investigated [182]. Regularisation

of the inversion helps to limit the nonuniqueness of a model by excluding features which

are not geologically reasonable [287]. This can be done either with statistical (Bayesian),

probabilistic (frequentist) or deterministic methods. Bayesian inversion is a more flexible

approach where model limits depend statistically on the data quality and are defined a

priori [285, 333]. This acts as a method for model appraisal or inference theory [286] and

can be employed in either local or global inversion schemes.

Shallow shear velocity resolution Parametric model appraisal tests of shallow surface

wave dispersion were considered in [262]. A synthetic dispersion curve based on a cross-

hole velocity model was generated and inverted under various distributions of data, with

both exact and assumed layer interface depths. One shortcoming, along with the limited

number of tests, was the neglect of data errors, thus it was more of a test of inversion

nonuniqueness. However, it was revealed that when the layers assumed in the inversion

span the true interfaces, the recovered shear velocity is about the average of the true shear

velocities. This can lead to considerable error when large contrasts are present, which in

that case was a LVL. Both distribution and number of data points had little effect on

model parameter recovery, but the longest wavelength (i.e. lower cutoff frequency) was


particularly influential, especially when the maximum wavelength does not penetrate to

the zones of interest.

A similar deterministic study was conducted in [303]. In that work, perturbations

of a base model were made to predict how the absolute RMS dispersion difference and

frequency band with largest sensitivity varied within a range of a two-parameter model

subspace. It was based on a single LVL case and moreover, even though dominant higher

modes were known to be generated, only the fundamental mode plane-wave dispersion was

analysed. However, data resolution limits due both to wavenumber resolution (variable)

and acquisition error (constant) were incorporated into the sensitivity analysis, but were

quite unrealistic. It was suggested that even thin LVLs at depth are resolvable and that

dispersion curves scale proportionally to thickness/depth/velocity variations of the LVL.

SASW error propagation An early study of inverted model uncertainty based on linear

inverse theory error propagation with statistically estimated data errors was applied in

[366]. A more thorough example study of error propagation from data through to final

model from an experimental SASW point of view is in [330]. The starting point was an

analysis of the different phase angle spectra from 30 repeated shots at a test site while the

source and receiver positions remained constant. As only high frequencies were collected

(60-120 Hz), the depth penetration was limited. Moreover, frequencies with large error

were removed from the input datasets. The ensemble of data was then inverted with both

4- and 8-layer assumed structures to create an statistical ensemble of models. While not a

thorough Monte Carlo analysis, both in the acquisition and inversion stages, the findings

were:

1. Data phase angle (and thus phase velocity) errors are within 7% and normally dis-

tributed;

2. Model shear velocity errors are within 8% and not normally distributed;

3. Model uncertainty increases with depth and/or number of layers.

All these findings are of limited use for practical inference of model accuracy as the results

are constrained the regularisation of the inversion. Conversely, without constraints, model

errors may be a manifestation of the lack of smoothness in the optimisation, even though

data fit may be better.

MASW error propagation A similar test was later conducted from MASW data in

[160], with a maximum of 15 repeated shots. They too revealed a normal distribution of

effective phase velocities and larger errors apparently associated with ambient noise bands.

However, the relative uncertainties showed strong frequency dependence in two separate

ranges and was strongly site dependent. The frequency of marked change in uncertainty

was about 10.5-12.5 Hz. below this, errors exceeded 10% and above this maximum errors

were only 1.25%. The distribution of inverted model parameters was calculated from the


assumption of Gaussian error propagation and local linearity. However, large uncertainties

at low frequency did not appear to correlate with the standard deviations from the model

covariance matrices, where the deepest layers had maximum standard deviation of only a

few percent. However, when correlated to crosshole data at one site, the mismatch was in

excess of 25%. This suggested that standard linearised inversion covariance analysis may

not be entirely reliable.

As stated in [160], an extensive Monte Carlo analysis is probably the best way to

infer uncertainty in model parameters from a linearised inversion. This was performed a

priori on the basin-scale reference model in [288] prior to field measurements. From the

likely errors in inverted shear wave models, an indication of the uncertainty envelope to

be expected in ground amplification could be estimated. On a smaller scale, this kind of

test was used a priori for generating starting models for an inversion in [172]. By defining

the increase of layer thickness with depth as an exponential function and scaling the shear

velocities to the reduced wavelength dispersion curve, the model which best reproduces

the observed dispersion is used to start the inversion process. While not covering the full

space of possible layer arrangements, features of the target model could be recovered.

A related method was used a posteriori in [81]. There, a ‘rubberband test’ on the

final inverted model was applied to provide a suite of models which still fit the observed

data within a given RMS error. Resolution decreased with depth, being poorest in the

homogenous half-space shear velocity and depth but VS of a low velocity layer (LVL) is

quite well constrained. VP of a LVL is not resolved at all by Rayleigh wave inversion,

which is also the case for body wave inversion. However, the larger scale studies of [335]

suggested the LVL VS is not resolvable at all from phase velocity dispersion alone.

LVL/HVL resolution While the resolution of a HVL is untested in shallow experi-

mental surface wave data, it was discovered to be quite poor at crustal scale in genetic

algorithm inversions of both group velocity dispersion [175] and receiver functions [300].

Numerically, high velocity layers (HVL’s) at shallow depth are poorly resolved with a lin-

earised inversion, but improved under a global optimisation scheme [43, 182]. However, a

plane-wave matrix dispersion was used so dominant higher modes were not incorporated.

On a similar scale, the error propagation from two different multiple filter analysis (MFA)

methods for group velocity to the final model is shown in [247]. Group velocity measure-

ments are subject to the resolution tradeoffs in the time-frequency domain [77, 295], which

were not considered in [186]. Nevertheless, the poorer resolution of a HVL as opposed to

a LVL is again quite evident from the Monte Carlo inversion ensembles.

Global optimisation In global optimisation, standard deviations of model parameters

are usually defined from the spread of acceptable models generated during the search.

This is an a posteriori model appraisal and estimates from a low number of iterations will

be inaccurate [22]. In spite of this, global inversions appear to show that there is a rapid


rise in uncertainty of the recovered shear wave velocities of deeper layers. In particular,

the resolution of the homogenous half-space shear velocity is almost zero, that is, can lie

anywhere within the range allowed for the global search [186].

In summary, the range in final model parameters (shear velocity) is a function of

three sources for error propagation: (a) The dispersion curve, incorporating additive and

equipment noise, source effects and dispersion measurement methods; (b) The model pa-

rameterisation and inherent assumptions; and (c) The inversion method (linearised or

global) and the inherent nonuniqueness. These issues, in a linearised framework, are in-

vestigated further in Chapter 7 for both synthetic and field models, again where dominant

higher modes are a common feature.

87

CHAPTER 3

Dispersion resolution and accuracy: Synthetic testing

3.1 Introduction

The objective of this chapter is to investigate how processing parameters, field layouts

and acquisition errors combine to contaminate the dispersion curve supplied to a surface

wave inversion algorithm. It is divided into the following parts:

1. Dispersion processing;

2. Source and receiver layouts;

3. Individual acquisition errors; and

4. Dispersion repeatability and robustness.

Item 1 (above) will employ simple, analytically defined dispersion curves. Remaining tests

(items 2 to 3) will employ the models of Tokimatsu et. al., described in Chapter 2, incor-

porating dominant higher modes. A brief indication of model parameter resolution in a

linearised inversion framework will follow item 4, where realistic repeatability is simulated.

88 Chapter 3. Dispersion resolution and accuracy: Synthetic testing

3.2 Dispersion processing

These experiments outline how given single mode dispersion curves are mapped by a

plane-wave decomposition and the inherent resolution limits therein. The following aspects

will be considered:

1. Spread layouts;

2. Spread length;

3. Trace padding; and

4. Spatial aliasing.

These theoretical aspects should not be confused with practical acquisition and processing

variables, discussed in Section 3.3, associated with modelling and dispersion observation

from Earth models. Here, the Earth model and specific acquisition parameters are ir-

relevant and the dispersion curves to be employed are ideal, analytically defined ‘modal’

curves. In these tests, discrete phase velocities available for the given t−x parameters, in

both f−k (c(f) = f/k(f))and f−p (c(f) = 1/p(f)) space, are generated and the approx-

imations of mapping an algebraically defined curve onto these planes will be measured.

Thus, accuracy in these tests is a limitation of the plane-wave discretisation.

The theoretical dispersion was calculated for the frequency vector from a 512 point

seismogram, sampled at 1 ms (1000 Hz sample rate, giving a 500 Hz Nyquist frequency).

The half-space dispersions are simply constant for all frequencies. The curves which show

velocity dispersion are modelled with an analytic Gaussian which is asymptotic at the

low- and high-frequency ends, representative of a single layer a few metres thick over a

homogenous half-space. In real data, even the simplest pure fundamental mode curve can

usually not be modelled by a high-order polynomial or other analytic form, thus they are

purely representative and do not incorporate any higher modes and modal superposition.

Note again that any low-frequency effects or detector rolloff and low cut filters are ignored.

The f − k transform covers a space of Nf frequencies by Nk = Nx/2 wavenumbers,

from (δf , δk) resolution to the Nyquist limits (fN , kN ), where aliased wavenumbers are

not included. Phase velocity resolution depends inversely on the wavenumber, which

for surface waves is highly frequency dependent for a given phase velocity. The f −p transform covers the same frequency band, with Nk slownesses linearly from 1.0-12.5

s/km, corresponding to phase velocities of 80-1000 m/s. The phase velocity resolution

from this discretisation depends purely on magnitude of the slowness and is frequency

independent for a given phase velocity.

By f−k, the number of dispersion points is strictly limited to the smaller dimension of

the transform, otherwise the wavenumber function may not be monotonically increasing.

This is a fundamental requirement analogous to SASW which requires the phase of the

3.2. Dispersion processing 89

cross spectrum to be a continuously increasing function. However, dispersion points for

all frequencies up to 100 Hz are shown, or the Nyquist wavenumber limit, whichever

is first. This correlates with the f − p equivalent where a slowness is available at all

frequencies. If only points for each f − k wavenumber were used, a better visual match

with the f − p dispersion and lower RMS error would result. However, in practise the full

frequency range evenly spaced dispersion points are generally used.

3.2.1 Spread layouts These tests will show the effects of both number of receivers

and receiver spacing. In these tests, no trace padding is applied. Thus the dimension of

the wavenumber (f − k) or slowness (f − p) axis is half the number of traces, assuming

use of unaliased wavenumbers or slownesses only.

100 m/s half-space In Figure 3.1 the inaccuracy of the f −k transform is immediately

evident when no trace padding is applied. Moreover, for the wider geophone spacings, the

f − k alias limit is reached at lower frequencies while the f − p dispersion is exact. This

is in part due to the coincidence of a matching slowness of the assumed dispersion and

slant stack range. Thus the error for the 48-channel gather (Tables 3.1 and 3.2) is zero.

Note however that the Nyquist slowness has not been considered here, to be discussed in

Section 3.2.4.

500 m/s half-space In Figure 3.2 errors are invariably larger, in f−k due to the higher

velocity and in f − p due to the lower resolution in slowness at higher phase velocity.

However, the rate that errors decrease with number of channels and geophone spacing is

similar to the 100 m/s half-space (Tables 3.1 and 3.2).

Normal dispersion In Figure 3.3, errors decrease with frequency (velocity) since wavenum-

ber resolution increases. One limitation with this dispersion can be seen in the 24-channel

gather at 1 m spacing (Figure 3.3(a)), where the f − k lower frequency limit is around 10

Hz. At this frequency, wavenumber becomes zero and no dispersion points are available,

but f − p transform this limit is not a concern. The lower frequency limit is calculated by

considering the first wavenumber (δk) and the Rayleigh wave velocity (c) in the half-space

(zero frequency), denoted cN . This velocity can calculated by the FSW method of by the

approximation of Equation 1.2 [3] in Chapter 1.

Since the wavenumber dimension is invariably less than or equal to the frequency

dimension, the lower cutoff frequency (fc) is wavenumber resolution limited. Since zero

wavenumber is not allowed, this minimum wavenumber will be the wavenumber resolution

(δk) and is given by

fc =cNδk

(3.1)

where cN is the maximum fundamental mode phase velocity, which in a normally dispersive

medium will be that of the half-space (layer N) and δk is the wavenumber resolution, given

by

δk =kN

Nk(3.2)


20 40 60 80 100

80

90

100

110

120

(a) 24 channels at 1.00 m spacingc

(m/s

)

f−k f−p Ideal

20 40 60 80 100

80

90

100

110

120

(b) 24 channels at 2.43 m spacing

c (m

/s)

f−k f−p Ideal

20 40 60 80 100

80

90

100

110

120

(c) 48 channels at 1.00 m spacing

c (m

/s)

f−k f−p Ideal

20 40 60 80 100

80

90

100

110

120

(d) 48 channels at 1.19 m spacing

c (m

/s)

f−k f−p Ideal

20 40 60 80 100

80

90

100

110

120

(e) 96 channels at 1.00 m spacing

c (m

/s)

Frequency (Hz)

f−k f−p Ideal

20 40 60 80 100

80

90

100

110

120

(f) 96 channels at 0.59 m spacingc

(m/s

)

Frequency (Hz)

f−k f−p Ideal

Figure 3.1: Dispersion accuracy limitations for an ideal 100 m/s half-space using 24, 48

and 96 channels. Geophone spacing is fixed at 1 m for (a), (c) and (e) and spread length

fixed at 56 m for (b), (d) and (f).

20 40 60 80 100

400

450

500

550

600

(a) 24 channels at 1.00 m spacing

c (m

/s)

f−k f−p Ideal

20 40 60 80 100

400

450

500

550

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

400

450

500

550

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

400

450

500

550

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

400

450

500

550

600


c (m

/s)

Frequency (Hz)

f−k f−p Ideal

20 40 60 80 100

400

450

500

550

600

(f) 96 channels at 0.59 m spacing

c (m

/s)

Frequency (Hz)

f−k f−p Ideal

Figure 3.2: Dispersion accuracy for an ideal 500 m/s half-space using 24, 48 and 96

channels. Geophone spacing is fixed at 1 m for (a), (c) and (e) and spread length fixed at

56 m for (b), (d) and (f).


20 40 60 80 100

200

400

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

200

400

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

200

400

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

200

400

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

200

400

600


c (m

/s)

Frequency (Hz)

f−k f−p Ideal

20 40 60 80 100

200

400

600


c (m

/s)

Frequency (Hz)

f−k f−p Ideal

Figure 3.3: Dispersion accuracy limitations for an ideal 500-100 m/s normal dispersion

curve using 24, 48 and 96 channels. Geophone spacing is fixed at 1 m for (a), (c) and (e)

and spread length fixed at 56 m for (b), (d) and (f).

20 40 60 80 100

200

400

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

200

400

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

200

400

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

200

400

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

200

400

600


c (m

/s)

Frequency (Hz)

f−k f−p Ideal

20 40 60 80 100

200

400

600


c (m

/s)

Frequency (Hz)

f−k f−p Ideal

Figure 3.4: Dispersion accuracy limitations for an ideal 100-500 m/s inverse dispersion

curve using 24, 48 and 96 channels. Geophone spacing is fixed at 1 m for (a), (c) and (e)

and spread length fixed at 56 m for (b), (d) and (f).


Table 3.1: RMS accuracy (δc, in m/s) limitations with various number of channels at fixed

trace spacing (1 m) for the four ideal dispersion curves of Figures 3.1 to 3.4 (left).

Channels 100 m/s 500 m/s Normal Inverse

f − k

24 10.7 55.4 60.1 34.7

48 9.5 39.3 59.2 17.1

96 5.2 30.4 57.2 8.5

f − p

24 3.9 11.1 9.1 14.4

48 0.0 0.0 4.9 6.3

96 0.5 5.4 2.5 6.4

Table 3.2: RMS accuracy (δc, in m/s) limitations at various trace spacing and fixed spread

length (56 m) for the four ideal dispersion curves of Figures 3.1 to 3.4 (right).

Channels 100 m/s 500 m/s Normal Inverse

f − k

24 9.5 54.6 76.4 22.5

48 6.8 37.7 62.8 16.8

96 4.9 26.4 44.4 12.2

f − p

24 3.9 11.1 9.1 14.4

48 0.0 0.0 4.9 6.3

96 0.5 5.4 2.5 6.4


where kN is the Nyquist wavenumber (m−1) andNk is the number of discrete wavenumbers

available, which is Nx/2, where Nx is the number of traces (zero padded or not). If

DC wavenumber was included, there would be Nx/2 + 1 wavenumbers available up to

and including the Nyquist. If the double Nyquist was used as the upper limit (that is,

including the k-aliased half of the f−k plane with an off-end shot gather) then by symmetry

there would be Nx wavenumbers. However the resolution would not be improved, merely

doubling the wavenumber wrapping frequency.

The lower cutoff frequency can be thought of intuitively as the frequency where the

picked mode reaches minimum wavenumber. In this case, when no trace padding is applied,

the minimum wavenumber in Figure 3.3(a) is equivalent to the wavenumber resolution

(Equation 3.2) and is 1/24 = 0.0417. Thus fc is 20.8 Hz. Phase velocities from any

lower frequencies are thus not physically possible and are not part of a monotonically

increasing wavenumber function. Note however that unlike the phase velocity resolution

limit (Equation 2.7) the lower cutoff frequency can be easily improved by trace padding

which reduces δk. Thus, padding 24 traces to 256 lowers it by over a factor of 10.

The upper cutoff frequency will be either where the picked spectral lobe reaches either

the Nyquist frequency, wavenumber or slowness, whichever comes first. In practise, it is

usually limited by signal/noise ratio, controlled mainly by limited source function band-

width. In Tables 3.1 and 3.2 the overall error and rate of error decrease with Nx is similar

to the 500 m/s half-space, except with f − k for a constant spacing of 1 m.

Inverse dispersion In this case (Figure 3.4), errors increase with frequency. However,

with increasing number of channels and wavenumber resolution, RMS errors decrease

rapidly (Tables 3.1 and 3.2). Since the half-space velocity is lower, the cutoff frequency is

less of a concern. In the worst case, Figure 3.4(a), it would be 100/24 or 4.2 Hz.

3.2.2 Spread length Using Equation 2.7 and the four ideal dispersion curves, the

theoretical resolution with frequency is shown in Figure 3.5. These envelopes mimic the

accuracy of the dispersion curve mapped with both f − k and f − p. Note that at a

certain low frequency, (proportional to the phase velocity), the resolution envelopes become

asymptotic indicating zero resolution. This is an important empirical result, since the

maximum depth of the inversion will be limited by this poor resolution of low-frequency

data. In Figures 3.5(a) to 3.5(c) we see that for half-space and normal dispersion, the

envelope narrows with frequency. However, for inversely dispersive curves, although it is

a maximum at lowest frequency, it reaches another local maximum at mid-frequency, and

tapers in between. This band of poor resolution is also around the point where higher

modes come into play.

Note too that with flat and normal dispersion, the onset of rapid resolution drop off

is around the cutoff frequency, plotted as a vertical line using the same line type as the

envelope. We can see that with the high velocity half-spaces, Figures 3.5(b) and 3.5(c), the


20 40 60 80 1000

50

100

150

200(a) 100m/s halfspace

c (m

/s)

Frequency (Hz)

24 m spread length 48 m spread length 96 m spread lengthIdeal dispersion

20 40 60 80 1000

200

400

600

800

1000(b) 500m/s halfspace

c (m

/s)

Frequency (Hz)


20 40 60 80 1000

200

400

600

800

1000(c) 500−100m/s normally dispersive

c (m

/s)

Frequency (Hz)


20 40 60 80 1000

200

400

600

800

1000(d) 100−500m/s inversely dispersive

c (m

/s)

Frequency (Hz)


Figure 3.5: Resolution limitation with spread length based on Equation 2.7 for the four

ideal dispersion curves of Figures 3.1 to 3.4.

resolution envelope becomes asymptotic then drops again once below the cutoff frequency.

This is a consequence of those frequencies not mapping the ideal dispersion at the given

layout and should be ignored. Cutoff frequency and velocity resolution must be considered

together to confirm the validity of the dispersion data at low frequency.

3.2.3 Trace padding The effects of trace padding prior to plane-wave decomposi-

tion is shown in Figures 3.6 to 3.9. It is obvious, both visually from the figures and in the

RMS errors of Table 3.3 that more trace padding reduces error in the mapped dispersion

curve. Moreover, f−p performs better than f−k over the same frequency band. However,

when decomposing real data, if the ratio of trace padding to number of traces is too large,

wavenumbers will be over-interpolated.

The strange rolloff of phase velocity even at maximum padding in Figure 3.8(d) ap-

pears very similar to the ‘near-field effects’ observed in [281]. This rolloff is explained in

Figure 3.10, where the ideal dispersion curve is mapped onto the f − k plane at both 64

and 512 (square) trace padding. At low padding, the misfit is large, since k resolution

is poor. However, even when the shot gather is padded square, the picked curve at low


20 40 60 80 100

80

90

100

110

120

(a) 48 channels padded to 64

c (m

/s)

f−k f−p Ideal

20 40 60 80 100

80

90

100

110

120

(b) 48 channels padded to 128

c (m

/s)

f−k f−p Ideal

20 40 60 80 100

80

90

100

110

120

(c) 48 channels padded to 256

c (m

/s)

Frequency (Hz)

f−k f−p Ideal

20 40 60 80 100

80

90

100

110

120

(d) 48 channels padded to 512

c (m

/s)

Frequency (Hz)

f−k f−p Ideal

Figure 3.6: Trace padding effects on accuracy for a 100 m/s half-space. 48 channels padded

to (a) 64, (b) 128, (c) 256 and (d) 512 have been tested.

20 40 60 80 100

400

450

500

550

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

400

450

500

550

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

400

450

500

550

600


c (m

/s)

Frequency (Hz)

f−k f−p Ideal

20 40 60 80 100

400

450

500

550

600


c (m

/s)

Frequency (Hz)

f−k f−p Ideal

Figure 3.7: Trace padding effects on accuracy for a 500 m/s half-space. 48 channels padded

to (a) 64, (b) 128, (c) 256 and (d) 512 have been tested.


20 40 60 80 100

100

200

300

400

500

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

100

200

300

400

500

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

100

200

300

400

500

600


c (m

/s)

Frequency (Hz)

f−k f−p Ideal

20 40 60 80 100

100

200

300

400

500

600


c (m

/s)

Frequency (Hz)

f−k f−p Ideal

Figure 3.8: Trace padding effects on accuracy for a normal dispersion curve. 48 channels

padded to (a) 64, (b) 128, (c) 256 and (d) 512 have been tested. Time dimension is 512

samples at 1ms. Key to lines: f − k - thick dotted, f − p - thick dashed, ideal - thin solid.

20 40 60 80 100

100

200

300

400

500

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

100

200

300

400

500

600


c (m

/s)

f−k f−p Ideal

20 40 60 80 100

100

200

300

400

500

600


c (m

/s)

Frequency (Hz)

f−k f−p Ideal

20 40 60 80 100

100

200

300

400

500

600


c (m

/s)

Frequency (Hz)

f−k f−p Ideal

Figure 3.9: Trace padding effects on accuracy for an inverse dispersion curve. 48 channels

padded to (a) 64, (b) 128, (c) 256 and (d) 512 have been tested.


Table 3.3: RMS accuracy (δc, in m/s) limitations with various trace padding for the four

ideal dispersion curves of Figures 3.6 to 3.9.

Trace padding 100m/s 500m/s Normal Inverse

f − k

64 4.6 34.5 58.0 12.5

128 2.3 21.4 36.1 6.3

256 2.3 21.3 39.8 3.5

512 1.2 12.0 22.4 1.8

f − p

64 1.0 26.7 3.6 25.6

128 0.6 22.7 2.5 21.7

256 0.4 1.0 0.9 1.8

512 0.2 2.0 0.4 2.0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

5

10

15

20

Fre

quen

cy(H

z)

(a)

250

500

750

1000

1250

1500

1750

2000

2250

2500

82

165

247

330

412

495

577

660

742

824

49

99

148

197

247

296

346

395

444

494

35

70

106

141

176

211

247

282

317

352

27

55

82

110

137

164

192

219

246

274

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

2

4

6

8

10

Wavenumber (m−1)

Fre

quen

cy(H

z)

(b)

2000

4000

6000

8000

10000

666

1332

1997

2663

3329

399

799

1198

1597

1997

285

570

856

1141

1426

222

444

666

887

1109

181

363

544

726

907

154

307

461

614

768

133

266

399

532

665

117

235

352

470

587

105

210

315

420

525

Figure 3.10: Illustration of the projection of the normal dispersion curve of Figure 3.8 onto

the f − k domain created by a 512 point, 1 ms sampled shot gather. Trace padding for

each is (a) 64 and (b) 512. The posted values show the discrete phase velocities available

for each f − k pixel. Key to lines: Mapped (dashed) and ideal (solid).


0 5 10 15 20 25

130

140

150

160

(a) 150 m/s to 100 m/s

c (m

/s)

f−k f−p Ideal

0 5 10 15 20 25200

220

240

260

280

300

320

(b) 300 m/s to 100 m/s

c (m

/s)

f−k f−p Ideal

0 5 10 15 20 25

300

350

400

450

500

(c) 450 m/s to 100 m/s

c (m

/s)

Frequency (Hz)

f−k f−p Ideal

0 5 10 15 20 25350

400

450

500

550

600

650

700

(d) 600 m/s to 100 m/s

c (m

/s)

Frequency (Hz)

f−k f−p Ideal

Figure 3.11: Low-frequency accuracy of four normal dispersion curves, based on Figure 3.8.

All curves have a high-frequency velocity of 100 m/s and low-frequency velocity of (a) 150,

(b) 300, (c) 450 and (d) 600 m/s. Time and trace padding is square to 512.

0 5 10 15 20 25

130

140

150

160

(a) 150 m/s to 100 m/s

c (m

/s)

f−k f−p Ideal

0 5 10 15 20 25200

220

240

260

280

300

320

(b) 300 m/s to 100 m/s

c (m

/s)

f−k f−p Ideal

0 5 10 15 20 25

300

350

400

450

500

(c) 450 m/s to 100 m/s

c (m

/s)

Frequency (Hz)

f−k f−p Ideal

0 5 10 15 20 25350

400

450

500

550

600

650

700

(d) 600 m/s to 100 m/s

c (m

/s)

Frequency (Hz)

f−k f−p Ideal

Figure 3.12: Low-frequency accuracy of four normal dispersion curves, based on Figure 3.8,

with time and trace padding is square to 1024.


frequency is at lower velocity than expected. The phase velocity values for each (f, k)

pixel is in Figure 3.10 show this.

As further tests of this low-frequency anomaly for normal dispersion curves, four other

models were tested. These are shown in Figure 3.11. The problem is only evident at higher

half-space velocities, and moreover the 500 m/s maximum test of Figure 3.8 was probably

an unlucky example to trial first. However, even though the rolloff is not apparent, the

inaccuracy at low-frequency is still present. By doubling resolution to 1024 time samples

and padding square to 1024 traces (Figure 3.12), accuracy at low frequency is improved,

not apparently by a factor of two as the padding would suggest.

3.2.4 Spatial aliasing The Nyquist wavenumber (kN ) is an important parameter

in f − k dispersion and there is a similar limit in f − p space, the Nyquist slowness, pN

which has been not been mentioned so far. It arises from the τ −p alias and in f −p space

is described by the curve [78]

pN =1

f∆x(3.3)

where f is the frequency (Hz) and ∆x is the trace spacing (m). Once in f − p space, any

frequency-slowness combinations above this curve are aliased and are unusable. This limit

will be a maximum bound on the usable frequency range.

If we were to transform an off-end shot gather by both f − k and f − p (with no

padding) then the maximum frequency possible, defined by the alias wavenumber and

slowness respectively, would be the same. However, in f−k, we could recover aliased phase

velocities by simply using the negative waveumbers, as long as there are no overlapping

wrapped features. Assuming the same off-end shot gather when transforming into f − p,

only use positive slownesses are allowed (no incoming waves). Since the slowness limits

are decided during the slant-stack, aliased slownesses are not recoverable and as such is

one disadvantage of the method. However, unless extremely high frequencies were desired

from a large spread, this would not be a problem. Proper field procedure would ensure two

or more geophone intervals would be recorded if both shallow, detailed and deep, broad

depth features were required.

3.2.5 Discussion and comparisons with previous work

Spread length The requirement of a larger array aperture for increased low-frequency

resolution thus depth penetration has been realised by many researchers [80, 173, 288, 301].

In both analytic [241] and experimental [85] frameworks, it was shown that longer spread

lengths give improved modal resolution. This is especially important in resolving higher

and superposed modes, with lower energy, which otherwise become smeared with poor

k resolution. This is at the expense of lateral variation problems, spatial aliasing and

signal to noise (S/N). S/N is especially important for higher frequencies which are vital

for resolving shallow layers.


In [241] it was suggested that for a given spread length, an increased number of channels

does not improve resolution, and this was also supported experimentally. By incorporating

wrapped wavenumbers, [85] supported this conclusion. That work also suggested that

with a fixed number of channels to use a non-constant geophone spacing, comprising three

separate shot gathers spliced together. Once separated, they can be processed individually

to gain the benefits of three overlapping frequency bands, similar to the SASW method, or,

with a trace padding chosen to provide equal k-resolution, the transforms can be stacked

if necessary.

In the work here, longer spread lengths only improved the accuracy for non- or inversely-

dispersive curves. For normal dispersion, the error actually increased with spread length

(Figure 3.3). Note too in all cases from Figures 3.1 to 3.4, the larger geophone spacing

(higher k-resolution) gave no improvement when using f−p dispersion. In reality, it would

have simply narrowed the unaliased band in τ − p space.

Trace padding An early multichannel analysis by [89] suggested that to employ f − k,

many channels (256) must be collected. Trace padding was considered in [85] with fewer

channels and showed that larger padding increased the accuracy of the wavenumbers of

spectral peaks, thus more accurate phase velocity. However, in that work, the reference

curve was one with very high padding and accuracy judged relative to it. In the results of

Sections 3.2.1 to 3.2.3 the reference curve is known analytically, thus the accuracy results

are not biased.

f − k versus f − p One addition made in this work is the comparison of multichannel

dispersion methods. Previous experimental work has qualitatively shown that f − k and

τ − p dispersion give equivalent results [82]. In terms of resolution with frequency, this

is analytically correct. However, with a well chosen slowness vector f − p dispersion will

produce more accurate results than f − k over the same frequency band for non- and

normally-dispersive curves. However, at low trace padding for inverse dispersion, f − k is

more accurate.

One disadvantage of the f − p method is at higher frequency, where aliased slownesses

can not be recovered beyond the 1/f∆x point (Equation 3.3), discussed in Section 3.2.4.

Thus, if very shallow information is required, a smaller trace spacing must be employed to

increase the alias slowness limit. Ideally, several different spacings should be collected in

the field and then stacked in f −k or f −p space to provide a broad bandwidth dispersion

curve. Discrepancies are larger at low frequency, which limits the the maximum depth

of investigation to be accurately interpreted. This is controlled by the phase velocity

resolution, based on Equation 2.7 and is a function of spread length.

f − k cutoff frequency A less important factor (only for normally dispersive sites) is

the f − k cutoff frequency. This may prohibit the measurement of low-frequency phase

velocities when little or no trace padding is applied. With even a modest trace padding


it can be reduced. Although trace padding effectively increases the spread length, it does

so by interpolation in the spectral domain and the physical spread length is unchanged.

Low-frequency observations In repeated SASW tests, lower and upper cutoff frequency

has previously been defined from the amplitudes of the cross-spectra and coherance func-

tions [121]. In standard MASW dispersion observation methods it is obvious that there

is a broadening of the spectral lobe, thus loss of resolution at low frequencies [237]. The

lower cutoff frequency is generally a heuristic observation of the frequency below which

the data is ‘unusable’. By f−k, there is a lower cutoff frequency dictated by Equation 3.1

but in τ − p it is does not exist. The lower cutoff frequency is the frequency below which

slowness (or phase velocity) resolution exceeds a user-defined threshold. Generally, this

point is higher than the f − k lower cutoff frequency and applies for data transformed by

either f − k or τ − p. Thus, the lower cutoff frequency is invariably spread length limited,

the resolution defined by Equation 2.7.

In the half-space and severe normal dispersion cases of Figures 3.6 to 3.9, the trend

of the mapped dispersion curve at low frequency seem to mimic the so-called ‘near-field

effects’ observed at low frequency in [281]. It appears that a possible contributor to

spurious dispersion observed at low-frequency effects is nothing more than a resolution

limitation at low frequency. In a simple 2-layer stiff-over-soft scenario, the theoretical low-

frequency asymptote of the dispersion curve is the Rayleigh wave velocity of the half-space.

However, due to poor resolution, it will be spurious with no predictable pattern. However,

for inverse dispersion the error envelope is not a simple decreasing function of frequency.

Instead, inaccuracy ‘bulges’ at mid-frequency where wavenumber resolution and velocity

combine to give the poorest resolution. However, at the very lowest frequencies, poor

wavenumber resolution will always dominate.

Maximum measurable wavelength There have been many publications giving optimum

field layout criteria for SASW, summarised well in [91, 373]. These are both in-field

recommendations and post-processing ‘filters’, for retaining wavelengths considering ‘near-

field effects’ and aliasing limits, devised both experimentally and numerically.

Figure 3.13 shows the same data as in Figure 3.5 but converted to phase velocity-

wavelength as in most of the original SASW presentations, thus phase velocity resolution

with approximate depth is more evident. Notice how the maximum surface wavelength is

a limitation of the half-space velocity. In addition, the cutoff frequency when converted

to wavelength is approximately equal to the spread length. Note how this is independent

of trace interval (Figures 3.1 to 3.4), where the spurious low-frequency onset is the same

for a 56 m spread length.

However, at long wavelength (low frequency) in Figure 3.13, the phase velocity reso-

lution envelope becomes infinite, the point where it ‘turns’. At the corresponding wave-

length, phase velocity resolution is zero. This wavelength is consistently 0.4 that of the


0 50 100 150 200

0

20

40

60

80

100

(a) 100m/s halfspace

Wav

elen

gth

(m)

c (m/s)

24 m 48 m 96 mIdeal

0 200 400 600 800 1000

0

20

40

60

80

100

(b) 500m/s halfspace

Wav

elen

gth

(m)

c (m/s)

24 m 48 m 96 mIdeal

0 200 400 600 800 1000

0

20

40

60

80

100

(c) 500−100m/s normally dispersive

Wav

elen

gth

(m)

c (m/s)

24 m 48 m 96 mIdeal

0 200 400 600 800 1000

0

20

40

60

80

100

(d) 100−500m/s inversely dispersiveW

avel

engt

h (m

)

c (m/s)

24 m 48 m 96 mIdeal

Figure 3.13: Wavelength resolution limitations with spread length based on Equation 2.7

for the four ideal dispersion curves of Figures 3.1 to 3.4. This is a simple transform of

Figure 3.5.

spread length. Thus, in terms of spread length (L) a new long-wavelength (low-frequency)

cutoff criteria can be specified as

λmax < 0.4L (3.4)

which means that the maximum wavelength (λmax) available is limited to 0.4 times the

spread length (L). This is similar to the rule in [322] of L > λ/3. In terms of relation to

SASW rules, it is between those of Heisey et. al. of 0.33λ and Gucunski and Woods of 0.5λ

[91, 373]. All these rules are related to the wavelength reduction factors for approximate

inversion (Equation 1.21), on the basis of a directly proportional λ− z relationship.

Another representation of this rule is

L > 2.5λmax (3.5)

which means that spread length should be at least 2.5 times the largest wavelength desired

to be measured. This rule is not entirely rigorous, however, since the upper phase velocity

envelope is still finite for much larger wavelengths and cannot be disregarded.


Combined with analytic studies of the approximate maximum depth-wavelength rela-

tion (eg. 0.5λ [92]) this correlates with a maximum investigation depth of around 0.2L.

For example, a 23 m spread will allow interpretation down to 4.6 m, which is similar to

the commonly assumed limitations in seismic refraction shooting of a maximum depth of

0.2-0.25 the spread length [340]. With fixed-channel equipment, trace spacing must be de-

cided considering aliasing limits, however, will usually be more constrained by the degree

of lateral heterogeneity, topography, room at the site and available source energy. Field

procedures with limited-channel equipment should include walkaway shooting to increase

channel density and/or spread length and trace padding to improve wavenumber/slowness

accuracy.

The high-frequency limit will invariably be an alias limitation, which is both field

layout and Earth velocity dependent. From a MASW perspective, a slower dominant

group velocity will mean that in f − k the effective mode dispersion lobe reaches the alias

wavenumber at a lower frequency unless trace spacing is lowered to counteract this. Note

that higher frequencies can be recovered by retaining and translating the aliased (negative)

wavenumbers to the positive limit [85] but wrapped lobes may interfere.


3.3 Spread layouts

In this and the following sections, observed dispersion curves will be investigated,

as opposed to simply the mapping of an ideal dispersion curve for various f − k and

f − p spectral dimensions as shown in Section 3.5.1. The observed dispersion curves

will be extracted from transformed, full-waveform, synthetic shot gathers, considering

variety of field layouts. This comprises a complete, automated simulation of the MASW

process. Thus, any effects that body waves, spherical spreading and frequency bands have

in addition to the fundamental discretisation limits can be deduced. A 40 Hz Berlage

wavelet from vertical impact, with low-frequency rolloff from 4.5 Hz, are employed in

these tests.

3.3.1 Homogenous half-space The homogenous half-space model and acquisition

arrangement employed in Section 2.4 is used to illustrate the trend of dispersion with near

offset. Shear velocity is maintained at 360 m/s with Poisson’s ratio as 0.25. The slant-

stack range is 243-405 m/s, thus excluding direct P -waves (624 m/s). Starting at 1 m

near offset and extending to 201 m, at 4 m steps, the PSV observed dispersion curves

are shown in Figure 3.14 for three spread lengths of 24, 48 and 96 m. The curves are

plotted at arbitrary scale and offset vertically for clarity. The trend which was noticed

in Section 2.4 is evident, where spurious low-frequency effects ar generated with shorter

spread lengths.

The measured phase velocity variation with near offset for four frequencies (5, 10,

15 and 20 Hz) for each array length is shown in Figure 3.15. The true phase velocity

is 331 m/s, which is the FSW frequency independent phase velocity. At 5 Hz, each

array measures similar dispersion with near offset. Missing phase velocities are where the

τ − p limits were exceeded. At higher frequencies, the trend for all arrays is to measure a

lower phase velocity at closer shot offsets. However, the 96 m array is less prone to this

problem, measuring a correct phase velocity for all near offsets.

A statistical analysis of the errors variation of measured phase velocity with near

offset is shown in Figure 3.16. On the left are standard deviations of phase velocity with

frequency (grey) with the theoretical phase velocity resolution overlain (black line). On

the right are RMS errors (solid) and χ2 (dashed) variation with near offset. The reference

curve is again 331 m/s . At around 5 Hz, the error introduced by the PSV method from

near offset alone is up to 40 m/s or over 10%. However, in the 5-50 Hz range the error

improves markedly with spread length. In terms of near-offset dependence, it appears that

error does improve for the larger source distances. However, the χ2 statistic shows clearly

that a constant minimum error is attained at about 20 m near offset for the 24 and 48 m

spreads. For the 96 m spread, the optimal near offset is at 50 m.

3.3. Spread layouts 105

1 21 41 61 81101121141161181201

Nea

r of

fset

(m

)

(a)

1 21 41 61 81101121141161181201

Nea

r of

fset

(m

)

(b)

1 21 41 61 81101121141161181201

0 10 20 30 40 50 60 70 80 90 100

Nea

r of

fset

(m

)

Frequency (Hz)

(c)

Figure 3.14: Variation of observed dispersion with near offset for a 360 m/s half-space

with vertical impact source. Trace windows are (a) 24, (b) 48 and (c) 96 channels at 1 m

spacing. Dispersion is by f − p with trace padding of 256 channels. The dispersion curves

are stacked with an arbitrary logarithmic scale and vertical offset for clarity.

3.3.2 Synthetic models The synthetic models of Tokimatsu et. al. [325] (Sec-

tion 2.2) are shallow, thus combined with the need for a high Nyquist wavenumber, spread

lengths can afford to be relatively short. With a dominant group velocity of 140 m/s, at

0.5 m trace spacing aliasing will occur at about 75 Hz, or a wavelength of about 2 m. By

the rule of thumb of Section 3.2.5, to resolve the basement at 14 m would require a spread

length of at least 35 m. For these tests, a 192-channel synthetic reflectivity shot gather at

0.5 m trace spacings with 1024 samples at 2 ms was generated. Then, by employing 24-,

48- and 96-channel windows, starting with a 0.5 m shot offset and sliding away from the

shot at 1 m intervals, MASW dispersion by both f − k and f − p was observed up to the

alias wavenumber/slowness. The f − k cutoff frequency with 256 trace padding at 0.5 m

spacing and based on model velocities VP=1400 m/s and VS=360 m/s is 2.7 Hz, where 4

Hz which was the lower limit presented in the original work of [325]. Although body waves

are modelled and the upper τ − p velocity of 380 m/s incorporates the direct waves, no

dispersion variation between the models from this wavefield can be ascertained, since the

P -wave velocities of the synthetic models are same. Although different velocity SV -waves


250

300

350

400c(

5 H

z) (

m/s

)(a) 24 m spread

250

300

350

400

c(10

Hz)

(m

/s)

(d)

250

300

350

400

c(15

Hz)

(m

/s)

(g)

50 100 150 200250

300

350

400

c(20

Hz)

(m

/s)

(j)

Near offset (m)

(b) 48 m spread

(e)

(h)

50 100 150 200

(k)

Near offset (m)

250

300

350

400

c(5

Hz)

(m

/s)

(c) 96 m spread

250

300

350

400

c(10

Hz)

(m

/s)

(f)

250

300

350

400

c(15

Hz)

(m

/s)

(i)

50 100 150 200250

300

350

400

c(20

Hz)

(m

/s)

(l)

Near offset (m)

Figure 3.15: Variation of observed dispersion with near offset for a 360 m/s half-space

with vertical impact source for four low frequencies of 5, 10, 15 and 20 Hz. Each column

shows measured phase velocity with dispersion 24, 48 and 96 m arrays.

0

50

100

σ c (m

/s)

(a) (i) 24 channels by τ−p

100

101

102

∆ c R

MS (

m/s

)

(a) (ii)

101

102

103

χ c2 (da

shed

)

0

50

100

σ c (m

/s)

(b) (i) 48 channels by τ−p

100

101

102

∆ c R

MS (

m/s

)

(b) (ii)

101

102

103

χ c2 (da

shed

)

0 50 1000

50

100

Frequency (Hz)

σ c (m

/s)

(c) (i) 96 channels by τ−p

50 100 150 20010

0

101

102

Near offset (m)

∆ c R

MS (

m/s

)

(c) (ii)

50 100 150 20010

1

102

103

χ c2 (da

shed

)

Figure 3.16: Error analysis of variation of observed dispersion with near offset for the

homogenous half-space (Figure 3.14). On the left are standard deviations of phase velocity

with frequency in grey, with the theoretical phase velocity resolution overlain in black. On

the right are RMS errors (solid) and χ2 (dashed) variation with near offset. RMS errors

are calculated from a reference phase velocity of 331 m/s for all offsets over the frequency

range 0-100 Hz.


are in effect, the near-offset tests are more an analysis of the effects of the cylindrically

spreading surface wavefield.

3.3.3 Case 1 near offset The variation of dispersion with near offset is shown

in Figures 3.17 and 3.18. By f − k we see that within the first few metres of the shot,

there are some unusual low-frequency effects. With short spreads there are ‘ripples’ in

the dispersion curves which increase in amplitude and migrate to lower frequency with

increasing offset. By f − p the curves are much smoother and the ripples are still evident,

thus they are not purely a function of the mapping of the dispersion into the discrete

f − k spectral space.

At low frequency (Figure 3.19) there is unusual behaviour in the dispersion curves

from the first three offsets (0.5-2.5 m). By f − k, there is no change in dispersion above

about 4 Hz with near offsets beyond 2.5 m. However, by f − p, the low-frequency nature

of the curve changes remarkably and becomes irregular at very far offset. Note that the

limits of the slant-stack were 64-1400 m/s, thus the maximum phase velocity is actually a

clipping effect. In both cases, however, below about 4 Hz the dispersion is erratic.

The error analysis of these plots is shown in Figure 3.20. In the left hand column,

the rise in standard deviation at low frequency appears to be a consequence of a rapid

decrease in resolution envelope. In the right hand column, the statistics with near offset

are presented. As a unique reference curve is unavailable, both the mean and median

curves were used, with little difference between the two. χ2 is calculated based on the

mean RMS. For the 24- and 48-channel spreads there is a rapid misfit improvement for

near offsets up to 2.5 m. However with 96 channels there is no improvement with near

offset. The overall variance decreases with spread length and appears to best when f−p is

used.

3.3.4 Case 2 near offset This model is inversely dispersive with a much higher

dominant group velocity than Case 1. Nevertheless, the same sampling was adopted. In

Figures 3.21 and 3.22 the prominent effect is the variation of modal osculation points with

near offset. This complies with theory, since the effective phase velocity is offset dependent

[159]. With 24 and 48 channels, (Figure 3.22) the osculations are largest at around 20 m

near offset. With 96 channels the jumps are lowest over this range. The low-frequency

response (Figure 3.23) is similar to Case 1.

The error analysis (Figure 3.24) shows as expected the large variation around the

modal osculation points, and with 24-channels, irrecoverably poor resolution is reached

at around 10 Hz. In terms of variance with offset, results are similar to Case 1, only

marginally larger. Of note is the larger error with 24 channels at around 20 m near

offset, since over a smaller spatial window the unusual effects seen around 15-20 m are

exacerbated. With the 48- and 96-channel data, by f − p, there is a definite lower error


0.5

47.5

Nea

r of

fset

(m

)

(a) 24 channels at 0.5m spacing, 256 trace padding by f−k

0.5

47.5

Nea

r of

fset

(m

)

(b) 48 channels at 0.5m spacing, 256 trace padding by f−k

0.5

47.5

0 10 20 30 40 50 60 70

Nea

r of

fset

(m

)

Frequency (Hz)

(c) 96 channels at 0.5m spacing, 256 trace padding by f−k

Figure 3.17: Variation of observed dispersion with near offset for Case 1. Trace windows

are (a) 24, (b) 48 and (c) 96 channels at 0.5 m spacing. Dispersion is by f − k with trace

padding to 256 channels.

0.5

47.5

Nea

r of

fset

(m

)

(a) 24 channels at 0.5m spacing, 128 trace padding by τ−p

0.5

47.5

Nea

r of

fset

(m

)

(b) 48 channels at 0.5m spacing, 128 trace padding by τ−p

0.5

47.5

0 10 20 30 40 50 60 70

Nea

r of

fset

(m

)

Frequency (Hz)

(c) 96 channels at 0.5m spacing, 128 trace padding by τ−p


are (a) 24, (b) 48 and (c) 96 channels at 0.5 m spacing. Dispersion is by f − p with trace



0.5

47.5

0 5 10 15

Nea

r of

fset

(m

)

Frequency (Hz)


0.5

47.5

0 5 10 15

Nea

r of

fset

(m

)

Frequency (Hz)


Figure 3.19: Zoom in on the 96-channel data in Figures 3.17(c) and 3.18(c) showing

variation of observed dispersion with near offset for Case 1 at low frequency. Dispersion

is by (a) f − k and (b) f − p with trace padding to 256 channels.

0

10

20

σ c (m

/s)

(a) (i) 24 channels by f−k

0.11

10100

∆ c R

MS (

m/s

) (a) (ii)

100

101

102

103

χ c2 (da

shed

)

0

10

20

σ c (m

/s)


0.11

10100

∆ c R

MS (

m/s

) (b) (ii)

100

101

102

103

χ c2 (da

shed

)

0

10

20

σ c (m

/s)

(c) (i) 48 channels by f−k

0.11

10100

∆ c R

MS (

m/s

) (c) (ii)

100

101

102

103

χ c2 (da

shed

)

0

10

20

σ c (m

/s)

(d) (i) 48 channels by τ−p

0.11

10100

∆ c R

MS (

m/s

) (d) (ii)

100

101

102

103

χ c2 (da

shed

)

0

10

20

σ c (m

/s)

(e) (i) 96 channels by f−k

0.11

10100

∆ c R

MS (

m/s

) (e) (ii)

100

101

102

103

χ c2 (da

shed

)

10 20 30 40 50 600

10

20

Frequency (Hz)

σ c (m

/s)

(f) (i) 96 channels by τ−p

10 20 30 400.1

110

100

Near offset (m)

∆ c R

MS (

m/s

) (f) (ii)

10 20 30 4010

010

110

210

3

χ c2 (da

shed

)

Figure 3.20: Error analysis of variation of observed dispersion with near offset for Case

1 (Figures 3.17 and 3.18). (Left) Standard deviations of phase velocity (grey), with the

theoretical phase velocity resolution overlain (black), and; (Right) RMS errors (solid) and

χ2 (dashed) variation with near offset, from both mean and median reference dispersion

over frequencies 5-65 Hz.


0.5

47.5

Nea

r of

fset

(m

)


0.5

47.5

Nea

r of

fset

(m

)


0.5

47.5

0 20 40 60 80 100 120

Nea

r of

fset

(m

)

Frequency (Hz)





0.5

47.5

Nea

r of

fset

(m

)


0.5

47.5

Nea

r of

fset

(m

)


0.5

47.5

0 20 40 60 80 100 120

Nea

r of

fset

(m

)

Frequency (Hz)


Figure 3.22: Variation of observed dispersion with near offset Case 2. Trace windows are

(a) 24, (b) 48 and (c) 96 channels at 0.5 m spacing. Dispersion is by f − p with trace



0.5

47.5

0 5 10 15

Nea

r of

fset

(m

)

Frequency (Hz)


0.5

47.5

0 5 10 15

Nea

r of

fset

(m

)

Frequency (Hz)





0

10

20

σ c (m

/s)


0.11

10100

∆ c R

MS (

m/s

) (a) (ii)

100

101

102

103

χ c2 (da

shed

)

0

10

20

σ c (m

/s)


0.11

10100

∆ c R

MS (

m/s

) (b) (ii)

100

101

102

103

χ c2 (da

shed

)

0

10

20

σ c (m

/s)


0.11

10100

∆ c R

MS (

m/s

) (c) (ii)

100

101

102

103

χ c2 (da

shed

)

0

10

20

σ c (m

/s)


0.11

10100

∆ c R

MS (

m/s

) (d) (ii)

100

101

102

103

χ c2 (da

shed

)

0

10

20

σ c (m

/s)


0.11

10100

∆ c R

MS (

m/s

) (e) (ii)

100

101

102

103

χ c2 (da

shed

)

20 40 60 80 1000

10

20

Frequency (Hz)

σ c (m

/s)


10 20 30 400.1

110

100

Near offset (m)

∆ c R

MS (

m/s

) (f) (ii)

10 20 30 4010

010

110

210

3

χ c2 (da

shed

)







0.5

47.5

Nea

r of

fset

(m

)


0.5

47.5

Nea

r of

fset

(m

)


0.5

47.5

0 10 20 30 40 50 60 70

Nea

r of

fset

(m

)

Frequency (Hz)





0.5

47.5

Nea

r of

fset

(m

)


0.5

47.5

Nea

r of

fset

(m

)


0.5

47.5

0 10 20 30 40 50 60 70

Nea

r of

fset

(m

)

Frequency (Hz)



are (a) 24, (b) 48 and (c) 96 channels at 0.5 m spacing. Dispersion is by f − p with trace

padding to 256 channels


0.5

47.5

0 2 4 6 8 10 12 14 16 18 20

Nea

r of

fset

(m

)

Frequency (Hz)


0.5

47.5

0 2 4 6 8 10 12 14 16 18 20

Nea

r of

fset

(m

)

Frequency (Hz)





0

10

20

σ c (m

/s)


0.11

10100

∆ c R

MS (

m/s

) (a) (ii)

100

101

102

103

χ c2 (da

shed

)

0

10

20

σ c (m

/s)


0.11

10100

∆ c R

MS (

m/s

) (b) (ii)

100

101

102

103

χ c2 (da

shed

)

0

10

20

σ c (m

/s)


0.11

10100

∆ c R

MS (

m/s

) (c) (ii)

100

101

102

103

χ c2 (da

shed

)

0

10

20

σ c (m

/s)


0.11

10100

∆ c R

MS (

m/s

) (d) (ii)

100

101

102

103

χ c2 (da

shed

)

0

10

20

σ c (m

/s)


0.11

10100

∆ c R

MS (

m/s

) (e) (ii)

100

101

102

103

χ c2 (da

shed

)

10 20 30 40 50 600

10

20

Frequency (Hz)

σ c (m

/s)


10 20 30 400.1

110

100

Near offset (m)

∆ c R

MS (

m/s

) (f) (ii)

10 20 30 4010

010

110

210

3

χ c2 (da

shed

)







for shot offsets beyond 5 m and the broad spatial window smooths out any anomalous

dispersion effects.

3.3.5 Case 3 near offset The dominant higher mode between about 9-16 Hz shows

a definite variation with offset (Figures 3.25 and 3.26). This is more pronounced in the 24-

and 48-channel data than the 96-channel data. The low-frequency response (Figure 3.27)

shows similarity at all near offsets above 5 Hz, but highly variable below 3 Hz band. Similar

to Case 1, the rise at low frequency associated with the high velocity half-space (3-4 Hz) is

only affected by near offsets up to 1 m. The error analysis (Figure 3.28), considering the

96-channel data shows almost no difference between f − k and f − p dispersion. However,

the f − k dispersion is marginally better around 10 Hz for long spreads (Figure 3.28(e)),

which is where the dominant higher mode occurs.

3.3.6 Near-offset error distributions Two simple analytic models have been fit-

ted to the phase velocity distributions. The Gaussian distribution is defined in [29] as

PG(x, µ, σ) =1

σ√

2πexp

[

−1

2

(

x− µ

σ

)2]

(3.6)

where µ is the mean and σ is the standard deviation. The half-width (Γ) is 2.354σ and

probable error (PE) is 0.6745σ. The Lorentzian distribution is defined in [29] as

PL(x, µ,Γ) ≡ 1

π

Γ/2

(x− µ)2 + (Γ/2)2(3.7)

where µ is the mean and Γ is the half-maximum half-width. A standard deviation does not

exist for this distribution, but is approximated by Γ/2.354, the broader shape accounting

for larger outliers.

Analysis of Figures 3.16(b), 3.20(b) and 3.24(b) and 3.28(b) are shown in Figures 3.29,

3.30, 3.31 and 3.32 respectively. Relative (percentage) errors over 10 Hz bands (20 Hz for

Case 2) respectively were averaged and binned into 1% histograms over a ±100% range

and normalised. The standard deviation of the Gaussian and half-width of the Lorentzian

are also shown as σG and ΓL respectively and RMS errors of the fits are shown as δG and

δL. At low frequency, the low probability outliers are better modelled by a Lorentzian

curve. However, below 20 Hz in Case 3 (Figure 3.32(a)), the mean dispersion is not

representative and the phase velocity scatter is bimodal. Above 20 Hz in the homogenous

half-space, Case 1 and Case 2 dispersion the phase velocity distribution with near offset

is adequately modelled with a Gaussian, but in Case 2 the Gaussian distribution is only

valid above 40 Hz. This is the assumed distribution in a linear inversion, and the average

ΓL/σG ratio is around 2.

3.3.7 Discussion and comparisons with previous work


00.20.40.60.8

1

P(δ

c)

0−10Hz

σG=0.95%

ΓL=2.04%

δG=5.05%

δL=3.84%

00.20.40.60.81

P(δ

c)

10−20Hz

σG=0.49%

ΓL=0.99%

δG=0.89%

δL=1.01%

00.20.40.60.8

1

P(δ

c)

20−30Hz

σG=0.29%

ΓL=0.51%

δG=0.36%

δL=0.49%

00.20.40.60.81

P(δ

c)

30−40Hz

σG=0.30%

ΓL=0.54%

δG=1.57%

δL=1.64%

00.20.40.60.8

1

P(δ

c)

40−50Hz

σG=0.29%

ΓL=0.51%

δG=1.44%

δL=1.51%

00.20.40.60.81

P(δ

c)

50−60Hz

σG=0.29%

ΓL=0.50%

δG=1.40%

δL=1.49%

−5 −2.5 0 2.5 50

0.20.40.60.8

1

P(δ

c)

60−70Hz

σG=0.22%

ΓL=0.29%

δG=0.47%

δL=0.50%

δ c (%)−5 −2.5 0 2.5 5

00.20.40.60.81

P(δ

c)

70−80Hz

σG=0.20%

ΓL=0.19%

δG=0.28%

δL=0.29%

δ c (%)

Figure 3.29: Statistical analysis of the homogenous half-space error distribution with near

offset, averaged over 10 Hz bands, based on Figure 3.14(b).

0.001

0.01

0.1

1

P(δ

c)

0−10Hz

σG=0.64%

ΓL=1.84%

δG=11.98%

δL=11.58%

0.001

0.01

0.1

1P

(δ c

)

10−20Hz

σG=1.50%

ΓL=3.13%

δG=3.23%

δL=2.29%

0.001

0.01

0.1

1

P(δ

c)

20−30Hz

σG=0.53%

ΓL=0.85%

δG=0.23%

δL=0.48%

0.001

0.01

0.1

1

P(δ

c)

30−40Hz

σG=0.42%

ΓL=0.46%

δG=0.10%

δL=0.17%

−20 −10 0 10 20

0.001

0.01

0.1

1

P(δ

c)

40−50Hz

σG=0.10%

ΓL=0.10%

δG=0.10%

δL=0.10%

δ c (%)−20 −10 0 10 20

0.001

0.01

0.1

1

P(δ

c)

50−60Hz

σG=0.51%

ΓL=0.80%

δG=0.29%

δL=0.40%

δ c (%)

Figure 3.30: Statistical analysis of the Case 1 error distribution with near offset, averaged

over 10 Hz bands, based on Figure 3.18(b).


0.001

0.01

0.1

1

P(δ

c)

0−20Hz

σG=1.43%

ΓL=2.96%

δG=2.10%

δL=3.47%

0.001

0.01

0.1

1

P(δ

c)

20−40Hz

σG=0.62%

ΓL=1.43%

δG=3.26%

δL=2.32%

0.001

0.01

0.1

1

P(δ

c)

40−60Hz

σG=1.64%

ΓL=3.28%

δG=2.74%

δL=3.73%

0.001

0.01

0.1

1

P(δ

c)

60−80Hz

σG=1.31%

ΓL=2.75%

δG=4.16%

δL=4.77%

−20 −10 0 10 20

0.001

0.01

0.1

1

P(δ

c)

80−100Hz

σG=1.09%

ΓL=2.15%

δG=1.31%

δL=1.46%

δ c (%)−20 −10 0 10 20

0.001

0.01

0.1

1

P(δ

c)

100−120Hz

σG=0.87%

ΓL=1.76%

δG=1.30%

δL=1.71%

δ c (%)



0

0.2

0.4

0.6

0.8

1

P(δ

c)

0−10Hz

σG=0.20%

ΓL=0.42%

δG=14.81%

δL=14.77%

0

0.2

0.4

0.6

0.8

1

P(δ

c)

10−20Hz

σG=0.33%

ΓL=0.85%

δG=4.61%

δL=4.16%

0

0.2

0.4

0.6

0.8

1

P(δ

c)

20−30Hz

σG=0.18%

ΓL=0.13%

δG=0.14%

δL=0.14%

0

0.2

0.4

0.6

0.8

1

P(δ

c)

30−40Hz

σG=0.23%

ΓL=0.34%

δG=0.43%

δL=0.24%

−5 −2.5 0 2.5 5

0

0.2

0.4

0.6

0.8

1

P(δ

c)

40−50Hz

σG=0.10%

ΓL=0.10%

δG=0.00%

δL=0.07%

δ c (%)−5 −2.5 0 2.5 5

0

0.2

0.4

0.6

0.8

1

P(δ

c)

50−60Hz

σG=0.37%

ΓL=0.75%

δG=3.12%

δL=3.20%

δ c (%)




Near offset Previous works which investigated homogenous half-space dispersion at-

tributed anomalous low-frequency data to body wave interference [281] and/or cylindrical

spreading [373]. However, [373] measured closer and further spacings than [281], at nor-

malised distances of 0.75, 1.5 and 3.0, as opposed to 1, 1.2, 1.5 and 2.0. Nevertheless,

both showed that transfer function measurements (between source and single receiver

waveforms) were more prone than traditional cross power spectra (between two distant

receivers) and that larger offsets generally improved the solution. The very far offsets of

3lambda best matched the theoretical frequency independent phase velocity.

The work here suggests that there is a similar improvement with near offset in half-

space velocity measured by MASW, however, only for shorter spreads which do not satisfy

the L > 2.5λmax rule (Equation 3.5). For example, at 10 Hz, the 360 m/s shear velocity

half-space has a 331 m/s phase velocity, thus surface wavelength of 33 m. The 24 and

48 m arrays do not satisfy the required length to measure this wave, however, achieve a

stable measurement once near offset exceeds about 25 m, which is less than one wavelength

(Figures 3.15 and 3.16). The 96 m array dispersion is much less affected at 10 Hz for all

near offsets. However, all arrays show similar large discrepancies at very low frequencies.

Near-field tests by SASW on the Tokimatsu et. al.Cases 1, 2 and 3 were conducted

in [322]. Those results showed that in Cases 2 and 3 that, for larger near offsets, modal

osculations increased in amplitude. This work generated effective mode dispersion curves

based on the Rayleigh eigenvalue solution with an active, harmonic point source. When

body waves were included, observed dispersion at different receiver spacings was markedly

different for Case 2, but not so for Case 1 and 3. In [82], a model similar to Case 2 showed

that dispersion at frequencies less than 25 Hz was much better reproduced with a 20 m

near offset, corresponding to just over one wavelength. However, MASW field tests in

[238, 234] showed that near offsets as little as 0.1λ suffice for plane wave propagation. For

example, 60 m wavelengths appear as plane wavefronts from a swept impulsive source at

about 10 m from the source.

In the MASW simulations of the synthetic models, near-offset dependence on low-

frequency dispersion only extends on average to about 5 m near offset. At larger near

offsets, the variations are negligible. This is contrary to the historical recommendations of

a large near offset to reduce near-field effects in irregularly dispersive sites. For example,

considering the errors in the 24 m spread length data by f − p, we see that in Cases 1, 2

and 3, the error drops rapidly at around 2, 5 and 8 m near offset, respectively. Considering

the dispersion at 5 Hz, this corresponds to surface wavelengths of about 64 m for Cases

1 and 2 (phase velocities of 255 m/s) and 31 m for Case 3 (155 m/s). The acceptable

near offsets in these tests are well within one wavelength and even less than the 0.5λ limit

proposed in [238] for higher mode resolution.

Near-field effects considered purely as body wave influence is not a consideration in


this work, due to the constant VP of the test models. Thus, the SASW observations in

[325] that near-field effects due to body waves are greater with a stiff layer overlying a

soft layer can not be ascertained. However, in normally dispersive sites, body waves are

reportedly only influential to a quarter of the surface wavelength [322].

Spread length A comparison of 6-channel f − k (at a single near offset) and 2-channel

SASW synthetic dispersion analysis of Case 2 at various offsets was also conducted in

[322]. The multichannel dispersion was a much smoother curve was less affected by ad-

dition of body waves into the solution. However, we have seen in the above tests that

even large multichannel spreads are prone to a non-linear error envelope as a function of

frequency. The envelope from the scatter in overlapping SASW dispersion curves from

overlapping receiver spacings is a similar effect. As shown in Section 3.2.5, both analytic

[241] and experimental [85] tests show that longer spreads provide better spectral resolu-

tion. However, dominant higher modes were not considered in those works. The scatter

in phase velocity around modal jumps observed here seems reasonable since the effective

phase velocity is offset dependent [325, 159].

‘Low-frequency effects’ From the results presented here, it appears that the traditional

near-field phenomena of body waves and cylindrical spreading have much less effect at low

frequency than that of plane-wave and spread length resolution limits. In addition, the

general decrease of phase velocity modelled in homogenous half-spaces only occurs at very

near offsets, where at very far offsets it shows an increase at low frequency, along with

‘ripples’. The low-frequency limit of the data as influenced by spread length can be seen

best in the homogenous half-space tests. For example, at 24, 48 and 96 m, if the largest

measurable wavelengths are 10, 19 and 38 m (L/2.5), this corresponds to frequencies of

35, 17 and 9 Hz. At these frequencies, near-offset variations are negligible (Figure 3.16)

and are Gaussian distributed (Figure 3.29).

The spread length resolution is the most important factor in accurately measuring

low-frequency dispersion [79]. In the field, the effects on dispersion with near offset will

be exacerbated where wavefield scattering occurs. However, in MASW, there will not be

errors associated with the different propagation path effects in standard SASW acquisition

when a coincident trace window in walkaway or CMP dispersion stacking is employed.

3.4. Individual acquisition errors 119

3.4 Individual acquisition errors

3.4.1 Geophone positioning The positioning errors are assumed Gaussian, with

a standard deviation equal to a fraction of the geophone interval. This is probable for

short spreads and spacings, since simple tape measures are used and errors will arise due

to difficulties of planting due to obstructions, gravel or merely poor field practice. If GPS

or other locating is used for larger spreads and spacings, the fractional error will probably

decrease. In any case, the radius of error is tested about several fractional values of the

nominal geophone spacing, representative of the typical limits encountered under different

field conditions.

3.4.2 Geophone tilt The geophone tilt error will again be taken from a Gaussian

distribution with a given standard deviation in degrees. A full 360◦ azimuth is allowed

for, however, only the apparent tilt in the x− z plane is measured, because the synthetic

seismograms are pure P -SV and confined to this plane. Love wave interference would of

course be preferable, since even for supposedly pure vertical, spherical or inline sources,

considerable Love waves are generated (Section 4.2).

3.4.3 Geophone coupling Geophone coupling problems are difficult to accurately

simulate. Field studies show great dependence on local geology and source type [66]. The

tests here will consider four coupling types of a lightweight spike geophone based on [150]:

1. ‘Best’ - long spike inserted into shallow pit;

2. ‘Good’ - short spike inserted into shallow pit;

3. ‘Fair’ - short spike with vegetation removed but no pit; and

4. ‘Bad’ - short spike without any vegetation removal.

For example, at the Hyden site in Section 4.2, wheat stubble was removed and a small pit

was scuffed into the fine, dry topsoil to reveal coarser, moist soil at an average depth of 5

cm, where short spike geophones achieved best maximum coupling (and possibly reducing

wind noise as well).

The effects of coupling on the amplitude spectra are based on the results in [150, 66]

and are approximated in Figure 3.33. Poor coupling acts as a bandpass filter which reduces

the very low frequencies and also those above about 200 Hz. As coupling becomes worse,

the bandwidth of the filter becomes narrower and rolloffs become broader. In [150] the

effects on the phase spectra are also shown. Poor coupling introduces a 90◦ phase shift

at higher frequencies. The onset of the this shift occurs at lower frequencies as coupling

becomes worse. ‘Best’ coupling is basically a flat amplitude and zero phase filter thus the

input seismograms are unaffected.


0

0.5

1

Am

plitu

de

(a) (i) Best coupling response

−90−4504590

Pha

se

−1

0

1(a) (ii) Best coupled seismogram

0

0.5

1

Am

plitu

de

(b) (i) Good coupling response

−90−4504590

Pha

se−1

0

1(b) (ii) Good coupled seismogram

0

0.5

1

Am

plitu

de

(c) (i) Fair coupling response

−90−4504590

Pha

se

−1

0

1(c) (ii) Fair coupled seismogram

0 100 200 300 400 5000

0.5

1

Frequency (Hz)

Am

plitu

de

(d) (i) Bad coupling response

0 100 200 300 400 500

−90−4504590

Pha

se

0 0.1 0.2 0.3 0.4 0.5−1

0

1(d) (ii) Bad coupled seismogram

Time (s)

Figure 3.33: Representative responses due to variable geophone coupling loosely based on

[150, 66]. From top to bottom are on the left: (a) Best, (b) Good, (c) Fair and (d) Bad

coupling amplitude and phase response filters. On the right are the effects they have on a

synthetic seismogram comprised of simple chirps.

Considering a surface wave bandwidth of 0-100 Hz only the ‘bad’ coupling will have a

major effect. Previous work has been focused on seismic reflection applications, which are

traditionally optimised for high frequency. Indeed, in Figure 3.33 it is the first breaks and

high frequencies which are mostly affected. However, poor low-frequency response and

bandwidth and the effects on P -wave reflection data have been recently reported [213].

While low-frequency surface waves are more likely to be affected by resolution limitations

(Section 3.5.1) than geophone rolloff, the coupling effects will still interfere in their useful

bandwidth.

3.4.4 Source parameters Each of the following is considered separately: Type

(vertical or explosive); Depth; Time-function (analytic), and; Frequency, phase and lag.

Different source type is implemented in the reflectivity method through different down-

going PP , PS, SP and SS reflection coefficients. In the field, an impulsive vertical source

might be a sledgehammer, weight drop or gun. An explosive source is usually buried and


may be either a blasting cap or gelignite. A shotgun with the barrel down a shallow hole

filled with water is a hybrid - mostly vertical muzzle force, but a spherical air bubble as

well. Source depth is an important parameter for static reductions in reflection profiling,

however in surface wave surveying it is generally not a consideration. Indeed, a surface

impact source is usually employed, however, various source depths within the top layer for

both vertical and explosive sources is numerically tested.

All the analytic source time functions of Figure 2.1 are used, except for the Green’s

function seismograms (Dirac-delta source). While not all source functions allow variations

of N or φ, centre frequency was varied to within extreme limits for field sources, namely

10-75 Hz. The damping factor was also varied between the recommended limits of 0.2-0.5

of the time window [214].

3.4.5 Equipment errors In these simulations, all geophones of the gather will be

given a random low-frequency cosine rolloff over the range 0 Hz (ideal) to 28 Hz (popular

reflection geophones). Ideally, each geophone of the gather should have then been given

an erroneous rolloff, however since geophone amplitude response curves usually have a low

specification tolerance this was neglected.

Another equipment error is trigger delay. In reflection work this is a vital parameter

but in dispersion observation is less important as frequency domain analysis is cyclic

invariant. If a very poor trigger was noted in the field, usually at very short shot offsets,

it would require correcting. Note that clipping has not been implemented in these tests

as well, since it is a basic field QC variable.

A final equipment parameter is A/D bit resolution. Here, it is randomly rescaled

between 8- and 24-bit. While the synthetics were generated with 24-bit resolution, which

is the current standard for digital seismographs, 12-bit machines are still in use.

3.4.6 Additive noise To simulate an airwave, a short, high-frequency chirp will be

added to the gather at around 343 m/s. Other coherant noise will be simulated by simple

sinusoidal wavefronts, propagating either with a positive or negative velocity across the

array, such as from heavy machinery or vehicles in the vicinity. Ambient noise is a simple

addition of a Gaussian random distribution, with either a constant standard deviation or

an amplitude relative to the seismic signal. Other problems which are invariably present

in the field, such as dead traces and DC shifts due to shot cables over the data lines or

nearby electrical noise are also simulated.

3.4.7 Methods for acquisition error testing

Synthetic acquisition and dispersion observation The basic field layout is a linear 48-

channel gather, at 1 m geophone spacings with a 5 m shot offset. Other than for source

parameter tests, the source used is the same as that for the layout and resolution tests of


Section 3.3, that is, a vertical impact at the surface with a 40 Hz Berlage wavelet. The

seismograms comprise 512 points at 2 ms sampling and are RMS-mean trace normalised.

In all tests, the f − p method will be used, with 256 trace padding. At this padding,

by using Equation 3.1 the theoretical f − k cutoff frequency is 2.7 Hz. For the models of

Cases 1 and 3, the τ − p slowness stack limits were 64-380 m/s. In Case 2, it was 110-380

m/s. These lower limits are approximately from 80% of the lowest phase velocity to 20

m/s over the highest model shear velocity. While a higher upper limit could have been

equally used with minimal resolution loss (it was 1400 m/s in the layout tests), this limit

keeps the observed dispersion to around the theoretical modal dispersion limits of each

model. When these broad τ − p limits are used, strong direct, refracted or guided waves

can fall within the slant-stack slowness range, since in the Tokimatsu et. al. models the

near-surface VP is 360 m/s.

To reduce these waves, a top group velocity mute at 200 m/s for Cases 1 and 3 and 300

m/s for Case 2, with a 50 ms cosine taper, can be applied prior to slant-stack. This also

excludes the simulated airwave, which is at 343 m/s. Alternatively, the upper slant-stack

velocity limits can be chosen to avoid P -wave slownesses. In practise, since there is usually

a lower cutoff frequency around 5-10 Hz, attempting to recover large phase velocities at

low frequencies is often of no benefit. Thus, an upper phase velocity limit can be chosen

to exclude any possible acoustic wave arrivals. In the Tokimatsu et. al. models, 317 m/s

was chosen for Cases 1 and 2 and 242 m/s for Case 2, the lower limit also ensuring better

slowness resolution in the slant-stack.

Statistical analysis From most small acquisition errors, the effective (observed) phase

velocity has only minor perturbations and it is these small perturbations about an ‘ideal’

dispersion curve which will be used for the statistical analysis. Most undesired wavefields

can be excluded by applying phase velocity limits to the plane wave transform. By f − p,

this is inherent in the slowness limits of the slant stack. By f − k, the same limits are

applied as a ‘fan’ or ‘pie slice’ filter to ensure the two methods can be directly comparable.

In Cases 2 and 3, dominant higher modes are present in the ‘ideal’ dispersion. However,

some other undesired wavefields may manifest, such as noise spikes in the 2D spectrum.

These points are not part of the surface wave dispersion and, since the automatic plane-

wave picking routine cannot discriminate between wavefields, these ‘mispicks’ contaminate

the statistics. Thus, the statistics should be considered as an overestimation of phase

velocity error. On the other hand, Love wave influence has not been accounted for in

the geophone tilt error analysis, so, in reality the uncertainty will be larger than in these

numerical tests.

3.4.8 Case 1 errors To save repetition, in the following figures, the following are

shown, relative to the median dispersion of 30 tests:

1. Standard deviation of phase velocity (grey);


2. Theoretical resolution limit for the 48 m spread from Equation 2.7 (solid dashed);

and

3. Pixel resolution limits from plane-wave transform (solid dotted).

The pixel resolution limits are the f−p equivalent of Figure 3.10, based on time-padding to

512 samples and trace padding to 256 channels, at 2 ms time sampling and 1 m geophone

spacing respectively.

Geophone positioning and tilt In Figure 3.34 position error radii of 0.1 m and 0.2m

and geophone tilt errors of 10◦ and 45◦ were tested. Based on field experience these

represent an average and high error, although a 45◦ tilt error is extreme. At less than 5

Hz, poor resolution prohibits an accurate statistic, however overall the larger parameter

errors produce a larger spread in observed dispersion. The geophone positional errors

produce an almost constant error of 1-2 m/s over the usable frequency range, bordering

on the resolution limit at high frequency. Geophone tilt error appear to more affect the

low usable frequencies. This is feasible, since although horizontal component seismograms

should theoretically produce the same dispersion as the vertical component, it differs at

low frequency due to a dominant higher modes. The distribution of these errors averaged

over 25 Hz bands are shown in Figure 3.35. At higher frequencies, the errors are closest

to zero mean Gaussian distributions, However, less than 25 Hz, outliers dominate and

distributions are again better modelled by a Lorentzian curve. In these tests, near offset

is fixed at 5 m so there is no influence from shot offset effects.

Geophone coupling and static errors In Figure 3.36, severe differences in coupling seem

to have little influence on the observed dispersion. Although the poor coupling produces

up to 2 m/s errors in the usable frequency range, they are only over narrow bandwidths.

The static shifts, on the other hand, produce quite severe dispersion errors. Even random

one-sample static shifts (2 ms) along the gather produce on average a 0.5% error. With

random 10 ms shifts, the low-frequency errors are twice as bad and the frequencies above

33 Hz are catastrophically affected. Although the modal dispersion lobe may still be

coherant at these frequencies, it is obvious that noise is more dominant power.

Source parameter ranges Figure 3.37 shows the effects various source parameter ranges

have on the variance of the observed dispersion curve. Both source centre frequency and

source type appear to have little effect over the usable frequency range of dispersion data.

Only at low frequency do errors exceed 1%.

Various source depth, however, causes severe error increase around the middle of the

usable frequency range. The actual dispersion curves from this analysis at arbitrary scale

are shown in Figure 3.38. All dispersion curves are plotted at arbitrary scale and vertically

offset for clarity. The lowest curve corresponds to the surface shot with remaining curves

from gradually deeper shots, at 0.1 m steps, with the highest curve corresponding to a 2 m

source depth (base of top layer). In both cases, there is a dominance of higher modes with


10−1

100

101

102

σ ∆ c (

%)

(a) Random geophone position error up to 0.1m by τ−p

10−1

100

101

102

σ ∆ c (

%)

(b) Random geophone position errors up to 0.2m by τ−p

10−1

100

101

102

σ ∆ c (

%)

(c) Random geophone tilt up to 10 degrees by τ−p

0 10 20 30 40 50 60 7010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)

(d) Random geophone tilt up to 45 degrees by τ−p

Figure 3.34: Effects of geophone positioning and tilt errors for Case 1. Maximum positional

errors are (a) 0.1 m and (b) 0.2 m and tilt errors are (c) 10◦ and (d) 20◦.

0.001

0.01

0.1

1

P(δ

c)

0−25Hz

σG=0.44%

ΓL=0.55%

δG=0.26%

δL=0.30%

0.001

0.01

0.1

1

P(δ

c)

25−50Hz

σG=0.46%

ΓL=0.62%

δG=0.28%

δL=0.39%

−10 −5 0 5 10

0.001

0.01

0.1

1

P(δ

c)

50−75Hz

σG=0.47%

ΓL=0.66%

δG=1.08%

δL=1.13%

δ c (%)

Figure 3.35: Statistical analysis of the error distribution of Figure 3.34(b), (up to 20 cm

geophone positioning errors).


10−1

100

101

102

σ ∆ c (

%)

(a) Random coupling of Best to Good by τ−p

10−1

100

101

102

σ ∆ c (

%)

(b) Random coupling of Fair to Poor by τ−p

10−1

100

101

102

σ ∆ c (

%)

(c) Random static errors up to 2ms by τ−p

0 10 20 30 40 50 60 7010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)

(d) Random static errors up to 10ms by τ−p

Figure 3.36: Effects of geophone coupling and static errors for Case 1. Coupling ranges

are (a) ‘Best-Good’ and (b) ‘Fair-Poor’ and static errors are (c) 2 ms and (d) 10 ms.

10−1

100

101

102

σ ∆ c (

%)

(a) Source centre frequency range 10−75Hz at 5Hz steps by τ−p

10−1

100

101

102

σ ∆ c (

%)

(b) Source time function effects by τ−p

10−1

100

101

102

σ ∆ c (

%)

(c) Explosion source depth range 0−2m at 0.1m steps by τ−p

0 10 20 30 40 50 60 7010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)

(d) Vertical source depth range 0−2m at 0.1m steps by τ−p

Figure 3.37: Effects of source parameter and depth ranges for Case 1. Source parameter

ranges are (a) centre frequency from 10-75Hz and (b) various 40 Hz time functions as per

Figure 2.1 and source depth ranges are from 0 m (surface) to 2 m (base of top layer) for

both (c) explosion and (d) vertical types.


(a) Explosion source depth range 0−2m at 0.1m steps

0 10 20 30 40 50 60 70Frequency (Hz)

(b) Vertical source depth range 0−2m at 0.1m steps

Figure 3.38: Effects of source depth for Case 1, for both (a) Explosion; and (b) Vertical

types.

10−1

100

101

102

σ ∆ c (

%)

(a) All random additive noise types by τ−p

10−1

100

101

102

σ ∆ c (

%)

(b) Random DC shifts of 1% by τ−p

10−1

100

101

102

σ ∆ c (

%)

(c) Random kill up to 12 traces by τ−p

10−1

100

101

102

σ ∆ c (

%)

(d) Random Gaussian noise of 5% by τ−p

0 10 20 30 40 50 60 7010

−110

0

101

102

Frequency (Hz)

σ ∆ c (

%)

(e) Additive sinusoidal propagating noise by τ−p

Figure 3.39: Effects of additive noise for Case 1. Noise types are: (a) All combined; (b)

DC shifts; (c) Dead traces; (d) Gaussian noise and; (e) Sinusoidal propagating wavefields.


0.001

0.01

0.1

1

P(δ

c)

0−25Hz

σG=0.58%

ΓL=1.07%

δG=1.02%

δL=0.73%

0.001

0.01

0.1

1

P(δ

c)

25−50Hz

σG=0.46%

ΓL=0.61%

δG=0.15%

δL=0.30%

−10 −5 0 5 10

0.001

0.01

0.1

1

P(δ

c)

50−75Hz

σG=0.46%

ΓL=0.62%

δG=0.97%

δL=1.00%

δ c (%)

Figure 3.40: Statistical analysis of the error distribution of Figure 3.39(a) (all additive

noise sources).

10−1

100

101

102

Tilt errors to 5 degCoupling errors Best/Good

Tilt errors to 10 degTilt errors to 20 deg

Coupling errors Best−BadKilled traces up to 12

Coupling errors Fair/BadSource frequency 10−75 HzVertical source depth 0−2 m

Tilt errors to 45 degGaussian noise of 1%Tilt to 45 deg at 12−bit

Static error to 1 msPosition errors to 10 cm

Source pulse typeSinusoidal noise of 5%

Position errors to 20 cm Static error to 2 ms

Gaussian noise of 2.5%Static error to 5 ms

Gaussian noise of 5%DC shifts errors to 1%

Explosive source depth 0−2 mAll additive noise types

(a)

Summed σ (%)

Figure 3.41: Ranking of error influences for the Case 1 dispersion, by summing relative

(%) standard deviation of scatter over 5-70 Hz.


(a) Explosion source depth range 0−2m

0 20 40 60 80 100 120Frequency (Hz)

(b) Vertical source depth range 0−2m


types.

(a) Explosion source depth range 0−2m

0 10 20 30 40 50 60 70Frequency (Hz)

(b) Vertical source depth range 0−2m


types.


10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Offset (m)

Tim

e (s

)

(a)

10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Offset (m)

Tim

e (s

)

(b)

10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Offset (m)

Tim

e (s

)

(c)

Figure 3.44: Full-waveform P -SV synthetic common shot gathers of the 1D models of

Tokimatsu et. al. [325] with 1.4 m deep explosive sources. An AGC window of 256 ms has

been applied: (a) Case 1; (b) Case 2; and (c) Case 3.


Fre

quen

cy (

Hz)

(a)

4 6 8 10 12 14

10

20

30

40

50

60

70


Fre

quen

cy (

Hz)

(b)

4 6 8

10

20

30

40

50

60

70

80

90

100

110


Fre

quen

cy (

Hz)

(c)

4 6 8 10 12 14

10

20

30

40

50

60

70

Figure 3.45: Frequency-slowness (f−p) transforms of the synthetic shot gathers of the 1D

models of Tokimatsu et. al. [325] with 1.4 m deep explosive sources, showing the picked

spectral maxima. Prior to τ −p transform, Cases 1 and 3 had a 200 m/s top mute applied

and all gathers were trace normalised: (a) Case 1; (b) Case 2 and; (c) Case 3.


deeper sources. The vertical source produces a regular pattern whereby higher modes

become preferentially generated at successively lower frequency with increasing depth.

However, the explosion source does not generate the decreasing frequency-onset of first

modal transition, remaining constant beyond a source depth of about 1.2m.

Additive noise The individual effects of several types of additive noise are shown in

Figure 3.39. In the usable frequency range, sinusoidal noise produces anomalies due to a

higher spectral power than the dispersive wavefield. Aside from these spikes, the overall

error is low. Additive white Gaussian noise (AWGN) produces the broadest error in phase

velocities, around 0.5% over the usable frequency range. DC shifts to the traces mainly

affect the low-frequency end of the usable frequency range, and indeed encroach on the

low-frequency poor resolution band. Dead traces seem to have the least affect of all.

When all these noise influences are combined (Figure 3.39(a)) they appear to sum con-

structively and narrow the usable frequency range considerably, the standard deviation

attaining the spread resolution limit at 15 Hz. The distribution of these errors averaged

over 25 Hz bands are shown in Figure 3.40. Above 25 Hz, the dispersion approximate

zero mean Gaussian distributions, and again at lower frequency are better modelled with

a Lorentzian distribution. These additive errors may be the major source of low-frequency

uncertainty in normal dispersion curves, aside from the spread length resolution limita-

tions.

Error ranking A ranking of all these various error sources, by summing the relative

standard deviations over 5-70 Hz is shown in Figure 3.41. An impact source with 48

channels from 5-52 m is used throughout and a heuristic threshold of a ‘strong’ influence

is a summed error over 10%. It is surprising to see that geophone coupling and tilt have

little effect on measured dispersion. Note again that geophone tilt is projected into the

x− z plane and y (Love wave) motion is not accounted for. The more important factors

to consider are additive noise (Gaussian or sinusoidal), trace-to-trace static errors and

geophone positioning. Source pulse type errors are mostly affected by the poor bandwidth

of the Kupper wavelet. Again, while topographic and lateral discontinuities are discounted,

these factors most influence the average phase differences between traces along the shot

gather, causing the phase velocity scatter. The large error due to explosive source depth

is discussed below.

3.4.9 Source depth influence Due to the large dispersion variation in source

depth errors in Case 1 (Figure 3.38) this was tested again for Case 2, shown in Fig-

ure 3.42. Again, all dispersion curves are plotted at arbitrary scale and vertically offset

for clarity. The lowest curve corresponds to the surface shot with remaining curves from

gradually deeper shots, at 0.1 m steps, with the highest curve corresponding to a 2 m

source depth (base of top layer). It seems that the irregularly dispersive case is not as

affected by a deeper vertical source since the effective dispersion is similar for all shots.


There is a minor frequency shift of the modal osculation points, but this is similar to the

shifts due to near-offset variation. Thus, it may merely be a consequence of increased shot

offset by trigonometry. The explosion source, however, generates quite a marked variation,

but the large shifts are due to overlapping guided waves.

The same process for Case 3 is shown in Figure 3.43. The results are similar to Case

1 (Figure 3.38), which may be expected as it has the same upper layer. Of note, however,

is the marked difference in the higher mode jump over 8-16 Hz between the vertical and

explosive shots. The explosion shot does not repeat the expected effective mode dispersion

curve at any depth, whereas the vertical source seems to reproduce the mode jump at all

depths. However, at higher frequency, both shots generate dominant higher modes, which

migrate towards lower frequency with increasing depth.

The reason for the unusual dispersion characteristics with source depth can be ex-

plained from the raw shot gathers and f − p transforms for an explosive shot at 1.4 m

depth. These are shown in Figures 3.44 and 3.45 respectively, which can be directly com-

pared for the case of a vertical impact source Figures 2.3 and 3.45. The shot gathers

(Figure 3.44) have a 256 ms AGC window applied to better reveal all the wavefields.

Guided wave arrivals dominate the record, the ratio of their amplitude to the surface wave

peak being about 2 at near offset and over 5 at the end of the spread. These waves are

the shingled arrivals at about 260 m/s but also overlap with the direct and refracted P -

wavefield. This velocity is within the slant-stack range thus manifests in the f−p spectra.

In Cases 1 and 3, a 200 m/s top mute effectively removes the guided wave influence.

Nevertheless, there are clear jumps to dominant higher modes in Figure 3.45(a) and (c),

different to those of the surface impact case . In the AGC shot gathers, this can also be

explained by the larger amount of splitting and evidence of higher modes in the surface

wavetrain. However, in Case 2 (Figure 3.45(b)), the guided and surface wavefields overlap,

thus cannot be separated by a group velocity window. Only a bow filter in the f − p or

f−k plane could distinguish the two wavefields. Thus, at the τ−p slowness limits chosen,

the guided waves are the dominant mode and are the reason for the unusual patterns in

Figure 3.42. With a suitable slowness range, the guided wave dispersion can be observed

and inverted in a similar manner to surface waves [273].

It is interesting to compare these with the experimental data from Bietigheim of For-

briger [78]. That site generated a dominant higher mode by hammer source, which per-

vaded from the point from about 30 Hz up to the alias limit of about 70 Hz. The funda-

mental mode was quite broad and did not pervade above 40 Hz. However, with a buried

explosive source, the fundamental and dominant higher mode both terminated at about

50 Hz. In that study, the inverted models show that the site has a steep velocity gradient

in the shallow zone, thus, the explosion source was in soil very different from the surficial

material. Synthetic modelling of that site showed that when a buried explosion is indeed


at, or near the base of, a steep gradient in surficial elastic parameters, dominant higher

modes are developed. Thus, the gradual onset of dominant higher modes with depth of

explosive sources (Figures 3.38 to 3.43) is only valid when the upper layer is homogenous.

3.5. Repeatability and model resolution 133

3.5 Repeatability and model resolution

All the parameters from Sections 3.3 and 3.4 are randomised for a full combined error

test in 1D surface wave acquisition. A partial derivative analysis of the observed dispersion

with respect to shear velocity perturbation of the models will then indicate how the model

sensitivities relate to the numerical noise envelopes.

3.5.1 Acquisition errors and processing parameters The acquisition param-

eter ranges in Table 3.4 were randomly employed for many repeated tests. In all tests,

source type was maintained as a vertical impact and sinusoidal noise was not added. This

is a reasonable assumption for most engineering site investigations outside extreme urban

noise areas, where the logistical difficulty of drilling shotholes into a road or hard caprock

would prohibit the use of explosives and a hammer would invariably be used. A repre-

sentative sample of random spread layouts and near offsets, including geophone positional

errors, is shown in Figure 3.46.

Table 3.5 shows the processing parameters used. By using aliased wavenumbers, the

usable frequency range is extended. While an f − k transform of a 1024x512 array is fast,

the τ − p transform is limited to 1024x256 and is far less efficient. As described in , there

is a fundamental phase velocity resolution limitation from the plane-wave discretisation.

The large trace padding and narrow slowness interval were aimed at minimising this to

ensure errors were mapped to the finest detail and these limits are plotted, along with the

resolution limitations from the finite spread length (Equation 2.7). While f − k provides

very large range of phase velocities (infinite if DC (f, k) are allowed), a velocity fan em-

ploying the τ − p transform phase velocity limits was applied to make a fairer comparison

between the two methods.

3.5.2 Case 1 repeatability The results with all parameters random and dispersion

observed by f−k is shown in Figure 3.47 and by f−p in Figure 3.48. To avoid repetition,

these (and following) figures, comprise:

(a) Mean and median dispersion curves, the latter on which error bars of standard devi-

ation are imposed. The grey bands are the modal dispersion curves calculated by

the FSW method; and

(b) Standard deviation of phase velocity (grey);

Theoretical resolution limits from Equation 2.7 for spread lengths employed, mini-

mum 6 m and maximum 96 m (solid dashed); and

Pixel resolution limits from plane-wave transform (solid dotted). Only f − k is

minimum and maximum geophone spacing dependent.


Table 3.4: Acquisition parameter error ranges used in the synthetic seismogram generation

for observed dispersion error envelope tests. ∗Percentage of mean trace maximum.

Parameter Minimum Maximum

Layout ranges

Channels 12 96

Shot offset 0.5 m 20 m

Trace spacing 0.5 m 1 m

Geophone positioning and tilt error

∆x 0 m 0.1dx (m)

∆y 0 m 0.1dx (m)

Tilt 0◦ 30◦

Coupling and static errors

Coupling ‘Poor’ ‘Best’

Static -2 ms 2 ms

Source parameter ranges

X-position -0.2 m 0.2 m

Y-position -0.2 m 0.2 m

Functions Five analytic (Except Dirac)

Frequency 10 Hz 60 Hz

Damping 0.2 0.5

Phase -90◦ 90◦

Lag -5 ms 5 ms

Equipment parameter ranges

Low rolloff 0 Hz 28 Hz

High rolloff 125 Hz 250 Hz

Trigger delay 0 ms 10 ms

Additive noise

Air wave 0% 10%∗

- frequency 200 Hz centre 50 Hz bandwidth

- time span 0 ms 30 ms

DC shifts 0% 5%∗

Kill traces 0 12 at random

High rolloff 125 Hz 250 Hz

Trigger delay 0 ms 10 ms

Gaussian noise 0% 1%∗


Table 3.5: Processing parameters using during synthetic seismogram generation and ob-

served dispersion processing for numerical error envelope tests.

Parameter Case 1 Case 2 Case 3

Time samples 1024 512 1024

Sample interval (ms) 2 2 2

Max. slownesses 750 750 750

Max. tests 80 120 115

Time padding 1024 1024 1024

Trace padding 512 512 512

Max. frequencies 512 512 512

Nyquist frequency (Hz) 250 250 250

Max. usable frequency (Hz) 80 120 80

Max. wavenumbers 512 512 512

Nyquist wavenumber (/m) 1/dx 1/dx 1/dx

Max slownesses (m/s) 256 256 256

Min. τ − p velocity (m/s) 62 115 62

Max. τ − p velocity (m/s) 317 242 317

0 10 20 30 40 50 60 70 80 90

5

10

15

20

25

30

35

40

45

50

Offset (m)

Ran

dom

test

no.

Random spread layouts with geophone positioning error up to 1/5 of trace interval

Figure 3.46: Representative random spread layouts and positional errors used in the com-

bined tests.


c(m

/s)

(a) All random acquisition parameters by f−k

50

100

150

200

250

300

350 Mean Median(with σ

c)

0 10 20 30 40 50 60 70 8010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)

(b) Standard deviations and resolution envelopes

Spread limitsPixel limits

Figure 3.47: Repeatability of the Case 1 dispersion for 80 tests by f − k observation.

c(m

/s)

(a) All random acquisition parameters by τ−p

50

100

150

200

250

300


c)

0 10 20 30 40 50 60 70 8010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 3.48: Repeatability of the Case 1 dispersion for 80 tests by f − p observation.


c(m

/s)


50

100

150

200

250

300


c)

0 10 20 30 40 50 60 70 8010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 3.49: Repeatability of the Case 1 dispersion for 80 tests by f −p observation where

errors outside the spread length resolution envelopes have been thresholded.

c(m

/s)


50

100

150

200

250

300


c)

0 20 40 60 80 100 12010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 3.50: Repeatability of the Case 2 dispersion for 120 tests by f − k observation.


c(m

/s)


50

100

150

200

250

300


c)

0 20 40 60 80 100 12010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 3.51: Repeatability of the Case 2 dispersion for 120 tests by f − k observation

where errors outside the spread length resolution envelopes have been thresholded.

c(m

/s)


50

100

150

200

250

300


c)

0 20 40 60 80 100 12010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 3.52: Repeatability of the Case 2 dispersion for 120 tests by f − p observation



c(m

/s)


50

100

150

200

250

300


c)

0 10 20 30 40 50 60 70 8010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 3.53: Repeatability of the Case 3 dispersion for 115 tests by f − p observation.

c(m

/s)


50

100

150

200

250

300


c)

0 10 20 30 40 50 60 70 8010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 3.54: Repeatability of the Case 3 dispersion by for tests by f−k observation where

errors outside the spread length resolution envelopes have been thresholded.


c(m

/s)


50

100

150

200

250

300


c)

0 10 20 30 40 50 60 70 8010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 3.55: Repeatability of the Case 3 dispersion for 115 tests by f − p observation


In Figure 3.47(a) the median dispersion curve matches best over almost the entire fre-

quency range shown, whereas the mean is corrupted by non-central outliers. In Fig-

ure 3.47(b) the 20-30 Hz band seems to be the most robust in terms of repeatability,

with an average relative error of 1-2%. Below 15 Hz and above 35 Hz errors are all over

10% and below about 6 Hz the observed dispersion departs rapidly from the expected

trend as a direct consequence of the poor resolution envelope. Note that by f − k with

fixed trace padding the wavenumber resolution depends on Nyquist wavenumber, which

is a function of trace spacing (Equation 3.2). Thus two curves are necessary to show the

best pixel resolution (largest trace spacing of 1 m) and poorest pixel resolution (smallest

trace spacing of 0.5 m).

Figure 3.48 shows the same results, using dispersion picked by f−p transform. Results

are similar to the f−k method, however, even though very low-frequency errors are smaller,

the median dispersion departs from the expected trend at a slightly higher frequency, about

7 Hz. In Figure 3.48(b)there is only one pixel resolution limit, based on the slant-stack

velocity range chosen, and are less than the lowest dispersion errors at all frequencies. In

both Figures 3.47(b) and 3.48(b), below 40 Hz, all errors are within the poorest spread

length resolution limit, but at higher frequency the large error may be the increased

number of erroneous picks which would be generated due to dominant noise in the plane-


wave transform. The automatic picking routine does not discriminate between modal

and noise points, which a user would do manually, so this analysis may not be a fair

representation of the modal errors.

Figure 3.49 shows the data but with errors for each trial thresholded to within the

spread length resolution limit for that layout. Even though the poorest spread resolution

limit may be over 100%, the errors are all confined to within a few percent. Only at less

than 10 Hz does the error exceed 10%. These results prove that a simple error estimation

of a constant proportion of the phase velocity (eg. 3%) is unsuitable.

3.5.3 Case 2 repeatability The raw f − k errors are shown in Figure 3.50 and

those thresholded by the spread length resolution limits are shown in Figures 3.51 and

3.52. Again the general fall in standard error with frequency is observed. Errors exceed

10% at frequencies less than 10 Hz and are on average around 1-2% above this frequency.

Frequencies less than 30 Hz appear to be the least affected by random picking errors,

possibly since those outside the τ − p limits (115-242 m/s) were cropped from both trans-

forms. The median phase velocity curve matches well the modal response to a low enough

frequency to allow interpretation of the basement half-space layer. Errors in the f −p dis-

persion (Figure 3.52(b)) at frequencies less than 10 Hz are much less than those in Case

1.

An important consideration here from Section 3.3.2 is the shot offset error envelope. In

Figure 3.21 the largest standard deviations occurred around the frequencies of transitions

to dominant higher modes. Here, there is a definite anomaly in the standard error around

the modal transitions of a few percent, up to 10 m/s in phase velocity. Another important

point is that the standard errors revealed here may cross modal curves at high frequency.

Thus, even if the mode number of a segment of the effective dispersion curve could be

identified, the error would prohibit assigning this number accurately.

3.5.4 Case 3 repeatability The raw f − p errors are shown in Figure 3.53, where

the most robust band is 20-30 Hz, but there is no indication of high phase velocity at

low frequency, where the dispersion below 7 Hz shows soft half-space. However, when

the distribution is thresholded with the spread length resolution limits (Figures 3.54 and

3.55), the dispersion shows the expected trend down to a lower cutoff frequency of about

5 Hz. Similar to Case 1 (Figure 3.49) the most robust data is above 20 Hz. The dominant

higher mode over 8-16 Hz shows good accuracy, yet standard error is quite high, where the

effective error bars in this band cross two modal edges. The f − k dispersion also supplies

useful data to a slightly lower frequency than with f − p and these few extra points would

be essential to accurately model the half-space.

3.5.5 Repeatability statistics


0.001

0.01

0.1

1

P(δ

c)

0−10Hz

σG=0.31%

ΓL=0.87%

δG=7.91%

δL=7.62%

0.001

0.01

0.1

1

P(δ

c)

10−20Hz

σG=0.55%

ΓL=1.16%

δG=1.51%

δL=0.67%

0.001

0.01

0.1

1

P(δ

c)

20−30Hz

σG=0.36%

ΓL=0.73%

δG=0.83%

δL=0.41%

0.001

0.01

0.1

1

P(δ

c)

30−40Hz

σG=0.36%

ΓL=0.73%

δG=0.68%

δL=0.41%

0.001

0.01

0.1

1

P(δ

c)

40−50Hz

σG=0.36%

ΓL=0.74%

δG=0.74%

δL=0.40%

0.001

0.01

0.1

1

P(δ

c)

50−60Hz

σG=0.37%

ΓL=0.74%

δG=0.72%

δL=0.46%

−10 −5 0 5 10

0.001

0.01

0.1

1

P(δ

c)

60−70Hz

σG=0.39%

ΓL=0.81%

δG=1.71%

δL=1.50%

δ c (%)−10 −5 0 5 10

0.001

0.01

0.1

1

P(δ

c)

70−80Hz

σG=0.50%

ΓL=1.03%

δG=2.25%

δL=2.06%

δ c (%)

Figure 3.56: Statistical analysis of the error distribution of Figure 3.48, Case 1 repeata-

bility.

0.001

0.01

0.1

1

P(δ

c)

0−10Hz

σG=0.23%

ΓL=0.39%

δG=2.78%

δL=2.70%

0.001

0.01

0.1

1

P(δ

c)

10−20Hz

σG=0.45%

ΓL=0.96%

δG=1.47%

δL=0.56%

0.001

0.01

0.1

1

P(δ

c)

20−30Hz

σG=0.35%

ΓL=0.72%

δG=0.82%

δL=0.41%

0.001

0.01

0.1

1

P(δ

c)

30−40Hz

σG=0.34%

ΓL=0.69%

δG=0.65%

δL=0.36%

0.001

0.01

0.1

1

P(δ

c)

40−50Hz

σG=0.32%

ΓL=0.62%

δG=0.62%

δL=0.28%

0.001

0.01

0.1

1

P(δ

c)

50−60Hz

σG=0.30%

ΓL=0.56%

δG=0.51%

δL=0.22%

−10 −5 0 5 10

0.001

0.01

0.1

1

P(δ

c)

60−70Hz

σG=0.29%

ΓL=0.52%

δG=0.91%

δL=0.75%

δ c (%)−10 −5 0 5 10

0.001

0.01

0.1

1

P(δ

c)

70−80Hz

σG=0.29%

ΓL=0.53%

δG=1.00%

δL=0.79%

δ c (%)

Figure 3.57: Statistical analysis of the error distribution of Figure 3.49, Case 1 thresholded

repeatability.


0.001

0.01

0.1

1

P(δ

c)

0−10Hz

σG=1.31%

ΓL=2.83%

δG=4.31%

δL=3.01%

10−20Hz

σG=0.40%

ΓL=0.87%

δG=1.54%

δL=0.91%

0.001

0.01

0.1

1

P(δ

c)

20−30Hz

σG=0.32%

ΓL=0.66%

δG=1.34%

δL=0.80%

0.001

0.01

0.1

1

P(δ

c)

30−40Hz

σG=0.71%

ΓL=1.49%

δG=1.78%

δL=1.82%

40−50Hz

σG=0.62%

ΓL=1.30%

δG=1.10%

δL=0.87%

0.001

0.01

0.1

1

P(δ

c)

50−60Hz

σG=0.58%

ΓL=1.21%

δG=1.09%

δL=0.87%

0.001

0.01

0.1

1

P(δ

c)

60−70Hz

σG=0.37%

ΓL=0.80%

δG=1.19%

δL=0.36%

70−80Hz

σG=0.42%

ΓL=0.88%

δG=0.98%

δL=0.37%

0.001

0.01

0.1

1

P(δ

c)

80−90Hz

σG=0.32%

ΓL=0.66%

δG=1.04%

δL=0.35%

−10 −5 0 5 10

0.001

0.01

0.1

1

P(δ

c)

90−100Hz

σG=0.31%

ΓL=0.63%

δG=0.85%

δL=0.21%

δ c (%)−10 −5 0 5 10

100−110Hz

σG=0.24%

ΓL=0.41%

δG=0.83%

δL=0.56%

δ c (%)−10 −5 0 5 10

0.001

0.01

0.1

1

P(δ

c)

110−120Hz

σG=0.23%

ΓL=0.33%

δG=0.50%

δL=0.32%

δ c (%)


repeatability.

0.001

0.01

0.1

1

P(δ

c)

0−10Hz

σG=0.20%

ΓL=0.25%

δG=2.98%

δL=2.96%

0.001

0.01

0.1

1

P(δ

c)

10−20Hz

σG=0.31%

ΓL=0.69%

δG=2.14%

δL=1.67%

0.001

0.01

0.1

1

P(δ

c)

20−30Hz

σG=0.35%

ΓL=0.70%

δG=0.71%

δL=0.27%

0.001

0.01

0.1

1

P(δ

c)

30−40Hz

σG=0.34%

ΓL=0.68%

δG=0.68%

δL=0.29%

0.001

0.01

0.1

1

P(δ

c)

40−50Hz

σG=0.33%

ΓL=0.65%

δG=0.74%

δL=0.24%

0.001

0.01

0.1

1

P(δ

c)

50−60Hz

σG=0.30%

ΓL=0.57%

δG=0.68%

δL=0.33%

−10 −5 0 5 10

0.001

0.01

0.1

1

P(δ

c)

60−70Hz

σG=0.29%

ΓL=0.55%

δG=0.86%

δL=0.61%

δ c (%)−10 −5 0 5 10

0.001

0.01

0.1

1

P(δ

c)

70−80Hz

σG=0.27%

ΓL=0.48%

δG=0.88%

δL=0.67%

δ c (%)


repeatability.


0 10 20 30 40 50 60 7010

−2

100

102

104

Frequency (Hz)

χ ν2

(a) Case 1

Reduced χ2 χν

2>0.05

0 10 20 30 40 50 60 70 80 90 100 11010

−2

100

102

104

Frequency (Hz)

χ ν2

(b) Case 2

Reduced χ2 χν

2>0.05

0 10 20 30 40 50 60 7010

−2

100

102

104

Frequency (Hz)

χ ν2

(c) Case 3

Reduced χ2 χν

2>0.05

Figure 3.60: Reduced χ2 fit of Gaussian distribution by to the observed phase velocity

ranges in (a) Case 1, (b) Case 2, and (c) Case 3.

Distributions An analysis of the the distribution of the relative errors of Figures 3.48

and 3.49 is shown in Figures 3.56 and 3.57. The errors are a zero mean distribution of

phase velocity variation relative to the median phase velocity. The actual phase velocities

could have been equally used, thus this analysis is a test of the statistical assumptions of

normally distributed data in linear inversions [161]. Errors over 10 Hz bands were averaged

and binned into 1% histograms over a ±100% range and normalised.

The analysis of the raw f − p phase velocity errors (Figure 3.56(b)) shows that in fact

a Lorentzian distribution is the more suitable model over all frequency bands, adequately

accounting for the low probability outliers. In individual error tests, the Lorentzian was

generally more suitable at low frequencies only. The visual improvement of the fit is

biased by the log probability scale however the RMS error proves the better fit than the

Gaussian. Even when the thresholded errors are analysed (Figure 3.57), a Lorentzian is

still the better model of the distribution. However, above about 50 Hz, the large phase

velocity outliers become less frequent. This suggests that indeed the raw error tests are

contaminated by a small number of outliers and the relative standard deviations are above

20 Hz are all within 1%. Only at frequencies less than 20 Hz are errors significant. The

f − k distributions showed similar results.

An analytic distribution fitting to the thresholded Case 2 repeatability (Figure 3.52(b))

is shown in Figure 3.58. Although averaging over 10 Hz intervals probably smooths out

the errors around modal transitions points, these are still evident in the Lorentzian half-


−200

−150

−100

−50

0

Pow

er (

dB)

(a) Tokimatsu Case 1

−200

−150

−100

−50

0

Pow

er (

dB)

(b) Tokimatsu Case 2

0 10 20 30 40 50 60 70 80

−200

−150

−100

−50

0

Pow

er (

dB)

(c) Tokimatsu Case 3

Figure 3.61: Dispersion power curves from the synthetic repeatability analysis, taken from

the 2D spectral values along the picked ridge in f − k space.

−0.8

−0.6

−0.4

−0.2

0

Pow

er (

dB)

(a) Tokimatsu Case 1

−0.1

−0.05

0

Pow

er (

dB)

(b) Tokimatsu Case 2

0 10 20 30 40 50 60 70 80−2

−1.5

−1

−0.5

0

Pow

er (

dB)

(c) Tokimatsu Case 3

Figure 3.62: Dispersion power curves from the synthetic repeatability analysis, taken from

the 2D spectral values along the picked ridge in f − p space.


widths, especially around 30-50 Hz. No bands show a Gaussian distribution. The error

distributions for Case 3 (Figure 3.55(b)) are shown in Figure 3.59 and are similar to those

for Case 1 (Figure 3.57). However, there is a broader spread over the 10-20 Hz band as to

be expected. Once again, all bands are best modelled with a Lorentzian.

χ2 test A further verification for a model of phase velocity distribution is the χ2 test.

This test was applied in [160], however, here the reduced χ2 [29] will be used. Assuming

n measurements at each frequency (that is, n repeated acquisition and processing runs),

the χ2 distribution is given as;

χ2 =n

∑

k=1

[ck(f) − ¯ck(f)]2

¯ck(f)(3.8)

where ck(f) is the observed phase velocity dispersion curve at each run (m/s) and ¯ck(f) is

the expected dispersion (m/s). In this analysis, the median will be used for the expected

dispersion as, unlike the mean, it does not incorporate large outliers. The reduced χ2 is

then

χν2 =

χ2

ν(3.9)

where ν is the degrees of freedom, which is the number of data samples minus the number

of free model parameters. In the case of a Gaussian, this is [29, p85]

ν = n− 2 (3.10)

Then, referring to tables of χ2 probability versus ν, the reduced χ2 gives an indication

of the degree of exceeding the certain probability. The reduced χ2 analysis for all the

random tests on the Tokimatsu et. al. models is shown in Figure 3.60. The dotted line

is the threshold χν2 at each frequency for a probability of exceeding 5%(0.05), as used in

[160]. It is mostly constant, except at very low frequency, due to some phase velocities not

being included in the summation (for example those at the τ −p limits) thus n is variable.

Because of this, of the average 120 tests for each case, the number of samples used in

the statistics was as low as 20 at some frequencies. This correlated to a threshold χν2 of

about 1.22, up to a maximum of 1.57 for a full 120 sample range. The solid line shows the

χν2 for the ensemble of tests. For the Gaussian model to be suitable, the observed χν

2

should be below the threshold χν2. In all cases, below 20 Hz the Gaussian model has poor

likelihood, which in Case 2 extends over the entire frequency range. Even at frequencies

above 20 Hz in Cases 1 and 3, the observed χν2 is within one decade of the threshold,

which is still quite a poor fit.

3.5.6 Effective dispersion power A useful measurement in surface wave disper-

sion measurement is the spectral power underlying the picked effective mode ridge in the

2D f−k or f−p transform. These approximate the average trace to trace power spectrum

but more readily show the power in each picked effective mode. The power curves for all


the random tests on the Tokimatsu et. al. models are shown in Figures 3.61 and 3.62. The

immediate impression is a broader nature for the normally dispersive Case 1 and narrower

bandwidths for the irregularly dispersive Cases 2 and 3. The dominant higher modes of

Case 2 show a rippled effect, where power is low at the transitions to dominant higher

modes.

Case 3 shows an abrupt loss of energy below 16 Hz, more so in the f −p power curves,

which is around the band where the dominant higher mode lies, due to the vibration of this

type of model not facilitating low-frequency transmission. It is not a source bandwidth

problem as a centre frequency range of 10-60 Hz was used, thus it is a significant problems

to interpreting deeper layers. A typical scenario may be sand over a hard, limestone

horizon, where layers below the limestone are difficult to interpret due to poor energy

transmission at low frequencies. The poor resolution by spread length is usually the

dominant factor, however, this physical signal-to-noise (S/N) limitation exacerbates the

problem.

3.5.7 Partial derivative analysis The matrix of partial derivatives is an MxN

array, where M is the number of datum points and N the number of model parameters.

Other names for it are the Jacobian or Frechet derivative matrix or sensitivity matrix. It

shows how the datum responds to perturbations of the model parameters and its integrity

is vital in a linear inversion framework to ensure model parameters are correctly updated at

each iteration. The matrix is only valid at the point in model space where it is calculated,

thus must also be readjusted during an iterative inversion.

Analytic modal Rayleigh wave partial derivatives are based on Hamilton’s variational

principal and are derived in [5, 161, 317]. When modal superposition occurs, such as in

the irregularly dispersive cases of Tokimatsu et. al.[325], effective mode partial derivatives

are required when a linearised optimisation is employed.

Numerical partial derivatives are estimated using the forward modelling algorithm,

then by perturbing each model parameter by a small amount, usually in both negative

and positive directions. A best-fit linear or parabolic regression then defines this finite-

difference partial derivative. This is known to be highly unstable and inaccurate. More-

over, this requires 2N extra synthetic runs during the inversion. In this work, the analytic

Jacobian will be considered, with M as the number of frequencies in the effective disper-

sion curve and N the layer shear velocities (VS). This is the conventional data/model

space construction in 1D surface wave inversion. However, the other elastic parameters

can have a marked influence in some scenarios [40, 86].

The standardised partial derivative [40] is a function of frequency and defined as

∂c′

∂β=∂c

∂β

σβn

σcm

(3.11)

where ∂c/∂β is the normal partial derivative, σβnis the a priori standard error for the true


0

200

400

c (m

/s)

(a) Observed dispersion

0

1

2

∂c/∂

β

(b) Layer 1

0

1

2

∂c/∂

β

(c) Layer 2

0

1

2

∂c/∂

β

(d) Layer 3

0 10 20 30 40 50 60 70

0

1

2

∂c/∂

β

(e) Layer 4

Frequency (Hz)

Figure 3.63: Analytic partial derivative analysis of Case 1. (a) Ideal observed dispersion

curve. (b) to (e) are the shear velocity partial derivatives (∂c/∂β).

0

200

400

c (m

/s)


0.010.1

110

100

∂c’/∂

β

(b) Layer 1, δβ=16 m/s

0.010.1

110

100

∂c’/∂

β

(c) Layer 2, δβ=24 m/s

0.010.1

110

100

∂c’/∂

β

(d) Layer 3, δβ=36 m/s

0 10 20 30 40 50 60 700.01

0.11

10100

∂c’/∂

β

(e) Layer 4, δβ=72 m/s

Frequency (Hz)

Figure 3.64: Standardised partial derivative analysis of Case 1 from the errors in Fig-

ure 3.49. (a) Ideal observed dispersion curve. (b) to (e) are the shear velocity standard-

ised partial derivatives (∂c′/∂β). The a priori layer shear velocity uncertainty values are

shown above each plot.


0

200

400

c (m

/s)


0

1

2∂c

/∂β

(b) Layer 1

0

1

2

∂c/∂

β

(c) Layer 2

0

1

2

∂c/∂

β

(d) Layer 3

0 20 40 60 80 100 120

0

1

2

∂c/∂

β

(e) Layer 4

Frequency (Hz)



0

200

400

c (m

/s)


0.010.1

110

100

∂c’/∂

β


0.010.1

110

100

∂c’/∂

β


0.010.1

110

100

∂c’/∂

β


0 20 40 60 80 100 1200.01

0.11

10100

∂c’/∂

β


Frequency (Hz)






0

200

400

c (m

/s)


0

2

4

∂c/∂

β

(b) Layer 1,

0

2

4

∂c/∂

β

(c) Layer 2,

0

2

4

∂c/∂

β

(d) Layer 3,

0 10 20 30 40 50 60 700

2

4

∂c/∂

β

(e) Layer 4,

Frequency (Hz)



0

200

400

c (m

/s)


0.010.1

110

100

∂c’/∂

β


0.010.1

110

100

∂c’/∂

β


0.010.1

110

100

∂c’/∂

β


0 10 20 30 40 50 60 700.01

0.11

10100

∂c’/∂

β


Frequency (Hz)






shear velocity of layer n (m/s) and σc the standard error in the phase velocity (m/s), which

has been the subject of the tests in this chapter. If the standardised partial derivative is

much greater than 1 then it can be assumed significant in an inversion. Alternatively, if

a parameter has standardised partial derivatives much less than 1 it can be ignored in an

inversion, which is the case for density, especially since it can usually be well constrained

in the near surface (eg. borehole data).

The forward algorithm and code for analytic partial derivative and effective mode phase

velocity calculation is from [161]. They only require an observed dispersion curve and all

model parameters. No forward dispersion modelling is performed, but the displacement-

stress then partial derivatives are calculated over user-defined depths then integrated

for the individual layer partial derivatives. In the following analysis of the Tokimatsu

et. al. models, the dispersion was measured by the PSV method from a 48-channel gather

at 1 m near offset. At low frequency, where the observed (effective) mode dispersion de-

parted obviously from the theoretical (plane-wave) dispersion, points were picked from the

dispersion images to ensure accurate extrapolation back to minimum frequency.

Case 1 sensitivity Figure 3.63 shows the raw partial derivatives about the observed

dispersion, which includes error bars from Figure 3.49. All the values are positive, whereby

variation in layer shear velocity scales the phase velocity accordingly. Layer 1 has quite

a strong influence, almost 1 : 1 over the entire usable frequency range. That is, a change

in shear velocity of this layer will affect the entire observed dispersion the same amount.

Layer 2 only influences frequencies around 10 Hz and layer 3 at frequencies around 5

Hz. Layer 4 has a 1 : 1 influence only at very low frequency, about 1 Hz. However, any

sensitivity of this layer is at frequencies less than the lower cutoff frequency and is thus is

a major factor in the increased uncertainty and equivalence in deeper layers.

In Figure 3.64, the standardised partial derivatives are presented, using the numerically

modelled data errors in Figure 3.49. Standard a priori shear velocity errors are assumed

as 20% of the layer shear velocity values. Layer 1 has a strong influence of all frequencies

above 10 Hz. In layer 2, the sensitivities are much lower, but there is still a modest

influence from 10-20 Hz. In layers 3 and 4, there is only minor influence over narrow,

low-frequency bands, indicating that at these a priorierrors they cannot be involved in

the inversion. The standardised partial derivatives are linearly related to the expected

error in the parameter, so if a 50% uncertainty in shear velocity is allowed, the large error

in the data (phase velocity) is compensated and could be incorporated in the inversion.

Case 2 sensitivity Raw analytic partial derivatives are shown in Figure 3.65. Even

though layer 1 has the same thickness as in Case 1 (normally dispersive) it actually has less

influence when it forms a hard caprock. Layer 2 (low velocity channel) particularly affects

the dispersion around modal transition points whereas layer 1 actually has a suppressed

sensitivity around modal transition points. Deeper layers 3 and 4 have markedly reduced


sensitivity over frequencies above 10 Hz.

Standardised partial derivatives are shown in Figure 3.66. The errors are those taken

from Figure 3.52 and again a 20% a priori layer shear velocity uncertainty is assumed.

Layers 1 and 2 have equally strong influence over all frequencies, due to the higher absolute

a priori assumed for the caprock. Layer 3 also has a strong influence over 10-60 Hz and is

probably better resolvable in this profile than the normally dispersive case 1. However, any

resolution of the layer 4 shear velocity is highly questionable, similar to Case 1. Accurate

interpretation of this layer would rely entirely on a narrow band over 5-10 Hz, assuming

spread length was long enough (with lateral discontinuity not an issue) to resolve the

low-frequency dispersion.

Although linearity is assumed in the calculation of these partial derivatives, this is

questionable around the modal transition points and may cause an inversion to become

unstable. Some regularisation may need to be applied if strong non-linearity is suspected,

or alternatively a global search optimisation employed.

Case 3 sensitivity The analytic partial derivatives shown in Figure 3.67 are much

stronger than in Case 1 and 2, reaching well over 3 : 1 at some frequencies. These are

around the 8-16 Hz band where a higher mode is dominant. Layer 3 has a small response,

only sensitive in a narrow band Again, layer 4 is only sensitive at frequencies less than

about 5 Hz which is outside the usable frequency range.

Figure 3.68 shows the standardised partial derivatives, from which it can be seen that

top layer again contributes the most. Layer 2 (hard, interbedded) is only resolvable from

the 8-25 Hz band, with a reduced sensitivity around 15 Hz, where the dominant higher

mode ceases and propagation returns to the fundamental. Resolution of layer 3 depends

on a narrow band 13-16 Hz and may be uninterpretable, along with the homogenous

half-space, due to the low sensitivity in the presence of realistic experimental errors.

3.5.8 Discussion and comparisons with previous work These results show

that a previously assumed constant relative error [43, 86] is an unsuitable estimate and

better methods are required. Large errors at low frequency were observed via the bootstrap

of [22] and in experimental data of [189, 288]. Linear array microtremor dispersion show

similar scatter [179]. While the repeated shooting in [330] did not include frequencies

below 60 Hz, the errors below 10 Hz in [160] showed large deviation from a constant

relative level.

In [288], one factor was thought to be the high pass filtering of the Earth. The high

pass nature of the Tokimatsu et. al. Case 3 is evident in Figure 3.62. The ellipticities of

the Rayleigh waves was also noted to be a vital measurement in [288], where the frequency

at minimal ellipticity was approximately the lower cutoff frequency. While this was not

tested here, the dominant factor is still thought to be dominated by poor resolution due to

finite array aperture, more so than low power of the observed dispersion. In the field, array


length is limited by topographic and lateral geological discontinuities, however practical

limitations of spatial aliasing and equipment are also important.

In addition, the commonly assumed Gaussian distribution of error may not be suit-

able. In the repeated shooting tests of [160, 330], this was shown to be true but only

distributions at higher frequencies were analysed. In addition, those tests only accounted

for additive noise and shot-to-shot repeatability. However, by SASW the results are not

directly comparable as different propagation paths were involved, as the site could not

be guaranteed as perfectly one-dimensional. Indeed, Figure 3.40 which only accounts for

additive errors, does show a Gaussian distribution, but only above 25 Hz. At low frequen-

cies, a Lorentzian is more suitable. Moreover, when all acquisition errors are incorporated,

including spread length permutations, all frequencies show larger outliers suggestive of a

Lorentzian distribution. These combined tests have probably overestimated the disper-

sion error envelope, however they are useful upper limit guides when conducting sensitivity

analyses and assigning standard deviations in an inversion scheme.

Under the assumption of local linearity, the theory of Gaussian error propagation states

that if the uncertainty in the data is normally distributed then the uncertainty in inverted

model parameters will also be so [201]. In nonlinear inversion this does not apply and

the model parameter distribution will not be the same as that of the data. While in

[160, 330] it appeared that the inverted shear velocities appeared normally distributed

in proportion to the data errors, the distribution of model parameters around the point

of maximum likelihood has not been rigorously studied with a directed or Monte Carlo

search. Moreover, the propagation of different data distributions has not been studied.

Incorporation of errors is usually accomplished by assigning a weighting matrix to the data

vector, with the diagonal comprising the inverse standard errors. From the uncertainties

of this section, higher frequencies will thus have larger weight and vice-versa [161]. Note

that this is only valid when errors are uncorrelated, as is usually assumed. Their effects

in a global inversion scheme are yet to be quantified.


3.6 Conclusions for this chapter

Many factors influence the repeatability of an ideal effective dispersion curve observed

by multichannel acquisition and processing. Error envelopes are generally estimated or

neglected, but play a major role in the inversion process to propagate into the final model

parameters.

3.6.1 Fundamental limitations The maximum depth of investigation is limited

by the lowest frequency that can be accurately measured. This is controlled by the phase

velocity resolution, based on Equation 2.7, which is a function of spread length.

The maximum measurable wavelength is 0.4 of the spread length; that is, spread length

should be at least 2.5 times the maximum desired wavelength. Assuming the approximate

depth of investigation as 0.3-0.5 the surface wavelength, this implies that the maximum

depth of investigation is only 0.1-0.2 of the spread length. However, this is only a very

rough guide, since the subsurface is actually disturbed to depths well over a wavelength.

A less important factor for low-frequency dispersion is the f−k cutoff frequency, which

may prohibit the measurement of low-frequency phase velocities when no trace padding is

applied. However, is is usually at or below the lower cutoff frequency, dictated by spread

length resolution.

For plane-wave transform, the f − p method is preferable to the f − k method in non-

and normally-dispersive sites, as long as sensible slant-stack limits and trace padding are

applied. Trace padding should, in general, be a minimum of 128, assuming a time-padding

in the order of 512 or 1024 points. In inversely dispersive sites, the f − k method almost

always more accurate discretisation of a mapped dispersion curve assuming trace padding

is applied.

3.6.2 Near offset and spread layout It seems that the ‘near-field effect’ is slightly

model dependent, but by synthetic modelling it only extends to a near offset of about 5

m, correlating to less than 0.15λ. Long spreads satisfying the L > 2.5λmax rule are not

subject to near-offset effects.

At any offset, the lower cutoff frequency is dictated by spread resolution. Traditional

low-frequency effects due to cylindrical spreading and body waves are less influential.

Thus, the preference to use the term ‘low-frequency effects’, which is somewhat of a

synonym to ‘near-field effects’ but does not imply that near offset is the dominant factor.

Low-frequency effects are not Gaussian distributed, larger outliers better modelled

with a Lorentzian model. Higher frequency scatter due to near-offset variations (for long

spreads) is Gaussian distributed. Dominant higher mode dispersion from a buried HVL

structure is especially prone to low-frequency effects.

3.6.3 Individual acquisition errors Depth of explosive source is a primary con-

sideration in dominant higher mode inversion. However, most field tests are conducted

3.6. Conclusions for this chapter 155

with a vertical impact source, for which dispersion in a LVL under caprock is independent

of source depth.

Random noise affects overall repeatability over a broader frequency range. DC shifts

to traces are a concern, as are trace-to-trace static time-shifts, however, in practise, can

usually be avoided.

Geophone position errors are have modest influence, almost an order of magnitude

than small geophone tile errors. However, Love and other scattered wavefields were not

modelled, which would be a concern in practice with large tilt errors. Geophone coupling

and dead traces are of little significance.

When error sources are considered individually, phase velocity error distributions ap-

pear to mimic the distribution of the error sources (Gaussian), but only above a certain

frequency. At low frequency, distributions are Lorentzian, dominated by the spread length

resolution limitation.

3.6.4 Repeatability and sensitivity The ‘background’ error at high frequency

in all cases is around 1%. Around the frequencies of modal transitions, errors were also

larger by a few percent. A large, nonlinear increase in error at low frequency occurs in

all cases. This is observed in both continuous dispersion mapping and active dispersion

processing, again the spread length limitation

The relative error can exceed 30% at low frequency. This is especially so for the

dominant higher mode generated in a buried HVL case. Even though there are physical

limits which describe the theoretical lower cutoff frequency, in practise it must be decided

upon by the operator, with primarily consideration of the L > 2.5λmax rule.

Distributions of dispersion affected by all error sources are not Gaussian distributed.

The Lorentzian distribution associated with low-frequency effects affects the entire usable

frequency range when random spread lengths and channel densities are incorporated. This

result will have implications in the study of error propagation through to the final model

parameters based on linearised inversion theory.

Dispersion power is high over a broad bandwidth in normally dispersive and buried

LVL cases. In a buried HVL, strong signal to noise is confined to very narrow bandwidth,

compounding the problems of accurate low-frequency dispersion measurement.

Sensitivity analysis shows the homogenous half-space shear velocity especially affected

by realistic dispersion error envelopes. Layers above the half-space in a buried LVL case

are least affected, whereas deeper layers in normally dispersive and buried HVL cases will

have poor resolution.

3.6.5 Noise model At logarithmic scale, the dispersion repeatability (standard de-

viation) shows a proportional relationship to the spread length resolution envelope (Equa-

tion 2.7), on average, about a decade less. Given a frequency-phase velocity dispersion


curve (f, c) acquired over a linear array of length L, an algorithm for a simple noise model

of the dispersion is developed:

1. Calculate slowness resolution, ∆p(f) = 1/fL;

2. Convert dispersion to slowness, p(f) = 1/c(f);

3. Calculate resolution in terms of phase velocity, ∆c(f) = abs(1/pmin − 1/pmax)

where {pmax, pmin}(f) = p(f) ± ∆p(f)/2;

4. Convert to a relative phase velocity error, δc(f) = ∆c(f)/c(f); and

5. Calculate logarithmic reduction of relative error, δc(f) = 10−aδc(f).

The factor a in Step 5 is a user-defined value which reduces the ∆c(f) curve by a proportion

of a decade, usually about 0.5, based on heuristic observations. One further step to

suppress very large errors is to set an upper limit, usually around 0.2-0.3 (20-30%), so

low-frequency errors are not necessarily asymptotic. However, the lower cutoff frequency is

usually above the frequency where this occurs.

Figure 3.69 shows how this noise model compares to the actual errors measured in

the random numerical tests of the Tokimatsu et. al. models, converted to percentages.

While the minor undulations such as those at the jumps to higher modes in Case 2 are

not reproduced, the overall trend in relative error compares well for each case, with the

reduction factor a between 0.5-0.75. This relative phase velocity error is converted back

to an absolute error by δc(f) ∗ c(f) and these uncorrelated values are used in the diagonal

weighting matrix in the inversion.

Where repeatability analyses are available, such as in the synthetic data of the Toki-

matsu et. al. cases or field data where repeated shooting was performed, the measured

scatter will be used for dispersion standard deviation weighting. However, in single shot

field data the noise model will be employed.

3.6.6 SASW perspective It should be noted that this entire work could have

been equally approached from a SASW perspective by considering errors in the phase

of the cross-spectra, which was the method adopted by [330]. With several channels,

the phase error would be better estimated by an RMS error of the best-fit regression of

phase lag with offset, calculated over a number of frequencies [160]. However, 2-channel

SASW errors would not be directly comparable those observed in the multichannel data

of Chapter 4, moreover, dominant higher modes would not be properly detected, since

fundamental mode propagation is assumed in standard SASW.


0 10 20 30 40 50 60 7010

−1

100

101

102

Frequency (Hz)

σ c (%

)

(a)

Factor = 0.25 Factor = 0.50 Factor = 0.75 Numerical errors

0 10 20 30 40 50 60 70 80 90 100 11010

−1

100

101

102

Frequency (Hz)

σ c (%

)

(b)


0 10 20 30 40 50 60 7010

−1

100

101

102

Frequency (Hz)

σ c (%

)

(c)


Figure 3.69: Dispersion noise model described in Section 3.6.5 compared to numerical

errors modelled in the random error testing. The maximum error is capped at 30%. (a)

Case 1; (b) Case 2; and (c) Case 3.


159

CHAPTER 4

Dispersion repeatability: Field tests

4.1 Introduction

Repeated shooting at two field sites provided a chance to measure dispersion repeata-

bility where the only contributing factors were source and receivers (positions, types, etc.)

and additive noise. The sites surveyed were:

1. Hyden fault scarp; and

2. Perth Convention Centre.

This chapter illustrates how the repeatability observed in the field compares to that from

the synthetic testing of Chapter 3.

160 Chapter 4. Dispersion repeatability: Field tests

4.2 Hyden fault scarp

See Section 6.2 for location, geology and significance of this site.

4.2.1 Data acquisition A 24-bit OYO Geospace DAS-1 receiver based on an MS-

DOS 486PC was used for acquisition. The system has 48 channels on-board, with an

extender box for another 48 channels, giving 96 channels overall. In reflection mode, roll-

along was accomplished with an Input/Output Inc. Rota-Long-Switch and a total cable

capacity of 120 takeouts was available. Cable takeouts were at 25 m, so with 1 m geophone

spacings great care was necessary to ensure correct receiver location.

Five sources and four geophone responses were available, where the source types used

were:

1. 12 gauge, 25 grain (12 g) shotgun blanks with wad (GB);

2. 12 gauge, 70 grain (30 g) shotgun blanks powder only (GS);

3. 25 kg cast iron drop-ball on 200 mm diameter, 25 mm thick UHMW plastic plate

and sand(BP/BS);

4. 10 kg sledge hammer on UHMW plate and sand (HP/HS); and

5. 30 kg steel shear source box (SN/SS/SE/SW).

Photographs of these are shown in Figure 4.1. UHMW stands for ‘ultra-high molecular

weight’ and is a stiff, high-impact, high-density polyethylene plastic, similar to nylon. The

geophones available were:

1. 28 Hz radial, transverse and vertical components (X,Y,Z); and

2. 8 Hz vertical component (Z).

All components could only be measured separately, the transition from radial and trans-

verse made by manually swivelling each geophone from the inline to crossline direction,

with the geophone spike bolt pointing to the west or south respectively. In the data

presented, a key has been devised to describe the source/geophone combinations. For

example, ‘GB08Z’ indicates shotgun (G), big shell (B), 8 Hz geophones (8) and vertical

component (Z). Note that the ‘big’ descriptor for shotgun shells (GB) describes purely the

length of the shell - the big shells (12 grain) actually contained less than half the powder

in the ‘small’ shells (25 grain). However, they included a plastic wad, where the small

shells were purely an ignition cap with powder. The second letter of the shear source data

indicate the direction from which the box was struck, either north, south, east and west.

For example, ‘SN28Y’ is the shear box (S), struck from the north side (N), recorded with

28 Hz geophones (28) in the transverse component (Y).

4.2. Hyden fault scarp 161

(a) Shotgun shells and gun barrel. (b) Gun in firing position.

(c) 25 kg drop ball on UHMW plate. (d) 10 kg sledgehammer.

(e) Underside of shear wave box. (f) Shear box in firing position.

Figure 4.1: Photographs of source types used at the Hyden fault scarp.


The recording trigger for the ball and hammer sources was a vertical component 8 Hz

geophone, inserted next to the plate or impact point. The trigger for the gun source was

a similar geophone taped vertically to the barrel, and set to record on a negative pulse

(geophone rise). The shear source trigger was through a short circuit between the steel

box and hammer head and appeared to be the most repeatable of the three.

The survey line was aligned west-east, perpendicular to the trend of the fault scarp.

The peak of the scarp was at about shotpoint 980W, where the line extended further

west to SP1150W and east to SP980W. However, the test data and production reflection

shooting were confined to shotpoints west of the scarp ridge. The terrain was gently

undulating, the survey line entirely in a recently harvested wheat field, with only short

stubble remaining in a thin veneer of dry, fine windblown material. The only portion

subject to obstructions was west of a track at SP1120W, where a scrubby treeline in low

ground was entered.

4.2.2 Test data and source analysis During the day 1 tests, 45 channels were laid

out from shotpoint locations 1051W to 1095W and a shot at 1050.5W was used. Usually

a 5 cm deep trough was scuffed through the dry, dusty layer to expose slightly darker,

moist soil and provide a better geophone plant. Surficial laterite sometimes prevented

good coupling and also introduced positional error, since the crew had to ‘search’ around

the desired point to find softer ground. This was up to 0.2 m in some cases.

Figure 4.2 shows vertical component shot gathers for three different source-receiver

type combinations. Visually, there is no major difference between the three. The fun-

damental mode Rayleigh wave group velocity is about 300 m/s and higher modes are

present. The unusual higher mode velocity drop in the last ten metres may be a lateral

discontinuity effect. The ball and hammer sources appear to generate stronger direct and

guided waves, contrary with the theory that a buried explosion should provide this. The

more banded nature of the hammer trace spectra suggest several modes. The hammer

also generates a broader frequency range and all show a shift to dominant lower frequency

with offset, due to attenuation of high frequencies.

Figure 4.3 shows inline component shot gathers for three different source-receiver type

combinations, all for 28 Hz geophones. The patterns are similar to that of Figure 4.2,

with one major difference being stronger higher ground-roll modes (higher velocity) at far

offsets. This is also shown by the more well defined peaks in the FFT of each trace. There

is also a slight power loss of frequencies below 50 Hz compared with the vertical component.

Overall, there is better modal ‘tuning’, but with narrower bandwidth. Another difference

is the saturation and clipping of the nearest geophone at 0.5 m offset.

Figure 4.4 shows the transverse component shot gathers, again only 28 Hz geophones.

The shear source generates a strong fundamental mode Love wave at somewhat higher

group velocity than the Rayleigh wave, and is around 300 m/s. Modest dispersion is


0

0.05

0.1

0.15

0.2

0.25

Tim

e (s

)

(a) BP, 8Hz

Offset (m)

Fre

quen

cy (

Hz)

0 10 20 30 40

0

50

100

150

200

(b) GB, 28Hz

Offset (m)0 10 20 30 40

(c) HP, 28Hz

Offset (m)0 10 20 30 40

Figure 4.2: Vertical component shot gathers from Hyden fault scarp from day 1 tests.

The shot was at 1050.5 m and 45 channels to the west are shown. (a) Ball on plate, 8

Hz geophones; (b) Big shotgun shell, 28 Hz geophones; and (c) Hammer on plate, 28 Hz

geophones.

0

50

100

150

200

250

Tim

e (m

s)

(a) BP, 28Hz

Offset (m)

Fre

quen

cy (

Hz)

0 10 20 30 40

0

50

100

150

200

(b) GB, 28Hz

Offset (m)0 10 20 30 40

(c) HP, 28Hz

Offset (m)0 10 20 30 40

Figure 4.3: Radial component shot gathers from Hyden fault scarp from day 1 tests. The

shot was at 1050.5 m and 45 channels to the west are shown. (a) Ball on plate, 28 Hz

geophones; (b) Big shotgun shell, 28 Hz geophones; and (c) Hammer on plate, 28 Hz

geophones.


0

0.05

0.1

0.15

0.2

0.25

Tim

e (s

)(a) HS, 28Hz

Offset (m)

Fre

quen

cy (

Hz)

0 10 20 30 40

0

50

100

150

200

(b) SN, 28Hz

Offset (m)0 10 20 30 40

(c) SS, 28Hz

Offset (m)0 10 20 30 40

Figure 4.4: Transverse component shot gathers from Hyden fault scarp from day 1 tests.

The shot was at 1050.5 m and 45 channels to the west are shown. All are 28 Hz geophones.

(a) Hammer on sand; (b) Shear source north; and (c) Shear source south.

SS28ZHS28ZSW28ZSE28ZHP28ZHS28YBS28ZSS28XSS28YBP28ZSN28XSN28YGS28ZSW28XHS28XGB28ZHS08ZSE28XHP08ZBP28XHP28XBS28XBS08ZBP08ZGS28XGS08ZGB08ZGB28X

Total shot energy

Energy (arbitrary) 10 20 30 40Offset (m)

Amplitude decay with offset

0 100 200

SS28ZHS28ZSW28ZSE28ZHP28ZHS28YBS28ZSS28XSS28YBP28ZSN28XSN28YGS28ZSW28XHS28XGB28ZHS08ZSE28XHP08ZBP28XHP28XBS28XBS08ZBP08ZGS28XGS08ZGB08ZGB28X

Total spectral power

Frequency (Hz)

Figure 4.5: Source type analysis at Hyden fault scarp from day 1 tests. All source-geophone

combinations are shown, where the shot was at 1050.5 m and 45 channels to the west have

been used in the statistics. (a) Total source energy, by a summed trace-to-trace RMS

mean; (b) Decay of energy with offset; and (c) Frequency spectra. See text for key to

source type labels.


evident as well as a broad frequency response, but without the banding as observed in

the vertical component FFTs. What is interesting is that the ‘hammer on sand’ shot

also generates appreciable Love waves. This is to be expected for other than a purely

vertical source and perfectly transverse geophones. While there may be a geophone tilt

error introducing this there may also be a tilt in the surface wave polarisation.

Figure 4.5 shows a statistical analysis of all the source-receiver combinations used

during the day 1 tests. Higher energy is received by the 8 Hz geophones and these receivers

also show a peak at their natural frequency. However, this resonance does not occur in

the radial component spectra. The big shotgun shells deliver more energy, due to the wad

providing better compression at the ignition of the powder before release. However, the

small shotgun shells seem to show a broader bandwidth than the big shells, comparable to

the hammer versus ball spectra. Considering the shear source, when data is recorded in

a component perpendicular to the striking direction the energy measured is low as would

be expected. Since the PSV method models both vertical and inline components, these

data may also be useful in dispersion inversion. However, the transverse component data

will not be considered at all in this thesis beyond these qualitative observations.

4.2.3 Fixed-offset variations Considering the vertical component geophones and

vertical source types guns (big and small) and ball and hammer (on plate and sand),

dispersion over the test spreads is measured. The 512-point, 45-trace shot gathers were

padded to 1024 time points and 512 traces before decomposition. By f − k, the aliased

wavenumbers were allowed (up to 1 m−1) extending the upper cutoff frequency. The

τ − p limits were from 250-761 m/s, and only 256 slownesses allowed. This gave an

optimum pixel resolution of 2 m/s, which is quite large, but unavoidable due to the large

observed phase velocity dispersion range. The pixel resolution should ideally be much less

than the spread resolution limit. Each curve will thus incorporate all the errors discussed

in Chapter 3. However, as the Earth is undoubtedly laterally discontinuous, dispersion at

various tests will reflect this moreso than offset related errors.

It should be noted that the source was not directly in-line with the spread, but about

0.25 m off-line. Thus, especially at near offsets, waves will arrive obliquely generating

artificially high phase velocities. Both near- and far-offset ranges are separately tested,

along with all channels, to identify if this is a problem. A 320 m/s top mute with 5

ms rolloff was also trialled to try and minimise the air and guided waves. However, this

interfered irrecoverably with the fundamental and higher mode ground-roll wavetrains.

Figure 4.6 shows the dispersion curves and standard deviations over the entire 45-

channel spreads by f − k. The dispersion is highly irregular and errors generally around

1-10%. Areas of no data are those where phase velocity exceeded the τ −p limits and have

been nulled. By f−p (Figure 4.7), the errors are almost identical, however pixel resolution

at low frequency is much better. Errors are slightly higher around modal transitions and


200

300

400

500

600

700

800

c(m

/s)

(a) Offsets 1−45m by f−k

0 10 20 30 40 50 60 70 80 90 10010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 4.6: Full spread dispersion of the day 1 test data at Hyden fault scarp by f−k where

the shot was at 1050.5 m and channels 1-45 have been used. (a) All dispersion curves. (b)

Standard deviations along with pixel and spread length resolution envelopes.

200

300

400

500

600

700

800

c(m

/s)

(a) Offsets 1−45m by τ−p

0 10 20 30 40 50 60 70 80 90 10010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 4.7: Full spread dispersion of the day 1 test data at Hyden fault scarp by f−p where

the shot was at 1050.5 m and channels 1-45 have been used. (a) All dispersion curves. (b)



0.0010.01

0.11

P(δ

c)

0−20Hzσ

G=1.20%

ΓL=3.01%

δG=7.34%

δL=6.34%

0.0010.01

0.11

P(δ

c)

20−40Hzσ

G=3.77%

ΓL=7.91%

δG=5.06%

δL=5.13%

0.0010.01

0.11

P(δ

c)

40−60Hzσ

G=1.10%

ΓL=1.82%

δG=0.57%

δL=0.91%

0.0010.01

0.11

P(δ

c)

60−80Hzσ

G=1.17%

ΓL=2.04%

δG=1.14%

δL=1.41%

−20 −10 0 10 20

0.0010.01

0.11

P(δ

c)

80−100Hzσ

G=0.99%

ΓL=1.51%

δG=0.55%

δL=0.35%

δ c (%)

Figure 4.8: Statistical analysis of the error distribution of Figure 4.7. Percentage errors

are averaged over 20 Hz bands, then both zero-mean Gaussian (dotted) and Lorentzian

(dashed) curves fitted. Results of the fits are shown as σG for Gaussian standard deviation

and ΓL for Lorentzian half width. RMS errors of the fits are shown as δG and δL.

100

200

300

400

500

600

700

800

c(m

/s)

(a) 28Hz geophones, channels 1−24 by τ−p

0 10 20 30 40 50 60 70 80 90 10010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 4.9: Near-field dispersion of the day 1 test data at Hyden fault scarp where the

shot was at 1050.5 m and channels 1-24 have been used, by f − p and 28 Hz geophones

alone. (a) All dispersion curves. (b) Standard deviations along with pixel and spread

length resolution envelopes.


100

200

300

400

500

600

700

800

c(m

/s)

(a) 8Hz geophones, channels 22−45 by f−k

0 10 20 30 40 50 60 70 80 90 10010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 4.10: Far-field dispersion of the day 1 test data at Hyden fault scarp where the

shot was at 1050.5 m and channels 22-45 have been used, by f − k and 8 Hz geophones

alone. (a) All dispersion curves. (b) Standard deviations along with pixel and spread

length resolution envelopes.

0

0.2

0.4

0.6

0.8

1

Pow

er (

Nor

m)

(a) Channels 1−45

0

0.2

0.4

0.6

0.8

1

Pow

er (

Nor

m)

(b) Channels 1−24

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Pow

er (

Nor

m)

(c) Channels 22−45

Figure 4.11: All power curves of the dispersion of the day 1 test data at Hyden fault scarp

by f − k. (a) All channels (1-45), (b) Near channels (1-24), (c) Far channels (22-45).


exceed 10% at frequencies less than 15 Hz and also around 33 Hz. This band (30-35 Hz)

shows an unusually high variation which source function or geophone positioning alone

would not be expected to generate. Although not discriminated in Figure 4.6(a), the

higher phase velocities over 30-35 Hz are measured with the 8 Hz geophones and impact

sources only. Another unusual feature is the modal transitions to a lower phase velocity

between 60-90 Hz. Normally, modal transitions at high frequency manifest as jumps to

higher dominant modes. The possible causes are either lateral discontinuity or air-wave

interference, since by coincidence the 65-85 Hz phase velocity is around 340 m/s with little

or weak dispersion.

Figure 4.8 shows the dispersion error analysis averaged over 20 Hz bands. Unlike the

numerical simulations, some bands are modelled well with a Gaussian error distribution.

The 40-80 Hz range comprises the smallest error envelopes and will only be due to random

noise, source types and geophone position. The larger errors are better modelled with a

Lorentzian curve to account for the large outliers.

Figure 4.9 shows the near-offset dispersion from only the 28 Hz geophone data. In

these figures, the horizontal dotted lines are the slant-stack limits and only 256 trace

padding is used. Also, phase velocities outside the slant-stack limits were not excluded in

the statistics. In both figures the phase velocity anomaly at 33 Hz does not occur and can

possibly be an artifact of the higher modes dominant at farther offsets. The dispersion

data are highly repeatable above 30 Hz, due mainly to there being no modal transitions.

The far-offset data (Figure 4.10) confirms the 33 Hz anomaly as a higher mode effect.

This is particularly high for the surface sources impacting on a plate, with the 8 Hz

geophone data. In spite of the the 23 m near- and far-offset windows having less resolution

than the full-spread data, the near-offset repeatability is of similar quality to the full-

spread repeatability. The far-offset dispersion is dominated by wavefield scattering and

mode conversions.

The normalised energy of the picked curve in the f − k plane is shown in Figure 4.11.

When all channels are used (1-45) almost all the energy is contained in the 30-90 Hz band

and in this zone, the errors in dispersion are very low. Over the frequencies of maximum

power, errors are generally lower. Note how higher frequencies are still favoured over the

near offsets (Figure 4.11(b)) and at far offsets (Figure 4.11(b)) power at lower frequencies

is maintained. Moreover, the large oscillation in power is a consequence of the modal

splitting and higher modes being preferably detected.

In all these dispersion curves, the average usable frequency range is about 10-80 Hz.

Higher frequencies could possibly be included with care in an inversion, however, lower

frequencies should be avoided. Of course, the large differences in dispersion between

the different trace windows will largely be due to lateral discontinuity. Observation of

reciprocal shots, or the CMP data as in the next section, should confirm this.


200

300

400

500

600

700

800c(

m/s

)(a) 48 channels, 1−48m near offset by τ−p

Mean Median (with σ

c)

0 10 20 30 40 50 60 70 80 90 10010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 4.12: Dispersion variation with near offset at Hyden fault scarp by f − p. 48

channels with near offsets from 1 m to 48 m have been used. Receiver locations remain

fixed from 1001W to 1048W. (a) All dispersion curves. (b) Standard deviations along with

pixel and spread length resolution envelopes.

0.0010.01

0.11

P(δ

c)

0−20Hzσ

G=1.90%

ΓL=4.06%

δG=2.99%

δL=1.33%

0.0010.01

0.11

P(δ

c)

20−40Hzσ

G=3.02%

ΓL=6.28%

δG=2.40%

δL=4.08%

0.0010.01

0.11

P(δ

c)

40−60Hzσ

G=3.46%

ΓL=7.07%

δG=5.77%

δL=4.41%

0.0010.01

0.11

P(δ

c)

60−80Hzσ

G=1.69%

ΓL=3.75%

δG=4.12%

δL=2.21%

−50 −25 0 25 50

0.0010.01

0.11

P(δ

c)

80−100Hzσ

G=3.49%

ΓL=7.68%

δG=9.86%

δL=7.63%

δ c (%)


are averaged over 20Hz bands, then both zero-mean Gaussian (dotted) and Lorentzian


and ΓL = Lorentzian half width. RMS errors of the fits are shown as δG and δL.


0

0.05

0.1

0.15

0.2

0.25

Tim

e (s

)

(a) East shot CSG

Offset (m)

Fre

quen

cy (

Hz)

0 10 20 30 40

0

50

100

150

200

(b) East shot CMP

Offset (m)0 10 20 30 40

(c) West shot CMP

Offset (m)0 10 20 30 40

Figure 4.14: Vertical component shot gathers from Hyden fault scarp from day 2 reflection

profiling. The shot was a large shotgun shell and 28Hz geophones were all 28Hz vertical.

(a) Trace decimated CSG with shot at 1050.0W, (b) CMP with shot to the east (same as

CSG orientation), (c) CMP with shot to the west (opposite to CSG orientation.

200

300

400

500

600

700

800

c(m

/s)

(a) CSG versus CMP dispersion by τ−p

CSGCMP

0 10 20 30 40 50 60 70 80 90 10010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 4.15: Full spread CSG and CMP dispersion of the day 2 reflection data at Hyden

fault scarp by f − pwhere all channels 1-45 have been used. (a) All dispersion curves. (b)



4.2.4 Near-offset variation On day 2, P -wave reflection data was collected with

the rollalong method. 96 channels were set from 1001W (channel 1) to 1096W (channel

96) and the shot rolled through from 1005W to 1097W at 1 m intervals. Big shotgun shells

were used, and the shotholes were on average 0.5 m north of the geophone line. From 1097

W onwards, the spread was ‘pulled’ along maintaining 1 m near offset (from channel 96)

until the last shot at 1157W.

The dispersion curve distribution over a 48 channel window from 1001W to 1048W,

starting from 1 m near offset (1049W) and ending at 48m near offset (1097W) are shown

in Figure 4.12. The range is large, often exceeding 10%. In ideal, 1D layering it was shown

that there is a modest offset-dependency on the observed dispersion (Section 3.3). With

48-channels, it is in the order of 10%. In this case, higher modes appear to dominate at

higher frequencies causing the large standard deviations. Moreover, lateral discontinuities

cause scattering and mode conversions, corrupting the plane-wave transform considerably.

Aside from a peak at 32 Hz due to a modal osculation point, the 20-40 Hz band is the

least sensitive to near offset variations.

Figure 4.13 shows the statistical dispersion error analysis averaged over 20 Hz bands.

Only the 20-40 Hz band is modelled well with a Gaussian error distribution, but some

positive outliers. The other bands are best modelled with a Lorentzian curve, indicating

many large outliers.

4.2.5 Common-shot gather versus common-midpoint gather The results

presented above are for common shot gathers (CSG). From the day 2 reflection data,

common midpoint (CMP) gathers over the same spatial range were compiled. However,

with the shot and geophone spacing both at 1 m, the CMP locations are actually mid-way

between the geophones, with a minimum offset of 1 m and trace spacing of 2 m. Moreover,

since there were 96-channels available, over the 45 m day 1 CSG offset range, CMP gathers

compiled from shots both west and east of the midpoint could be compiled. Considering

the day 1 shotpoint range of 1051W to 1095W, 1072.5W was chosen as the CMP location

nearest to the centre of the day 1 CSG data.

Figure 4.14 shows the CSG and CMP gathers (west and east) centred on 1072.5W. The

CSG data have been trace decimated to 2 m for better comparison. The west CMP gather

shot shows a greater similarity to the CSG, obviously since the shot-receiver orientation

was the same. The east shot is shooting up-dip of the supposed fault plane and the stronger

higher mode ground roll is the most obvious difference. However, lateral discontinuities

are also in effect.

The dispersion for both CSG and CMP data by f−p is shown in Figure 4.15. The CSG

dispersion curves are from subsequent shotpoints or 1 m and zero near x-offset (y-offset is

0.5 m in all cases) and both agree well with each other. However, the CSG dispersion show

many differences especially at higher frequency. The major differences are for the west shot


CMP, which is in the opposite orientation to the CSG data. Undoubtedly, this is caused

by a lateral discontinuity. The standard deviations are best over the range 10-50 Hz, but

are all less than 10% over the entire usable frequency range. The major difference arises at

low frequency, where the CSG phase velocity rises rapidly and the CMP dispersion drops.

This occurs by f − k as well. This probably outside the usable frequency range, however,

if that data was to be incorporated in an inversion, it would have a major effect on the

accuracy of the deeper layer parameters. This error is also observed in the CSG analysis

of Figure 4.7 and also from comparing these figures, the anomalously high phase velocity

at around 33 Hz is not recorded with the gun source and 28 Hz geophones, as used in the

P -wave profiling.


4.3 Perth Convention Centre

See Section 6.2 for location, geology and significance of this site.

4.3.1 Acquisition equipment An 8-channel, 24-bit National Instruments acqui-

sition card in a standard Windows 32-bit PC was used. Geophones were 2 Hz Mark

Products velocity transducers with an 8-channel Sercel cable at 15 m takeouts. A 10

kg sledge hammer and 50 kg rectangular-based steel weight were used as sources. The

hammer was impacted on a 200 mm diameter, 25 mm thick UHMW (ultra-high molecular

weight) plastic plate and weight dropped from an average of 2.0-2.5 m onto the sand from

a hydraulic crane mounted on a trayback vehicle.

The seismic line was located parallel to the southern boundary fence of the site. Outside

this fence and also parallel to it was a temporary freeway off-ramp. The line itself was

clear of building obstructions and topography, however, the nearest row of foundation

piles, emplaced at 10 m intervals, was only several metres north of the spread. Geophone

coupling in the dry, yellow, loamy reclaim soils was not usually an issue except for some

secondary cemented patches, inducing a position error up to 20 cm. The survey was done

on a Sunday, since construction equipment is stopped and road traffic is minimal at this

time. While other natural noise was minimal, some traffic noise was unavoidable.

4.3.2 Walkaway data To gather many channels, a shot walkaway procedure was

used. With fixed receiver locations, this creates a common receiver gather (CRG), by

concatenating the individual 8-channel common shot gathers (CSG). At 1 m geophone

intervals the hammer source was moved at 8 m intervals to give end-to-end shot shot

gathers with no trace overlap. In the westerly direction, near offsets from 1-49 m gave

a total of 7 gathers or 56 channels. In the easterly direction the near offset range was

only 1-25 m, limited by the boundary fence, giving a total of 4 gathers or 32 channels.

At 3 m geophone intervals the drop source was moved at 24 m intervals, with near offsets

of 3-51 m to give 3 gathers or 24 channels. This procedure is shown in Figure 4.16 and

representative walkaway CRG’s are shown in Figure 4.17.

This type of walkaway shooting is usually only employed for refraction analysis. In

the more sophisticated interpretations, several reciprocal traveltimes are required at each

geophone to properly resolve refractors. In seismic reflection, walkaway shooting is con-

ducted for source generated noise analysis. Usually the source position remains fixed and

the spread moved away to very long offsets. Close geophone spacings are used to ensure

no wavefields are spatially aliased. This would be preferable for surface wave surveying,

but is limited by space at the site and topographic or other obstructions. In addition,

survey time is increased severalfold.

4.3.3 Observed dispersion curves Usually 5 repeated shots were made at each

walkaway position. Thus, there is a vast combination of possible walkaway CRGs. First,

4.3. Perth Convention Centre 175

0 10 20 30 40 50 60 70 80 90Distance (m)

1m spacing

3m spacing

10kg hammer on UHMW plate

50kg drop source 2m AGL

Figure 4.16: Schematic source and receiver locations for walkaway shooting at the Perth

Convention Centre with an 8-channel seismograph. West is to the right.

1 11 21 31 41 51

0

0.2

0.4

Tim

e (s

)

(a) 1m spacing, 8m shot walkaway to west

1 11 21 31 1 11 21

0

0.2

0.4

Tim

e (s

)

(b) 1m spacing, 8m shot walkaway to east

3 13 23 33 43 53 63 73

0

0.5

1

Offset (m)

Tim

e (s

)

(c) 3m spacing, 24m shot walkaway to west

Figure 4.17: Representative shot walkaway gathers at the Perth Convention Centre with

an 8-channel seismograph. The 1 m geophone spacing array was left in place with shot

walkaways to the west (a) and east (b). The 3m geophones were only used for a shot

walkaway to the west (c).


0 5 10 15 20 25 30 35 40 45 5050

100

150

200

250

300

350

Pha

se v

eloc

ity (

m/s

)(a) f−k

Frequency (Hz)

3 m 27 m51 m

0 5 10 15 20 25 30 35 40 45 5050

100

150

200

250

300

350

Pha

se v

eloc

ity (

m/s

)

(b) τ−p

Frequency (Hz)

3 m 27 m51 m

Figure 4.18: Dispersion analysis of 3 m geophone spacing shot gathers at three near offsets:

3, 24 and 51 m where a 50 kg drop source was used. (a) f − k and (b) f − p transform

dispersion.

50

100

150

200

250

300

350

400

c(m

/s)

(a) West walkaway, 1m spacing, 56 channels by τ−p


c)

0 5 10 15 20 25 30 35 40 45 5010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 4.19: Walkaway dispersion analysis with 1 m geophone spacing at the Perth Con-

vention Centre by f − p, where a 5 kg sledgehammer was used for each CSG. (a) Mean

and median dispersion curves. (b) Standard deviations along with pixel and spread length

resolution envelopes.


50

100

150

200

250

300

350

400

c(m

/s)

(a) 3m geophone spacing, west walkaway by τ−p


c)

0 5 10 15 20 25 30 35 40 45 5010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 4.20: Walkaway dispersion analysis with 3m geophone spacing at Perth Convention

Centre by τ − p, where a 50 kg drop source was used for each CSG. (a) Mean and median

dispersion curves. (b) Standard deviations along with pixel and spread length resolution

envelopes.

0

0.2

0.4

0.6

0.8

1

Pow

er (

Nor

m)

(a) West walkaway, 1m spacing, 56 channels by τ−p

0 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

0.8

1

Pow

er (

Nor

m)

(b) West walkaway, 3m spacing, 24 channels by τ−p

Frequency (Hz)

Figure 4.21: Maximum power of the dispersion curves at the Perth Convention Centre by

τ − p. (a) Hammer source, (b) Drop source.


0.001

0.01

0.1

1

P(δ

c)

0−13Hz

σG=1.88%

ΓL=3.75%

δG=4.26%

δL=2.71%

0.001

0.01

0.1

1

P(δ

c)

13−25Hzσ

G=0.79%

ΓL=1.57%

δG=0.27%

δL=1.34%

0.001

0.01

0.1

1

P(δ

c)

25−38Hz

σG=0.64%

ΓL=1.16%

δG=0.45%

δL=1.06%

−20 −15 −10 −5 0 5 10 15 200.001

0.01

0.1

1

P(δ

c)

38−50Hzσ

G=0.66%

ΓL=1.24%

δG=0.41%

δL=1.02%

δ c (%)


are averaged over 12.5 Hz bands, then both zero-mean Gaussian (dotted) and Lorentzian


and ΓL for Lorentzian half width. RMS errors of the fits are shown as δG and δL.

0 500 1000

0

5

10

15

20

25

30

35

40

α (m/s)

Dep

th (

m)

(a)

0 200 400β (m/s)

(b)

1 1.5 2ρ (g/cc)

(c)

0.4 0.45 0.5σ

(d)0

5

10

15

20

25

30

35

40

Sand

Silty clay

Silty sand

Silty clay

Siltstone

Lithology

Dep

th (

m)

(e)

Figure 4.23: Layered model based on borehole logs near the Perth Convention Centre

surface wave array.


c(m

/s)

(a) 56 channels, 1m spacing by τ−p

50

100

150

200

250

300

350

400Mean Median (with σ

c)

0 5 10 15 20 25 30 35 40 45 5010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)



Figure 4.24: Effects of random acquisition parameters applied to synthetic seismograms

of the SC2 model (Figure 4.23) at the Perth Convention Centre. (a) Mean and median

dispersion curves. (b) Standard deviations along with pixel and spread length resolution

envelopes.

0 10 20 30 40 500.01

0.1

1

10

100

∂c’/∂

β

(a) Layer 1, δβ=29 m/s

0 10 20 30 40 50


0 10 20 30 40 500.01

0.1

1

10

100

∂c’/∂

β(c) Layer 3, δβ=18 m/s

0 10 20 30 40 500.01

0.1

1

10

100

∂c’/∂

β


0 10 20 30 40 50


0 10 20 30 40 500.01

0.1

1

10

100

∂c’/∂

β

(f) Layer 6, δβ=44 m/s

0 10 20 30 40 500.01

0.1

1

10

100

∂c’/∂

β

(g) Layer 7, δβ=90 m/s

0 10 20 30 40 50

(h) Layer 8, δβ=73 m/s

0 10 20 30 40 500.01

0.1

1

10

100

∂c’/∂

β

(i) Layer 9, δβ=47 m/s

0 10 20 30 40 500.01

0.1

1

10

100

Frequency (Hz)

∂c’/∂

β

(j) Layer 10, δβ=27 m/s

0 10 20 30 40 50Frequency (Hz)

(k) Layer 11, δβ=40 m/s

0 10 20 30 40 500.01

0.1

1

10

100

Frequency (Hz)

∂c’/∂

β

(l) Layer 12, δβ=120 m/s

Figure 4.25: Standardised partial derivative analysis of the SC2 model from the errors in

Figure 4.24. The a priori layer shear velocity uncertainty values are shown above each

plot.


each 8-channel segment will be considered for a measure of the effects of shot offset, within

the shot-to-shot repeatability envelope. Then, the full 56- and 24-channel spreads will be

transformed for a repeatability envelope independent of near offset.

At 1 m geophone spacing, the f − k alias frequency was 67 Hz. At 3 m spacing it was

only 27 Hz, but could be extended to 54 Hz by incorporating the aliases f−k wavenumbers.

For comparison purposes, an upper frequency of 50 Hz has been used for both datasets.

The τ − p limits were 100-355 m/s with 256 trace padding. In all cases, any errors outside

the spread length resolution envelopes have been excluded in the statistics.

Near offset windows Only the 3 m geophone spacing data provided a suitable array

length for plane wave transform of the individual 8-channel gathers. For shot offsets of

3, 24 and 51 m, the scatter in dispersion between repeated shots was measured, and

the median dispersion curve with error bars for each shot offset is shown in Figure 4.18.

Wavefield scattering is evident in the longest shot offset (51 m) and probably prohibits

any comparison. Moreover, below about 4 Hz, the poor resolution dictated by the spread

length is much larger than any measurable uncertainty. Overall, dispersion from the

nearest shot offset is both smoother and somewhat lower than that of the further offsets.

This is similar to that observed in the f − k, f − p and SASW experimental dispersion

variation with near offset in [82].

Random channel windows A Monte Carlo test with 150 trials of randomly generated

56-channel CRGs (Figure 4.17(a)) were analysed and shown in Figure 4.19. The repeata-

bility is around 0.5% over much of the usable frequency range. At less than about 12

Hz errors exceed 1%. Results of same trials conducted on the 24-channel CRGs (Fig-

ure 4.17(c)) are shown in Figure 4.20. The maximum frequency was only 50 Hz due to

τ − p alias. Although the average error is around 1%, this is only exceeded at frequencies

lower than about 5 Hz. It is clear that the longer array has better low-frequency accuracy

for deeper layer interpretation.

The average spectral power of the dispersion curves are shown in Figure 4.21. There

is slightly more low-frequency power as well as a reduction in cutoff frequency (vertical

black bar) in the 72 m array data (Figure 4.21(b)). However, these alone do not explain

the large improvement in dispersion errors and can only be attributed to the longer spread

length. This result is predicted in the theoretical models, where an array length of at least

twice the target depth is required. The drop source does not generate appreciably higher

energy at low frequencies, but was merely employed to ensure adequate signal to noise at

far offsets for the longer array.

The distributions of errors from the 56-channel gather, averaged over frequency bands,

are shown in Figure 4.22. Above 13 Hz, the distributions are all Gaussian. At low

frequency, where errors and outliers are larger, a Lorentzian fit is better. The narrowest

distribution is around the band where the source function appears to be centred and


suggests that a broadband, sweep source would be preferable to the band limited impact

sources generally employed.

4.3.4 Numerical error estimation From the seismic cone penetrometer log data

(SC2) a 1D model was constructed for synthetic seismogram generation. Figure 4.23 shows

this model, along with the geological interfaces, where above 3 m and below 30 m depth

the velocities were estimated and the shale layer at 35.9 m was estimated as 300 m/s.

Density was backcalculated from the log of shear moduli (Gmax) and Poisson’s ratio was

assumed high throughout the section, since considerable saturation was expected below

about 2 metres. Quality factors for Qα and Qβ were set at 70 and 40 respectively. Fifty

tests were run using a 56-channel gather with 1 m near offset with the following acquisition

errors applied:

1. Geophone positions - up to 0.2 m radius;

2. Geophone angles - up to 20◦;

3. Coupling - ‘Best’ to ‘Poor’;

4. Statics - up to ±4 ms between 8-channel groups;

5. Source location - up to 0.2 m;

6. Source frequency - centre range 25-45 Hz;

7. Geophone response - low range of 1-3 Hz;

8. Propagating noise - maximum (1%, 125 m/s, 25 Hz); and

9. Random noise - maximum 1% of data 3σ.

The source was a vertical impact at the surface and all processing parameters were the

same as for the real 56-channel data, except the τ − p limits were lowered to 80-335m/s.

Figure 4.24 shows the results of 50 trials. The overall dispersion is similar to the field

data, except that modest dominant higher modes at 5 Hz intervals can be seen as a result

of strong low velocity layers. Moreover, the overall velocity is about 20-30 m/s lower

than that observed in the field. Differences may be due to the several metre difference in

locations and lateral discontinutities along the survey line. It was also noticed that the

left and right hammer strikes gave velocity differences up to 50% in some layers, which

is quite large considering that they should give almost equal results, only different due

to arrival time picking error. In any case, above 15 Hz the average background error is

under 1%. Less than 15 Hz, errors rise and reach 10% at less than 5 Hz. The errors in the

field data (Figure 4.19) are marginally higher at these low frequencies. Possibly various

propagating noise in each of the 8-channel segments of the SWG are a cause, which was


not accounted for in the numerical data. The larger errors around 9-12 Hz may be due to

low-frequency, heavy traffic noise.

4.3.5 Partial derivative analysis From the assumed layered model of Section 4.3.4,

the phase velocity standardised partial derivatives were analysed as per Section 3.5.7.

Assuming an inversion error of 20% for all layer shear velocities and using the errors of

Figure 4.24, standardised partial derivatives were calculated and are shown in Figure 4.25.

Where no curve is plotted indicates an amplitude less than the lowest tick on the y-axis.

Of the 12 layers, only the top three will be reliably resolvable. The sensitivity of layers

4, 5 and 6 suggests that they should not be resolvable. The maximum depth of detection

is thus probably to the top of the interbedded hard layer, however, it probably would

not be inverted accurately. The deeper low velocity layer and basement appear to not be

interpretable from this field layout and are further discussed in the inversion analysis of

Chapter 5.



4.4.1 Hyden fault scarp The error envelopes observed here match well with those

revealed through the synthetic testing. The dispersion curve is highly offset dependent, as

to be expected in an obviously laterally varying earth. However, CSG and CMP dispersion

over coincident trace locations match well. For this reason, dispersion for inversion should

be preferred from shorter spreads and/or near offsets. Only to ensure adequate depth res-

olution should farther offsets be employed. Although shorter spreads do not provide good

low-frequency resolution (thus deeper depth penetration) this is offset by less interference

from 2D and 3D effects. The dispersion curves suggest a hard interbedded horizon under

a thin, soft overburden which is likely to be a buried laterite, which outcrops near the

surface near the top of the fault.

4.4.2 Perth Convention Centre Shot walkaway common receiver gathers (CRG)

are a robust method to increase trace density and/or volume for surface wave dispersion,

with minimal Gaussian-distributed error inclusion. Although any trigger errors would

only manifest between CSG, bad traces would persist repetitively. (In a CMP gather,

bad traces would only occur once, but trigger errors would be evident between traces).

In order to adequately detect high phase velocity at low frequency (assumed to be the

shale layer at 35 m depth) the 69 m spread (3 m geophone spacing) is necessary, aided

by the lower frequency of the heavier drop-source. A high velocity layer at depth would

not be interpretable with the 55 m spread (1 m geophone spacing) data. Thus, multiple

geophone spacing and shot offsets (with redundant shooting) are recommended.

It is true that the repeatability envelopes and the accuracy of the dispersion curve

are somewhat related. However, it can be seen that even when errors are high (up to

30%) in Figure 4.20 the dispersion still follows the ‘expected’ trend. This expectation was

only from the a priori borehole information. At a similar error level in Figure 4.20 the

dispersion shows almost no indication of a high-velocity basal layer. The inaccuracies in

interpreting deep layers are qualitatively very obvious from these walkaway datasets.


185

CHAPTER 5

Dispersion inversion: Methods and synthetic tests

5.1 Introduction

In nearly all reported shallow surface wave inversion applications, the dispersion is

modelled in terms of normal modes. This is a direct translation of the methods used in

earthquake seismology where dispersion is invariably dominated by the fundamental mode

and plane wave matrix methods are suitable. Unlike shallow environments, when reversals

in elastic parameters occur, the contrasts are small, usually only a few to several percent

[81, 302]. When higher modes are incorporated into the inversion, it is usually a multi-

modal framework, where they are of less power and well separated from the fundamental

mode and mode number is easily identifiable. In these cases, they can be separated if

desired, usually from filtering the raw data prior to dispersion curve extraction [126]. This

filtering is vital for inversion procedures which analyse the fundamental mode only [235].

However, even with well separated modes, it is possible that mode number is not clear

and/or the modes are only observable over limited frequency ranges, due to negligible

power outside their band of propagation. In more severe cases, dominant higher modes

occur, which are the types that were outlined in Chapter 2 and observed in active source

synthetic tests of Chapter 3. For these types of dispersion curves, an inversion procedure

which does not rely on individual mode isolation is required. The PSV forward modelling

described in Chapter 2 and employed in the numerical tests of Chapter 3 simulates effective

dispersion, removing the need for mode identification, and incorporating dominant higher

modes for a more accurate inversion.

186 Chapter 5. Dispersion inversion: Methods and synthetic tests

5.2 Inverse theory review

Inversion is the procedure by which a solution is sought to describe an observed phe-

nomena, which in geophysics usually comprises the structure and properties (parameters)

of the Earth under investigation, based on a measurement made at the surface, from

the air, in the sea or down a borehole. Some synonyms for inversion are parameter fit-

ting (physics), ridge regression (statistics) or systems identification (engineering). In all

cases, to achieve the optimisation most algorithms aim to minimise the misfit between

the observed (measured in the field or laboratory) and model (defined by a mathematical

representation) data. The misfit is known as the objective function, also called a cost

or penalty function. Usually, various constraints known as regularisation are applied to

ensure the solution has some sense of realism.

In this thesis, the observed data are the effective phase velocity dispersion points picked

manually from the plane wave transform of a multichannel shot gather. Their theoretical

counterparts are picked automatically from transforms of P -SV reflectivity generated shot

gathers and the model comprises the shear velocities of the geological layering (assumed

flat) with thicknesses set at the start of the inversion (Section 5.3.3). Before describing

the scheme, a brief overview of linear inverse theory will be presented.

5.2.1 Ill-posed problems The scope of inverse theory is very broad, but it is

enough to appreciate that the solution of any geophysical inverse problem is never exact,

even if there are infinite data, no errors and perfect parameterisation [286]. The problem

is merely worsened due to finite number of data and errors in the measurements, coupled

with the the simplified mathematical model of the process. There are three considerations

to take into account [371]:

1. Existence;

2. Uniqueness; and

3. Stability.

Non-existence of a solution implies that the inverse operator from data to model is singular,

that is, the forward operator cannot be inverted. Uniqueness of a solution is the degree

of equivalence, that is, how many other models can reproduce the observed data equally

well. Instability of a solution implies the problem is poorly-conditioned, that is, small

changes in model parameters do not induce relative changes in the data. When any of the

above three conditions fail, the problem is ill-posed, which is the case for all geophysical

problems.

5.2.2 Linear problem In a purely linear, infinite dimensional and error-free sys-

tem, the equation d = Gm describes the model to data mapping, where d are the M data

5.2. Inverse theory review 187

values, m are the N model parameters and G is the M×N operator which accomplishes

the transform. The model space is thus N -dimensional and the operator may not be ex-

actly linear, but vary smoothly enough so that changes in model parameters cause equally

proportional changes in the data.

When the system is overdetermined (more data than model parameters) an objective

function is often defined by the L2 norm

χ2 = ‖d − Gm‖ = U (5.1)

The L2 norm is the simplest misfit measure to be used as an objective function, where

norm in general is defined as

‖x‖p =

{

1

N

N∑

i=1

|xi|p}1/p

(5.2)

Minimisation of the misfit function (∇U = 0) gives the well known unconstrained least

squares solution [127, 170, 200]

m = (GTG)−1GTd (5.3)

This assumes GTG is square symmetric and invertible, the conditions for existence and

that the minimum is global.

Often, the forward operator is not known explicitly, but the partial derivatives are and

can be written in matrix form. When an initial model is perturbed, the new datum values

can be calculated by Taylor series expansion to first order as

d2 = d1 +∂d1

∂m1∆m (5.4)

where ∆m is the model perturbation (m2 −m1) and ∂d1/∂m1 is the Jacobian or Frechet

derivative matrix at the initial model, hereafter denoted as J. It describes how each

data point, di, i = 1 : M changes with the perturbation of each model parameter, mj ,

j = 1 : N . In linear problems the Jacobian is weakly model dependent, which means

in some cases the solution can be determined in one iteration from a starting model by

the step ∆m = (JTJ)−1JT∆d. Here ∆d is the residual between experimental and initial

model data.

5.2.3 Linearised problem In geophysics, the model to data mapping is ubiqui-

tously nonlinear thus the Jacobian is highly model dependent. The problem is linearised

when Equation 5.4 is applied with the assumption of linearity over small changes in model

space. We can iteratively improve model updates by creeping or jumping throughout

model space, each time recalculating the sensitivity matrix. With a good starting model,

the solution will converge towards an optimum model which is hoped to be at the global

misfit error minimum.


In the inversion of seismic data, a suitable transform from the fundamental data (wave-

forms) can help reduce the nonlinearity of the problem. Observation of dispersion partially

accomplishes this, but with effective dispersion, a strong non-linearity still remains. This

is recognised from the discontinuities in the picked dispersion curves and solution curves

which converge irregularly.

The forward calculation is defined as

d = F[m] (5.5)

where d and m are described as above, but F is a nonlinear operator, usually only available

numerically. In this work it is the recursive reflectivity algorithm followed by plane wave

transform. Allowing for systematic error in the forward modelling, s and the observation

errors, e, Equation 5.5 should be written d = F[m] + s + e [287]. The deficiencies of

the modelling are generally ignored, although a strong awareness of them should be kept

foremost. The misfit defined by least squares is then

χ2 = ‖Wd − WF[m]‖ = U (5.6)

Here W weights the data, usually based on the standard deviation (either observed or

estimated) as

W = diag{1/σ1, 1/σ2, . . . , 1/σM} (5.7)

Here the diagonal can only be used when errors are uncorrelated, as is usually assumed,

as well as normally distributed. Any linear combinations of uncertainty would require a

more complex weighting matrix.

The use of generalised inverses and iteratively updating the Jacobian is often called

Gauss-Newton or local inversion. One problem is when the search arrives at a local

minimum in model space, iterative updates may not improve the residuals. This is usually

only for a broad minimum and will be illustrated in the Case 2 inversion tests. This is

where global inversion schemes are preferable, which are derivative free and search much

wider throughout the model space, irrespective of the nonlinearity of the problem.

5.2.4 Regularisation Methods to overcome the ill-posedness of local, linear(ised)

inverse problems in geophysics are collectively known as regularisation. In essence, they

constrain the inversion to prevent structure in the solution (model) which is not necessary

to fit the data. This is especially important when the model is very finely discretised.

Without constraints, geologically unreasonable features must be subjectively discredited

by the interpreter.

The simplest regularisation method is a minimum error criterion, which stops the

inversion when an acceptable limit or range is reached [306]. Thus, even hard-limiting

the number of iterations allowed is a form of regularisation. One method uses a damping

5.2. Inverse theory review 189

parameter β to add an amount to the diagonal of the generalised inverse (GTG)−1 in

Equation 5.3 to give the solution [371]

m = (GTG + βI)−1GTd (5.8)

In a linearised system it is the matrix (JTJ)−1 which is damped. In either case, the damp-

ing parameter improves the conditioning by increasing the difference between smallest and

largest eigenvalue. It does not usually control the final solution, but usually the rate of

convergence [170, 352]. This is known as Levenberg-Marquardt inversion and is not valid

for highly nonlinear systems, where it is the second derivative of the objective function

(Hessian) which ultimately defines the minimisation [170].

Another method is to force a solution to be smooth. If the objective function is a

combination of data misfit and model smoothness, the objective function to be minimised

may be [284]

U = ‖d − Gm‖ + β‖∂m‖ (5.9)

where ∂ is a differencing operator and β is a Lagrange multiplier. In this case, the solution

becomes

m = (GTG + β∂T∂)−1GTd (5.10)

The Occam’s inversion algorithm employed in this work (Section 5.3.2) uses this type of

regularisation. When the damping parameter (Lagrange multiplier) is zero, the solution

and confidence limits are unconstrained. Other constrained (regularised) solutions em-

ploying a priori information are defined in [200], which have been known from the early

rigorous treatments of linear inversion [129].


5.3 Proposed inversion procedure

The only reports to date which address the inversion of dominant higher modes by

linear optimisation are summarised in Table 5.1. The new scheme can be seen as a hybrid,

where the reflectivity seismograms are used to observe a dispersion curve thus allowing

rapid analytic partial derivatives calculation. Moreover, realistic data errors are incorpo-

rated (synthetically generated and field verified in Chapters 3 and 4 respectively) which

better show the uncertainty propagation from field data to final model.

Table 5.1: Outline of recent full-simulation inversion methods and also the method pro-

posed here showing its hybrid nature. The primary new addition is a better understanding

of the uncertainty of the ‘effective’ dispersion curve, which enables the use of rapid, ana-

lytic partial derivatives and thus more realistic inversion resolution kernels.

Reference Forward Dataset Dispersion Partial Data

engine method derivatives errors

Ganji et al [91] Active f-c SASW Numerical Neglected

stiffness curve

Lai and Rix [161] Active f-c SASW or Analytic Percentage

propagator curve f-k

Forbriger [78] Reflectivity f-p Fourier- Numerical Statistical

plane Bessel

Here Reflectivity f-c f-k or Analytic True or

curve f-p modelled

5.3.1 Data and noise The input data are the effective phase velocity dispersion

curves picked manually from the plane-wave transform of the observed shot gather. Their

theoretical counterparts are those picked automatically from similarly transformed, full-

waveform synthetic shot gathers. This is a nonlinear transform of the measured quantities

(seismograms) as explained in Chapter 2. The data are real values and not scaled to

unit variance, thus the residual error criterion is also a real value with the same units as

the data (m/s). Moreover, the model vector is real, thus partial derivatives are also real,

removing the need for any complex arithmetic.

The data standard deviations were tested and proven in Chapters 3 and 4 respectively.

When they are available from synthetic or field repeatability tests they are applied as-is.

Otherwise, the noise model of Section 3.6.5 can be used. Error propagation through the

inversion with percentage, true and modelled noise envelopes of the dispersion will also

be tested. Note that in earthquake seismology, it is reported that group velocity (U)

may be more sensitive to shear velocity structure than phase velocity (c) and errors are

approximately 1% [335]. However, due to the derivative nature of U with c, there is a

5.3. Proposed inversion procedure 191

larger non-uniqueness in the inversion [248] and partial derivative calculation also requires

differentiation with frequency [267].

5.3.2 Regularised optimisation algorithm Occam’s inversion uses a Lagrange

multiplier as a regularisation parameter, by providing the smoothest model which best fits

the data [54]. Due to the expected non-uniqueness in 1D dispersion inversion, it would

appear suitable to reject unlikely models and this was realised by previous workers who

also supplied computer code [161]. The model roughness is defined as

R1 = ‖∂m‖2 (5.11)

where ∂ is the NxN difference matrix

∂ =

0 0 0 0 0 . . .

−1 1 0 0 0 . . .

0 −1 1 0 0 . . .. . .

. . . . . .

0 0 0 −1 1 . . .

(5.12)

A regularising parameter, µ−1, damps the data misfit from a predefined limit, ε, as

χ2 = µ−1{‖Wd − WF[m]‖2 − ε2} (5.13)

so the functional to be minimised is the sum of model roughness and data misfit, given as

U = ‖∂m‖2 + µ−1{‖Wd − WF[m]‖2 − ε2} (5.14)

Linearisation of the forward problem, F[m] means that in line with Equation 5.4

F[m1 + ∆m] = F[m1] + J1∆m + ε (5.15)

where J1 is the Jacobian at the start model m1, ∇m1F[m1] and ε contains the error of

the linearisation. By writing

∆m = m2 − m1 (5.16)

Equation 5.15 becomes

F[m2] = F[m1] + J1(m2 − m1) + ε (5.17)

thus assuming a perfect linearisation (zero error),

F[m2] = F[m1] + J1m2 − J1m1 (5.18)

Substituting Equation 5.18 into the functional of Equation 5.14, it becomes a linear com-

bination of the initial model m1, to be solved for the new model, m2, as

U = ‖∂m2‖2 + µ−1{‖W(d − F[m1] + J1m1) − WJ1m2‖2 − ε2} (5.19)


where the expression in brackets is a new data vector with only experimental and start

model dependence, written

d = d − F[m1] + J1m1) (5.20)

is simply the experimental data, d minus the current forward data d1 plus the local per-

turbation. Now, the optimum updated model will be when the gradient of Equation 5.19

with respect to m2 is zero. Combing the gradients of both the model roughness

∇m2

[

(∂m2)T

∂m2

]

= 2∂T∂m2 (5.21)

and similarly the data misfit

∇m2

[

µ−1(

Wd − WJ1m2

)T (

Wd − WJ1m2

)

]

= −2µ−1WJ1Wd + 2µ−1(WJ1)TWJ1m2 (5.22)

when rearranged for m2 gives

m2 =[

µ∂T∂ + (WJ1)

T(WJ1)]−1

WJ1TWd (5.23)

This is the solution for the updated model vector (m2) about a trial model (m1), regu-

larised by the damping parameter µ. Note that the model parameters themselves are not

damped as in [81], merely their first derivative (shear velocity differences between layers).

However, a second derivative may be more suitable [284]. When µ = 0 this becomes the

unconstrained least squares solution [200]. Usually a number regularisation parameters

are iteratively tested [54], implemented as follows:

1 Set start model layering and shear velocity

2 Set a range of damping parameters, usually logarithmic

3 Iteration loop

a Forward calculate datum at current model

b Calculate current partial derivatives

4 Damping loop

a With current damping parameter, solve Equation 5.23

b Forward calculate datum at solved model

c Re-calculate partial derivatives

d Repeat for all damping parameters (Step 4a)

5 Check best damped model (least datum error) for convergence

6 If not converged, pass model to next iteration (Step 3) until maximum iterations reached


The method is more computationally expensive than other schemes [284], however, de-

pendence on the assumed number of layers will be minimal [161] and an optimal µ will

be semi-objectively ascertained. Other linear least squares methods set the regularisation

parameter according to the error state [170]. For example, starting with a high value to

mimic the steepest descent, then later reducing it for a less constrained solution. It could

also be increased following a divergence. Note too that the kernel in Occam’s inversion

(WJ) is not scaled during optimisation, so the solution comprises absolute values of shear

velocity.

5.3.3 Model parameters It is well accepted that VS has the most influence on the

surface wave datum (Rayleigh and Love wave dispersion) and thus will be the primary

unknown in the inversion. This has been shown by numerical partial derivatives at both

crust-mantle scale [40] and local site scale [352, 86]. While these sensitivity analyses

were based on fundamental mode dispersion curves, it has been shown that higher modes

show similar dependencies [356]. Based on this, the effective dispersion will show similar

sensitivities and the same model vector can be used. This VS is the shear body wave

velocity in the layer, assumed frequency independent, and since all layers are assumed

homogenous and isotropic, there is no distinction between VSH and VSV. Moreover, since

perfect elasticity is assumed, they are real values.

Studies based on a common synthetic model have been calculated for the fundamental

mode [352], higher mode [356] and damping ratio [355] sensitivity to elastic parameter

assumptions. In the phase velocity studies, aside from VS being the dominant parameter

and thickness the next dominant, it was stated that a 25% error in the VP or ρ of all layers

only induces a 10% perturbance of the fundamental mode dispersion, much less for higher

modes, and is reasonable to estimate it to this accuracy. The model used, however, had a

high average Poisson’s ratio of 0.45 and a ±25% change in VP correlated to only +4% and

−10% perturbations in Poisson’s ratio.

However, wrong assumptions of Poisson’s ratio (σ) due to the unknown water table

depth can result in gross misinterpretation [86]. For example, if saturated sediments

are not expected and σ is estimated too low (0.2), layers below the water table with

high σ (0.49) will be inverted with shear velocities too high, as a result of low-frequency

phase velocities being modelled as too high. Capillary effects above a water table would

should also be taken into account. Even though higher mode sensitivities to σ were

studied in [356], these were continuous, modal dispersion curves. It has been reported

that the effective phase velocity comprising dominant higher modes is expected to be

more susceptible to parameter assumptions [86], but not been investigated. These effects

will be studied in this chapter.

While density has been shown to have a small influence on modal dispersion in terms

of RMS difference, certain frequencies around 20 Hz in shallow (10-15 m) layered stacks


can be affected, in excess of 20% [352, 86]. While density contrasts between layers were

small (or absent) in those studies, at a larger (crust-mantle) scale with large contrasts

a modest influence exists [40]. In shallow environments, however, density contrasts are

invariably within several percent. However, one example of an abrupt contrast, along with

Poisson’s ratio, is at the water table [86].

Determining the layer interfaces is a common problem, mainly since analytic partial

derivatives for thickness do not exist. Even if the Earth under investigation could be

perfectly represented by flat layers, the recovered shear velocity is highly dependent on

the assumed layer interfaces. This was realised in early parametric tests on how various

starting model layer thickness affect how the inverted model varies from the ‘true’ scenario

[262]. More recently, by using exponentially increasing thicknesses, recommendations on

layering were made [172]. In general all of layer thickness, density, Poissons’s ratio and

damping remain fixed in linearised inversion, one exception is the inclusion of layer thick-

ness in [366]. In global inversions, layer thicknesses along with shear velocity are usually

optimised. Various layer interface parameters will be tested to give an idea of inverted

model resolution, rather than just perturbations of the dispersion datum as in [303].

One way to overcome the depth dependency is to employ continuous shear velocity

functions with depth, thereby representing the layered system with a high-order polyno-

mial, thus only the polynomial coefficients need be optimised [81]. In a derivative free

global inversion this may be particularly useful [22]. However, since the forward modelling

algorithms require a layered stack, a discretised model must be estimated from the smooth

curves at each iteration in order to calculate the theoretical model response. The advan-

tages of using polynomials is that the dimensionality of model space is highly reduced and

nonlinear gradients can be modelled. These gradients have been noted in both onshore

[348] and offshore [105] geotechnical studies. One model for the shear velocity gradient is

the Gibson halfspace, where shear modulus (µ) increases linearly with depth and fits both

refraction [340] and surface wave inversion [353] models. However, the first metre or so of

a site often shows an extreme second order gradient [81] and has a large influence on the

surface wave effective dispersion over a broad bandwidth.

Intrinsic attenuation does not appear in dispersion or partial derivative theory and will

only affect waveforms. It has been shown that surface wave attenuation is primarily due to

the compressional wave quality factor (Q) at high Poisson’s ratio [355]. Methods exist for

estimating near-surface Q include [259, 261, 260]. Moreover it can have a large frequency

dependence, much more so than elastic moduli, and is usually overestimated, (that is,

damping underestimated). Q values are used in the PSV forward modelling merely to

ensure at least partially realistic waveforms and a narrow lobe in 2D spectral space.

A more influential parameter, however, is anisotropy. In strong, transverse anisotropy,

the dispersion arising from anisotropic effects may be of similar magnitude to that of the


vertical stiffness variations, that is, the layering which is being investigated [369]. Never-

theless, in this thesis and in all practical examples to date, anisotropy is not incorporated

into the inversion.

5.3.4 Partial derivatives In a linear inversion, one requirement is the Jacobian

or Frechet derivative matrix, which describes how each datum point changes with small

changes in each model parameter, denoted J in the above equations. It is convenient that

for our unknown vector, VS, they exist in analytic form. This is preferred to numerical

calculation of derivatives both for accuracy and speed, since they require at least one

extra forward calculation per model parameter per iteration, the accuracy depending on

the differencing method used. Additionally, effective dispersion automatically picked off

a plane wave transform is limited to the pixel resolution of the image. Thus, numerical

derivatives would be far less accurate than those from modal dispersion curves, where

phase velocity resolution is as fine as the root-bracketing employed.

While several methods exist to derive analytic phase (and group) velocity partial

derivatives [267, 226], the procedure here utilises the variational technique, which is the

application of Hamilton’s principle to the average Lagrangian density of the surface wave

boundary value problem. It was first suggested around the time of the first numerical

dispersion inversions in earthquake seismology [5] and is often used to support numerical

sensitivity analysis [352, 86]. It is employed in field data inversion where the number of

layers is large [366]

Essentially, a wavenumber perturbation caused by a change in medium parameters is

calculated, as described in [5], for a continuous function of depth but a more useful form

from [161] is[

∂c

∂β

]

i

=ρiβi

2UI1k2

∫ zi

zi−1

[

(

kr2 −dr1dz

)2

− 4kr1dr2dz

]

dz (5.24)

[

∂c

∂α

]

i

=ραi

2UI1k2

∫ zi

zi−1

[

kr1 −dr2dz

]2

dz (5.25)

These are the same results as in [115] who also provided a density partial derivative

[

∂c

∂ρ

]

i

=1

2ρi

[

αi∂c

∂α i+ βi

∂c

∂β i

]

− c2

2UI1

∫ zi

zi−1

[

r21 + r22]

dz (5.26)

These explicit forms are deceptive, since the datum c is actually an implicit function of

model parameters and (ω, k). Note that the layer partial derivatives of Equations 5.24

to 5.26 are integrals at each frequency of the continuous function over depths zi−1 to zi

where i is the layer number, 1 to N . The depth vector used for the continuous function

is discretised from zero to zmax, where zmax is a stretched wavelength, that is, the inverse

of approximate depth inversion (Equation 1.21). This is to ensure that a sufficient depth

is attained, to a point where the eigenfunctions have substantially decayed. Moreover, if


this depth does not exceed the depth to the half-space, the derivative at that frequency

can not be used.

Considering the shallowest layers, the shear velocity partial derivatives usually have

sensitivities of 1 or sometimes more, except at very low frequencies. Over the same frequen-

cies, compressional velocity partial derivatives are usually less than 1% of these. Although

the analytic density partial may show some sensitivity, numerical tests with both the

FSW and PSV codes show it has negligible influence in a broad range of models.

5.3.5 Statistical analysis measures

Data misfit Usually a the Euclidean L2 norm is converted to a relative (percentage)

error based on Equation 5.1 as

∆c = 100

∥

∥

∥

∥

∥

Wcfwd − Wcexp

cexp

∥

∥

∥

∥

∥

% (5.27)

where δc is the relative RMS error (in %) and cfwd and cexp are the forward and exper-

imental phase velocities (m/s). The data weight matrices W are usually based on the

experimental error estimation or analysis. When the forward dispersion is not similar to

the experimental trend, it should actually require a different set of weights. However, the

assumption cfwd ≈ cexp will be maintained, which will remove the need for estimating

weight matrices at each iteration.

Parameter covariance While measures of model error can only be made if the true

model is known, as is the case in synthetic tests, with field data only estimated uncertainty

can be made. Moreover, the measures are dependent on the regularising parameter. The

usual covariance of model parameters from a linear inversion solved by Equation 5.3 is

[200]

Cm = B Cd BT (5.28)

where

B = (GTG)−1GT (5.29)

and Cd is the covariance matrix of data, which for the uncorrelated M phase velocities is

Cd = diag{σ12, σ2

2, . . . , σ2M} (5.30)

This theory assumes that the data are uncorrelated and Gaussian distributed and, thus,

so will the model parameters. When the forward kernel (G) is linearised to WJ and the

generalised inverse (GTG)−1 is regularised by Occam’s inversion, B at the last iteration

becomes

B =[

µ∂T∂ + (WJ)T(WJ)

]−1WJTW (5.31)

where µ is that which provides the lowest fit at the final iteration. The standard deviations

of each model parameter are then√

diag{Cm}.


The correlation matrix of covariances is [114]

Xij =Cij

√

CiiCjj

(5.32)

and it’s mean spread

s =

√

√

√

√

1

N(N − 1)

N∑

i=1

N∑

j=1

(Xij − Iij)2 (5.33)

Model resolution kernels The covarariance matrix above is defined statistically, but

may also be known a priori. In either case, it can be improved by adding the covariance

of random errors ‘propagated’ into the model parameters from the data. This is defined

in [129] as

C = (GTC−1d G + C−1

m )−1 (5.34)

where Cd and Cm are the data and model covariances, defined in Equations 5.30 and 5.28.

The matrix of resolving kernels is [129]

R = CGTC−1d G (5.35)

The rows or R are then the resolution kernels for each parameter [127] and the difference

from the identity matrix illustrate the degree of non-uniqueness of each parameter.

Another unconstrained measure is defined in [200] as

R = (GTG)−1GTG (5.36)

where R will be the identity matrix I if all parameters are perfectly resolved (uniquely

determined). Similar to Equation 5.31, in Occam’s inversion R becomes

R =[

µ∂T∂ + (WJ)T(WJ1)

]−1WJTW · WJ (5.37)

To reduce this matrix to a representative scalar, two useful measures will be the norm of

the diagonal defined as

r = ‖diag{R}‖ (5.38)

and it’s mean spread

s =

√

√

√

√

1

N(N − 1)

N∑

i=1

N∑

j=1

(Rij − Iij)2 (5.39)

Before applying Equation 5.39 the rows of R can be rescaled over [0, 1], so the spread will

be zero if R = I and asymptote to 1 in zero resolution.

Method sensitivity and resolution The sensitivity of any geophysical method describes

the degree of datum change with respect to model parameter perturbation. Resolution is

then the smallest perturbation which could be detected in an inversion. Single scalars for

these in the vicinity of a model m0 are defined in [371] as

S = ‖Gm0‖ (5.40)

198 Chapter 5. Dispersion inversion: Methods and synthetic testsLa

yer

1

Case 1

Laye

r 2

Laye

r 3

−20 −10 0 10 20∆β(%)

Laye

r 4

Case 2

−20 −10 0 10 20∆β(%)

10−2

100

102

δcR

MS(m

/s)

Case 3

10−2

100

102

δcR

MS(m

/s)

10−2

100

102

δcR

MS(m

/s)

−20 −10 0 10 2010

−2

100

102

∆β(%)

δcR

MS(m

/s)

Figure 5.1: Synthetic model solution subspaces for Cases 1, 2 and 3 (left to right). For

each layer (1 to 4, top to bottom), shear velocity is perturbed over a ±20% range under

various data weights for relative data misfit.

and

R =1

‖G−1m0

‖(5.41)

where G is the forward operator. If this does not exist, by the assumption of linearity about

m0 it can be replaced with the Jacobian (J) [284], or more appropriately the weighted

Jacobian (WJ). If the noise of the data is δ, a model m will only be detectable if

δS = ‖m − m0‖ ≥ δ

S(5.42)

Similarly, two models m1 and m2 in the vicinity of m will be resolved if

δR = ‖m1 − m2‖ ≥ δ

R(5.43)

Thus, both S and R are required to be large for good sensitivity and resolution, that is,

ability to both detect and discriminate small model variations.

5.3.6 Convergence misfit The nonlinearity of convergence can be shown by 1D

subspace solution curves. By fixing all other parameters and perturbing each layer shear


velocity over a modest range, the weighted RMS misfit is calculated (relative to the un-

perturbed model) from Equation 5.27. An example of this for the synthetic Cases 1, 2

and 3 over a ±20% range of shear velocity perturbation for each layer is shown in Fig-

ure 5.1. The solid curve (no dots) assumes no weighting (W = I), the dashed curve has a

constant 2% disperison error and the solid curve (with dots) assumes realistic numerical

error envelopes, developed in Chapter 3. Although plotted at logarithmic scale, we can

see than layer 1 converges almost linearly in all cases. In Cases 1 and 2, with progressively

deeper layers, convergence is less rapid and moreover varies depending on the approach

direction. For layer 4 especially, accurate convergence may not occur since the fit is good

over a very large perturbation. This implies that if realistic error envelopes are used, the

parameter covariance will be high. For Case 3, however, there is an obvious nonlinearity

in the system, which may become worse with larger parameter subspaces.


5.4 Numerical summary

The numerical procedure comprises both a forward calculation to simulate the data,

and an inverse calculation to iteratively arrive at the optimum solution. The PSV forward

calculation described in Chapter 3 and the Occam’s inverse procedure are summarised in

Figures 5.2 and 5.3.

From initial tests of the standard Occam’s procedure, some parametric observations

were made:

• A suitable range of µ−1 might be [100, 101, 102, 103, 104, 105]

• This µ−1 range is then numerically used as µ, thus starts from large smoothness (1),

working through to gradually increasing roughness (10−5)

• Hard-coded a priori limits, eg. velocity must be positive

• Least datum error is the L2 norm, but others may be suitable

• Convergence may be either when datum error and/or model change arrive at a preset

limit

A useful modification for monitoring convergence was to use an unweighted absolute RMS

error, or rather, constant weights of 1 applied to the misfit calculation of Equation 5.27.

This seemed to improve the robustness, especially at low frequency for recovering deep

layer parameters. Two modifications which proved of little benefit were:

1. Narrowing the regularisation window by one decade at each iteration around the

optimal µ;

2. Weighting the partial derivatives to artificially increase layer sensitivity with depth

The first did not allow a large enough variety in trial models. The process with the second

involved calculating a reduced wavelength (Equation 1.21) and the average data error

about the reduced wavelengths between consecutive layer interfaces (λ(f)±σc(f)). These

average errors were then used to rescale the partial derivatives so the shallowest layers

have weight of 1 and those of the deepest layer has a weight of 2. This greatly decreased

the stability of deeper layers.

5.4.1 Layer parameterisation As already stated, the models to be employed are

1D stacks and no re-parameterisation into polynomials will be applied. However, various

functions to define the layering can be applied, such as the exponential families in [172].

In this work, simple algebraic functions for thicknesses will be employed, as shown in

Figure 5.4, where each is defined as:

(a) True thicknesses;

5.4. Numerical summary 201

Read currentmodel

��

�� - Read field

parameters��

�� - Generate

shot gather- Plane-wave

transform- Auto-pick

dispersion

Figure 5.2: Flowchart showing the forward dispersion calculation by the PSV method.

Read fieldparameters

��

�� - Read field

dispersion, d

��

�� - Set start

model, m1

?��

��

Startiterating

?Forward

model, F[m1]

and partials, J1

�Update

dampingparameter, µ

?Solve Occam’s

for m2

?Forward

model, F[m2]

and partials, J2

?

��

��

@@

@@

��

��

@@

@@

All µ

tested?

N

Y

?Selectbest µ

and m2

-��

��

@@

@@

��

��

@@

@@

Converged?N

Y

?Save m2,

J2 and F[m2]��

�� -

��

��

Finishiterating

6

m1 = m2

�-

‘Occam’sloop’

Figure 5.3: Flowchart showing the Occam’s inversion procedure, employing the forward

model of Figure 5.2. After trialling all µ, the best model is chosen as that which gives

the lowest RMS error. If two or more trial models give an error below the threshold, the

largest µ (smoothest) is preferred.


0 100 200 300 400

0

2

4

6

8

10

12

14

16

β (m/s)

z (m

)

(a)

0 100 200 300 400β (m/s)

(b)

0 100 200 300 400β (m/s)

(c)

0 100 200 300 400

0

2

4

6

8

10

12

14

16

β (m/s)

z (m

)

(d)

Figure 5.4: Various types of layering thickness parameterisations demonstrated about the

Case 3 true model. (a) True model; (b) Constant thicknesses; (c) Geometrically increasing

thicknesses; and (d) Linearly increasing thicknesses.

(b) Constant thicknesses: h = n;

(c) Geometrically increasing thickness: hi = hni−1, then rescaled to fit given [h1,hN ]; and

(d) Linearly increasing thicknesses: hi = zi/n;

where h is the layer thickness (h1 the top layer and hN the last layer atop the homogenous

half-space), zi the depth to the top of layer i and n the factor defining the (increasing)

thickness function with depth.

In Figure 5.4, all layering other than the true thicknesses are chosen to fit N layers

between the surface (z0) and half-space depth (zN ), here 10 layers ending at 14 m depth.

Constant thicknesses (Figure 5.4(b)) are simply zN/N . Geometrically increasing thick-

nesses (Figure 5.4(c)) require some manipulation, since in addition to fitting into into the

depth range specified (14 m), user-defined starting (h1) and ending (hN ) thicknesses (here

0.25 m and 2.5 m respectively) mean the rate of increase must be calculated by optimisa-

tion. With the assumption that thicknesses monotonically increase, this is accomplished

with a simple root-bracketing search. Since linearly increasing thicknesses (Figure 5.4(d))

form an infinite sum, they must start from a finite surface thickness, here 0.5 m.

Note that some combinations of N , zN , h1 and hN will not allow a solution and must

be properly accounted for. For example, if in Figure 5.4(c), N is reduced to 5 layers, the

geometrically increasing thicknesses with given end limits cannot be created.


10−5

100

10−4

10−2

100

102

Damping parameter (µ)

RM

S e

rror

%(a)

0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)

Figure 5.5: Optimal damping parameter (µ) range about the Case 1 true model. (a)

Relative RMS misfit, (b) Smoothest to roughest model range (grey) with optimum solution

(bold).

5.4.2 Damping parameter range With the Case 1 dispersion and a starting

model as the true model, data misfit for a range of µ from 101 to 10−8 is plotted in

Figure 5.5. When µ is large, the data misfit has little influence on the solution and when

µ is small, the solution is good for any roughness. The steep gradient from 10−6 to 100

shows the best range over which to scan, where the data misfit has the most dependence

on model smoothness. The grey area in Figure 5.5(b) shows the range of smoothest and

roughest models, where the true model lies between these limits.

5.4.3 Experimental dispersion curves ‘Experimental’ dispersion curves for the

synthetic Cases 1, 2 and 3 of Tokimatsu et al [325], which were used in Chapter 3, were

observed by the PSV method. The P -SV reflectivity parameters used were:

1. Source function - Berlage wavelet, φ = −90◦;

2. Source frequency - 40 Hz;

3. Source type - vertical point impact at surface;

4. Time window - 512 points at 2 ms (1.022 s);

5. Channels - 48 (1 m spacing);

6. Offsets - 5 to 52 m;

7. Receiver types - vertical geophones;


8. Receiver response corners - 4.5 to 250 Hz; and

9. Slownesses for integration - 400;

The dispersion processing methods used were:

1. Sample padding - 1024;

2. Trace padding - 256;

3. Transform method - f − p;

4. Slowness range - 62-317 m/s (Cases 1 and 3), 110-237 m/s (Case 2);

5. Minimum frequency - 0.4883 Hz; and

6. Maximum frequency - 80 Hz (Cases 1 and 3), 120 Hz (Case 2).

Since these low-frequency limits are in the zone of low-frequency effects where the disper-

sion deviates from the plane-wave matrix values, the phase velocities are changed to fit

the FSW dispersion. Thus, the experimental dispersion curves are an effective hybrid of

FSW and PSV phase velocities. Note too that f − k trace padding implies wavenumber

interpolation in the f − k plane and f − p trace padding implies the number of slownesses

sampled in the τ − p stack. If only wavenumbers up to the Nyquist are allowed (0.5/∆x),

the actual number of interpolated wavenumber pixels is half this. For this reason, when

f − p is used, the trace padding value is halved prior to setting the slant-stack slowness

vector.

5.4.4 Inversion parameters For the inversion, when the PSV kernel is employed

all acquisition parameters are necessarily duplicated. The FSW method is also used to

illustrate how it is not suitable for inverting dominant higher modes. In that case, the

only necessary acquisition parameter to match is the frequency interval (∆f). This is

calculated by 1/(Nt∆t), where Nt is the number of time samples (padded) and ∆t is the

sample interval (s). Other general parameters for the inversions employed in all these

synthetic tests are:

1. Slownesses for integration - 200;

2. Trace padding - 128;

3. Minimum frequency - 2 Hz (Cases 1 and 3), 5 Hz (Case 2);

4. Maximum frequency - 70 Hz (Cases 1 and 3) and 110 Hz (Case 2).

5. Maximum iterations - 5;

6. Maximum damping parameters - 5;


7. Largest damping - 10−0.5;

8. Smallest damping - 10−4.5 (Cases 1 and 3), 10−5 (Case 2);

9. Error for convergence - 0.001 m/s;

The reason for the the number of reflectivity acquisition slownesses and dispersion pro-

cessing trace padding being reduced by half is to assist speed. Another time saving process

can be made by halving the number of time (padded) samples and doubling the sample

rate to maintain the same frequency interval. This is not done for the synthetic tests, but

can be employed for field data collected at 0.5 ms or less sample rate.

5.4.5 Some concerns There may appear to be some serious flaws in applying a

Gauss-Newton (linearised) procedure to the inversion of dominant higher modes due to

the following problems:

1. Data - discontinuous and highly non-linear in complex cases;

2. Uncertainties - non-Gaussian, especially at low frequency; and

3. Partial derivatives - based on continuous, modal dispersion assumption.

Discontinuities in the data are due to jumps to dominant higher modes. It has been sug-

gested that linearised inversion of complex layered cases, such as those with large stiffness

contrasts between layers [43]. While the linearised scheme has been shown applicable to

the discontinuous dispersion from Case 2 style models [159], it was suspected that the

procedure would not be applicable to the more difficult Case 3. However, as the following

results show, accurate convergence occurs even with large data uncertainty and poor as-

sumptions of fixed model parameters. Discontinuities can also arise in forward modelled

data, when the automatic picking routine may find a maximum in plane-wave space which

is not part of the dispersion lobe. Although rare in synthetic data, this may occur at

low frequency where resolution is poor (as seen in Chapter 3) or at high frequency, where

if a poor slowness range is specified, aliased or P -wavefields may be picked. This is a

systematic error due to shortcoming of the simple ridge detection used.

In addition, when a (constrained) minimum is found via linearised optimisation, it

is normally assumed to be the global minimum and the operator from the last iteration

used to calculate sensitivities. However, with a constraint or weights, the operator may

be corrupted. A simple a posteriori measure of model resolution is a Monte-Carlo search

of the data likelihood (or unconstrained data misfit) function around the final solution to

see if all are in the same minimum. This gives an idea on whether the final solution is

indeed at the global minimum (as is usually assumed) and, if so, indicates the features

common to all successful models. This process is investigated further in Chapter 7. In

Figure 5.1, only Case 3 (HVL model) showed evidence for a solution space (albeit only a


small subset) with multiple, closely spaced minima. This indicates that nonuniquness is

likely to be greater than in other cases and convergence may get ‘stuck’ in local minima.

Gaussian error distribution is an assumption used in the theory of error propagation,

for example in calculating covariance matrices, which also assume system linearity. It is

usually only at low frequencies where data distribution showed a more Lorentzian shape

with larger outliers. In the field tests of Chapter 4, where spread length was incorporated,

repeatability distribution at higher frequencies was definitely Gaussian. As for improving

misfit, minimising the L1 norm rather than L2 norm (least squares) is a suggested measure

for increasing the robustness of the inversion [52]. At low frequency especially, this would

apparently provide better deep layer parameter accuracy. However, it is suspected that

the physical limitations on low-frequency accuracy (array length, Earth filter effects and

source bandwidth) would override any optimisation measure. Geophone low-frequency

response is not a concern, but any A/D acquisition filters will certainly reduce the lower

cutoff frequency.

The modal partial derivatives in [161] are also extended to effective partial derivatives,

which are offset dependent. They usually do not vary from modal sensitivities and espe-

cially since effective dispersion is a path average over the recording array (when multichan-

nel plane wave transform is used), the effective partial derivatives are necessarily smeared.

For dominant higher modes due to a low velocity layer, the modal partial derivatives have

been used effectively in an inversion in [159]. Initial trials showed that the analytic partial

derivatives sufficiently guided the optimisation in the direction of convergence, even at

models far from the solution.

5.4.6 Issues to be tested In synthetic testing, both the ‘experimental’ and ‘syn-

thetic’ data are generated by the same forward calculation, that is, the PSV method.

However, the ‘experimental’ data at low frequency matches the FSW dispersion, thus is

unaffected by low-frequency effects. Nevertheless, this chapter is essentially a test of the

optimisation procedure, but the following effects are incorporated:

1. Data uncertainty - standard Gaussian errors such as 5% [86] or 3% [43] versus more

realistic envelopes, synthetically tested in Chapter 3 and field verified in Chapter 4;

and

2. Model parameterisation - various assumptions on layer interface depth, density, Pois-

son’s ratio and starting shear velocity model.

In the initial inversion tests, the model parameterisation and assumptions (layer interfaces,

Poisson’s ratios and densities) will be maintained at the true values. Thus, the unknown

vector comprises only the shear velocities of the four layers. The starting shear velocity

model is a constant 150 m/s half-space. First, the effects of different resolution kernels


will be considered, with a constant 3% uncertainty in the experimental dispersion. This

will be followed by the influence of experimental error on the ideal inversion.

In addition, it is well known that in ideal, homogenous-layer surface wave inversion,

the recovered shear velocity is highly dependent on the assumed layer interfaces, Pois-

son’s ratios and densities (in that order), when shear velocity is the only free parameter.

Following the trials of inversion kernels and error influence, the effects of various layer

parameterisation and assumed elastic constant will be investigated. In addition, the effect

of starting shear velocity will also be investigated.

In all tests, to reduce the influence of regularisation, both the damping parameter

range and number of iterations will be kept constant. The RMS error for convergence is

set extremely low to ensure that the maximum number of iterations is reached. Even if

an acceptable data misfit is attained at an intermediate iteration, the inversion is allowed

to continue. If the solution is indeed in the global minimum, divergence should not occur

- but the ‘jumping’ nature of the optimisation will allow further searching beyond local

minima.


5.5 Case 1 inversion

The Case 1 dispersion is dominated by the fundamental mode, so traditional plane

wave matrix methods should be applicable in the inversion. Thus, the PSV method can

be directly compared to the FSW method.

5.5.1 Effects of inversion kernel and dispersion uncertainty For the figures

in this section, to avoid repetition in the captions, each sub-plot of each figure is defined

here:

(a) Dispersion curves with experimental error on experimental dispersion. Plot title has

the weighted RMS percentage error at the final iteration;

(b) True (grey) and iterated shear velocity models, including starting (dotted), interme-

diate (solid) and final (thick dashed) shear velocities; and

(c) Relative standard deviations (grey) and errors (solid line) in inverted layer shear

velocities, where values less than 1% are neglected.

Figure 5.6 shows the inversion results employing the PSV method and Figure 5.7 employ-

ing the FSW method, with a 3% dispersion standard deviation assumption. The final

RMS error is slightly less by FSW and inverted shear velocity errors are negligible, only

exceeding 1% in the half-space. By PSV, the error in inverted shear velocity is much

larger with depth, as well as the standard deviations, reflected by a larger RMS error at

convergence. On inspection, it is apparent that the poorest fit is at low frequency, where

low-frequency effects of surface waves exist. Thus, the inversions were repeated neglecting

dispersion points less than 8 Hz, excluding the band affected by low-frequency effects. The

results are shown in Figures 5.8 and 5.9. While the RMS errors are different, the errors

in final shear velocities are very comparable. The half-space shear velocity has not been

recovered with either forward method, indicating that the few low-frequency dispersion

points between 2 and 8 Hz are vital for interpreting the half-space at 14 m depth.

The same settings used in the inversions of Figures 5.6 and 5.9 are again employed,

but with more realistic error envelopes on the dispersion curve. These were numerically

simulated in Chapter 3. The results over the frequency band 2-70 Hz are shown in Figures

5.10 and 5.11, for the PSV and FSW methods respectively. Both methods show similar

errors in the recovered shear velocity of the half-space, while the PSV method is still

affected by low-frequency effects. Again, the results neglecting frequencies less than 8

Hz are shown in Figures 5.12 and 5.13. The PSV inversion still has larger errors than

the FSW method in the intermediate layers. However, the half-space shear velocity is

recovered with much larger error than when only 3% dispersion errors are assumed, and

indeed a stiff substratum is not interpreted in these inversions.

5.5. Case 1 inversion 209

0 10 20 30 40 50 600

200

400

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.47 %

MeasuredInitial Final

0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)

True InitialFinal Inter

100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β

Figure 5.6: Inversion of Case 1 with the PSV method and 3% dispersion errors. Starting

model has the true depth interfaces, densities and Poisson’s ratios.

0 10 20 30 40 50 600

200

400

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.00 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β

Figure 5.7: Inversion of Case 1 with the FSW method and 3% dispersion errors. Starting



0 10 20 30 40 50 6050

100

150

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.31 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β


model has the true depth interfaces, densities and Poisson’s ratios. Frequencies below 8

Hz have been neglected.

0 10 20 30 40 50 6050

100

150

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.02 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β


model has the true depth interfaces, densities and Poisson’s ratios. Frequencies below 8

Hz have been neglected.


0 10 20 30 40 50 600

200

400

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.86 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β

Figure 5.10: Inversion of Case 1 with the PSV method and realistic dispersion errors.

Starting model has the true depth interfaces, densities and Poisson’s ratios.

0 10 20 30 40 50 600

200

400

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.03 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β

Figure 5.11: Inversion of Case 1 with the FSW method and realistic dispersion errors.



0 10 20 30 40 50 6050

100

150

200

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.90 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β


Starting model has the true depth interfaces, densities and Poisson’s ratios. Frequencies

below 8 Hz have been neglected.

0 10 20 30 40 50 6050

100

150

200

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.01 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β


Starting model has the true depth interfaces, densities and Poisson’s ratios. Frequencies

below 8 Hz have been neglected.


5.5.2 Effects of assumed model parameters and layer interfaces Tests of

how the assumed parameters (layer interfaces, Poisson’s ratio and density) influence the

inversion were tested with the same input data and inversion parameters as the above

section. Again, for the figures of this section, to avoid repetition in the captions, each

sub-plot of each figure is divided into three rows, each comprising the following results:

Row 1 True (grey) and iterated shear velocity models, including starting (dotted), in-

termediate (solid) and final (thick dashed) shear velocities. The title indicates the

parameter and its assumed value, either relative to the true values (%) or absolute

values;

Row 2 Relative standard deviations (grey) and errors (solid line) in inverted layer shear

velocities. The title indicates both the relative RMS errors between true and inverted

and shear velocities (∆β) parameter and standard deviation from covariance matrix

(σβ); and

Row 3 Shows two error measures at the same scale. The dashed line is the RMS error

at each iteration (∆c(freq)) with the current data weights (Equation 5.27). The

grey are the residuals between the input phase velocity dispersion and final iterated

dispersion (∆c(freq)). The title is the relative RMS dispersion error at the final

iteration (∆c).

For each parameter, usually four or five trials are conducted spanning realistic ranges.

Poisson’s ratio The effects of Poisson’s ratio assumptions, relative to the true values,

with both 3% and realistic dispersion errors are shown in Figures 5.14 and 5.15 respectively.

Note that at 150% it exceeds the acoustic limit of 0.5, thus 0.499 is employed for all layers.

When 3% dispersion errors are assumed (Figure 5.14), it can be seen that although the

inversion converges almost equally well in all cases, to recover the top three layers to

better than 10% requires Poisson’s ratio to be estimated to within 50% accuracy. At

90% accuracy, all layers are within 10% and again it can be seen that the inversion standard

deviation (diagonal of covariance matrix) does not provide a true measure of inversion

accuracy. When realistic dispersion errors are assumed, (Figure 5.15) the results are

similar, where on average, ∆β is slightly smaller but σβ is larger. One difference with

Figure 5.14 are the residuals ((k) to (o)) which are higher at high frequency, due to a

smaller weight applied to these frequencies, but smaller at low frequencies, due to a smaller

weight. It is interesting to note that the smallest residuals occur consistently around 40

Hz, which is also the synthetic source function centre frequency.

The effects of various absolute Poisson’s ratio underestimates and overestimates are

shown in Figures 5.16 and 5.17 respectively (3% dispersion errors) and Figures 5.18 and

5.19 (realistic dispersion errors). The effects mimic the differences for relative errors in

Poisson’s ratio, where ∆β dereases with larger σ. Poisson’s ratios within the range 0.35 to


0 200 400 600

0

5

10

15

(a) σ= 25%

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=42.3%, σβ=3.8%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(k) ∆c=0.7%

0 1 2 3 4 5

0 200 400 600

(b) σ= 50%

0.1 1 10 100 1000

(g) ∆β=19.0%, σβ=1.8%

0 20 40 60

(l) ∆c=0.5%

0 1 2 3 4 5

0 200 400 600

(c) σ= 75%

0.1 1 10 100 1000

(h) ∆β=20.4%, σβ=1.0%

0 20 40 60

(m) ∆c=0.7%

0 1 2 3 4 5

0 200 400 600

(d) σ= 90%

0.1 1 10 100 1000

(i) ∆β=3.9%, σβ=1.4%

0 20 40 60

(n) ∆c=0.3%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) σ=150%

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=17.1%, σβ=1.3%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(o) ∆c=0.6%

∆c(freq) ∆c(iter #)

0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)


model has the true depth interfaces and densities, with various Poisson’s ratio estimates

relative to the true values.

0 200 400 600

0

5

10

15

(a) σ= 25%

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=42.1%, σβ=2.5%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(k) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

(b) σ= 50%

0.1 1 10 100 1000

(g) ∆β=21.6%, σβ=5.7%

0 20 40 60

(l) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

(c) σ= 75%

0.1 1 10 100 1000

(h) ∆β=8.1%, σβ=4.0%

0 20 40 60

(m) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

(d) σ= 90%

0.1 1 10 100 1000

(i) ∆β=2.8%, σβ=4.6%

0 20 40 60

(n) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) σ=150%

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=16.0%, σβ=4.6%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(o) ∆c=0.8%


0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)


Starting model has the true depth interfaces and densities, with various Poisson’s ratios

relative to the true values.


0 200 400 600

0

5

10

15

(a) σ=0.20

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=49.6%, σβ=1.9%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(i) ∆c=0.6%

0 1 2 3 4 5

0 200 400 600

(b) σ=0.25

0.1 1 10 100 1000

(f) ∆β=20.0%, σβ=1.6%

0 20 40 60

(j) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

(c) σ=0.30

0.1 1 10 100 1000

(g) ∆β=12.9%, σβ=1.5%

0 20 40 60

(k) ∆c=0.5%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) σ=0.35

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=6.3%, σβ=0.6%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(l) ∆c=0.4%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)


model has the true depth interfaces and densities, with various constant, absolute Poisson’s

ratio underestimates.

0 200 400 600

0

5

10

15

(a) σ=0.40

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=10.4%, σβ=0.9%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(i) ∆c=0.5%

0 1 2 3 4 5

0 200 400 600

(b) σ=0.45

0.1 1 10 100 1000

(f) ∆β=6.9%, σβ=1.5%

0 20 40 60

(j) ∆c=0.4%

0 1 2 3 4 5

0 200 400 600

(c) σ=0.475

0.1 1 10 100 1000

(g) ∆β=10.3%, σβ=1.3%

0 20 40 60

(k) ∆c=0.4%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) σ=0.495

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=16.8%, σβ=1.3%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(l) ∆c=0.6%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)



ratio overestimates.


0 200 400 600

0

5

10

15

(a) σ=0.20

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=28.0%, σβ=2.0%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(i) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

(b) σ=0.25

0.1 1 10 100 1000

(f) ∆β=26.3%, σβ=5.7%

0 20 40 60

(j) ∆c=1.0%

0 1 2 3 4 5

0 200 400 600

(c) σ=0.30

0.1 1 10 100 1000

(g) ∆β=12.3%, σβ=4.6%

0 20 40 60

(k) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) σ=0.35

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=9.3%, σβ=4.0%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(l) ∆c=0.8%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)


Starting model has the true depth interfaces and densities, with various constant, absolute

Poisson’s ratio underestimates.

0 200 400 600

0

5

10

15

(a) σ=0.40

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=6.3%, σβ=3.7%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(i) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

(b) σ=0.45

0.1 1 10 100 1000

(f) ∆β=3.5%, σβ=5.0%

0 20 40 60

(j) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

(c) σ=0.475

0.1 1 10 100 1000

(g) ∆β=14.4%, σβ=4.7%

0 20 40 60

(k) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) σ=0.495

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=14.4%, σβ=4.6%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(l) ∆c=0.9%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)



Poisson’s ratio overestimates.


0 200 400 600

0

5

10

15

(a) ρ= 25%

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=14.2%, σβ=1.3%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(i) ∆c=0.5%

0 1 2 3 4 5

0 200 400 600

(b) ρ= 75%

0.1 1 10 100 1000

(f) ∆β=12.3%, σβ=1.2%

0 20 40 60

(j) ∆c=0.4%

0 1 2 3 4 5

0 200 400 600

(c) ρ=125%

0.1 1 10 100 1000

(g) ∆β=12.3%, σβ=1.2%

0 20 40 60

(k) ∆c=0.4%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) ρ=175%

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=12.3%, σβ=1.2%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(l) ∆c=0.4%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)


model has the true depth interfaces and densities, with various density errors, relative to

the true values.

0.495 all produce acceptable solutions, at least for the upper 3 layers. This occurs when

both 3% and realistic dispersion errors are employed.

Density The effects of both under- and overestimating density are shown in Fig-

ure 5.20. The only discrepancy arises in the case of a large underestimate (25%) where the

error in inverted shear velocity of layers 2 and 3 is marginally larger than that with other

density errors, but such a large underestimate would be unusual in practise. Otherwise,

errors in assumed density have no effect on the inversion result, as expected from previous

sensitivity studies of density on surface wave propagation. The effects of density will not

be shown for the remaining cases as other parameters are deemed more important.

Starting shear velocity The effects of various starting half-space shear velocities are

shown in Figures 5.21 and 5.22. In Section 5.5.1, 150 m/s is arbitrarily chosen as the

starting value, but it can be seen that this parameter has little effect for the normally

dispersive Case 1. There is, however, a minor improvement in the recovery of the half-

space shear velocity from the 250 m/s starting model, but only when 3% dispersion error is

used (Figure 5.21(d)). With realistic dispersion errors, a 100 m/s starting model provides

the most accurate shear velocities (Figure 5.21(b)), but in all cases residuals are higher at

high frequency due to the larger weighting of realistic error at these frequencies, which is

in the order of 1%.

Layer interfaces By fixing the homogenous half-space at its true depth (14 m), both

subdividing the true layering and setting various numbers of constant thickness layers is


0 200 400 600

0

5

10

15

(a) β= 75m/s

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=8.6%, σβ=1.2%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(i) ∆c=0.3%

0 1 2 3 4 5

0 200 400 600

(b) β=100m/s

0.1 1 10 100 1000

(f) ∆β=10.2%, σβ=1.3%

0 20 40 60

(j) ∆c=0.3%

0 1 2 3 4 5

0 200 400 600

(c) β=200m/s

0.1 1 10 100 1000

(g) ∆β=12.4%, σβ=1.3%

0 20 40 60

(k) ∆c=0.4%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) β=250m/s

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=7.1%, σβ=1.2%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(l) ∆c=0.3%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)


model has the true depth interfaces and densities, with various starting half-space shear

velocity models.

0 200 400 600

0

5

10

15

(a) β= 75m/s

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=9.0%, σβ=4.3%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(i) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

(b) β=100m/s

0.1 1 10 100 1000

(f) ∆β=1.2%, σβ=4.8%

0 20 40 60

(j) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

(c) β=200m/s

0.1 1 10 100 1000

(g) ∆β=7.7%, σβ=5.2%

0 20 40 60

(k) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) β=250m/s

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=8.8%, σβ=5.2%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(l) ∆c=0.9%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)


Starting model has the true depth interfaces and densities, with various starting half-

space shear velocity models.


0 200 400 600

0

5

10

15

(a) Divisions= 2

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=9.6%, σβ=3.6%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(i) ∆c=0.3%

0 1 2 3 4 5

0 200 400 600

(b) Divisions= 3

0.1 1 10 100 1000

(f) ∆β=11.4%, σβ=5.8%

0 20 40 60

(j) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

(c) Divisions= 4

0.1 1 10 100 1000

(g) ∆β=13.7%, σβ=2.3%

0 20 40 60

(k) ∆c=0.6%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) Divisions= 5

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=20.6%, σβ=8.9%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(l) ∆c=1.9%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)


model has the true depth interfaces and densities, with various subdivisions of the true

layer thicknesses.

0 200 400 600

0

5

10

15

(a) Divisions= 2

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=14.4%, σβ=5.0%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(i) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

(b) Divisions= 3

0.1 1 10 100 1000

(f) ∆β=32.6%, σβ=2.5%

0 20 40 60

(j) ∆c=18.4%

0 1 2 3 4 5

0 200 400 600

(c) Divisions= 4

0.1 1 10 100 1000

(g) ∆β=16.1%, σβ=20.9%

0 20 40 60

(k) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) Divisions= 5

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=20.0%, σβ=3.4%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(l) ∆c=2.7%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)


Starting model has the true depth interfaces and densities, with various subdivisions of

the true layer thicknesses.


0 200 400 600

0

5

10

15

(a) Layers= 1

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=67.4%, σβ=0.2%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(k) ∆c=4.8%

0 1 2 3 4 5

0 200 400 600

(b) Layers= 3

0.1 1 10 100 1000

(g) ∆β=71.9%, σβ=1.2%

0 20 40 60

(l) ∆c=3.0%

0 1 2 3 4 5

0 200 400 600

(c) Layers= 5

0.1 1 10 100 1000

(h) ∆β=22.5%, σβ=1.6%

0 20 40 60

(m) ∆c=1.2%

0 1 2 3 4 5

0 200 400 600

(d) Layers= 7

0.1 1 10 100 1000

(i) ∆β=13.9%, σβ=4.2%

0 20 40 60

(n) ∆c=0.3%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) Layers= 9

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=17.4%, σβ=2.3%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(o) ∆c=0.5%


0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)


model has the true densities, but constant Poisson’s ratio of 0.45 and various numbers of

constant thickness layers.

0 200 400 600

0

5

10

15

(a) Layers= 1

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=63.3%, σβ=0.9%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(k) ∆c=2.7%

0 1 2 3 4 5

0 200 400 600

(b) Layers= 3

0.1 1 10 100 1000

(g) ∆β=60.1%, σβ=5.6%

0 20 40 60

(l) ∆c=3.1%

0 1 2 3 4 5

0 200 400 600

(c) Layers= 5

0.1 1 10 100 1000

(h) ∆β=26.6%, σβ=1.5%

0 20 40 60

(m) ∆c=1.5%

0 1 2 3 4 5

0 200 400 600

(d) Layers= 7

0.1 1 10 100 1000

(i) ∆β=15.3%, σβ=16.5%

0 20 40 60

(n) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) Layers= 9

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=24.7%, σβ=7.0%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(o) ∆c=0.9%


0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)


Starting model has the true densities, but constant Poisson’s ratio of 0.45 and various

numbers of constant thickness layers.


0 200 400 600

0

5

10

15

(a) Layers= 12 D

epth

(m

)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=18.5%, σβ=5.9%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(k) ∆c=0.5%

0 1 2 3 4 5

0 200 400 600

(b) Layers= 14

0.1 1 10 100 1000

(g) ∆β=14.9%, σβ=6.9%

0 20 40 60

(l) ∆c=0.4%

0 1 2 3 4 5

0 200 400 600

(c) Layers= 14

0.1 1 10 100 1000

(h) ∆β=22.6%, σβ=7.7%

0 20 40 60

(m) ∆c=0.1%

0 1 2 3 4 5

0 200 400 600

(d) Layers= 19

0.1 1 10 100 1000

(i) ∆β=15.0%, σβ=0.8%

0 20 40 60

(n) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) Layers= 28

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=23.3%, σβ=8.9%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(o) ∆c=0.5%


0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)



constant thickness layers. Note: (c) employs the FSW method.

0 200 400 600

0

5

10

15

(a) Layers= 12

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=25.4%, σβ=23.2%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(k) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

(b) Layers= 14

0.1 1 10 100 1000

(g) ∆β=22.8%, σβ=23.0%

0 20 40 60

(l) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

(c) Layers= 14

0.1 1 10 100 1000

(h) ∆β=26.5%, σβ=20.8%

0 20 40 60

(m) ∆c=0.0%

0 1 2 3 4 5

0 200 400 600

(d) Layers= 19

0.1 1 10 100 1000

(i) ∆β=22.1%, σβ=24.6%

0 20 40 60

(n) ∆c=1.2%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) Layers= 28

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=22.4%, σβ=10.7%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(o) ∆c=0.9%


0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)



numbers of constant thickness layers. Note: (c) employs the FSW method.


tested. Only the true density is maintained (1.8 g/cc) and Poisson’s ratio of all layers

assumed as 0.45.

The effects of subdividing the true layering is shown in Figures 5.23 and 5.24. On

average, with 3% dispersion errors (Figure 5.23), the lower the number of layers the more

accurate the inversion result. However, when realistic errors are applied (Figure 5.24),

the patter is not the same, where in inversion is more likely to diverge, such as in (j) and

(l). There is an obvious smoothing of the shear velocity profile with increasing number of

layers, which becomes erratic when this becomes large.

The effects of number of constant thickness layers is shown in Figures 5.25 and 5.26.

Here, only when certain layer interfaces correspond with the true depths, as is the case

with 7 layers, does the recovered shear velocity mimic the true profile. There is again the

smoothing effect of the recovered shear velocity profile with increasing number of layers.

Nevertheless, the top layer is always recovered within about 1% accuracy no matter what

interface assumptions are made. Note how when realistic dispersion errors are applied, the

RMS minimum is achieved by the second iteration and further iterations do not improve

the solution.

The effects of much larger numbers of layers are shown in Figures 5.27 and 5.28.

In these figures, the 14 layer case is tested twice: (b) with the PSV method and (c)

employs the FSW method. The interesting point to note here is that in (c), even though

the FSW method has smaller residuals and better RMS error of convergence than the

PSV method, the PSV method in (b) provides a smaller RMS error in inverted shear

wave velocity. Nevertheless, the nonuniqueness of the method is well illustrated, both

highly dependent on the assumed layering, the data errors and the forward modelling

procedure.

One interesting point with the layer thickness assumptions is that with increasing

numbers of thinner layers, the standard deviation from the inversion covariance matrix

better approximates the error between the inverted and true models. The correlation

is best layers less than 2 m thick, that is, less than 15% the depth to the assumed half-

space. This provides an estimate for interpreting solution error directly from the covariance

matrix.



The Case 2 dispersion shows gradually dominant higher modes with increasing fre-

quency, due to the shallow LVL, thus traditional plane-wave matrix methods are not be

applicable. However, the FSW method will be compared to the PSV method solution to

illustrate the systematic error propagated into the final model.

5.6.1 Effects of inversion kernel and dispersion uncertainty Figures 5.29 and

5.30 show the inversion results employing the PSV and FSW methods respectively, with

a 3% dispersion standard deviation assumption. The full waveform method has properly

modelled the jumps to higher modes, albeit minor shifts in frequencies, and recovered the

LVL and over- and underlying layers to about 1% accuracy. However, since the plane wave

matrix method can not properly model the jumps to higher modes, by fitting a smoothing

dispersion to these causes the LVL to remain undetected. The large overestimate of the

homogenous half-space is due to the low-frequency cutoff. At the lowest frequencies of

5-10 Hz, the dispersion is still increasing, suggesting a very stiff layer, whereas the 0 Hz

phase velocity is only 340 m/s. Thus, similar to Case 1, the lowest frequencies are vital in

order that the homogenous half-space is neither under- or overestimated in the inversion.

The inversion results with more realistic dispersion error envelopes are in Figures 5.31

and 5.32. While the PSV method has detected the top of the LVL, the deeper, stiffer layers

are not correctly recovered and a LVL is not interpreted. The reason can be inferred from

the shape of the final dispersion curve in Figure 5.31(a). It appears that the optimisation

has arrived at a fit where the weighted RMS is a minimum, but apparently not the global

minimum. The solution at local minimum is a model with a smooth, inversely dispersive

curve. It is a result of the low data weights at low frequencies causing the higher phase

velocities corresponding to the deeper layers to be neglected in the optimisation. This is

a peril of all local linear inverse problems and is very clear here when the true solution is

known. However, the poor solution when the FSW method is used (Figure 5.32) is due

to the failure to properly model the dominant higher modes. One point of note is the

discontinuity at about 70 Hz in the final dispersion curve which appears as a jump to a

higher mode. It is purely a numerical error, where the roots of the f−c dispersion function

are closer than the initial bisecting step size chosen. This phenomena occurs especially

for cases with an LVL where many dispersion roots exist, as seen in Figure 2.6(b), and by

coincidence correlates with the osculation points of modal dispersion curves.

However, repeating the inversion but with a 200 m/s starting model (as opposed to

150 m/s) reveals a much different solution. The PSV method (Figure 5.33) has better

recovered the LVL shear velocities, albeit a large overestimate of the half-space shear

velocity. Again, the solution with the FSW method is incorrect (Figure 5.34). For this

reason, a 200 m/s starting model will be used for the inversion parameter assumption

tests.


0 10 20 30 40 50 60 70 80 90 100100

200

300

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.55 %


0 200 400 600 800 1000

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β



0 10 20 30 40 50 60 70 80 90 100100

200

300

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 1.46 %


0 200 400 600 800 1000

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β




0 10 20 30 40 50 60 70 80 90 100100

200

300

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 1.00 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β



0 10 20 30 40 50 60 70 80 90 100100

200

300

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 2.90 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β




0 10 20 30 40 50 60 70 80 90 100100

200

300

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.67 %


0 200 400 600 800 1000

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β


Starting model has the true depth interfaces, densities and Poisson’s ratios. Same as

Figure 5.31 but with 200 m/s starting model.

0 10 20 30 40 50 60 70 80 90 100100

200

300

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 3.05 %


0 200 400 600 800 1000

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β





5.6.2 Effects of assumed model parameters and layer interfaces

Poisson’s ratio Unlike Cases 1 and 3, this case incorporates a very low Poisson’s ratio,

being 0.22 for the stiff surface layer. At depth σ is high (0.46-0.49), typical of dry sands

overlying saturated clays. The effects of various absolute Poisson’s ratio underestimates

and overestimates with 3% dispersion errors are shown in Figures 5.35 and 5.36. At all

Poisson’s ratios, the top layer is recovered well and only the deeper layers are misinter-

preted, especially at low Poisson’s ratio assumptions. Although the average Poisson’s

ratio of the true model is 0.41, the best compromise for all layers is achieved with a ratio

of 0.47. This suggests that a slight overestimate is preferable in irregularly dispersive

profiles. Similar results are produced when realistic dispersion errors are used (Figures

5.37 and 5.38). However, in both cases, if the Poisson’s ratio is grossly overestimated,

close to the acoustic limit of 0.5 (Figure 5.38(d)), the LVL is not properly recovered. The

dispersion residuals in this case show a larger number of spikes exceeding 1% error which

is a consequence of more jumps to higher modes with increasing frequency.

Starting shear velocity The influence of various starting half-space shear velocity ap-

peared significant in this case, as seen in Section 5.6.1, where 150 m/s converged to a local

minimum. Various starting shear velocity models are shown with both 3% and realistic

dispersion standard deviations in Figures 5.39 and 5.40 respectively. At 3% dispersion

error all upper layers are recovered well almost independent of starting model, however,

the homogenous half-space is overestimated. With realistic dispersion errors, the 125 m/s

starting model does not converge to the true solution, similar to the local minimum re-

sult in Figure 5.31, suggested by the width of the dispersion residuals in Figure 5.40(i).

However, the solution improves markedly with gradually higher starting shear velocities,

suggesting that nonlinearity of the solution subspace space is less at higher shear veloci-

ties, allowing a smoother convergence. In addition, with realistic errors the homogenous

half-space accuracy is much better, around 1% or less.

Layer interfaces Again by fixing the homogenous half-space at its true depth (14 m),

using various numbers of constant thickness layers is tested. Again, only the true density

was maintained (1.8 g/cc) and the Poisson’s ratio of all layers was assumed as 0.45. It was

observed that constant velocity starting models converged when all parameters are set as

their true values (layer thicknesses and Poisson’s ratio). However, under various assumed

stacks of constant thickness layered models, convergence to local minima occurred when

realistic dispersion errors are incorporated, even when the 175 m/s minimum starting shear

velocity is applied. This is partially due to more free parameters in the inversion but only

occurred when realistic dispersion errors are used. The convergence to a local minimum for

16 layers is shown in Figure 5.41. However, to overcome this problem, the starting model

is discretised from the approximate inversion method (β(z) = 1.1c(z), where z = λ/2.5)

and convergence is achieved. Hereafter, this procedure will be referred to as ‘automatic’


0 200 400 600

0

5

10

15

(a) σ=0.20

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=21.5%, σβ=2.0%

Dep

th (

m)

0 20 40 60 80 10010

−1

100

101

102

∆c (

%)

(i) ∆c=0.4%

0 1 2 3 4 5

0 200 400 600

(b) σ=0.25

0.1 1 10 100 1000

(f) ∆β=27.4%, σβ=3.2%

0 20 40 60 80 100

(j) ∆c=0.7%

0 1 2 3 4 5

0 200 400 600

(c) σ=0.30

0.1 1 10 100 1000

(g) ∆β=26.4%, σβ=3.8%

0 20 40 60 80 100

(k) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) σ=0.35

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=59.5%, σβ=9.9%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 80 100 Freq.(Hz)

(l) ∆c=0.6%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)




0 200 400 600

0

5

10

15

(a) σ=0.40

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=25.6%, σβ=3.8%

Dep

th (

m)

0 20 40 60 80 10010

−1

100

101

102

∆c (

%)

(i) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

(b) σ=0.45

0.1 1 10 100 1000

(f) ∆β=145.6%, σβ=4.0%

0 20 40 60 80 100

(j) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

(c) σ=0.475

0.1 1 10 100 1000

(g) ∆β=11.3%, σβ=1.1%

0 20 40 60 80 100

(k) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) σ=0.495

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=20.2%, σβ=1.4%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 80 100 Freq.(Hz)

(l) ∆c=1.3%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)





0 200 400 600

0

5

10

15

(a) σ=0.20

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=39.4%, σβ=7.8%

Dep

th (

m)

0 20 40 60 80 10010

−1

100

101

102

∆c (

%)

(i) ∆c=1.2%

0 1 2 3 4 5

0 200 400 600

(b) σ=0.25

0.1 1 10 100 1000

(f) ∆β=73.5%, σβ=6.6%

0 20 40 60 80 100

(j) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

(c) σ=0.30

0.1 1 10 100 1000

(g) ∆β=23.6%, σβ=0.9%

0 20 40 60 80 100

(k) ∆c=1.3%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) σ=0.35

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=33.7%, σβ=2.3%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 80 100 Freq.(Hz)

(l) ∆c=0.8%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)




0 200 400 600

0

5

10

15

(a) σ=0.40

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=39.1%, σβ=1.8%

Dep

th (

m)

0 20 40 60 80 10010

−1

100

101

102

∆c (

%)

(i) ∆c=0.7%

0 1 2 3 4 5

0 200 400 600

(b) σ=0.45

0.1 1 10 100 1000

(f) ∆β=7.7%, σβ=4.4%

0 20 40 60 80 100

(j) ∆c=0.7%

0 1 2 3 4 5

0 200 400 600

(c) σ=0.475

0.1 1 10 100 1000

(g) ∆β=8.6%, σβ=1.0%

0 20 40 60 80 100

(k) ∆c=0.6%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) σ=0.495

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=26.7%, σβ=0.8%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 80 100 Freq.(Hz)

(l) ∆c=1.1%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)





0 200 400 600

0

5

10

15

(a) β=125m/s

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=67.3%, σβ=7.4%

Dep

th (

m)

0 20 40 60 80 10010

−1

100

101

102

∆c (

%)

(i) ∆c=0.6%

0 1 2 3 4 5

0 200 400 600

(b) β=175m/s

0.1 1 10 100 1000

(f) ∆β=84.9%, σβ=5.8%

0 20 40 60 80 100

(j) ∆c=0.6%

0 1 2 3 4 5

0 200 400 600

(c) β=225m/s

0.1 1 10 100 1000

(g) ∆β=59.0%, σβ=9.3%

0 20 40 60 80 100

(k) ∆c=0.4%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) β=250m/s

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=59.0%, σβ=8.9%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 80 100 Freq.(Hz)

(l) ∆c=0.6%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)



velocity models.

0 200 400 600

0

5

10

15

(a) β=125m/s

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=37.3%, σβ=1.0%

Dep

th (

m)

0 20 40 60 80 10010

−1

100

101

102

∆c (

%)

(i) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

(b) β=175m/s

0.1 1 10 100 1000

(f) ∆β=8.4%, σβ=3.5%

0 20 40 60 80 100

(j) ∆c=0.3%

0 1 2 3 4 5

0 200 400 600

(c) β=225m/s

0.1 1 10 100 1000

(g) ∆β=5.2%, σβ=14.3%

0 20 40 60 80 100

(k) ∆c=0.4%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) β=250m/s

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=3.6%, σβ=0.7%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 80 100 Freq.(Hz)

(l) ∆c=0.6%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)





0 10 20 30 40 50 60 70 80 90 100100

200

300

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.91 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β




0 10 20 30 40 50 60 70 80 90 100100

200

300

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.78 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β



numbers of constant thickness layers. Same as Figure 5.41 but with automatic starting

model.


0 200 400 600

0

5

10

15

(a) Layers= 1

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=28.3%, σβ=1.0%

Dep

th (

m)

0 50 10010

−1

100

101

102

∆c (

%)

(k) ∆c=1.8%

0 1 2 3 4 5

0 200 400 600

(b) Layers= 3

0.1 1 10 100 1000

(g) ∆β=58.7%, σβ=1.5%

0 50 100

(l) ∆c=1.9%

0 1 2 3 4 5

0 200 400 600

(c) Layers= 5

0.1 1 10 100 1000

(h) ∆β=312.3%, σβ=3.1%

0 50 100

(m) ∆c=2.3%

0 1 2 3 4 5

0 200 400 600

(d) Layers= 7

0.1 1 10 100 1000

(i) ∆β=27.2%, σβ=2.3%

0 50 100

(n) ∆c=0.6%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) Layers= 9

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=20.5%, σβ=2.6%

∆β (%)

Dep

th (

m)

σβ∆β

0 50 100 Freq.(Hz)

(o) ∆c=1.2%


0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)




0 200 400 600

0

5

10

15

(a) Layers= 12

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=21.0%, σβ=1.1%

Dep

th (

m)

0 50 10010

−1

100

101

102

∆c (

%)

(k) ∆c=0.6%

0 1 2 3 4 5

0 200 400 600

(b) Layers= 14

0.1 1 10 100 1000

(g) ∆β=56.3%, σβ=9.0%

0 50 100

(l) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

(c) Layers= 14

0.1 1 10 100 1000

(h) ∆β=21.5%, σβ=0.8%

0 50 100

(m) ∆c=1.7%

0 1 2 3 4 5

0 200 400 600

(d) Layers= 19

0.1 1 10 100 1000

(i) ∆β=36.4%, σβ=1.7%

0 50 100

(n) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) Layers= 28

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=56.9%, σβ=4.7%

∆β (%)

Dep

th (

m)

σβ∆β

0 50 100 Freq.(Hz)

(o) ∆c=0.7%


0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)





0 200 400 600

0

5

10

15

(a) Layers= 1

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=24.7%, σβ=16.2%

Dep

th (

m)

0 50 10010

−1

100

101

102

∆c (

%)

(k) ∆c=2.0%

0 1 2 3 4 5

0 200 400 600

(b) Layers= 3

0.1 1 10 100 1000

(g) ∆β=35.5%, σβ=1.1%

0 50 100

(l) ∆c=1.4%

0 1 2 3 4 5

0 200 400 600

(c) Layers= 5

0.1 1 10 100 1000

(h) ∆β=40.3%, σβ=0.5%

0 50 100

(m) ∆c=1.0%

0 1 2 3 4 5

0 200 400 600

(d) Layers= 7

0.1 1 10 100 1000

(i) ∆β=77.4%, σβ=1.1%

0 50 100

(n) ∆c=1.1%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) Layers= 9

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=39.6%, σβ=1.5%

∆β (%)

Dep

th (

m)

σβ∆β

0 50 100 Freq.(Hz)

(o) ∆c=1.0%


0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)




0 200 400 600

0

5

10

15

(a) Layers= 12

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=27.1%, σβ=26.8%

Dep

th (

m)

0 50 10010

−1

100

101

102

∆c (

%)

(k) ∆c=1.0%

0 1 2 3 4 5

0 200 400 600

(b) Layers= 14

0.1 1 10 100 1000

(g) ∆β=46.3%, σβ=1.6%

0 50 100

(l) ∆c=1.2%

0 1 2 3 4 5

0 200 400 600

(c) Layers= 14

0.1 1 10 100 1000

(h) ∆β=36.1%, σβ=1.2%

0 50 100

(m) ∆c=6.8%

0 1 2 3 4 5

0 200 400 600

(d) Layers= 19

0.1 1 10 100 1000

(i) ∆β=49.8%, σβ=0.8%

0 50 100

(n) ∆c=1.1%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) Layers= 28

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=30.4%, σβ=2.3%

∆β (%)

Dep

th (

m)

σβ∆β

0 50 100 Freq.(Hz)

(o) ∆c=0.8%


0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)





starting model, and the inversion results for 16 layers shown in Figure 5.42. The trend of

the LVL is recovered well.

Figures 5.43 and 5.44 show the convergence for various stacks of constant thickness

layers with 3% dispersion error. The results are similar to Case 1, where a coincidence with

the true layering is required for the LVL to be properly recovered. In addition, the top

layer is only correctly recovered when a minimum 7 layers is assumed. In Figure 5.44 the 14

layer case is tested twice: (b) with the PSV method and (c) employs the FSW method.

The FSW method does not interpret the large contrasts associated with the LVL and

dispersion residuals and RMS error of convergence are larger than the PSV method.

However, even the PSV method with 14 layers does not produce an accurate solution

as when 12 or 19 layers are used. Nevertheless, all PSV invesions converge to RMS

misfits less than 1%, but RMS error in shear velocity is quite varied, which is a further

illustration of the nonuniqueness of the method. It is interesting to note that with a nearly

smooth profile of 0.5 m thick layers (Figure 5.44(e)), both the LVL true shear velocity and

upper and lower interfaces are well recovered. The most error is in the transition to half-

space shear velocity, which is equivalent to poor constraint on the resolution of half-space

depth. Again, as noticed with Case 1, with increasing numbers of thinner layers, the

standard deviation from the inversion covariance matrix approximates the error between

the inverted and true models. However, this is not true when the FSW method is used.

Figures 5.45 and 5.46 show the results for various layering with realistic dispersion

errors. As mentioned above, the starting model is generated by the approximate inversion

method to avoid convergence to local minima. However, with 5 layers, this has not occurred

and a minimum of 7 layers is essential for mapping the trend of the LVL, albeit with larger

error in shear velocity at depth. With larger numbers of thinner layers, the solutions are

more varied. One point of note is weakly changing RMS error with iteration number,

which also diverges in cases, which is suggestive of a very broad global minimum. With

12 or 14 layers, the solution trend is approximately correct, however there is again the

smoothing effect in the transition from layer 3 to 4. When the FSW method is used,

the solution is again systematically incorrect. However, unlike Figure 5.44, with larger

numbers of layers the solution is an overall inversely dispersive structure. This is very

similar to the local minimum solution of Figure 5.41 and can be explained as such. It can

be seen that solutions of the LVL case are especially dependent on layer parameterisation

and convergence to local minima can occur, most likely with thin layers and realistic

dispersion errors. In those cases, the partial derivatives of deeper, thinner layers are

insufficient to influence the optimisation direction.



The Case 3 dispersion shows a jump to the second, and partly the third, higher modes

which are dominant over the frequency range 8-16 Hz. This is common when a HVL is

present, but also for a simple stiff, homogenous half-space at shallow depth. Inversion of

these types of dominant higher modes has not been reported previously, and the influence

of the thickness of the HVL on the dominant higher modal structure will be essential in

resolving the correct shear velocity of the underlying LVL. The modal dispersion predicted

by the FSW method will also be compared to the full waveform PSV method in the

inversion.

5.7.1 Effects of inversion kernel and dispersion uncertainty Figures 5.47

and 5.48 shows the inversion results employing the PSV and FSW methods respectively,

with a 3% dispersion standard deviation assumption. The full waveform method has

recovered the HVL and the overlying layer shear velocities to within 1% accuracy. The

underlying layers, however, are less accurately recovered, but the jumps to higher modes

are properly modelled and RMS error is low. The plane-wave matrix method too has

apparently recovered the general HVL structure. However, the jumps to higher modes are

not properly modelled and, as a result, the final RMS error is high and the HVL shear

velocity is overestimated.

The inversion results with more realistic dispersion error envelopes are in Figures 5.49

and 5.50. While the FSW method has recovered the layer 1 and 2 shear velocities, the

underlying LVL is undetected, thus a HVL structure is not interpreted. This can be seen

from the smearing effect of the final dispersion through the dominant higher mode(s)

from 8-16 Hz, which have been modelled as the fundamental. The solution with the

PSV solution is similar. However, the unusual modal structure of the final dispersion

curve suggests the global minimum has not been found.

Similar to Case 2, the inversion is repeated with a 200 m/s starting model, as opposed

to 150 m/s. The PSV method (Figure 5.51) has better recovered the HVL structure while

the solution with the FSW method is incorrect, and apparently independent of the starting

model (Figure 5.52). Thus, while the PSV method is more accurate when interpreting

a buried HVL, the starting model must be carefully considered when realistic errors are

applied.

5.7.2 Effects of assumed model parameters and layer interfaces

Poisson’s ratio The effects of various absolute Poisson’s ratio underestimates and over-

estimates with 3% dispersion errors are shown in Figures 5.53 and 5.54. At low Poisson’s

ratio assumptions, only the deeper layers are misinterpreted. Although the average Pois-

son’s ratio of the true model is 0.41, the best compromise for all layers is achieved with

a σ of 0.47. This suggests that a slight overestimate is preferable in irregularly dispersive


0 10 20 30 40 50 600

200

400

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.48 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β



0 10 20 30 40 50 600

200

400

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 2.43 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β




0 10 20 30 40 50 600

200

400

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 1.10 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β



0 10 20 30 40 50 600

200

400

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.52 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β




0 10 20 30 40 50 600

200

400

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.58 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β




0 10 20 30 40 50 600

200

400

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.52 %


0 100 200 300 400

0

5

10

15


Dep

th (

m)

(b)


100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ∆β





0 200 400 600

0

5

10

15

(a) σ=0.20

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=43.8%, σβ=1.4%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(i) ∆c=1.9%

0 1 2 3 4 5

0 200 400 600

(b) σ=0.25

0.1 1 10 100 1000

(f) ∆β=36.4%, σβ=2.2%

0 20 40 60

(j) ∆c=1.7%

0 1 2 3 4 5

0 200 400 600

(c) σ=0.30

0.1 1 10 100 1000

(g) ∆β=183.4%, σβ=1.3%

0 20 40 60

(k) ∆c=4.3%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) σ=0.35

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=20.4%, σβ=1.2%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(l) ∆c=1.5%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)




0 200 400 600

0

5

10

15

(a) σ=0.40

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=12.6%, σβ=1.2%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(i) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

(b) σ=0.45

0.1 1 10 100 1000

(f) ∆β=9.3%, σβ=1.3%

0 20 40 60

(j) ∆c=0.7%

0 1 2 3 4 5

0 200 400 600

(c) σ=0.475

0.1 1 10 100 1000

(g) ∆β=19.1%, σβ=1.2%

0 20 40 60

(k) ∆c=0.7%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) σ=0.495

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=10.0%, σβ=1.2%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(l) ∆c=0.4%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)





0 200 400 600

0

5

10

15

(a) σ=0.20

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=63.3%, σβ=1.1%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(i) ∆c=1.4%

0 1 2 3 4 5

0 200 400 600

(b) σ=0.25

0.1 1 10 100 1000

(f) ∆β=59.0%, σβ=2.8%

0 20 40 60

(j) ∆c=1.2%

0 1 2 3 4 5

0 200 400 600

(c) σ=0.30

0.1 1 10 100 1000

(g) ∆β=20.1%, σβ=6.6%

0 20 40 60

(k) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) σ=0.35

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=24.4%, σβ=5.0%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(l) ∆c=0.7%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)




0 200 400 600

0

5

10

15

(a) σ=0.40

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(e) ∆β=67.7%, σβ=4.4%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(i) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

(b) σ=0.45

0.1 1 10 100 1000

(f) ∆β=24.0%, σβ=4.9%

0 20 40 60

(j) ∆c=0.6%

0 1 2 3 4 5

0 200 400 600

(c) σ=0.475

0.1 1 10 100 1000

(g) ∆β=26.1%, σβ=4.9%

0 20 40 60

(k) ∆c=0.6%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(d) σ=0.495

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(h) ∆β=77.5%, σβ=7.0%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(l) ∆c=1.1%


0 1 2 3 4 5

10−1

100

101

102 Iter.#

∆c (

%)





0 200 400 600

0

5

10

15

(a) β= 75m/s

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=8.0%, σβ=1.3%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(k) ∆c=0.4%

0 1 2 3 4 5

0 200 400 600

(b) β=125m/s

0.1 1 10 100 1000

(g) ∆β=9.5%, σβ=1.3%

0 20 40 60

(l) ∆c=0.4%

0 1 2 3 4 5

0 200 400 600

(c) β=175m/s

0.1 1 10 100 1000

(h) ∆β=8.5%, σβ=1.2%

0 20 40 60

(m) ∆c=0.4%

0 1 2 3 4 5

0 200 400 600

(d) β=225m/s

0.1 1 10 100 1000

(i) ∆β=10.1%, σβ=1.3%

0 20 40 60

(n) ∆c=0.4%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) β=250m/s

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=10.8%, σβ=1.3%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(o) ∆c=0.7%


0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)



velocity models.

0 200 400 600

0

5

10

15

(a) β= 75m/s

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=36.7%, σβ=8.6%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(k) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

(b) β=125m/s

0.1 1 10 100 1000

(g) ∆β=30.9%, σβ=8.1%

0 20 40 60

(l) ∆c=0.6%

0 1 2 3 4 5

0 200 400 600

(c) β=175m/s

0.1 1 10 100 1000

(h) ∆β=34.9%, σβ=1.7%

0 20 40 60

(m) ∆c=1.0%

0 1 2 3 4 5

0 200 400 600

(d) β=225m/s

0.1 1 10 100 1000

(i) ∆β=37.3%, σβ=1.0%

0 20 40 60

(n) ∆c=1.0%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) β=250m/s

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=8.8%, σβ=4.6%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(o) ∆c=0.5%


0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)





0 200 400 600

0

5

10

15

(a) Layers= 1

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=61.6%, σβ=1.6%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(k) ∆c=6.0%

0 1 2 3 4 5

0 200 400 600

(b) Layers= 3

0.1 1 10 100 1000

(g) ∆β=37.7%, σβ=0.7%

0 20 40 60

(l) ∆c=4.7%

0 1 2 3 4 5

0 200 400 600

(c) Layers= 5

0.1 1 10 100 1000

(h) ∆β=56.3%, σβ=3.8%

0 20 40 60

(m) ∆c=2.9%

0 1 2 3 4 5

0 200 400 600

(d) Layers= 7

0.1 1 10 100 1000

(i) ∆β=28.4%, σβ=6.4%

0 20 40 60

(n) ∆c=0.7%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) Layers= 9

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=35.8%, σβ=2.0%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(o) ∆c=0.9%


0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)




0 200 400 600

0

5

10

15

(a) Layers= 11

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=29.4%, σβ=6.7%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(k) ∆c=0.9%

0 1 2 3 4 5

0 200 400 600

(b) Layers= 14

0.1 1 10 100 1000

(g) ∆β=43.8%, σβ=7.9%

0 20 40 60

(l) ∆c=0.4%

0 1 2 3 4 5

0 200 400 600

(c) Layers= 14

0.1 1 10 100 1000

(h) ∆β=43.0%, σβ=1.3%

0 20 40 60

(m) ∆c=1.7%

0 1 2 3 4 5

0 200 400 600

(d) Layers= 19

0.1 1 10 100 1000

(i) ∆β=34.9%, σβ=1.4%

0 20 40 60

(n) ∆c=1.0%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) Layers= 28

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=31.8%, σβ=3.4%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(o) ∆c=0.6%


0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)





0 200 400 600

0

5

10

15

(a) Layers= 1

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=56.9%, σβ=3.8%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(k) ∆c=3.9%

0 1 2 3 4 5

0 200 400 600

(b) Layers= 3

0.1 1 10 100 1000

(g) ∆β=51.0%, σβ=6.1%

0 20 40 60

(l) ∆c=3.8%

0 1 2 3 4 5

0 200 400 600

(c) Layers= 5

0.1 1 10 100 1000

(h) ∆β=39.6%, σβ=1.3%

0 20 40 60

(m) ∆c=3.2%

0 1 2 3 4 5

0 200 400 600

(d) Layers= 7

0.1 1 10 100 1000

(i) ∆β=34.8%, σβ=18.2%

0 20 40 60

(n) ∆c=0.5%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) Layers= 9

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=32.1%, σβ=5.0%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(o) ∆c=1.2%


0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)




0 200 400 600

0

5

10

15

(a) Layers= 11

Dep

th (

m)

0.1 1 10 100 1000

0

5

10

15

(f) ∆β=46.6%, σβ=24.1%

Dep

th (

m)

0 20 40 6010

−1

100

101

102

∆c (

%)

(k) ∆c=0.8%

0 1 2 3 4 5

0 200 400 600

(b) Layers= 14

0.1 1 10 100 1000

(g) ∆β=40.1%, σβ=8.8%

0 20 40 60

(l) ∆c=0.6%

0 1 2 3 4 5

0 200 400 600

(c) Layers= 14

0.1 1 10 100 1000

(h) ∆β=57.1%, σβ=9.1%

0 20 40 60

(m) ∆c=0.5%

0 1 2 3 4 5

0 200 400 600

(d) Layers= 19

0.1 1 10 100 1000

(i) ∆β=40.6%, σβ=4.6%

0 20 40 60

(n) ∆c=0.7%

0 1 2 3 4 5

0 200 400 600

0

5

10

15

(e) Layers= 28

β (m/s)

Dep

th (

m)


0.1 1 10 100 1000

0

5

10

15

(j) ∆β=36.9%, σβ=6.0%

∆β (%)

Dep

th (

m)

σβ∆β

0 20 40 60 Freq.(Hz)

(o) ∆c=0.8%


0 1 2 3 4 5

10−1

100

101

102Iter.#

∆c (

%)





profiles. Similar results are produced when realistic dispersion errors are used (Figures

5.55 and 5.56). However, in both cases, if the Poisson’s ratio is grossly overestimated,

close to the acoustic limit of 0.5 (Figure 5.56(d)), the LVL is not properly recovered. Al-

though the dispersion curves are not shown, the residuals show a larger number of spikes

exceeding 1% error which is a consequence of more jumps to higher modes with increasing

frequency.

Starting shear velocity The influence of various starting half-space shear velocity ap-

peared significant in this case, as seen in Section 5.7.1, where 150 m/s converged to a local

minimum. Various starting shear velocity models are shown with both 3% and realistic

dispersion standard deviations in Figures 5.57 and 5.58 respectively. At 3% dispersion

error all upper layers are recovered well almost independent of starting model, however,

the homogenous half-space is overestimated. With realistic dispersion errors, the 125 m/s

starting model does not converge to the true solution, similar to the local minimum re-

sult in Figure 5.49, suggested by the width of the dispersion residuals in Figure 5.58(i).

However, the solution improves markedly with gradually higher starting shear velocities,

suggesting that nonlinearity of the solution subspace space is less at higher shear veloci-

ties, allowing a smoother convergence. In addition, with realistic errors the homogenous

half-space accuracy is much better, around 1% or less.

Layer interfaces Again by fixing the homogenous half-space at its true depth (14 m),

both setting various numbers of constant thickness layers is tested. Again, only the true

density is maintained (1.8 g/cc) and the Poisson’s ratio of all layers is assumed as 0.45.

As for Case 2 (with realistic errors), to avoid possibilities of convergence to local minima,

the starting model is generated by the approximate inversion method. While this is not

necessary when 3% dispersion errors are used, it will be employed throughout for better

comparisons.

Figures 5.59 and 5.60 show the convergence for various stacks of constant thickness

layers with 3% dispersion error. The results are similar to Case 1, where a coincidence

with the true layering is required for the LVL to be properly recovered. In addition,

the top layer is only correctly recovered when a minimum 7 layers is assumed, but 9

layers provides an erratic solution. Unlike Case 2, when thinner layers are used, the

solutions are quite smooth and scatter in the intermediate iterated models is small. This

is supported by the small change in RMS error with iteration number, but in all cases the

HVL structure is properly inverted with the PSV method. In Figure 5.60 the 14 layer

case is tested twice: (b) with the PSV method and (c) employs the FSW method. While

the overall shear velocity trend with FSW method represents a true structure, the HVL

is interpreted as deeper and thicker. This is again due to the smoothing effect by the

fundamental mode assumption of the input data and illustrated by the large dispersion

residuals (Figure 5.60(m)). When the number of layers exceeds 14, the structure of the


LVL underlying the HVL is not properly recovered, showing a smooth curve fit between

the HVL and stiff basement.

Figures 5.61 and 5.62 show the results for various layering with realistic dispersion

errors. The starting model is again automatically generated by the approximate inversion

method. Again a minimum of 7 layers is required to recover the trend of the shear velocity

structure but with larger numbers of layers, the smearing effect between the base of the

HVL and top of the stiff half-space is present to a larger degree than when 3% dispersion

errors were used. Nevertheless, the HVL Case 3 appears less prone to convergence to

local minima than the LVL Case 2, which was unexpected, based on the single parameter

solution curves shown in Figure 5.1, which suggested a highly undulating solution space

and broad local minima.


5.8 Other layer thickness settings

In reality, the layer interfaces are best estimated from a priori data (eg. nearby

boreholes). Figure 5.4 shows various methods for statistically setting the layer thicknesses,

however, the homogenous half-space depth is in practise also free. Thus, while complete

suite of possible layering combinations can not be exhaustively individually checked, two

indications are given here for the case of geometrically increasing layers.

Figures 5.63 and 5.64 show the inversion of Case 2 by the PSV and FSW methods

respectively with realistic dispersion errors. Here, 20 layers with thicknesses geometrically

increasing from 0.5 m to 5 m and a homogenous half-space at 20 m depth are used, with

the starting VS model automatically generated. The structure of the LVL is recovered

well to within about 10% accuracy, however, there is again the smoothing of shear ve-

locity between layers 3 and 4, as noticed in the constant thickness layer stack tests. The

FSW method fails systematically and results in an inversely dispersive model, with large

RMS misfit.

Figures 5.65 and 5.66 show a similar inversion for Case 3. Here, 20 layers with thick-

nesses geometrically increasing from 0.25 m to 2.5 m and a homogenous half-space at 19

m depth are used, and the starting VS model automatically generated. The structure of

the HVL is recovered well to within about 10% accuracy, however, the underlying LVL is

not properly recovered. This was noticed in the constant thickness layer stack tests, where

the inverted shear velocity is smoothed between the HVL and stiff homogenous half-space.

It is interesting to note that the FSW method converges to a better RMS misfit, albeit

the smoothing effect of the dispersion through the dominant higher modes over 8-16 Hz.

Although the top of the HVL is inverted accurately, the centre of the HVL (and underlying

LVL) as well as homogenous half-space depth are all inverted as about 5 m too deep on

average.

The effect of layer thickness parameterisation will be more thoroughly evaluated in the

Monte Carlo testing of Chapter 7.

5.8. Other layer thickness settings 247

0 10 20 30 40 50 60 70 80 90 100100

200

300

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.70 %


0 100 200 300 400 500

0

5

10

15

20

25


Dep

th (

m)

(b)


100

101

102

0

5

10

15

20

25

Percent (%)

Dep

th (

m)

(c)

σβ∆β


Poisson’s ratios and densities are assumed constant at 0.4 and 1.8 g/cc respectively and

20 layers with thicknesses geometrically increasing from 0.5 m to 5 m and a homogenous

half-space at 20 m depth are used.

0 10 20 30 40 50 60 70 80 90 100100

200

300

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 4.54 %


0 100 200 300 400 500

0

5

10

15

20

25


Dep

th (

m)

(b)


100

101

102

0

5

10

15

20

25

Percent (%)

Dep

th (

m)

(c)

σβ∆β

Figure 5.64: Inversion of Case 2 with the FSW method and realistic dispersion errors. All

other parameters are the same as Figure 5.63.


0 10 20 30 40 50 600

200

400

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 1.05 %


0 100 200 300 400 500

0

5

10

15

20


Dep

th (

m)

(b)


100

101

102

0

5

10

15

20

Percent (%)

Dep

th (

m)

(c)

σβ∆β


Poisson’s ratios and densities are assumed constant at 0.4 and 1.8 g/cc respectively and

20 layers with thicknesses geometrically increasing from 0.25 m to 2.5 m and a homogenous

half-space at 19 m depth are used.

0 10 20 30 40 50 600

200

400

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.43 %


0 100 200 300 400 500

0

5

10

15

20


Dep

th (

m)

(b)


100

101

102

0

5

10

15

20

Percent (%)

Dep

th (

m)

(c)

σβ∆β

Figure 5.66: Inversion of Case 2 with the FSW method and realistic dispersion errors. All

other parameters are the same as Figure 5.65.



The 1-parameter (shear velocity only) subspaces shown in Figure 5.1 suggest a larger

degree of nonuniqueness (multiple minima) and slower rate of convergence in Case 3 (HVL)

over Case 2 (LVL). The normally dispersive Case 1 shows linearity, however, all cases show

a broader solution space for deeper layers, in particular the half-space. This immediately

suggests poor resolution for these parameters, especially when realistic dispersion error is

incorporated.

Although the same synthetic modelling methods were used for the ‘experimental’

and ‘theoretical’ effective dispersion, and the solutions are influenced by the inherent

nonuniqueness of the regularised linear optimisation routine, several interesting conclu-

sions can be made on the effects of various parameter assumptions in the inversion of

dominant higher modes.

5.9.1 Modelling kernel The first conclusion is that the full-waveform reflectiv-

ity dispersion modelling method (PSV ) is accurate and comparable to traditional plane

wave matrix forward dispersion methods (FSW ) when the input dispersion is normally

dispersive. However, when the input dispersion contains dominant higher modes due to

irregularly dispersive LVL and HVL cases, there are systematic errors manifested when

the FSW method is used.

In Case 2, either a normally or inversely dispersive profile results when the FSW method

is used to interpret a LVL dispersion, either when layer interfaces are known exactly or

assumed. In Case 3, the smearing effect of fitting a modal dispersion through dominant

higher modes also causes the VS trend to be misinterpreted as normally dispersive.

One shortcoming of the PSV method is interpretation of a plane wave dispersion curve

at low frequency. In that case, the theoretical dispersion is affected by low-frequency effects

and error in the shear velocities of deeper layers can occur. However, this is not a concern

when field data is from engineering investigation scales, as it too will be affected and

usually not included in the inversion.

5.9.2 Dispersion errors The introduction of dispersion errors invariably causes

the deeper layers to be misinterpreted, since the lower frequencies are more affected, but

normally dispersive cases appear unaffected, at least when all other parameters are set

correctly. One problem with the irregularly dispersive cases (Cases 2 and 3) is convergence

to local minima, but can be recognised from the final phase velocity curve fit and addressed

with a higher velocity or more accurate starting model. The HVL of Case 3 appears less

prone to convergence to local minima than the LVL of Case 2, which was unexpected, based

on the single parameter solution curves shown at the end of Chapter 3, which suggested

a highly undulating solution space and broad local minima.


5.9.3 Poisson’s ratio (σ) In Case 1 (normally dispersive), relative σ perturbations

of up to -10% correlate well with [352, 356] for at most a 10% dispersion sensitivity to

±25% perturbations of VP (which in Case 1 correspond to only -4% and +2% σ variation)

correlate well. However, if a constant σ is assumed for the inversion, as is usually the case

in practice, the inverted VS is within 10% accuracy if the RMS error in σ is within -25% ,

corresponding to a -40%error in VP. The assumption of perfect compressibility (σ=0.25)

will generate 20% VS error in a normally dispersive, high σ structure.

Case 2 (LVL) is the only model with a large σ contrast, between the surficial layer

and underlying LVL of 0.22 and 0.49 respectively. This may be the case for dry sands

or rock overlying saturated sands or muds. At low constant σ assumptions, the results

correlate well with [86], where the deeper, saturated layers are interpreted with too high

shear velocities, even though RMS misfit of dispersion is low. However, the work of [86]

did not trial higher constant σ assumptions. The results for Case 2 show that this will

produce acceptable solutions for all layers thus best to overestimate σ. However, when

close to the liquid limit (0.5), a LVL may not be properly interpreted.

In Case 3 (HVL), like Case 1, σ ranges between 0.46-0.49 thus no large contrasts exist.

However, at very low σ, the HVL is inverted with too high shear velocity while layer 3 is

usually recovered well. With 3% dispersion errors, when a constant σ within the range

0.4-0.495 is assumed for all layers, VS solutions are within about 10% (at least for the

upper three layers). However, when realistic dispersion errors employed, the RMS errors

in inverted VS exceed 20% for all assumptions. This shows that Case 3 HVL models will

be very sensitive to assumed Poisson’s ratio in practice. Only when the true parameters

were assumed are the shear velocities of all layers recovered to less than 10%.

5.9.4 Starting model Normally dispersive cases appear independent of the shear

velocity of a constant velocity starting model within the limits of the dispersion. This is

for both constant and realistic dispersion errors with true and assumed layer interfaces,

correlating with the experimental trials in [352]. However, in strongly irregularly dispersive

cases, the inversion can converge to local minima if the starting half-space shear velocity

model is too low. This is when realistic dispersion errors are assumed and either the true

or assumed layer interfaces are used. In these cases, where dominant higher modes are

present, the starting shear velocity should be either estimated too high (if constant for all

layers) or with the approximate inversion method.

This contradicts misfit curves of Figure 5.1, where convergence is generally better

starting from low shear velocity. However, this contradiction shows that 1-parameter

solution subspaces should be interpreted with care, since do not represent a broad space

of parameter interaction.

5.9.5 Layer interfaces While VS has the largest influence on the forward dispersion

curve, layer thicknesses have the largest influence in inversion of dispersion. With field


data, it is expected that this parameter will be the most likely cause of misinterpretation

and nonuniqueness, compounded by the fact that the depth to homogenous half-space is

also unknown. In all these tests, the half-space depth was assumed known and there is

a thickness range within which the solution is feasible. A minimum number of 7 layers,

corresponding to 2 m thick or 15% of the depth to the homogenous half-space is invariably

required to recover the correct VS trend with depth. Another perspective is that the layer

thickness chosen should not exceed that of the thinnest layer, which in these tests is layer

1. In practise, the uppermost soil usually requires a steep velocity gradient comprising

many thin layers, often starting from less than 0.1 m in thickness [79]. In general, more

layers cause more smearing and inaccuracy at depth.

In Case 1 (normally dispersive), while the uppermost layer is always recovered well, as

well as the ‘depth’ to the second layer, using large numbers of layers can introduce incorrect

VS gradients at depth. The shear velocity of the second layer is invariably only correct in

its uppermost region. When realistic errors are introduced, incorrect VS undulations occur

at shallower depths and smearing from layer 3 to 4 is increased. In general, the method

could no be expected to correctly interpret the depth to a stiff homogenous half-space.

Seismic refraction would be better suited, but only when no velocity reversals occur.

Case 2 (LVL) is best interpreted with between 7 and 12 layers, that is, between about

8% and 15% the depth to homogenous half-space. However, with larger number of thinner

layers, the LVL can be grossly misinterpreted, with severe smoothing between layers 3

and 4. The problem is compounded when realistic dispersion errors are introduced, where

a stack of layers, each less than 1 m thick or about 5% the depth to the homogenous

half-space, will lead to an LVL not being interpreted. In addition, the RMS misfit is not

an accurate guide to the quality of the solution.

Case 3 (HVL) was expected to produce more erratic solutions, due to the anticipated

roughness of solution space. However, with 3% dispersion errors, even large numbers of

thin layers recovers the VS structure well. The results are much poorer with realistic

dispersion errors, where the largest errors in inverted VS occur in the LVL layer 3. This is

due to the smearing effect, which also noted in the inversion of experimental data of [258].

When other layering parameterisations are employed, including an assumed depth to

the half-space, results are generally good with the PSV method where the trend of both

LVL and HVL structures is recovered well. By the FSW method, the LVL model is

interpreted as weakly inversely dispersive, due to the overall dispersion trend being so

above about 10 Hz. Dominant higher modes associated with a HVL are fitted as an

average, the result being that the HVL is interpreted too deep and too thick. This error

propagates into the underlying LVL thickness and depth to homogenous half-space. While

only an introduction to the infinite possible number of layer interface parameterisations

possible, these will be more rigorously tested in Chapter 7.


It is expected that a HVL structure will be the most difficult structure to invert

from field data. The dominant higher modes occur at low frequency, which is where

dispersion errors are highest and partial derivatives lowest. The combination of these

factors (realistically weighted Jacobian matrix) will make deeper LVLs difficult to detect.

253

CHAPTER 6

Dispersion inversion: Field applications and pitfalls

6.1 Introduction

In Chapter 5, the ‘experimental’ and theoretical datasets were generated by the same

forward calculation, thus the phenomena is perfectly modelled and errors in inverted mod-

els are due purely to errors in the data and model parameterisation and assumptions,

compounded by the nonuniqueness of the inversion. A true test of the new inversion pro-

cedure is to observe dominant higher modes in field (or physical modelled) datasets and

invert to a physically correct model. A ‘correct’ solution can best be determined from

comparison to downhole or crosshole shear velocity data, since this is the sole parameter

being sought from the surface. However, the degree of lateral inhomogeneity cannot be

determined from a single borehole. Other methods of correlation, which also provide bet-

ter information of 2D and 3D effects, can be made with data from geological (borehole or

trench logs), geotechnical (penetrometer tests) or geophysical (radar, body wave seismic,

electrical etc.) techniques but will be less quantitative. All these sorts of comparisons are

methods of model appraisal. Numerical techniques for this which exclude any sort of a

priori information will be discussed in Chapter 7.

This chapter discusses the sites which were surveyed with surface wave seismic and their

significance, the seismic acquisition and surface wave processing methods and the inverted

models are appraised based on available information. In addition, some pitfalls associated

with inversion of dominant higher modes in sites with severe contrasts and/or gradients

in vertical shear wave velocity variations are also illustrated with real data examples.

254 Chapter 6. Dispersion inversion: Field applications and pitfalls

6.2 Site locations and significance

The sites investigated were all in Western Australia (WA). These are listed in Table 6.1

and the locations shown in Figure 6.1, where the road cutting site is in the Perth eastern

outskirts. The other sites where some pitfalls are illustrated in Section 6.7 are in the Perth

metro area (Narrows foreshore and railway tunnel excavation).

6.2.1 Telfer gold mine The Telfer mine is a sedimentary hosted gold deposit, in

the Great Sandy Desert about 450 km southeast inland from Port Hedland. The geology

at and around the mine is mostly steeply dipping sandstone beds, unaltered but with a

thin weathering layer of unconsolidated sands.

Since the gold ore undergoes preliminary processing on site, suitable locations were

sought for the construction of new heavy infrastructure, including vibrating milling equip-

ment. Both static and dynamic site responses are required to be calculated for the design

of structures housing (and in the vicinity of) large vibrating machinery and any ground

improvement which may be necessary. Site response is well known in earthquake engi-

neering but even on smaller scales is an important factor to consider, to avoid resonant

frequencies which may occur. The vital parameter for site response modelling is the small-

strain shear modulus (Gmax), which is related nonlinearly to VS through Gmax = ρVS2,

where ρ is the density. Thus shear velocity is required to be accurately known due to the

squared relationship.

The sites investigated were: (1) Two tunnel pits, approximately 200 m long, 20 m wide

and 4 m deep, dug parallel about 30 m apart to run perpendicularly across the regional

strike, reclaimed with rockfill material and artificially compacted; (2) in situ weathered

sandstone, covered by a thin, weathering veneer; and (3) A waste dump of ore refuse,

surficially compacted from the movement of mine trucks and other vehicles, less homoge-

neously than the tunnel pit sites. All were required to be surveyed for the variation of

shear velocity with depth and approximate depth to fresh rock (which was known almost

exactly for the tunnel pit). This work was initially proposed to be surveyed with shear

wave vertical seismic profiling (VSP). However, there was difficulty in locating a suitable

drilling rig so surface wave inversion was proposed as an alternative for estimating the

shear velocity. This was unique since no other methods were available for comparison,

other than the expected depths and approximate shear velocities.

In spite of the expected lateral discontinuities due to the dipping sandstone, surface

wave inversion was intended to provide a representative average with depth over the mea-

sured spreads, along with an indication of anisotropy by shooting perpendicularly across,

and parallel to, the strike of the buried sandstone formations. Moreover, at this site,

large contrasts in elastic constants and velocity reversals were expected, which indeed was

the case as will be shown from the dominant higher modes observed. Fortunately, when

this survey was requested, the inversion of dominant higher modes had been extensively

6.2. Site locations and significance 255

Table 6.1: Field sites investigated for surface wave inversion applications.

Site name Investigation Engineering Acquisition

purpose field date

Telfer gold mine Mining and milling Geotechnical Apr, 2003

infrastructure

Perth Convention Centre Multi-storey building Civil May, 2002

in soft sediment

Road cutting Caprock rippability Transportation Dec, 2002

Hyden fault scarp Palaeo-seismological Earthquake Dec, 2001

shallow mapping

Perth Hyden

Telfer

112oE 116oE 120oE 124oE 128oE 35oS

30oS

25oS

20oS

15oS

Figure 6.1: Map of Western Australia showing the sites where seismic surveys were con-

ducted for surface wave inversions.


trialled with synthetic data (Chapter 5) and the method was ready for verification with

field data.

6.2.2 Perth Convention Centre Perth City is located in the Perth Basin, on the

north shore of the Swan River. This basin is a thick sequence of Phanerozoic sediments,

up to 15 km deep, delineated to the east by the Darling fault and extending offshore

west into the Indian Ocean. In the Perth area, the shallow geology is dominated by

deposits of unconsolidated sands and muds and secondary cemented aeolian dune systems

of calcareous quartz sands (calcarenite). These are arranged in approximately north-south

trending bands, mainly deposited during marine transgressions (marine and lacustrine

formations) and regressions (aeolian sand dunes) from eustatic sea level changes during

the Quaternary ice ages [46].

The Perth Convention Centre is located on the Swan River foreshore, between the

CBD and north shore of the river. It is a large, multi-level structure located entirely

over land reclaimed from the Swan River in the 1950’s, the fill comprising mostly yellow,

loamy Quaternary sands. There is no occurrence of limestone, even though it outcrops

as a large plateau (King’s Park) to the west of the site. While this rock is used as a

building material, both historical and contemporary buildings, and its stratigraphy well

documented, little study has been made of its geotechnical properties, despite the fact

that most large developments, including the nearby CBD, are constructed on it.

However, the problem at the Perth Convention Centre is the low stiffness of a thick

sequence of organic rich, silty clays and sands, resting on siltstone which is part of the

younger formations of Phanerozoic sedimentary rocks. The siltstone is apparently quite

undulating, since it varies from 23 m to 36 m depth over the approximately 200 m by

100 m site. Due to the low strength of the soft, muddy formations, the foundations

for the structure comprises piles, covering the site over a 10 m square grid. These are

precast concrete of square cross-section and driven hydraulically to resistance, without

any vibrations and minimal noise, known commercially as ‘G-pile’. The foundation for the

structure is designed on the end support of piles driven to siltstone. The piles almost free-

fell through the softer layers, although an interbedded silty sand layer often presented some

resistance. The borehole logs near the site chosen for surface wave inversion are shown in

Figure 6.2 and summarised in Table 6.2. The shear velocity (VS) was measured by seismic

cone penetrometer (SCPT) which also measures cone tip resistance (qC) simultaneously.

The horizon of concern from the engineering point of view is the silty-sand at 20 m

depth. It is markedly stiffer than the over- and underlying silty clays and often presented

a resistance to pile driving. If a shallow refusal of the piles is encountered, it may be due

to this layer, which does not offer the same load support as the target siltstone. In this

case, the natural settlement may not provide sufficient pile stiffness criteria. In addition,

‘negative skin friction’ is more of a problem than when the pile ends rest on siltstone,


Figure 6.2: Borehole logs near the Perth Convention Centre surface wave survey spread

centre, including CPT (CC20), geology (CB5) and SCPT (SC2). The stiffness (Gmax) is

calculated from the VS log and estimated densities using ρβ2.

Table 6.2: Summary of borehole logs near the Perth Convention Centre surface wave

survey spread centre.

Borehole: CC20 CB5 SC2

Depth Tip resistance Geological Shear velocity

(m) qC (MPa) core log VS (m/s)

0-5 3-30 Sand (fill) 140

5-20 <1 Silty-clay 85-145

20-27 2-12 Silty-sand 220-450

27-36 2-3 Silty-clay 135-200

36 No data Siltstone No data


since as the plastic clays settle, the load can be released causing an oscillating stiffness-

settlement variation. Indeed, some piles refused at about 20 m depth due to the silty sand

layer, but this refusal exceeded the capability of the ‘G-pile’. For this site, the design

load of the structure was less than the maximum piling load (which is in itself a load

test which exceeds the design load) so it was allowed. However, for heavier structures, to

prevent uneven settlement, negative skin friction or even ‘punch-through’, a better option

would be to insert more piles around those where the basement is not reached. Thus,

rapid mapping of the depth, thickness and stiffness of this layer would be an advantage

for planning the piling and predict where shallow refusal may occur.

6.2.3 Road cutting This site is about 30 km east of Perth City, just inland from

the top of the Darling Scarp lineament. This is the surface expression of the Darling fault

which is the western edge of the Yilgarn Craton, a block containing several world-class

gold and nickel deposits. The general geology is an Archaean granite-greenstone terrane,

with a thickly weathered mantle (overburden) comprising mainly alluvium and colluvium

and weathered crystalline bedrock (saprolite) which in turn grades into fresh basement. A

common feature of this terrane is lateritic duricrust, a hard, cemented horizon of iron rich

pisoliths, caused by secondary cementation at the weathering front during humid to dry

climatic fluctuations of the Tertiary period. Hard horizons of silcrete or calcrete also occur,

which may outcrop or remain buried, and occur over both both felsic rocks or more iron-

rich mafics. The resistance to weathering of these hard horizons often forms large plateaus,

with steep edges called ‘breakaways’. There is, however, economic potential associate with

these formations, as supergene enrichment of gold or nickel sourced from shear zones in

the basement rocks can concentrate at lithological boundaries in the overburden.

The site is a reconstruction of a major highway into the Perth city. At the edge of the

Darling Scarp, before the road descends into the Perth Basin, it currently winds through

thickly vegetated terrane, dominated by undulating plateaus of laterite caprock. The

planned construction is both a widening of the road into a dual carriageway, along with a

reduction of steeper curves and gradients. This involves cutting of the hard caprock and

filling the overburden valleys, to depths of up to several metres. Traditionally, cutting is

done with special machinery and the rippability is evaluated from P -wave velocities [298].

While tables of rippability do not usually include values for lateritic duricrust, conglomer-

ate is the assumed closest sedimentary rock. For example, with a D-8 tractor, it is rippable

when VP is less than about 2600 m/s and non-rippable when velocity exceeds 3100 m/s. In

between these limits, it may be rippable, depending on degree of fracturing, but machinery

maintenance will be required at greater frequency. In the case of non-rippable material,

blasting is the only option, but since costs rise several-fold, it is important to evaluate this

accurately.

As the site was expected to be inversely dispersive and the thickness of the caprock was


also required, surface wave surveying was chosen to complement P -wave seismic refraction

for rippability evaluation. Several solid auger boreholes were previously drilled along the

surveyed cutting, with geological descriptions only and no wireline logs, but these were

used to to aid and support the surface wave interpretations. Other cases in this terrane

where laterite caprock is an economic concern is in bauxite mining, where it must be

extensively stripped off to expose the aluminium ore.

6.2.4 Hyden fault scarp Hyden is located about 340 km east-southeast of Perth,

Western Australia. The Hyden fault scarp, located about 70 km northeast of the town-

ship, is the result of a prehistoric earthquake. It is a 30 km long north-south trending,

topographic lineament over 2 m high for most of its length. The general geology is some-

what similar to the road cutting site, that is, laterite and calcrete, overlying saprolite

which grades into fresh crystalline basement, most likely felsic gneisses. However, in this

area, the surface is covered by a thin veneer of aeolian and fluvial material, comprising

fine sands and clays. The water table is expected to be quite deep and possibly saline,

and numerous salt claypans exist in the region.

An earthquake in the South-West Seismic Zone (SWSZ) in 1969 was a west-over-

east thrust with a maximum vertical throw of 3.5 m. However, the recent earthquake

history is too sparse for an accurate seismic risk assessment for the region. Since surface

rupturing events are rare and the older faults are mostly buried, airborne geophysics is

used for reconnaissance geological mapping to identify possible fault lines. From there,

ground geophysics is used to rapidly assess targets which are potential sites for drilling or

trenching. Previous interpretations of the Hyden fault were of a steeply dipping strike-slip

fracture. However, a 35 m long, 4 m deep trench dug approximately 200 m north of the

seismic line in 2001 revealed reverse thrust structures dipping at low angle to the west.

As part of a combined palaeoseismic study at the Hyden site, employing Ground Pene-

trating Radar (GPR), trench logging, geochronology and geodesy, a P - and S-wave seismic

reflection survey was conducted. The geophysical survey lines were aligned east-west, per-

pendicularly crossing the fault scarp. The seismic time window was allowed long enough

to record all the surface waves and the nature of the site was very conducive for ground-

roll generation. In fact, the surface wave data provided the only useful images from the

seismic data where the near surface layering caused masking of reflections. In general,

the wave methods (seismic and GPR) have proven most useful for palaeoseismology but

surface wave inversion has not been previously reported in this application.


6.3 Telfer gold mine

6.3.1 Data acquisition A 24-bit Geometrics SmartSeis with was used for acqui-

sition, with 24-channel recording. Geophones were undamped, 8 Hz vertical component,

with 2x12-channel cables at 5 m takeout spacings. The geophone spacing used throughout

was 1 m. The source was initially an 8 kg sledge hammer on UHMW plate, triggered by

an acoustic switch taped to the handle near the hammer head. The switch failed after

a few spreads where the trigger was replaced by circuit between the hammer head and a

thin steel plate. The only disadvantage of a steel plate as opposed to plastic (aside from

weight) is the intense air wave generated, however, surface waveforms are identical.

The three areas investigated near the mine were:

• Tunnel pits;

• Mill site; and

• Waste dump;

The spread numbers (lines) for each area are summarised in Table 6.3. Over the tunnel

pit backfill, both 0.5 m and 1 m geophone spacings were used, with time windows of

255.75 ms and 511.75 ms respectively (1024 and 2048 samples at 0.25 ms sample interval).

Elsewhere, 1 m geophone spacings were used and sample rate reduced to 0.5 ms for 1024

samples (511.5 ms). Over ground that was anticipated to be softer (slower), 2048 samples

at 0.5 ms (1023.5 ms) were used. These parameters are summarised in Table 6.4.

Since 48-channel shot gathers were preferred, the alternative was walkaway shooting,

with the geophones replanted for the further offsets. The shooting procedure shown in

Figure 6.3 allows each 24-channel shot gather to be concatenated to emulate a true 48-

channel gather at 1 m spacings. For each walkaway gather, forward and reverse shots at

5 m near offset become the final geometry and all surface wave arrivals were contained

within 512 ms.

6.3.2 Data quality As the surveying was done over a weekend, a large proportion

of mining traffic noise was absent. Large milling infrastructure was a few kilometres away

so cultural noise was minimal. The various conditions where the surveying was conducted

allow an interesting gallery of shot gathers to be compiled. These are helpful in later

interpretations and provide a reference for field practitioners to identify various quality

‘ground roll’.

Tunnel pits Over and near the backfilled tunnel pits, side scattering from the vertical

contrast of fill and in situ rock was expected, which was the reason for trialling two

different spread lengths. The lines across the pits were not expected to allow layered

interpretations, merely providing an indication of steep lateral heterogeneity. Alongside

6.3. Telfer gold mine 261

Table 6.3: Summary of surface wave spreads locations at Telfer gold mine.

Line Location and

no. orientation

1 Tunnel pit 1, short spread

2 Tunnel pit 1, long spread

3 Perpendicular across pit 1

4 Parallel between pits 1 and 2

6 Tunnel pit 2, short spread

7 Mill site, in furrow parallel to strike

8 Mill site, in furrow perpendicular to strike

9 Mill site, on natural soil, parallel to Line 8

10 Waste dump

11 Waste dump, perpendicular to Line 10

Table 6.4: Summary of surface wave recording and processing parameters at Telfer gold

mine. ∆x and ∆t are the geophone and sample intervals, tmax the time window and

(cmin,cmax) the slant-stack limits used.

Line ∆x ∆t tmax Shot cmin cmax

no. (m) (ms) (ms) plate (m/s) (m/s)

1 0.5 0.25 256 Plastic 260 1283

2 1.0 0.25 512 Plastic 260 1283

3 1.0 0.25 512 Plastic 260 1283

4 0.5 0.5 256 Plastic 260 2307

6 0.5 0.5 512 Plastic 285 1308

7 1.0 0.5 512 Steel 400 3000

8 1.0 0.5 512 Steel 400 3000

9 1.0 0.5 512 Steel 200 2800

10 1.0 0.5 1024 Steel 170 681

11 1.0 0.5 1024 Steel 210 721


−30 −20 −10 0 10 20 30

Spread 1

Spread 2

−23m to 0m

1m to 24m

Shot 1 Shot 2

Shot 3Shot 4

−28m 29m

29m−28m

Move geophones...

Distance (m)

Figure 6.3: Walkaway shooting procedure used at Telfer gold mine to create forward and

reverse 48-channel shot gathers at 1 m spacing.

the tunnel pits, the aim was to compare the effects of artificial compaction versus the

natural stiffness gradient in the upper 4 metres.

Shot gathers for Lines 1 and 2 are shown in Figures 6.4 and 6.5. The data quality is

exceptional with only marginal variations between shot directions. Only in Figure 6.5(b)

is there some evidence for side-scattering arrivals, the sub-horizontal arrivals at times

after the main surface wavetrain. Line 3 crosses tunnel pit 1 (Figure 6.6), the lateral

inhomogeneity evident in the broken surface wavetrain and variations with shot directions.

Line 4 is between the two tunnel pits over in situ weathered sandstone (Figure 6.7).

Lateral inhomogeneity is evident from the poor coherancy of the main surface wavetrain

and forward and reverse shot patterns. Line 6 employed the same geometry as Line 1 and

the waveforms are very similar (Figure 6.8). The minor differences are a slightly faster

group velocity and less dispersion at far offsets, where frequencies appear to be higher.

Mill site Here, an experiment was made into whether removing the topsoil can improve

the reliability of the buried basement rock stiffness estimate. Two furrows were dug by a

front end loader, perpendicular to each other about 1 metre deep and 2 metres wide. The

base of the furrow partially revealed some fractured sandstone and was smoothed back

with about 0.1 m of weathered material to allow geophone planting. Another line was

shot in soil at the natural ground level, perpendicular to Line 7, with channel 1 about 10

metres away from the nearest furrow edge.

Lines 7 and 8 are shown in Figures 6.9 and 6.10. Only the first 256 ms are shown,

exemplifying the high velocity arrivals and strong scattering. A coherant, dispersive sur-

face wavetrain only partially evident as the shingled arrivals at low velocity in the first

half of Figure 6.10(b). Due to the expected high stiffness and low Poisson’s ratio, the P

and surface wavefield will have a small difference in group velocity, which when funnelled

along the spread by the furrow walls causes the planar arrival pattern. Low frequencies are


5 10 15 20 25

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(a)

5 10 15 20 25

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(b)

Figure 6.4: Shot gathers from Line 1 at Telfer gold mine. (a) Forward; and (b) Reverse

(flipped).

10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Offset (m)

Tim

e (s

)

(a)

10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Offset (m)

Tim

e (s

)

(b)


(flipped).


10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Offset (m)

Tim

e (s

)

(a)

10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Offset (m)

Tim

e (s

)

(b)


(flipped).

5 10 15 20 25

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(a)

5 10 15 20 25

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(b)


(flipped).


5 10 15 20 25

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(a)

5 10 15 20 25

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(b)


(flipped).

10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(a)

10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(b)


(flipped).


10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(a)

10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(b)


(flipped).

10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Offset (m)

Tim

e (s

)

(a)

10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Offset (m)

Tim

e (s

)

(b)


(flipped).


10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Offset (m)

Tim

e (s

)

(a)

10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Offset (m)

Tim

e (s

)

(b)


(flipped).

10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Offset (m)

Tim

e (s

)

(a)

10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Offset (m)

Tim

e (s

)

(b)


(flipped).


especially affected, since surface wavelengths are long and are both guided and scattered

by the furrow walls at very short distance from the shot.

Line 9 (Figure 6.11) was emplaced in loose, in situ soil and shows coherant surface

wave arrivals at about 200 m/s. However, there is strong wavefield splitting and higher

mode generation, as expected in a thin, dry waveguide over hard rock. The double arrivals

at later times in Figure 6.10(a) are due to the sledgehammer bouncing.

Waste dump The surface material here was variably compacted by mining vehicle

traffic with fine, surficial material secondary cemented into a clay cap. Pre-hammered

holes were required for geophone spike insertion. Lines 10 and 11 are shown in Figures 6.12

and 6.13. Line 10 shows a definite inverse dispersion, with lower frequencies arriving later

in the ground-roll wavetrain. Line 11 shows similar patterns and both suggest a degree

of lateral discontinuity and/or anisotropy with different forward and reverse wavefield

patterns. However, a ‘mirroring’ effect occurs for the main surface wavetrain in both

lines. In the forward shots it is a convex-up shape, with a large positive amplitude leading

the main surface wave arrivals at near offsets. In the reverse shots it is a concave-down

shape, with a large negative amplitude leading the main surface wave arrivals. This implies

that the geology is quite different at either end of the 48-channel spread, separated by a

discontinuity near the middle.

6.3.3 Dispersion processing The tunnel pit 1 (Lines 1 and 2), mill site (Line 9)

and waste dump (Line 11) data are now the focus for dispersion processing. All data was

downsampled to 0.5 ms where necessary and zero padded to 1024 samples. This provides

a Nyquist frequency of 1000 Hz with frequency interval of 1.95 Hz.

Tunnel pits The f − p transforms using all 48 channels for Lines 1 and 2 are shown in

Figure 6.14. The dispersion is normally dispersive overall, except at very low frequency,

with a jump to a high velocity guided wave mode at 170 Hz. When the dispersion curves

are manually picked (Figure 6.15), the correlation is exceptional indicating good lateral

homogeneity. The maximum frequency for the effective surface wave dispersion is extended

to about 175 Hz, where a gentle rise occurs due to a slightly stiffer surface. Below this

frequency, no jumps to dominant higher modes are evident, the discontinuity at about 40

Hz suspected to be due to side scattering off the buried pit walls. There are slightly larger

undulations in the longer Line 2 dispersion at low frequency, due to side scattering, as was

observed in [299].

Above 170 Hz, the guided wave dispersion minimum phase velocity is about 700 m/s.

In the data, this wavefield can be recognised immediately from the shingled direct arrivals,

which pervade over the entire spreads seen in the AGC gathers of Figure 6.16. A refracted

arrival occurs at about 20 m offset, recognised by the strong negative first breaks in

Figure 6.16(b). The direct arrival downbreaks at near offsets are more difficult to pick

since the vertical component geophones do not respond well to grazing P -waves. If surficial



Fre

quen

cy (

Hz)

(a)

1 1.5 2 2.5 3 3.5

50

100

150

200


requ

ency

(H

z)

(b)

1 1.5 2 2.5 3 3.5

50

100

150

200

Figure 6.14: Frequency-slowness transforms from Telfer gold mine tunnel pit 1 showing

automatically picked dispersion. (a) Line 1 (0.5 m geophone spacing); and (b) Line 2 (0.5

m geophone spacing).

Line 1Line 2

0 50 100 150 200 250200

300

400

500

600

700

800

900

1000

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

(a)

Line 1Line 2

200 400 600 800 1000

0

2

4

6

8

10

12

14

16

18

20

Phase velocity (m/s)

Wav

elen

gth

(m)

(b)

Line 1Line 2

Figure 6.15: Phase velocity dispersion from Telfer gold mine tunnel pit 1. (a) f − c; and

(b) c− λ for Lines 1 and 2.


5 10 15 20 25

0

0.02

0.04

0.06

0.08

0.1

0.12

Offset (m)

Tim

e (s

)

(a)

10 20 30 40 50

0

0.02

0.04

0.06

0.08

0.1

0.12

Offset (m)

Tim

e (s

)

(b)

Figure 6.16: Early time shot gathers from Telfer gold mine tunnel pit 1 with 16 ms AGC

window. (a) Line 1; and (b) Line 2.


Fre

quen

cy (

Hz)

(a)

1 2 3 4 5

50

100

150

200


Fre

quen

cy (

Hz)

(b)

1.5 2 2.5 3 3.5 4 4.5

50

100

150

200

Figure 6.17: Frequency-slowness transforms from Telfer gold mine mill and waste dump

sites showing automatically picked dispersion. (a) Line 9; and (b) Line 11.



Fre

quen

cy (

Hz)

(a)

1 2 3 4 5

50

100

150

200


requ

ency

(H

z)

(b)

2.5 3 3.5 4 4.5

50

100

150

200

Figure 6.18: Processed frequency-slowness transforms from Telfer gold mine mill and waste

dump sites with showing automatically picked dispersion. (a) Line 9; and (b) Line 11.

Line 9 Line 11

0 50 100 150 2000

200

400

600

800

1000

1200

1400

1600

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

(a)

Line 9 Line 11

200 300 400 500 600

0

5

10

15


Wav

elen

gth

(m)

(b)

Line 9 Line 11

Figure 6.19: Phase velocity dispersion from Telfer gold mine mill and waste dump sites.

(a) f − c; and (b) c− λ for Lines 9 and 11.


shear velocity is estimated as 0.9 of the high frequency phase velocity of 300 m/s and

VP assumed as 700 m/s, the surficial Poisson’s ratio is calculated as 0.41.

The f−p transforms using all 48 channels for Lines 9 and 11 are shown in Figure 6.17.

These show strong interference from scattered wavefields and side lobes so some simple

pre-processing and different slant-stack options were trialled to provide smoother images

and allow better dispersion picking. For both lines, spatial windows of channels 1 to 24

and time windows to 256 ms maximum were applied. For In addition, for Line 9 top and

bottom mutes of 1200 m/s and 80 m/s (with 20 m/s cosine tapers) were applied and for

Line 11 an 80 m/s bottom mute (with 20 m/s cosine taper) with reduced upper slant-stack

velocity of 465 m/s were applied. The final f − p images are shown in Figure 6.18, and,

when the dispersion is picked manually, the final curves are plotted in Figure 6.19.

The discontinuities in these dispersion curves are due to dominant higher modes. They

are the best examples observed among the sites investigated and correlate well with the

synthetic modelling examples (Section 2.2.6). The character of the Line 9 dispersion is

very similar to the HVL Case 3 of [325] and of Line 11 very similar to the LVL Case 2 (See

Figures 2.4 and 2.6). These show dominant higher mode(s) confined to a low-frequency

band (Case 3) and gradually dominant higher modes at higher frequencies (Case 2), and

were successfully inverted in Chapter 5.

6.3.4 Inversion models The only synthetic seismogram parameters required to be

estimated for inversion by the PSV method are the source function and centre frequency,

where a causal Berlage wavelet with 45 Hz centre frequency is used. All other parameters

are maintained as per the acquisition and processing, with the dispersion uncertainty

estimated with the noise model of Section 3.6.5, using a factor of 0.5 and maximum of

0.2. Usually 5 iterations, each with 5 trials of damping parameter from 10−1 to 10−6 were

employed. In all cases, Poisson’s ratio and density were maintained for all layers as 0.4

and 1.8 g/cc respectively. The tunnel pit inversion (fundamental mode) is presented first,

followed by the more challenging dispersion of dominant higher modes observed at the

mill site and waste dump (Lines 9 and 11). No information on water table depths was

available, the only details known a priori being the tunnel pit depth of 4 m.

Tunnel pits Inversion of the Line 1 data (forward and reverse shots) by the PSV method

with no a priori information is shown in Figures 6.20 and 6.21. Here, 12 layers with thick-

ness geometrically increasing from 0.5 m to 1 m and a homogenous half-space at 9 m

are used. The same inversion using the FSW method (fundamental mode assumption) is

shown in Figure 6.22. The results are equivalent to about 4 m, showing the stiffer cap of

nearly 400 m/s and softer zone between 1-2 m of 300 m/s. Shear velocity at the base of

the fill (known to be at 4 m) is about 450 m/s in both cases. The PSV method shows an

increasing gradient below this, not exceeding 600 m/s while the FSW method predicts a

shear velocity in excess of 700 m/s. This is the range of stiffnesses which can be expected


0 20 40 60 80 100 120 140 160200

400

600

800

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.10 %


0 200 400 600 800

0

2

4

6

8

10


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

10

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.20: Inversion of the Telfer gold mine tunnel pit Line 1 forward shot dispersion

with the PSV method and modelled dispersion errors. Starting model is automatically

generated, with 12 layers of geometrically increasing thickness from 0.25 m to 1 m and a

homogenous half-space at 9 m.

0 20 40 60 80 100 120 140 160200

400

600

800

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.33 %


0 200 400 600 800

0

2

4

6

8

10


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

10

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.21: Inversion of the Telfer gold mine tunnel pit Line 1 reverse shot dispersion

with the PSV method and modelled dispersion errors.


0 20 40 60 80 100 120 140 160200

400

600

800

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.08 %


0 200 400 600 800

0

2

4

6

8

10


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

10

Percent (%)

Dep

th (

m)

(c)

σβ


with the FSW method and modelled dispersion errors. Starting model is automatically



0 20 40 60 80 100 120 140 160200

400

600

800

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.14 %


0 200 400 600 800

0

1

2

3

4

5


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

1

2

3

4

5

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.23: Shallow inversion of the Telfer gold mine tunnel pit Line 1 forward shot

dispersion with the PSV method and modelled dispersion errors. Starting model is au-

tomatically generated, with 12 layers of geometrically increasing thickness from 0.1 m to

0.5 m and a homogenous half-space at 4 m, the know thickness of the fill.


0 20 40 60 80 100 120200

400

600

800

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.65 %


0 200 400 600 800

0

2

4

6

8

10


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

10

Percent (%)

Dep

th (

m)

(c)

σβ





0 20 40 60 80 100 120 140 160 180 2000

1000

2000

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.31 %


0 200 400 600 800 1000

0

2

4

6

8

10


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

10

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.25: Inversion of the Telfer gold mine mill site Line 9 forward shot dispersion





0 20 40 60 80 100 120 140 160 180 2000

1000

2000

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.29 %


0 200 400 600 800 1000

0

2

4

6

8

10


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

10

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.26: Inversion of the Telfer gold mine mill site Line 9 forward shot dispersion with

the FSW method and modelled dispersion errors.

0 20 40 60 80 100 1200

1000

2000

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.22 %


0 200 400 600 800 1000

0

2

4

6

8

10


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

10

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.27: Inversion of the Telfer gold mine mill site Line 9 reverse shot dispersion with

the PSV method and modelled dispersion errors.


0 20 40 60 80 100 120 140 160 180 2000

1000

2000

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.32 %


0 200 400 600 800 1000

0

2

4

6


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.28: Shallow inversion of the Telfer gold mine mill site Line 9 forward shot disper-

sion with the PSV method and modelled dispersion errors. Starting model is automatically

generated, with 12 layers of constant 0.5 m thickness and a homogenous half-space at 6

m.

0 50 100 150 2000

1000

2000

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.34 %


0 200 400 600 800 1000

0

5

10

15

20

25

30


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

5

10

15

20

25

30

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.29: Deep inversion of the Telfer gold mine mill site Line 9 forward shot dispersion





0 20 40 60 80 100200

250

300

350

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.34 % MeasuredInitial Final

150 200 250 300 350 400 450

0

2

4

6

8

10


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

10

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.30: Inversion of the Telfer gold mine waste dump site Line 11 forward shot

dispersion with the PSV method and modelled dispersion errors. Starting model is auto-

matically generated, with 12 layers of geometrically increasing thickness from 0.25 m to 1

m and a homogenous half-space at 9 m.

0 20 40 60 80 100200

250

300

350

Frequency (Hz)

Pha

se v

eloc

ity (

m/s


150 200 250 300 350 400 450

0

2

4

6

8

10


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

10

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.31: Inversion of the Telfer gold mine waste dump site Line 11 forward shot

dispersion with the FSW method and modelled dispersion errors.


0 20 40 60 80 100200

250

300

350

Frequency (Hz)

Pha

se v

eloc

ity (

m/s


150 200 250 300 350 400 450

0

2

4

6


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.32: Shallow inversion of the Telfer gold mine waste dump site Line 11 forward

shot dispersion with the PSV method and modelled dispersion errors. Starting model is

automatically generated, with 12 layers of constant 0.5 m thickness and a homogenous

half-space at 6 m.

0 20 40 60 80 100200

250

300

350

Frequency (Hz)

Pha

se v

eloc

ity (

m/s


150 200 250 300 350 400 450

0

5

10

15


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.33: Deep inversion of the Telfer gold mine waste dump site Line 11 forward


automatically generated, with 12 layers of geometrically increasing thickness from 0.25 m

to 2 m and a homogenous half-space at 12 m.


0 10 20 30 40 50 60 70 80 90

200

300

400

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.21 %


150 200 250 300 350 400 450

0

2

4

6

8

10


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

10

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.34: Inversion of the Telfer gold mine mine waste dump site Line 11 reverse


automatically generated, with 12 layers of geometrically increasing thickness from 0.25 m

to 1 m and a homogenous half-space at 9 m.

10 20 30 40 50

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Offset (m)

Tim

e (s

)

(a)

10 20 30 40 50

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Offset (m)

Tim

e (s

)

(b)

Figure 6.35: Data from the Telfer gold mine waste dump Line 11 with a 32 ms AGC window

showing direct and refracted arrivals. (a) Forward; and (b) Reciprocal shot (flipped).


in the weathered to fresh transition of the sandstone.

To better constrain the fill material, an inversion setting the homogenous half-space at 4

m with much finer overlying layers (12 from 0.1 m to 0.5 m thick) is shown in Figure 6.23.

The soft horizon is modelled around 0.5-1.5 m depth with shear velocity around 250

m/s. For this data, the measured seismograms were excellent quality and would be good

candidates for attempting a waveform match with synthetic shot gathers modelled from

the inversion results. Inversion of the Line 2 data using the same layering assumptions

as for Line 1 is shown in Figure 6.24. While the final shear velocity model is similar,

the RMS fit is poorer than Line 1. The dispersion anomaly at 40 Hz due to scattered

wavefields as well as a higher lower cutoff frequency greatly affect the convergence. With

shorter spreads, profiling homogenous fill with surface wave inversion is a useful method

for detecting vertical compaction variations.

Mill site (HVL) Inversion of the Line 9 data by the PSV method with no a pri-

ori information is shown in Figure 6.25. Here, 12 layers with geometrically increasing

thicknesses from 0.25 m to 1 m with a homogenous half-space at 9 m are used. The 30-60

Hz range is possibly influenced by more than one higher mode and the return to the fun-

damental mode is less abrupt, but occurs by 80 Hz. The PSV method does not require

mode identification and has properly modelled the mode jump at 30 Hz, followed by a

smooth transition back to the fundamental. The large shear velocity contrast to 700 m/s

at about 2 m depth. This is interpreted to be sandstone, of which a rubbly, weathered

portion was revealed in the nearby furrows dug for Lines 7 and 8, less than 50 m away.

In the overburden, there is some suggestion of a slightly stiffer cap but the average shear

velocity is about 250 m/s. At depth, a LVL is interpreted, but large scatter in intermediate

models at depths below 4 m, along with large standard deviations in shear velocity, are

not encouraging for its correctness.

When the FSW method is used (Figure 6.26), the stiff interface is interpreted but the

dispersion fit at low frequency is poor. The fundamental mode can only be ascertained

above about 80 Hz thus there is a systematic error in the optimisation and the shear

velocity of the stiff layer appears to be an overestimate.

The dispersion from the reverse shot is markedly different, however an inversion em-

ploying the same parameters is shown in Figure 6.27. Dominant higher mode(s) pervade

between about 35-55 Hz but are still modelled by a structure with a large shear velocity

contrast at about 1.5-2 m depth. In this case, the shear velocity is only 550 m/s with a

gentle gradient up 700 m/s, with no indication of a LVL at depth. Part of the difference

may be since only channels 1 to 24 were used of the 48-channel walkaway shot gathers,

the forward and reverse shot ‘sounding points’ (spread centres) are actually 24 m apart.

Two more inversions of the Line 9 forward shot dispersion with different layer param-

eterisations and/or frequency bands are shown in Figures 6.28 (shallow) and 6.29 (deep).


Note that layers deeper than about 4-5 m are actually outside the realistic maximum in-

vestigation depth, since the 23 m spread has a maximum resolvable wavelength of only

9.2 m. In the shallow inversion (Figure 6.28), layers of constant 0.5 m thickness provided

better convergence than geometrically increasing thicknesses starting from 0.1 m. Here,

the overburden is interpreted as a constant 250 m/s with the rise to 650 m/s at 2 m depth

over a steep gradient, due to the smearing effect which occurs between layers with a large

contrast in stiffness. The slight velocity reversal below 2 m may be real, a result of the

upper weathered sandstone becoming secondary cemented. In the deep inversion (Fig-

ure 6.29), when a homogenous half-space at 24 m depth is used, the shear velocity of the

sandstone is again in the order of 700 m/s. While there is again a slight velocity reversal

at depth, it is more likely a gentle gradient to fresh sandstone by about 8-10 m depth.

However, this type ‘heuristic interpretation’ will be appraised by Monte Carlo analysis in

Chapter 7.

Waste dump (LVL) Inversion of the Line 11 data by the PSV method with no a

priori information is shown in Figure 6.30. As per Line 9, 12 layers with geometrically

increasing thicknesses from 0.25 m to 1 m with a homogenous half-space at 9 m are

used. Although the abrupt jumps to dominant higher modes are not exactly modelled,

the resulting profile shows a thick LVL with minimum shear velocity of 250 m/s under

a higher velocity cap of 300 m/s. At depth, the maximum shear velocity of 350 m/s

does not suggest fresh sandstone, probably weathered basement or more likely the natural

compaction gradient of the fill. The same inversion by the FSW method (Figure 6.31)

shows a gentle LVL of low contrast with the surficial material. An interesting point is

the ‘apparent’ mode jump in the final dispersion curve at about 90 Hz, which is actually

a numerical artefact. It arises from the closest approach of the fundamental and first

higher modes near an osculation point being less than the initial step size chosen for root

bracketing, causing a higher mode root to be found.

Two more inversions of the Line 11 forward shot dispersion with different layer param-

eterisations are shown in Figures 6.32 (shallow) and 6.33 (deep). The shallow inversion

shows an almost homogenous cap of shear velocity 300 m/s and 2 m thick, underlain by

a 4 m thick LVL of minimum 200 m/s. This correlates quite well with Figure 6.30, but

interpreting a sharper contrast at the base of the LVL. This sharp contrast is supported by

the deep inversion, however, the shear velocity gradient in the near surface is more gentle

with depth to a minimum of 225 m/s at 6 m. A half-space shear velocity of 400 m/s is,

however, more suggestive of sandstone (or the weathering front) rather than compacted

fill material.

An inversion of the reverse shot data is shown in Figure 6.34. The upper 8 m is

markedly different to the forward shot (Figure 6.30), with shear velocities between 50-100

m/s lower than the equivalent depths at the other ‘sounding point’, only 24 m away. The


absence of a markedly stiffer cap and overall normally dispersive trend was expected since

no dominant higher modes were generated. At depth, however, the interpretation is similar

to the forward shot. This suggests that the shallow compaction and/or fill lithology here

is highly variable.

The lateral variability at Line 11 is also evident in the different wavefields between the

the first 24-channels of the forward and reverse shots (Figure 6.35). The early time data

both not show prominent direct arrivals at near offsets, at the assumed P -wave velocity

of about 700 m/s. This is expected, since the vertical component geophones should not

respond well to horizontally propagating compressional waves. However, refracted arrivals

can be seen in the reverse shot (Figure 6.35(b)) from offsets 10-30 m and times 0.04-0.10 s.

A second first arrival break possibly occurs at 45 m offset at a time of 0.11 s. The presence

of refractions supports the increasing velocity with depth interpreted in Figure 6.34 and

it was merely fortunate that the discontinuity occurred around at the midpoint of the 48-

channel spread, with minimal wavefield scattering. This allowed each half to be interpreted

separately, remembering that each represents an average of the propagation path. These

models will be thoroughly appraised in the inference tests of Chapter 7.



6.4.1 Data acquisition See Section 4.3.1 for a description of the data acquisition

equipment and methods, as well as site conditions. In addition, trace normalised walkaway

shot gathers are shown in Figure 4.17.

6.4.2 Dispersion processing An example of an f−p transform for the 56-channel

(1 m geophone spacing) and 24-channel (3 m spacing) walkaway gathers is shown in

Figure 6.36. The upper slant-stack velocity is 1133 m/s and the discontinuity observed

at low frequency in the 56-channel data was suspected to be a dominant higher mode,

similar to the Tokimatsu et. al. Case 3. However, it does not exist in the 24-channel data

and is thus assumed to be from an undesired wavefield. Repeating the slant-stack with

upper phase velocity limit of 232 m/s (Figure 6.37) neglects this mode, which is at the

lower frequency resolution limit of both spreads. Strong side lobes are evident and the

24-channel dispersion shows discontinuities at high frequency due to these effects. They

are caused by Gibbs phenomenon, the ‘ringing’ in wavenumber caused by data truncation

in the spatial dimension. However, application of an offset window with cosine tapers to

the first and last 5 metres of the spread did suppress these. In any case, the alias slowness

is attained by about 50 Hz, thus this was chosen as the upper cutoff frequency.

The input dispersion curves for inversion of the two walkaway datasets were taken as

the median of the scatter of a Monte Carlo test, which randomly combined the individual

shot gathers (5 at each shot offset) for concatenation. Differences between individual shot

gathers arose due to cultural noise (vehicle traffic) and variations in source pulse and en-

ergy. These propagated through the plane wave processing (essentially calculating phase

differences) to corrupt the dispersion curves, the distribution quantified in Section 4.3.3.

However, the low-frequency repeatability was much better for the 24-channel data (69 m

spread length), opposed to the shorter 55 m array. The dispersion curves and standard

deviation of repeatability processed by f − p in Figures 4.19 and 4.20 are repeated here

in Figures 6.38 and 6.39, but with f − k transform observation. The two methods are

essentially equivalent, the important point to note here is in the 24-channel dispersion

(Figure 6.39(a)), where at low frequency an increase in phase velocity is predicted. This

correlates with the 56-channel low-frequency dispersion but not with the single transform

image of Figure 6.37(b). This is a good reason to stack several shots, especially in walk-

away shooting where the geophone array remains fixed, to ensure a consistent and better

representative dispersion curve for the site.

6.4.3 Inversion models Utilising the nearby borehole information nearby, two

classes of layer parameterisation can be trialled:

1. ‘Borehole inversion’ - employing lithological interface depths identified from the bore-

holes; or



Fre

quen

cy (

Hz)

(a)

2 4 6 8

5

10

15

20

25

30

35

40

45


Fre

quen

cy (

Hz)

(b)

2 4 6 8

5

10

15

20

25

30

35

40

45

Figure 6.36: Example frequency-slowness transforms from Perth Convention Centre show-

ing automatically picked dispersion. (a) 56-channel (1 m geophone spacing); and (b)

24-channel (3 m geophone spacing) walkaway shot gathers.


Fre

quen

cy (

Hz)

(a)

5 6 7 8 9

5

10

15

20

25

30

35

40

45


Fre

quen

cy (

Hz)

(b)

5 6 7 8 9 10

5

10

15

20

25

30

35

40

45

Figure 6.37: Frequency-slowness transforms from Perth Convention Centre with upper

slant stack velocity reduced. (a) 56-channel (1 m geophone spacing); and (b) 24-channel

(3 m geophone spacing) walkaway shot gathers.


50

100

150

200

250

300

350

400

c(m

/s)

(a) West walkaway, 1m spacing, 56 channels by f−k

Mean Median(with σ

c)

0 10 20 30 40 50 60 70 80 90 10010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)




vention Centre by f − k. (a) Dispersion curves; and (b) Repeatability envelopes.

50

100

150

200

250

300

350

400

c(m

/s)

(a) West walkaway, 3m spacing, 24 channels by f−k

Mean Median(with σ

c)

0 5 10 15 20 25 30 35 40 45 5010

−1

100

101

102

Frequency (Hz)

σ ∆ c (

%)




vention Centre by f − k. (a) Dispersion curves; and (b) Repeatability envelopes.


0 5 10 15 20 25 30 35 40 45

100

200

300

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 1.24 %


0 100 200 300 400 500

0

10

20

30

40


Dep

th (

m)

(b)


100

101

102

0

10

20

30

40

Percent (%)

Dep

th (

m)

(c)

σβ∆β

Figure 6.40: Borehole inversion of the Perth Convention Centre 1 m geophone spacing, by

PSV with dispersion errors taken from repeated walkaway shooting analysis.

0 5 10 15 20 25 30 35 40 45100

200

300

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 4.82 %


0 100 200 300 400 500

0

10

20

30

40


Dep

th (

m)

(b)


100

101

102

0

10

20

30

40

Percent (%)

Dep

th (

m)

(c)

σβ∆β


PSV with dispersion errors taken from repeated walkaway shooting analysis.


0 5 10 15 20 25 30 35 40 45100

200

300

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 22.04 %


0 100 200 300 400 500

0

10

20

30

40


Dep

th (

m)

(b)


100

101

102

0

10

20

30

40

Percent (%)

Dep

th (

m)

(c)

σβ∆β


FSW with dispersion errors taken from repeated walkaway shooting analysis.

0 5 10 15 20 25 30 35 40 45100

200

300

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 2.03 %


0 100 200 300 400 500

0

10

20

30

40


Dep

th (

m)

(b)


100

101

102

0

10

20

30

40

Percent (%)

Dep

th (

m)

(c)

σβ∆β

Figure 6.43: Blind inversion of the Perth Convention Centre 1 m geophone spacing, by

PSV with dispersion errors taken from repeated walkaway shooting analysis. Constant

thickness layers of 1 m are assumed with basement at 36 m.


0 5 10 15 20 25 30 35 40 45100

200

300

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 4.48 %


0 100 200 300 400 500

0

10

20

30

40


Dep

th (

m)

(b)


100

101

102

0

10

20

30

40

Percent (%)

Dep

th (

m)

(c)

σβ∆β

Figure 6.44: Blind inversion of the Perth Convention Centre 3 m geophone spacing, by

PSV with dispersion errors taken from repeated walkaway shooting analysis. Constant

thickness layers of 1 m are assumed with basement at 36 m.

2. ‘Blind inversion’ - using layer interfaces with no a priori information.

A borehole inversion based on a shear velocity log model also allows quantitative error

analysis. For all inversions, a 40 Hz causal wavelet from a surface impact was used as

the source function, with low and high frequency cosine tapers from 2 Hz and 125 Hz.

Otherwise, all acquisition parameters employed matched those used in the field, albeit for

single common shot gathers, not repeated shot walkaways. The dispersion uncertainty

was taken as that observed by Monte Carlo tests of the walkaway shooting, illustrated in

Figures 6.38 and 6.39.

Borehole PSV inversions of both 56- and 24-channel dispersion curves are presented

in Figures 6.40 and 6.41. The only liberty taken is subdividing the 10.1 m thick clay layer

at 3.9 m depth into 3 sub-layers, to allow more flexibility in the inversion. Inversion of the

1 m spacing data (Figure 6.40) provides good fit to the observed dispersion above about 5

Hz, but the final model does not match that of the true model, derived from the VS data

of the SCPT log, SC2. The only similarity is the trend around the stiff layer at 2 m depth,

albeit with a level difference of over 50%.

However, the 3 m spacing dispersion inverted over the same bandwidth provides much

better results (6.41). Again, the trend with depth in the upper 15 m is interpreted,

but with the same positive shift of around 50%. A possible reason is since the SCPT

log measures arrival times at higher strains (during cone insertion), the observed shear

velocities will be lower than surface seismic results. However, it was also noticed that there


0 5 10 15 20 25 30 35 40 4580

100

120

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 1.55 %


0 100 200 300 400 500

0

10

20

30

40


Dep

th (

m)

(b)


100

101

102

0

10

20

30

40

Percent (%)

Dep

th (

m)

(c)

σβ∆β

Figure 6.45: Borehole inversion of a synthetic dispersion curve based on the shear velocity

model of SC2 at the Perth Convention Centre, with a 56-channel spread at 1 m geophone

spacing and near offset.

0 5 10 15 20 25 30 35 40 4580

100

120

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 2.27 %


0 100 200 300 400 500

0

10

20

30

40


Dep

th (

m)

(b)


100

101

102

0

10

20

30

40

Percent (%)

Dep

th (

m)

(c)

σβ∆β

Figure 6.46: Borehole inversion of a synthetic dispersion curve based on the shear velocity

model of SC2 at the Perth Convention Centre, with a 24-channel spread at 3 m geophone

spacing and near offset.


was a large difference between the two arrival times measured by alternate left and right

impacts of the surface SCPT source. While a mean was taken as the final traveltime, this

statistic may not have been correct. In any case, the surface waves are low strain, averaged

over a broader propagation path, thus provide an absolute maximum of in situ stiffness.

Inversion of the same data by the FSW method is shown in Figure 6.42. Although the site

appears overall normally dispersive, the inversion diverged resulting in an invalid solution.

A blind inversion employed 36 layers of 1 m thickness. This (nearly) coincides with

the true half-space depth of 35.9 m, which would not normally be known but was chosen

to better illustrate solutions for the shallower layers. The inversion results of both the 1

m and 3 m spacing geophone dispersion are shown in Figures 6.43 and 6.44. With the 1

m spacing data (Figure 6.43), a similar trend in the upper 10 m is recovered, below which

the solution is erratic. The 3 m spacing data (Figure 6.44) is more accurate to about 15

m, but with a monotonically increasing shear velocity gradient at depth.

6.4.4 Resolution limitations None of the inversion solution show clear detection

of the stiff silty-sand layer at 20 m depth. One explanation is that since the borehole

was over 20 m away from the surface wave spread centre, it is possible that this layer has

pinched out or changed facies. However, it was intersected in nearly every borehole over

the entire site so this is unlikely, and the nearly constant intersection depth excludes any

possible lateral discontinuity effects due to the dip of this layer. It is more likely that

a resolution limitation of the method has prevented detection of this layer. One way to

test this is through synthetic modelling of the borehole data. Indeed, this should be done

prior to the fieldwork as an initial sensitivity analysis. A synthetic dispersion curve for

a model based on the SC2 shear velocity model (with siltstone assumed as 300 m/s) was

created by the PSV method, then inverted using the same forward engine, with modelled

dispersion errors and the true layering. This process was applied for both 56- and 24-

channel spreads, at 1 m and 3 m geophone spacings respectively

The synthetic inversion results are shown in Figures 6.45 and 6.46. In both cases, the

shallow zone is inverted to high accuracy. However, even with an underestimate assumed

for the ‘experimental’ dispersion uncertainty (in the case of the 1 m spacing data only),

the inversion fails to recover the true thickness and stiffness of the stiff layer at 20 m

depth. Instead, a gradation to a homogenous half-space of 225 m/s is interpreted. With

the 3 m spacing data and realistic dispersion errors, a HVL is interpreted at the correct

depth, but with both thickness and shear velocity less than half the true value, with the

very deep layer and half-space shear velocities also underestimated. Note too how jumps

to dominant higher modes are more distinct with this acquisition layout.

The only consolation of these tests is that the method can interpret the approximate

depth to the top of an interbedded horizon, whose stiffness is larger than the clays. With

this information, possible zones of shallow pile refusal could be predicted beforehand and


the piling designed to accommodate a load supported by the sand layer. Alternatively,

other piling methods employed to ensure penetration to the deeper, stiffer siltstone could

be considered. Further directed grid search tests on the limitations of detecting a buried

stiff layer around an extended range of depth-thickness-stiffness tradeoffs expected for the

silty-sand layer are analysed in Chapter 7.

6.5. Road cutting 293

6.5 Road cutting

6.5.1 Data acquisition A 24-bit Seistronix RAS-24 exploration seismograph was

used for acquisition. Acquisition settings and data storage are done via the Windows 98

interface on a laptop. Geophones were 8 Hz vertical component, with 2x12-channel cables

at 5 m takeout spacings. The geophone spacing used throughout was 1 m. The source was

a 8 kg sledge hammer on UHMW plate, triggered by a geophone next to the plate. Since

the data were also to be used for P -wave refraction analysis, two procedures were used

at each spread. For refraction, shots at 1 m and then 25 m from the nearest geophone

were used. This was similar to the walkaway shooting employed at the Perth Convention

Centre, where the geophones remained planted. Since only first arrivals were desired, the

time window was 64 ms with 0.25 ms sample interval (256 samples). For the surface wave

data, reciprocal near offsets of 5 m were used with a time window of 511 ms at 1 ms sample

interval (512 samples).

The cutting surveyed was over a laterite plateau, the seismic lines at most about 50

m north of the present highway. Vegetation was very dense, however the line chosen was

along a corridor mostly cleared by the drill trucks. Nevertheless, this and the gravelly

surface demanded extra care in geophone planting and up to 20 cm placement errors often

occurred due to obstructions. While the often poorly coupled geophones were mostly

protected from the wind, low-frequency noise was generated by the roots of large trees,

even in light winds. In addition, traffic noise was observed, but shooting was suspended in

periods of heavy trucks passing. Five points were surveyed, from west to east, with spread

centres positioned as close as possible to the borehole positions. This was completed in

one day with three people, although a rate of up to 12 points per day has been achieved.

6.5.2 Dispersion processing Two shot gathers are shown in Figure 6.47, both

from reverse shots off the eastern end of each spread. The wavefield in Line 1 (Fig-

ure 6.47(a)) shows strong surface waves with some evidence for side- or back-scattering

at late times. The character of the dispersion is difficult to judge, but in Line 5 (Fig-

ure 6.47(b)) it is very clearly inversely dispersive. There, the early surface wave arrivals

are markedly higher frequency than later arrivals, suggesting a stiff near surface, which

is expected in laterite caprock terrane. The plane wave transforms (Figure 6.48) clearly

show the different dispersion patterns between the two sites, the picked dispersion curves

shown together in Figure 6.49. Line 1 (Figure 6.48(a)) is overall normally dispersive (phase

velocity decreasing with frequency), but with some irregularly dispersive effects at high

frequency. Line 5 (Figure 6.48(b)) is clearly inversely dispersive, with undesired wavefields

dominating above 140 Hz.

Both appear affected by low-frequency effects, more so for Line 1, where he effect of

spread length on low-frequency resolution is particularly important in high phase velocity

environments due to large surface wavelengths. For example, the phase velocity at 40 Hz


5 10 15 20 25 30

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(a)

5 10 15 20 25 30

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(b)

Figure 6.47: Trace normalised shot gathers from the road cutting site. (a) Line 1 (BH9);

and (b) Line 5 (SAB23).


Fre

quen

cy (

Hz)

(a)

0.5 1 1.5 2 2.5 3

20

40

60

80

100

120

140

160

180


Fre

quen

cy (

Hz)

(b)

0.5 1 1.5 2 2.5 3 3.5 4

20

40

60

80

100

120

140

160

180

Figure 6.48: Frequency-slowness transforms from the road cutting site showing picked

dispersion. (a) Line 1 (BH9); and (b) Line 5 (SAB23).


Line 1Line 5

0 50 100 150 2000

500

1000

1500

2000

2500

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)(a)

Line 1Line 5

200 400 600 800 1000

0

5

10

15

20


Wav

elen

gth

(m)

(b)

Line 1Line 5

Figure 6.49: Phase velocity dispersion from the road cutting site. (a) f − c; and (b) c− λ

for Lines 1 and 5.

0 20 40 60 80 100 120 140 1600

1000

2000

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.39 %


0 500 1000 1500

0

2

4

6

8

Sandy gravel

Silty sand

Saprolite

Granite EOH


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.50: Inversion of the road cutting site Line 1 (BH9) dispersion with the

PSV method and modelled dispersion errors. Starting model has the depth interfaces

identified in BH9.


0 20 40 60 80 100 120 140 1600

1000

2000

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.56 %


0 500 1000 1500

0

2

4

6

8

Sandy gravel

Silty sand

Saprolite

Granite EOH


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

Percent (%)

Dep

th (

m)

(c)

σβ



identified in BH9, with the surficial sandy gravel horizon subdivided into 3 layers.

0 20 40 60 80 100 120 140 1600

1000

2000

3000

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.99 %


0 500 1000 1500

0

2

4

6

8

Sandy gravel

Silty sand

Saprolite

Granite EOH


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

Percent (%)

Dep

th (

m)

(c)

σβ


PSV method and modelled dispersion errors. Starting model has 13 layers, all 0.5 m

thick and a homogenous half-space at 6.5 m, the nearby depth to granite.


0 20 40 60 80 100 1200

500

1000

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.18 %


0 200 400 600 800

0

2

4

6

8

Laterite rock

Silt

Clay

EOH


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.53: Inversion of the road cutting site Line 5 (SAB23) dispersion with the


identified in SAB23.

0 20 40 60 80 100 1200

500

1000

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.48 %


0 200 400 600 800

0

2

4

6

8

Laterite

Silt

Clay

EOH


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.54: Inversion of the Road cutting site Line 5 (SAB23) dispersion with the


identified in SAB23, with the surficial laterite horizon subdivided into 3 layers.


0 20 40 60 80 100 1200

500

1000

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.42 %


0 200 400 600 800

0

2

4

6

8

Laterite

Silt

Clay

EOH


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

Percent (%)

Dep

th (

m)

(c)

σβ


PSV method and modelled dispersion errors. Starting model has 15 layers, all 0.5 m

thick and a homogenous half-space at 7.5 m.

39100 39200 39300 39400 39500 39600 39700 39800285

290

295

300

305

310

Chainage (m)

Red

uced

leve

l AM

SL

(m)

ExistingDesign

500 1000

0

2

4

6

8

Sandy gravel

Silty sand

Saprolite

Granite EOH

VS (m/s)

BH9

500 1000

0

2

4

6

8

10

Gravel

Laterite

Silty sand

Silty clay

EOH

VS (m/s)

SAB9

600 700 800

0

5

10

15

Gravel

Laterite

Clayey gravel

Silty sand

Silty gravel

EOH

VS (m/s)

SAB14

400 600 800

0

2

4

6

8

10

12

GravelLaterite

Gravel

Gravelley sand

Clayey silt

EOH

VS (m/s)

SAB18

400 600 800

0

2

4

6

8

Laterite

Silt

Clay

EOH

VS (m/s)

SAB23

Figure 6.56: Road cutting site cross section showing the PSV inversions overlain on nearby

borehole lithologies.


(2000 m/s) corresponds with a surface wavelength of about 50 m. However, with a 23

m spread length, the maximum measurable wavelength is only 9.2 m (according to the

λmax = 0.4L rule). In terms of frequency, this wavelength correlates with about 50 Hz,

where a phase velocity of 500 m/s gives a surface wavelength of 10 m. In Line 5, this is less

a problem, since at the lowest measured phase velocity (320 m/s) the frequency is about

16 Hz, correlating to a surface wavelength of 20 m. However, this is unchanging with

increasing frequency, thus a low-frequency limit needs to be more heuristically chosen.

Since the PSV modelling incorporates this effect, it is less of a concern than when using

the FSW method in the inversion.

While not shown, the first arrivals picked from AGC refraction shot gathers were 1100

m/s (direct) and 4500 m/s (refracted) in Line 1 and 1850 m/s (direct only) in Line 5. The

direct waves were difficult to pick, since in stiff surficial material, the angle of emergence

of diving waves is very low and are not well detected by vertical component geophones.

The refracted arrivals, only detected in Line 1, were much sharper. Modelling and later

field tests showed conclusively that horizontal component direct wave first breaks are

much sharper than the vertical component. This ensures more accurate P -wave velocity

estimation of the near surface caprock which is important for conventional rippability

estimates.

6.5.3 Inversion models Like the Perth Convention Centre site, both borehole and

blind inversions were performed. In the borehole inversions, Poisson’s ratio is set at 0.3

for laterite and rock and 0.4 for unconsolidated material. Density is assumed as 1.8 g/cc.

A borehole inversion for Line 1 is shown in Figure 6.50. While the low-frequency fit

is poor, it shows some correlation with geotechnical expectation. When the uppermost

layer is subdivided into 3 layers, each 0.9 m thick, the results are shown in Figure 6.51.

While the basement velocity is similar, the overburden structure has changed appreciably.

A blind inversion is shown in Figure 6.52. Here 13 layers each 0.5 m thick are used,

with the basement depth has been set as identified in BH9 at 6.5 m, so is not entirely

‘blind’. Nevertheless, the overburden structure is again markedly different. Such layer

parameterisation dependent results are not supportive of an accurate inversion and this

would be an ideal candidate for Monte Carlo model appraisal (Chapter 7).

However, Line 5 shows good correlation between the raw (Figure 6.53) and subdivided

laterite layer (Figure 6.54) inversions. A blind inversion with homogenous half-space fixed

at the EOH depth of 7.5 m is shown in Figure 6.55. Albeit the smoothing effects, a similar

trend is observed. With this information, the laterite can be concluded to have a minimum

shear velocity of 600 m/s, thus the depth of the base of the laterite can be assumed as the

point where this is exceeded.

A cross section of all points surveyed is shown in Figure 6.56. In general, the ‘borehole’

inversions are shown, where the recovered shear velocities can be used as indicators for


each horizon. On average, the laterite shear velocity is in the order of 750 m/s. The data

at Line 4 (SAB18) revealed some difficulties, which are discussed in Section 6.7.



6.6.1 Data acquisition

Test data The test array comprised common shot gathers (CSG) of 45 channels be-

tween SP1051 and SP1095 with source at SP1050.5. See Section 4.2.1 for a description

of the data acquisition equipment and methods, as well as site conditions and example

shot gathers, for vertical, inline and transverse components. Data from the full recording

window with trace normalisation to enhance surface waves is in Figures 4.2 to 4.2, which

showed the fundamental mode wavefields to be very similar, with minor differences in the

higher mode surface wavefield.

Early time waveforms with AGC of the vertical and inline 28 Hz geophone components

and ‘big’ shotgun shells are shown in Figure 6.57. The first breaks of the vertical com-

ponent suggest there is a refractor, possibly with a vertical velocity gradient. There is a

definite later arrival of these direct waves, which is interpreted as a possible trigger error.

However, this problem is not a concern for surface wave analysis, as long as it is a constant

shift to all traces, since plane wave transform is unaffected by time translation. This is

another advantage over refraction and reflection methods where trigger delays must be

negligible or known exactly. Notice how the first breaks are of opposite polarity for the

horizontal component, due to the geophone base pointing away from the source. Strong

guided waves are not evident in the raw data and at later times the wavefields differ,

possibly due to varying amounts of SV -wave contribution.

Rollalong data Both the P - and S-wave data were collected in rollalong fashion to

allow high-fold, common midpoint (CMP) reflection processing. The P -wave data source

was the shotgun with ‘big’ shells, in 100 mm diameter augered holes, on average 0.3 m

deep, the line of which was about 0.5 m north of the geophone line. The S-wave source

was the shear box struck from both the north and south, inserted over the open auger

holes. Transverse component records were saved individually to allow later stacking for

Love wave removal. While transverse components showed strong Love waves, the focus of

this work is purely on the vertical component P -wave gathers.

For recording P -wave data, the spread was first fixed from SP1001 to SP1096 (96

channels at 1 m spacing), with shooting starting at SP1005. Part of the reason for this

was an impenetrable laterite horizon, under a thin veneer of windblown sand and clay,

which prevented shothole augering. Indeed, shotpoints up to about SP1010 were less than

the average 0.3 m depth due to this laterite. Shooting continued in increasing shotpoint

direction (west) at 1 m intervals until SP1097 was reached, known as ‘shooting through’

the spread. Beyond that point, rollalong of the active channels was required, ‘pulling’ the

spread in the ‘end-on’ (or ‘off-end’) geometry. SP1156 was the last shotpoint and for that

spread (SP1060 to SP1155), a 15 m offset shot was also recorded (SP1170), along with a

1 m and 15 m offset shot at the reverse end.


For the S-wave data, the spread was first fixed from SP1086 to SP1155 (70 channels

at 1 m spacing), with shooting starting at SP1156. Shooting continued in decreasing

shotpoint direction (east) at 1 m intervals, ‘pushing’ the spread in end-on geometry until

until SP1096. There, the spread stayed fixed between SP1026 and SP1095 and was ‘shot

through’ to SP1025, where again, far offsets of 15 m were shot from both ends. At SP1010,

the shear box legs only penetrated to about 0.1 m, resting on the hard laterite horizon.

All cases, channel 1 was always at the low shotpoint (eastern) end of the active spread.

The maximum fold (F ) of each dataset in this configuration is calculated by [232]

F = 0.5

(

Nx∆x

∆s

)

(6.1)

where Nx is the number of active channels and ∆s and ∆x are the shot and geophone

spacings respectively. Maximum fold only occurs where a particular shotpoint has been

recorded by each and every channel and results in a nominal 48-fold for the P -wave

data and 35-fold for the S-wave data. During ‘shooting through’, fold builds up linearly,

maximum fold attained at the middle of the fixed spread.

6.6.2 Dispersion processing

Test data The f − p transforms using all 45 channels for both vertical and inline

components are shown in Figure 6.58. The velocity ranges for the slant stack of each were

220-731 m/s and 180-691 m/s respectively. At high frequency, there is much interference

from side-lobes, which are FFT artefacts due to abrupt truncation in the x-direction. To

suppress these, a cosine taper to the first and last 5 traces is applied. This also has the

effect of increasing the near offset by allowing full trace amplitude only after 4.5 m and the

resulting transforms shown Figure 6.59. Here, the discontinuity at about 115 Hz appears

to be a jump to a dominant high mode and not a side lobe effect. The manually picked

dispersion curves are shown in Figure 6.60. The large peak at about 35-50 Hz may be an

effect of scattering from lateral discontinuity and was more noticeable in the 8 Hz data

with surface impact shots. One marked difference between the two components is the

discontinuity in the inline dispersion at 50 Hz. A similar jump to a higher dominant mode

in inline component dispersion was also noticed in the synthetic modelling of Figure 2.27.

However, at low frequency there is merely a smooth, inversely dispersive trend.

The dominant higher mode appears to be a surface wave mode, not a guided wave

since it does not attain the P -wave velocity of the near surface, measured from the direct

wave arrivals of Figure 6.57 as 1000 m/s. The higher surface wave mode is also evident

in the trace normalised gathers of Figures 4.2 and 4.3, by the large degree of wavefield

splitting at far offsets. This type of large discontinuity in the dispersion curve is further

discussed with other examples in . However, it will not be included in the input data for

inversion at this site.


0 10 20 30 40

0

0.01

0.02

0.03

0.04

0.05

0.06

Offset (m)

Tim

e (s

)

(a)

0 10 20 30 40

0

0.01

0.02

0.03

0.04

0.05

0.06

Offset (m)

Tim

e (s

)

(b)

Figure 6.57: Early time data with the ‘big’ shotgun shells from the Hyden fault scarp test

array with a 16 ms AGC window. (a) Vertical; and (b) Inline component.


Fre

quen

cy (

Hz)

(a)

1.5 2 2.5 3 3.5 4 4.5

20

40

60

80

100

120

140

160

180


Fre

quen

cy (

Hz)

(b)

2 3 4 5

20

40

60

80

100

120

140

160

180

Figure 6.58: Frequency-slowness transforms from the Hyden fault scarp test array showing

automatically picked dispersion. (a) Vertical; and (b) Inline component 28 Hz geophones.



Fre

quen

cy (

Hz)

(a)

1.5 2 2.5 3 3.5 4 4.5

20

40

60

80

100

120

140

160

180


Fre

quen

cy (

Hz)

(b)

2 3 4 5

20

40

60

80

100

120

140

160

180

Figure 6.59: Processed frequency-slowness transforms from the Hyden fault scarp test

array, with cosine x-taper windows applied to the first and last 5 traces, showing auto-

matically picked dispersion. (a) Vertical; and (b) Inline component 28 Hz geophones.

VerticalInline

0 50 100 150 200150

200

250

300

350

400

450

500

550

600

650

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

(a)

VerticalInline

200 300 400 500 600 700

0

2

4

6

8

10

12

14

16

18

20


Wav

elen

gth

(m)

(b)

VerticalInline

Figure 6.60: Phase velocity dispersion from the Hyden fault scarp test array. (a) f − c;

and (b) c− λ for vertical and inline component 28 Hz geophones.


k (/m)f (

Hz)

(a) SP1061

0.1 0.2 0.3 0.4 0.5

20

40

60

80

100

k (/m)

(b) SP1062

0.1 0.2 0.3 0.4 0.5

k (/m)

(c) SP1063

0.1 0.2 0.3 0.4 0.5

k (/m)

f (H

z)

(d) SP1064

0.1 0.2 0.3 0.4 0.5

20

40

60

80

100

f (H

z)

(e) SP106520

40

60

80

100

(f) SP1066 (g) SP1067

f (H

z)

(h) SP1068 20

40

60

80

100

k (/m)

f (H

z)

(i) SP1069

0.1 0.2 0.3 0.4 0.5

20

40

60

80

100

k (/m)

f (H

z)

(j) SP1070

0.1 0.2 0.3 0.4 0.5

20

40

60

80

100f−k

0

0.2

0.4

0.6

0.8

1

k (/m)f (

Hz)

(k) Stack

0.1 0.2 0.3 0.4 0.5

20

40

60

80

100

Figure 6.61: Frequency-wavenumber stacking of rollalong data at the Hyden fault scarp,

where 50 channels between SP1011 and SP1060 are employed in all transforms. (a)-(j)

Individual f − k spectra with picked maxima for shots 1 m to 10 m near offset; and (k)

Stack of all individual f − k planes with final dispersion for spread centre at SP1035.5.

Rollalong data The multi-fold data proved useful for surface wave analysis, since bet-

ter path-averaged dispersion could be achieved by stacking in the plane wave transform

domain. A similar average could be achieved by measuring dispersion from the CMP

gathers, however, only 48 channels at 2 m spacing are available and static, trigger and

source function variations can contaminate the phase differences in the individual plane

wave spectra. Stacking of the rollalong data in the frequency domain was the preferred ap-

proach, similar to the method of [370]. However, no trace cross-correlations were applied,

as in [108, 110].

Trial and error revealed 50 channels provided the longest array with minimal lateral

discontinuity effects and, to further minimise scattered wavefield influence while ensuring

high frequency data is not severely attenuated, source near offsets to a maximum of 10 m

were used. For plane wave transformation, f −k was used, primarily to increase the speed

of processing and also remove any need to set slant-stack phase velocity limits. However,

it is also more or equally valid than f − p in inversely dispersive sites, as suggested by the

test array inversions, assumed to be the norm in the overburden at this site.

As an example, to calculate the stacked dispersion for SP1035.5, 50 channels between

SP1011 to SP1060 are employed. The first shot gather to be used is from SP1061 (1 m

near offset) and the process looped as follows:


1. Window the required 50 channels;

2. Preprocess windowed gather (eg. spatial cosine tapers);

3. Plane wave transform to f − k;

4. Normalise amplitude spectrum at each frequency;

5. Load next rollalong CSG, thus 50 channel window be displaced by 1 trace;

6. Repeat steps 1-5 until a 10 m source offset is reached; and

7. Stack (average) all normalised f−k spectra and pick effective dispersion off the final

image.

All the while, the geophones used for plane wave transform are common to SP1011 to

SP1060. The intermediate and stacked f − k images from this process are shown in Fig-

ure 6.61. The obvious shot offset effects above 50 Hz have been markedly smoothed in the

final dispersion for a more unique curve. This process was employed for all data, starting

with spread SP1001-SP1050 and ending with spread SP1096-SP1145, each displaced by

5 m. This provides dispersion curves for spread centres SP1025.5 to SP1120.5 at 5 m

spacing.

6.6.3 Inversion models

Common shot dispersion All PSV acquisition parameters were set as used in the field

(45 channels at 1 m spacing and 0.5 m near offset) and the vertical or inline horizontal

component calculated as required. The source used was a 50 Hz causal wavelet from a

vertical impact at 0.3 m depth, since it was assumed that the energy from the gun muzzle

would be mostly in this direction. Even though the shotgun muzzle is in water and a

gas bubble similar to marine air guns is expected, it is not a true spherical source, also

confined by the shot hole cylindrical walls. The processing parameters were the same as

used for the field data, except the upper slant-stack phase velocity reduced to 600 m/s.

The usable frequency range was chosen to be 8-116 Hz. The data standard deviations are

incorporated from the scatter in the dispersion observed by repeated shooting along this

array, analysed in Section 4.2. The dispersion peak at 35 Hz was of higher phase velocity

with 8 Hz geophones, but, the large standard deviation will lead to a lower weight for

those points.

The layering used is 12 layers, with thicknesses increasing at twice the depth to the layer

top (starting with 1 m thick surficial layer) to a half-space at 20 m depth. However, realistic

depth penetration is in the order of 10 m, based on the longest wavelength measured of

20 m (correlating well with 0.4 of the 44 m spread length and the 0.5λ rule for depth).

Poisson’s ratio was estimated at 0.45 and density at 1.8 g/cc for all layers. The actual

near-surface Poisson’s ratio is higher, based on a direct P -wave velocity of 1000 m/s and


0 20 40 60 80 100200

400

600

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.68 %


0 200 400 600

0

5

10

15

20

25


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

5

10

15

20

25

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.62: Inversion of the Hyden fault scarp test array raw vertical component dis-

persion with the PSV method and measured dispersion errors. Starting model is auto-

matically generated, with 12 layers of thickness as twice the depth to top, starting with a

surficial layer of 1 m thickness and a homogenous half-space at 20 m.

0 20 40 60 80 100200

400

600

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.83 %


0 200 400 600

0

5

10

15

20

25


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

5

10

15

20

25

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.63: Inversion of the Hyden fault scarp test array raw horizontal component

dispersion with the PSV method and measured dispersion errors.


0 20 40 60 80 100200

400

600

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.41 %


0 200 400 600

0

5

10

15

20

25


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

5

10

15

20

25

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.64: Inversion of the Hyden fault scarp test array processed vertical component


0 20 40 60 80 100200

400

600

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.57 %


0 200 400 600

0

5

10

15

20

25


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

5

10

15

20

25

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.65: Inversion of the Hyden fault scarp test array processed horizontal component



S-velocity of 270 m/s (high frequency phase velocity of 300 m/s), but expected to be

lower at depth. An inversion of the vertical and horizontal component raw dispersion

(unwindowed shot gathers) is shown in Figures 6.62 and 6.63. The fit in the horizontal

component dispersion is poor at low frequency due to a higher mode in the best fitting

dispersion curve.

Inversion results of the processed dispersion from both vertical and inline component

trace windowed data (Figures 6.59 and 6.60) are shown Figures 6.64 and 6.65. However,

an explosive source used, as well as an x-taper to the first and last 5 channels of the

synthetic shot gathers, as applied to the field data. In addition, the upper slant-stack

phase velocity was further reduced to 550 m/s. The inline component synthetic dispersion

does not properly model the discontinuity around 47 Hz and the vertical component final

data misfit is better, albeit the upper slant-stack velocity being attained between 22-35

Hz. Nevertheless, the inverted models are similar, with the inline component interpreting

an apparently shallower and thinner buried high velocity layer.

The modelled high velocity layer at about 5 m depth is interpreted as a laterite horizon,

which appears to be associated with the fault movement. It was noticed in the shotholes

up to about SP1010W, and also uncovered at the top of the fault scarp around SP980.

Below it, a velocity reversal occurs and there is no stiff basement, where the geology can

only be assumed as unconsolidated sediments or friable saprolite. There was no indication

of an increase in shear velocity with depth in the dispersion curves and it is also possible

that a water table is present below the laterite. The apparent thickness of the HVL

is highly dependent on the assumed layering, due to the increasing layer thicknesses in

the model and shear velocity nonuniqueness of layers at depth. In the shallow zone, the

steep shear velocity gradient may be either a true geological feature or a smoothing effect

when fine layer interfaces are assumed in the PSV inversion. However, it appears to

be the former, the reason being that a large, abrupt stiffness contrast should generate

a prominent discontinuity in the dispersion curve at low frequency. Here the dispersion

shows a gradational increase in phase velocity with decreasing frequency.

Rollalong stacked dispersion To invert the stacked dispersion, the following synthetic

parameters were chosen, where those used/observed for stacking the field data are shown

in parentheses:

1. 50 channels at 1m spacing and 5.5 m near offset (near offsets of 1 m to 10 m);

2. 50 Hz causal wavelet as source function (average peak of gun source trace spectra);

3. Vertical impact source at 0.3 m depth (barrel held vertically with muzzle at base of

hole);

4. Particle velocity seismograms with a cosine low-frequency rolloff filter at 28 Hz ap-

plied (28 Hz vertical component geophones).


0 10 20 30 40 50 60 70 80350

400

450

500

550

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.25 %


200 300 400 500 600 700

0

5

10

15

20

25


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

5

10

15

20

25

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.66: Inversion of the Hyden fault scarp stacked rollalong dispersion by the

PSV method.

0 10 20 30 40 50 60 70 80350

400

450

500

550

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.41 %


200 300 400 500 600 700

0

5

10

15

20

25


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

5

10

15

20

25

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.67: Inversion of the Hyden fault scarp stacked rollalong dispersion by the

FSW method.


338.2

341.9

345.7

Ele

v. (

m)

Shotpoint (m)T

ime

(ns)

900950100010501100

0

50

100

326.9

330.7

334.4

338.2

341.9

345.7

Ele

v. (

m)

Shotpoint (m)

Tim

e (n

s)

900950100010501100

0

50

100

150

200

250

Figure 6.68: Ground Penetrating Radar (GPR) profiles over the Hyden fault scarp seismic

line, with 100 MHz (top) and 25 MHz (bottom) antennas.

Shotpoint (m)

Ele

v. (

m)

900950100010501100

325

330

335

340

345

VS (m/s)

200

300

400

500

600

700

Figure 6.69: Inversion of the stacked dispersion over the Hyden fault scarp for a ‘pseudo’

shear wave velocity profile. Each ‘sounding’ is shown as a 5 m thick column (greyscale)

with the final model overlain (black line).


Shotpoint (m)

TW

T (

ms)

900950100010501100

0

50

100

150

200

250

Figure 6.70: Seismic P -wave reflection profile over the Hyden fault scarp. NMO velocity

is 1300 m/s with no static corrections applied.

Again, x-tapers with cosine rolloff to the first and last 5 traces are applied. Using the

same layer thicknesses as the single fold data, an inversion of the the stacked dispersion

of Figure 6.61 by both the PSV and FSW methods are shown in Figures 6.66 and 6.67.

While both predict a stiff layer at 4-5 m depth, the PSV method provides a better fit to

the observed dispersion and the final model is much smoother, showing the similar steep

shear velocity gradient of the near-surface. Considering a 550 m/s shear velocity, the stiff

layer is almost twice as thick here (SP1035.5 centre) than inverted from the test array

dispersion (SP1073 centre).

The interpretation of a thicker laterite horizon closer to the top of the fault scarp

is supported by the Ground Penetrating Radar (GPR) data. Figure 6.68 shows the 100

MHz (top) and 25 MHz (bottom) GPR profiles over this line. The 100 MHz penetration

limit is about 50 ns (less than 5 m) while the 25 MHz shows reflections up to 250 ms

(almost 20 m). The discontinuity in the overburden stratigraphy at SP970 is where the

fault plane is interpreted, supported by a diffraction in the 100 MHz data, sourced from

the edge of a reflecting horizon. Modelling this diffraction provided a radar velocity of

0.15 m/s, by which the data are depth converted. West of the fault plane (increasing

shotpoint direction) a distinct ‘wedge’ of sediments is evident, better seen in the 25 MHz

data. In the 100 MHz data, the top of this wedge can be noticed dipping to the west. This

type of structure is generated with a west-over-east thrust, which is also supported by the

topography. Further west, the overburden is flat-lying, with increased radar attenuation,

possibly from a higher clay and/or salinity content of moist overburden.

Each stacked dispersion curve was inverted and the final models aligned in a topo-

graphically corrected section Figure 6.69. Each model is shown as a 5 m thick column


(greyscale), overlain with the final model (black line). This type of display is better called

a ‘pseudo’ section, since each model is 1D inverted from stacked dispersion averaged over

a 50-channel path. The interpretation of a gridded or smoothed shear velocity image,

such as in [207] must be made with care. Nevertheless, the same wedge-shape identified

in the 25 MHz data is interpreted as a high velocity zone. In addition, the far west of the

line also shows a HVL, correlating with GPR reflections. The zone of radar attenuation

between SP1060 and SP1110 correlates with a low shear velocity zone with smooth gra-

dations, interpreted as a more clay-rich zone. No basement is interpreted, which may be

a limitation of the spread length, and a LVL underlies the entire line.

6.6.4 Seismic reflection profiles While most of the field effort at this site was

put into the rollalong P - and S-wave acquisition, no reflections were observed in the

raw data. However, shot gathers pre-processed with a 2D Harmonic Wavelet Transform

(HWT) showed some possible events with hyperbolic moveout and high frequency. The

2D HWT is similar to the f − k transform, but groups frequency and wavenumber into

octave (dyadic) bands. Statistical thresholding is used to retain only desired coefficients

within each band. This was applied within a velocity fan, where below 1750 m/s only

the lower 1% of coefficients in each f − k band were retained, and above 1750 m/s all

were retained. The reason for this was the expected low RMS velocity of the overburden,

combined with the overlapping surface, guided and body waves.

A brute stack of the processed vertical component data with a NMO velocity of 1300

m/s is shown in Figure 6.70. The first CMP was SP1005.5 and maximum fold (48) attained

by SP1030. Although the section shows coherant events, these are all interpreted as stacked

ground roll and/or processing artifacts. In addition, no elevation or refraction statics are

applied and were anticipated to be up to several milliseconds. Even though the events

are assumed to be from the surface wavefield, there are some correlations with the GPR

and VS sections. For example, dips in the westerly direction occur due to the ground-roll

velocity of CMP gathers decreasing in this direction as the loose overburden thickens. In

addition, the interpreted clay-rich zone around SP1060-SP1110 shows evidence of planar,

low-frequency events associated with slower ground-roll in laterally homogenous material.

The surface wave interpretations of a buried HVL help to explain the lack of reflec-

tions, due to the ‘masking effect’ of the near-surface waveguide and underlying HVL [367].

Figure 6.71 shows the expected wavefields from a simple 2-layer case, with basement at

45 m. The basement reflection can be seen in both the full-waveform (all internal and

surface multiples to include surface waves) and body-wave only (PP , PS, SP and SS

reflections). It is strongest at far offsets, the time at the furthest geophone (96 m) about

0.14 s, where the deeper hyperbola (after 0.15 s) is P to S conversion. However, in Fig-

ure 6.72, the strong surface wavefield splitting dominates the record and the body-wave

only seismograms are dominated by the P to S converted reflection from the top of the


HVL. The basement PP reflection at far offsets is present but of very small amplitude. In

practise, even the PS reflection is of very small amplitude and emphasises the fact that

even if deeper reflectors are present, the wavefields generated shroud all deeper body wave

returns to prohibit CMP stacking.

In summary, the best images at this site were provided with combined GPR and

surface wave inversion. Although the multi-fold seismic P -wave data did not reveal useful

reflections, the rollalong method allowed improved surface wave imaging by dispersion

stacking.


0 20 40 60 80

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(a)

0 20 40 60 80Offset (m)

(b)

0 1000α (m/s)

(c)

0 500

0

5

10

15

20

25

30

35

40

45

50

β (m/s)

Dep

th (

m)

(d)

Figure 6.71: Reflectivity modelling of a representative model for the Hyden fault scarp

with homogenous overburden. (a) Full-waveform and (b) Body wave only seismograms,

with 32 ms AGC windows; and (c) VP and (d) VS models.

0 20 40 60 80

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(a)

0 20 40 60 80Offset (m)

(b)

0 1000α (m/s)

(c)

0 500

0

5

10

15

20

25

30

35

40

45

50

β (m/s)

Dep

th (

m)

(d)

Figure 6.72: Reflectivity modelling of a representative model for the Hyden fault scarp

with shallow HVL. (a) Full-waveform and (b) Body wave only seismograms, with 32 ms

AGC windows; and (c) VP and (d) VS models.


6.7 Pitfalls in dominant higher mode inversion

While the PSV method overcomes the requirement of a continuous dispersion curve (in-

variably assumption of the fundamental mode only) when plane wave matrices (FSW method)

are used in the inversion, there are still instances where the PSV method fails to correctly

invert dominant higher modes. These are due to large discontinuities in the observed

dispersion curve which may be due to non-Rayleigh wavefields or where the effective

dispersion transitions back and forth between the fundamental and/or dominant higher

surface wave modes.

The many pitfalls and challenges in P -wave reflection [210, 309, 113] and S-wave

reflection [208] and refraction [359, 360] have been outlined, but so far not reported for

surface wave inversion. This section will outline a few of these from the full-waveform

PSV surface wave inversion perspective.

6.7.1 ‘Apparent’ guided waves

Narrows Bridge foreshore The data here were collected with the same equipment and

procedure as for the Telfer gold mine site. The site is over grassland not far from the

Perth Convention Centre, near the Narrows Bridge. The near surface geology comprises

the same reclaimed fill material and the water table is expected at a few metres depth,

since the spread was less than 50 metres from the retaining wall of the Swan River north

shore.

The data and dispersion for this site are shown in Figures 6.73 and 6.74. The dispersion

is quite smooth up to about 60 Hz (except for very low-frequency effects) but above that,

several large discontinuities occur. In Figure 6.74(b), above about 80 Hz at low slowness

are possible aliased lobes. A definitely aliased portion is at high slowness between about

168-178 Hz, which exceeds the slowness alias limit (curved black line in lower right). A

further undesired portion is the constant slowness above about 180 Hz which appears to

correlate with the air wave (vertical black line at 0.0029 s/m).

The discontinuity in the manually picked dispersion of Figure 6.74(a) at 60 Hz was

first suspected to be a transition to a guided wave mode, which are multiply reflected

P -waves in a shallow surface layer [273], similar to those observed at the Telfer gold mine

tunnel pits (Figures 6.14 and 6.15). However, the overlapping nature of the surface wave

modes (usually later time) and direct P -waves (earlier time), suggested a low Poisson’s

ratio. For separation in the time domain of these wavefields to produce shingled, dispersive

guided waves, a high Poisson’s ratio is required [273]. The full spread of early time data

(Figure 6.75), shows refracted arrivals from about 12-13 m near offsets at a velocity of

about 1200 m/s. Arrivals after this do not suggest the shingled nature of guided waves.

The high-frequency air wave (due to the steel baseplate) pervades across the spread at 335

m/s.

This mode is more like that observed in the vertical and inline component shot gathers

6.7. Pitfalls in dominant higher mode inversion 317

5 10 15 20 25 30

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Offset (m)

Tim

e (s

)(a)

Slowness (s/m)

Fre

quen

cy (

Hz)

(b)

0.002 0.004 0.006

20

40

60

80

100

120

140

160

180

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 6.73: Data from the Narrows Bridge foreshore showing strong surface waves. (a)

Raw shot gather; and (b) Frequency-slowness transform with automatically picked disper-

sion curve.

0 50 100150

200

250

300

350

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

(a)

150 200 250 300 350

0

2

4

6

8

10

12

14

16

18

20

Velocity (m/s)

Dep

th (

m)

(b)

c (m/s), z=λ/2.5 (m)

Figure 6.74: Dispersion from the Narrows Bridge foreshore showing dominant higher

modes. (a) Frequency-phase velocity; and (b) Reduced depth-phase velocity dispersion

over the band 0-160 Hz.


5 10 15 20 25 30 35 40 45 50

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Offset (m)

Tim

e (s

)

(a)

Figure 6.75: Data from the Narrows Bridge foreshore with 32 ms AGC window to highlight

direct and refracted arrivals.

Table 6.5: Model parameters for Bietigheim, a site investigated in [78].

h (m) VP (m/s) VS (m/s) σ

0.058 136.08 74.25 0.288

0.068 150.55 85.40 0.263

0.081 167.35 98.22 0.237

0.098 186.93 112.96 0.212

0.119 209.82 129.93 0.189

0.148 236.76 149.46 0.169

0.189 268.76 171.95 0.154

0.255 307.55 197.90 0.147

0.343 353.74 226.30 0.154

0.509 407.59 253.71 0.184

0.319 447.56 266.89 0.224

0.477 472.73 265.14 0.271

2.651 505.89 278.23 0.283

3.048 555.13 319.96 0.251

3.505 611.76 367.96 0.217

4.031 676.87 423.15 0.179

0.482 715.86 456.20 0.158

Inf 3702.40 2150.80 0.245


5 10 15 20 25 30

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Offset (m)

Tim

e (s

)(a)

Slowness (s/m)

Fre

quen

cy (

Hz)

(b)

0.002 0.004 0.006

10

20

30

40

50

60

70

80

90

100 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 6.76: Synthetic data of the Bietigheim model of [78]. (a) Raw shot gather; and (b)

Frequency-slowness transform with automatically picked dispersion curve.

0 50 100100

200

300

400

500

600

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

(a)

0 200 400 600

0

2

4

6

8

10

12

14

16

18

20

Velocity (m/s)

Dep

th (

m)

(b)

β (m/s) c (m/s), z=λ/2.5 (m)

Figure 6.77: Synthetic dispersion of the Bietigheim model of [78]. (a) Frequency-phase

velocity; and (b) Reduced depth-phase velocity dispersion over the band 0-100 Hz.


0 10 20 30 40 50 60 700

200

400

600

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 5.27 %MeasuredInitial Final

0 200 400 600 800

0

5

10

15

20


Dep

th (

m)

(b)


100

101

102

0

5

10

15

20

Percent (%)

Dep

th (

m)

(c)

σβ∆β

Figure 6.78: Inversion of the synthetic Bietigheim dispersion with the PSV method and

modelled dispersion errors. Starting model is automatically generated, with the true depth

interfaces, Poisson’s ratios and densities.

0 20 40 60 80 100150

200

250

300

350

Frequency (Hz)

Pha

se v

eloc

ity (

m/s


0 100 200 300 400

0

5

10


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

5

10

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.79: Inversion of the Narrows Bridge dispersion with the PSV method and mod-

elled dispersion errors. Starting model is automatically generated, with 12 layers geomet-

rically increasing in thickness from 0.1 m to 2 m and a homogenous half-space at 12 m

depth. Poisson’s ratios and densities assumed constant at 0.45 and 1.5 g/cc respectively.


at the Hyden fault scarp (Figures 6.59 and 6.60). There, a large discontinuity at 115

Hz was observed, but not attaining the P -wave velocity of the near surface thus not a

guided wave mode. Even though Poisson’s ratio in the near surface was high (over 0.45),

the vertical gradation of stiffness does not provide a suitable waveguide for propagating

multiply reflected P -waves. The higher surface wave mode is also evident in the raw Hyden

data (Figures 4.2 and 4.3).

A base for comparison from previous workers is the dispersion observed at Bietigheim in

[80]. The shear velocity structure of that site (Table 6.5, provided by [79]) was synthetically

modelled with the same acquisition parameters as used at the Narrows foreshore and

results shown in Figures 6.76 and 6.77. The PSV effective dispersion matches well the

FSW modal dispersion curves, with a transition to a dominant higher mode at about 40

Hz. In Forbriger’s data it occurred at 25 Hz with the hammer source and 35 Hz with a

buried explosive source, where in the latter it only pervaded to about 60 Hz. The cause is

a steep velocity gradient in the uppermost soil, where Poisson’s ratio is low [79]. It can be

appreciated that a similar stiffness structure exists at the Narrows Bridge site, the main

difference being the rapid drop in phase velocity at low frequency due to the expected

fully saturated sediments below the water table. The river level was only a few metres

below level of the seismic line.

An inversion for these dispersion curves was trialled with the input dispersion from the

original full-length spreads, not just 24-channel windows. In the Bietigheim case it was

66 channels (1 to 66 m offsets) and in the Narrows Bridge case, 48 channels (2x24-channel

walkaway gathers from 5 to 52 m offsets). Moreover, they were manually picked to prevent

preferential detection of aliased portions and cropped to maximum frequencies of 80 Hz

and 120 Hz respectively. Note too how the jump to the dominant higher mode for the

Bietigheim case occurs at a lower frequency (32 Hz) in this configuration, exemplifying

the effect of spread length on effective phase velocity. The PSV method failed to correctly

invert this type of dispersion, first in synthetically modelled dispersion (Figure 6.78),

thus the results are not expected to be accurate in the experimental case (Figure 6.79).

However, intermediate dispersion curves during the ‘Occam’s loop’ (damping parameter

updates) of the inversion did match the experimental trend of both these cases quite well

in some iterations. However, the optimisation did not converge due to the highly nonlinear

nature of the solution space and the jumping nature of the regularised inversion, which

the analytic partial derivatives are not tuned for [284].

Forbriger’s method [81] successfully addressed this type of dispersion which illustrates

the merit of his work. A possible solution with the PSV method would be to use a global

optimisation, such as the genetic algorithm employed in guided wave dispersion (albeit

with continuous dispersion curves) in [273].


5 10 15 20 25 30

0

0.02

0.04

0.06

0.08

0.1

0.12

Offset (m)

Tim

e (s

)

(a)

Slowness (s/m)

Fre

quen

cy (

Hz)

(b)

0.001 0.002 0.003

50

100

150

200

250 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 6.80: Data from the road cutting site Line 4 (SAB18) which contains Lamb waves.

(a) Raw shot gather; and (b) Frequency-slowness transform with automatically picked

dispersion curve.

20 40 60 80 100

400

600

800

1000

1200

1400

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

(a)

500 1000 1500

0

2

4

6

8

10

12

Velocity (m/s)

Dep

th (

m)

(b)

c (m/s), z=λ/2.5 (m)

Figure 6.81: Dispersion from the road cutting site Line 4 (SAB18) showing Lamb waves

interference from 50 Hz. (a) Frequency-phase velocity; and (b) Reduced depth-phase

velocity dispersion over the band 10-100 Hz.


0 20 40 60 80 100 120 140 160 180

500

1000

1500

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.48 %MeasuredInitial Final

0 500 1000 1500

0

2

4

6

8

10

12

GravelLateriteGravel

Gravelley sand

Clayey silt

EOH


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

2

4

6

8

10

12

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.82: Inversion of the the road cutting site Line 4 (SAB18) dispersion with the


identified in SAB18, with the laterite horizon subdivided into 2 layers.

6.7.2 Lamb waves

Road cutting caprock At one position along the road cutting site (Section 6.5) the

nearby solid auger borehole (SAB18) revealed a 1.3 m thick laterite horizon, which was

the thinnest caprock identified in all the boreholes of that cutting. It was sandwiched

between a surficial 0.2 m veneer and underlying 0.5 m layer of gravel.

The data and dispersion for a spread centred at that point are shown in Figures 6.80

and 6.81. The dispersion below 50 Hz is an inversely dispersive trend, however, at 50 Hz

a discontinuity to a normally dispersive trend occurs. This pattern is very similar to the

trends observed in [243], where Lamb wave dispersion was the dominant mode at higher

frequency. This is very feasible considering the almost unsupported, stiff laterite layer,

similar to the modelling of free plates and investigated geology of [243]. The dispersion

below 50 Hz appears to be of the asymmetric wave motion (A0) and above 50 Hz of the

symmetric (S0) motion.

An estimate of the asymptote of convergence of these modes at high frequency is in the

order of 750 m/s. Considering a P -wave velocity (from direct arrivals) of 1850 m/s and a

shear velocity 0.9 of the Rayleigh wave asymptote, this gives a Poisson’s ratio of 0.4. Lamb

wave dispersion modelling of a 1.3 m thick plate with these parameters did not exactly

reproduce the high frequency dispersion, thus the observed mode is not an ‘ideal’ Lamb

mode, due to coupling of the laterite with underlying gravels, otherwise called a ‘leaky’

Lamb mode [274]. Nevertheless, while the PSV method can model flexural waves, the

inversion shown in the cross-section of Section 6.5 did not properly model the dispersion


5 10 15 20 25

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(a)

Wavenumber (m−1)

Fre

quen

cy (

Hz)

(b)

0.1 0.2 0.3 0.4 0.5

10

20

30

40

50

60

70

80

90

100 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 6.83: Data from the railway tunnel excavation site showing strong overlapping

modes. (a) Raw shot gather; and (b) Frequency-wavenumber transform with automatically

picked effective dispersion curve.

around 50 Hz, but instead fitted an average. Those results are shown in Figure 6.82, and

is a combination of the partial derivatives not being valid for the Lamb wavefield and only

a few high velocity dispersion points, with large experimental error. While the inverted

shear wave velocities correlate approximately with geotechnical expectation, there is a

systematic error and cannot be taken as correct.

6.7.3 Oscillating mode transitions

Railway tunnel excavation This site is in a suburb of Perth investigated for a railway

tunnel. A silty clay layer with secondary iron cementation and nodules was proving

troublesome for the excavation of the ramp leading to the underground tunnel, where its

thickness was the desired parameter. The seismic data here were collected with the same

equipment and procedure as for the road cutting site.

One shot gather and automatically picked dispersion (in f − k space) for this site are

shown in Figure 6.83. In the 30-40 Hz range, the dispersion appears to oscillate between

two modes. While this may be a lateral discontinuity or other wavefield noise effect, at 70

Hz the dispersion dropped to a much lower phase velocity (larger wavenumber), suspected

to be the dominant higher mode. The same spread was shot in the reverse direction and

showed a much stronger fundamental mode (Figure 6.84(a)). This was manually picked,

which was only problematic around the weak energy band of 45-50 Hz (Figure 6.84(b)).


5 10 15 20 25

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)(a)

Wavenumber (m−1)

Fre

quen

cy (

Hz)

(b)

0.1 0.2 0.3 0.4 0.5

10

20

30

40

50

60

70

80

90

100 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 6.84: Data from the reciprocal shot at the railway tunnel excavation showing

similar strong overlapping modes. (a) Raw shot gather; and (b) Frequency-wavenumber

transform with manually picked fundamental mode dispersion curve.

0 20 40 60 800

100

200

300

400

500

600

700

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

)

(a)

100 200 300 400

0

5

10

15

20

25

30


Wav

elen

gth

(m)

(b)

Figure 6.85: Dispersion from the railway tunnel excavation site of the manually picked

fundamental mode. (a) Frequency-phase velocity; and (b) Wavelength-phase velocity dis-

persion.


0 10 20 30 40 50 600

200

400

600

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 0.82 %


0 200 400 600

0

5

10

15

?855593515U−tube3921

EOH


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.86: ‘Borehole’ inversion of the railway tunnel excavation site dispersion with the

FSW method and modelled dispersion errors. Starting model is automatically generated,

with layer interfaces identified in a nearby borehole (BH3) used. Poisson’s ratios and

densities assumed constant at 0.4 and 1.8 g/cc respectively.

0 10 20 30 40 50 600

200

400

600

Frequency (Hz)

Pha

se v

eloc

ity (

m/s

) (a) RMS error: 2.36 %


0 200 400 600

0

5

10

15

?855593515U−tube3921

EOH


Dep

th (

m)

(b)

InitialFinal Inter

100

101

102

0

5

10

15

Percent (%)

Dep

th (

m)

(c)

σβ

Figure 6.87: ‘Blind’ inversion of the railway tunnel excavation site dispersion with the

FSW method and modelled dispersion errors. Starting model is automatically generated,

with 12 layers geometrically increasing in thickness from 0.2 m to 2 m and a homogenous

half-space at 10 m depth. Poisson’s ratios and densities assumed constant at 0.4 and 1.8

g/cc respectively.


The resulting dispersion curves representing the assumed fundamental mode are shown in

Figure 6.85. While it shows some quite large discontinuities (Figure 6.85(a)), these are

assumed not to be due to dominant higher mode effects.

This dispersion can be inverted by the FSW method, since the fundamental mode is

being assumed. The results are shown in Figures 6.86 and 6.87, where both ‘borehole’

and ‘blind’ layer interfaces are respectively used. ‘Borehole’ interfaces are those identified

by lithological changes in a nearby borehole (BH3) and ‘blind’ interfaces are statistically

set. In both cases, the inversion recovers the approximate trend of SPT N -values, which

are overlain on Figure 6.87(b). It was fortunate in this case that the fundamental mode

retained enough energy to allow accurate picking and inversion with plane wave matrix

methods. In general, however, this is rare and the full-waveform PSV method is required,

since filtering the shot gather to enhance the fundamental mode is invariably not possible

when dominant higher modes are present.



In most cases, although the longest measured wavelength exceeded the theoretical

measurable limit (0.4L), recovery of shear velocity structure down to over a half (or more)

of the spread length was attained. Shallow structure appears, heuristically, well resolved

in nearly all cases.

6.8.1 Successful inversions

Telfer gold mine Clear dominant higher modes were observed, typical of HVL and

LVL structures. The PSV inversion properly modelled the dispersion discontinuities to

recover the expected profiles. The LVL structure is more dependent on layer interface

assumptions than the HVL dispersion. Dispersion over backfill material shows excellent

lateral homogeneity, but spread lengths longer than the width of the trench are affected by

retaining wall reflections. Guided waves are also generated above 170 Hz. Data from fur-

rows dug to remove overburden effects are contaminated by scattering and the dispersion

not interpretable.

Perth Convention Centre A 5 m thick stiff layer at 20 m depth is not interpreted with

either 56-channel (at 1 m spacing) or 24-channel (at 3 m spacing) data. The stiffness

trend in the upper 20 m is recovered well with either configuration. Deeper layers are

best interpreted with the 3 m spacing data. Synthetic modelling and inversion based on

a borehole model shows similar poor detection of a deep HVL.

Road cutting Thickness and shear velocity of laterite caprock is interpreted with the

PSV method. However, unconsolidated sediment over shallow basement shows strong

dependence on layer interface assumptions, with dispersion from a 24-channel spread (at

1 m spacing).

Hyden fault scarp Inline component dispersion shows dispersion discontinuities not

identified in the vertical component. The inverted models from each component differ in

thickness of the interpreted buried HVL. Stacking dispersion over near offsets of 1 m to

10 m from rollalong data smooths offset-dependencies above 50 Hz. Inverted models from

the stacked dispersion are also much smoother, allowing construction of ‘pseudo’ shear

velocity sections. Shear velocity variations correlate with undulations identified in GPR

profiles and the shallow laterite horizon prohibits reflectors being identified.

6.8.2 Inversion pitfalls Large dispersion discontinuities generated by steep shear

velocity gradients in the very shallow zone are not inverted by the PSV method. Similarly,

transition of an inversely dispersive effective phase velocity to normally dispersive due to

flexural modes are not interpreted. Dispersion which transitions between dominant higher

and fundamental modes is best inverted as the fundamental mode alone if sufficient signal

to noise ratio allows its identification.

329

CHAPTER 7

Model parameter resolution and appraisal

7.1 Introduction

Chapters 5 and 6 revealed that inverted models can show a large difference from the

true or expected solution. This is due to a combination of both resolution limitations of

the data and nonuniqueness of the inversion which is generally known as error propagation

from data through to the model. Thus model resolution and accuracy will depend on all

of data resolution and accuracy and inversion kernel, parameterisation and optimisation.

The SASW tests by [330] are probably the best reported study of error propagation to

date, but with only 30 dispersion curves and 2 starting models were far from exhaustive.

Data sensitivity (model dispersion curve response to parameter changes) was studied in

[303] and inverted model recovery with different layering assumptions and optimisation

schemes in [182]. These methods were mostly deterministic and based on a plane wave

dispersion modelling code, thus even though the LVL and HVL models used should have

generated dominant higher modes they were not considered. Moreover, data errors were

neglected.

While the dominant higher mode dispersion is quite nonlinear, the inversions have

shown that linearisation is still acceptable. However, several tests will be required to truly

evaluate the final model resolutions. The only real investigation into resolution of models

where dominant higher modes are generated was made by Forbriger [81]. However, the

unconstrained ‘rubberband test’ used there did not account for both: (1) Realistic error

envelopes; and (2) Full range of possible layer interfaces. This chapter aims to quantify

the resolution of model parameters in typical engineering layered earth structures obtained

from surface wave inversion of dominant higher modes. According to the improved view of

inverse problems in [286], this chapter is the appraisal or inference problem and Chapters

5 and 6 comprise the estimation problem. The methods are purely numerical and only

linearised inversion will be considered.

330 Chapter 7. Model parameter resolution and appraisal

7.2 Numerical methods for model inference

Many methods exist for evaluating an inverted model within known noise limits. The

more rigorous procedures employ statistical measures of model resolution a priori and,

when incorporated into the inversion, indicate how data errors contaminate the final model

parameters. Bayesian inversion is a popular framework for this analysis [285] which also

incorporates the nonuniqueness of inversion. Less rigorous methods simply scan the pa-

rameter space around a true or final model and evaluate the features which could be

adequately detected within a defined data misfit and include:

1. Deterministic grid search; and

2. Monte Carlo ensemble inference.

A deterministic grid search simply spans the model space over user-defined limits with a

regular spacing between model parameters and is only viable when model spaces are small.

It was introduced in Chapter 3 in one dimension for scanning shear velocity perturbations

of one layer and is more closely aligned with a sensitivity analysis or data likelihood.

A Monte Carlo ensemble inference is a random implementation of the procedure and is

more useful when a broad model space is required to be scanned. For each point in model

space, the data is forward calculated and the misfit between the current and initial datasets

is calculated. The initial model and dataset will be either known, as in the case of a a

synthetic test, or from the final iteration, in the case of field data. This type of practical

inference theory allows us to see what features are common to all successful models a

typical inversion might arrive at. ‘Successful’ is defined as those models which provide a

data misfit within the user-defined limit, thus, this test combines both the data and model

likelihood for a thorough model appraisal. In crustal and basin scale seismology, ensemble

analysis is regularly reported [280, 288].

In most linear inversions, as with the Occam’s inversion used in this thesis, the reg-

ularisation parameter reduces nonuniqueness and helps the solution converge. However,

the model covariance matrix is affected by this constraint, so the confidence limits can

become as small as desired by simply increasing regularisation. With excess weighting,

the influence of the data on the model can become negligible. For this reason, the essential

measure in conducting these tests is the unconstrained data misfit or data likelihood.

7.2.1 Monte Carlo model generation The starting model for a Monte Carlo

inference analysis may be either true (synthetic) or estimated (experimental). In the case

of a field model, however, ‘true’ interfaces may be defined from borehole logs or other

geophysical results. For any starting layered shear velocity structure, generation of a

random trial model comprises two steps:

1. Layering perturbation; and

7.2. Numerical methods for model inference 331

2. Shear velocity perturbation.

Layering perturbation defines a new set of layer interfaces, followed by perturbation of

shear velocity, and the dispersion forward modelled and data misfit calculated. If the

starting model is synthetic, the misfit is relative to the numerically calculated dispersion.

If it is an estimated model, the misfit is relative to the best-fitting dispersion curve of the

inversion. The misfit can be weighted by either estimated or modelled/measured standard

deviations, to assess how dispersion uncertainty affects the scatter of trial model shear

velocities.

Layering perturbation The styles of layer thickness estimation are shown in Figure 5.4

of Section 5.4. In addition, simple subdivision of the starting layers is also employed, thus,

there are four types of layering definitions to be drawn from. A layered profile is then

generated by equally distribute random selection of:

1. Layering style;

2. Half-space depth (between half and twice that of the starting model);

3. Layering-dependent parameters, which are:

(a) Constant thicknesses - between 0.2 and 4 of the average thickness of the starting

model;

(b) Geometric factors - between 0.5 and 5 (for both nonlinear and linearly increasing

thicknesses), and, for the select minimum

(c) Geometric thicknesses - minimum (shallowest) and maximum (deepest) layer

thickness between half and twice the respective minimum and maximum thick-

nesses of the starting model; and

(d) Layer subdivisions - between 1 and 6 of the starting layer thicknesses;

If the layering is invalid (eg. root bracketing for geometric thicknesses not solved), the

process is repeated until a valid model is generated. The shear velocities of the trial

layered structure are then fitted to those of the starting model. Where a trial layer crosses

an interface of the starting model, the average shear velocity of the true layers crossed is

assumed.

Shear velocity perturbation The shear velocities of the trial model are then statistically

perturbed. The amount of perturbation is randomly generated between within equally

distributed limits or within a normally distributed range, where the respective limits or

standard deviation for each are either:

1. User-defined for each (true) layer;

2. Linearly increasing with depth; or


0 100 200 300 400 500

0

5

10

15

20

25

VS (m/s)

z (m

)

(a)

0 100 200 300 400 500V

S (m/s)

(b)

0 100 200 300 400 500V

S (m/s)

(c)

0 100 200 300 400 500V

S (m/s)

(d)

0 100 200 300 400 500

0

5

10

15

20

25

VS (m/s)

(e)

Dep

th (

m)

Figure 7.1: Monte Carlo model generation demonstration for Case 1. (a) Starting model,

including approximate inversion with uncertainty (solid line) and standard deviation for

shear velocity perturbation, σβ′(h) (dotted lines), where trial model shear velocities are

chosen from a normal distribution for: (b) Constant thickness layers; (c) Nonlinearly in-

creasing thicknesses; (d) Linearly increasing thickness; and (e) Starting layer subdivisions.

3. Approximate inversion curve.

Only option 3 (above) incorporates measurement error propagation thus was used through-

out, whereby the amount of depth-dependent shear velocity perturbation is calculated by:

1. Converting starting dispersion (f, c ± σc) to approximate shear velocity structure

(β′ ± σ′β, z′) by β′ = 1.1c and z′ = 0.4λ; and

2. For each trial layer (hi) calculate the average σβ′(z′) for all approximate depths (z′)

over trial layer depth extents zi−1 to zi (and within minimum and maximum limits

of z′).

The result is an average shear velocity standard deviation for each trial layer (). Shear

velocity perturbation of each layer of the trial model is made by random selection from a

distribution either:

1. Normal – with standard deviation bσβ′(h); or

2. Equal – over a range [−bσβ′(h),+bσβ′(h)].

where b is a user-defined factor to alow a broader (>1) or narrow (<1) range of trial shear

velocity models. Usually, b = 1 ensures that very large perturbations (poor likelihood)

7.2. Numerical methods for model inference 333

models are not generated, where phase velocities may occur outside the pre-defined τ−p or

f − k fan range for PSV automatic picking. In addition, more consistent results were

achieved when b is kept constant for the entire model, as opposed to it varying randomly

(within limits) between the layers of a trial model.

The process of Monte Carlo trial model generation for Case 1 is shown graphically

in Figure 7.1 for the four possible types of random layering parameterisation. In Fig-

ure 7.1(a), the dispersion curve with uncertainty is transformed to approximate shear

velocity-depth (with equivalent uncertainty), and standard deviation of shear velocity

perturbation shown by bounding dotted lines about the starting shear velocities.


7.3 Synthetic model resolution and appraisal

Before appraising field models, the synthetic Cases 1, 2 and 3 of Tokimatsu et al [325]

will be considered.

7.3.1 Deterministic parameter resolution In these tests only one layer is con-

sidered at a time and a grid scan of perturbations in both thickness (h/h0) and shear

velocity (β/β0) over a range from 0.5 to 2 is conducted. This is essentially a sensitivity

analysis of a small subspace (2-parameter) of possible models, merely adding a dimension

to the 1-parameter solution subspaces (shear velocity only) shown in Figure 5.1 (Sec-

tion 5.3.6).

Data misfits are calculated from Equation 5.27 and model resolutions and sensitivities

are calculated for all unconstrained operators, that is WJ. The weight matrix is con-

structed from an estimated dispersion error (here 3%) or a realistic/true error envelope

(numerically evaluated in Chapter 3). This comprises 2-parameter solution subspaces cor-

relate well with the 1-parameter (shear velocity only) subspaces shown in Figure 5.1 in

Section 5.3.6.

The essential measures in Figures 7.2 to 7.9 are:

(a) ∆c is the relative data misfit (RMS error) from the starting model dispersion, defined

in Equation 5.27;

(b) ‖∂β‖ is the L2 norm of shear velocity roughness, independent of layer thicknesses;

(c) R is the resolution measure defined by Equation 5.41;

(d) S is the sensitivity measure defined by Equation 5.40;

(e) δR is the absolute resolution of model shear velocity with assumed data errors, cal-

culated from Equation 5.43 (plotted as the log value for clarity);

(f) δS is the absolute sensitivity of the model shear velocity with assumed data errors,

calculated from Equation 5.42; and

(g) to (j) R11, . . . , R44 are the diagonal elements of the covariance matrix at each pa-

rameter perturbation, calculated from Equation 5.35. Log values are used for layers

3 and 4, where the range is small.

The most useful measures is probably (a), the data misfit. This shows within any given

RMS error contour the range of h and β which can be allowed and which of h or β is least

constrained. One limitation of applying true realistic weights are that the trial dispersion

curve should be relatively close to the starting dispersion, especially if different dominant

higher modes come into effect. Thus for h/h0 and β/β0 much different from the starting

7.3. Synthetic model resolution and appraisal 335

values, the weighted RMS misfit will not be correct, as the weights for each dispersion will

be markedly different.

The δR and δS images show real shear velocity values of the overall resolution and

sensitivity of the method. To resolve models which are close together, both should be small,

that is, sensitive to small model perturbations. It is of no benefit to have good resolution

but poor sensitivity, or vice versa. The covariance matrix diagonals (R11, . . . , R44) should

be considered remembering that perfect resolution occurs when R = I. In all cases, the

shallow layers are better resolved, however, not always the uppermost layer dominating.

The starting β/β0 and h/h0 location (1,1) is shown as an encircled ‘x’ in the plots.

When the shear velocity of the layer in question becomes a LVL, this point is shown by

the vertical dotted line. When it becomes a HVL, a vertical dashed line is used. Thus

between these two limits are where the layer fits into a normally dispersive profile. In all

images, low values are shown in white with increasing values as darker greys. Note that

the traditional FSW method was used when the range chosen for β/β0 only encompassed

normally dispersive models, which was true only for the Case 1 half-space. For the irregu-

larly dispersive Cases 2 and 3, the PSV method was used to incorporate dominant higher

modes. When interpreting these plots, it must be kept in mind that all other parameters

are fixed at the correct values, thus they only represent a small subspace of the full model

space.

Case 1 half-space (layer 4) The Case 1 half-space showed the largest variability in

the inversion tests of Chapter 5, usually when the number of layers was small. When a

constant 3% error envelope is considered, Figure 7.2(a) shows that β is poorly resolved.

Nonuniqueness increases at higher shear velocity and the rate of convergence (misfit gra-

dient) is better at low shear velocity. In this case, the h/h0 plots are actually the depth

to to the half-space, which too is quite poorly constrained. It is interesting to note in Fig-

ures 7.2(e) and (f) that resolution and sensitivity improves when the half-space is lower

velocity. This supports the general observation of an underestimate of half-space shear

velocity in Section 5.5. Resolution of layer 3 (Figure 7.2(i)) is better when it is thicker

(that is, half-space is deeper) but resolution of the half-space naturally better when it is

shallower (Figure 7.2(j)).

When realistic data errors are used (Figure 7.3), similar patterns emerge, but the am-

plitudes are markedly different. The poor constraint in half-space depth-velocity is shown

in Figure 7.3(a), where even at a 0.5% misfit it is allowed anywhere within the parameter

limits shown. The absolute resolution is much poorer (Figure 7.3(e)) as are sensitivities

(Figure 7.3(f)), which show little variation over the entire parameter range, often larger by

an order of magnitude than in Figures 7.2(e) and (f). There are improvements, however,

in the resolutions of the upper two layers (Figures 7.2(g) and (h)). This is understandable,

as realistic errors at higher frequencies are around 1%.


h/h 0

(a) ∆c (%)

0.50.5

1

11

1.5

1.52

1

2

h/h 0

(b) ||∂β|| (m/s)

50

100

150

200 250

1

2

h/h 0

(c) R

1

1 1

2

22

3

3 45

61

2

h/h 0

(d) S

0.191

0.191

0.1915 0.1920.19250.1930.19350.194

1

2

h/h 0

(e) log10

(δR) (m/s)

−0.2

−4.9343e−0170.2

0.2

0.4

0.4 0.40.6 0.6

0.60.8 1 1

1

2

h/h 0

(f) δS (m/s)

16

1820

22

1

2

h/h 0

(g) R11

0.980.9820.9840.9860.988

0.99

0.992

0.9941

2

h/h 0

(h) R22

0.2

0.3

0.3

0.4

0.4

0.50.6

0.70.8

1

2

β/β0

h/h 0

(i) log10

(R33

)

−3 −3−2.5 −2.5−2 −2

−1.5

0.5 1 1.5 2

1

2

β/β0

h/h 0

(j) log10

(R44

)

−5

−4.5

−4

−3.5−3

0.5 1 1.5 2

1

2

Figure 7.2: Resolution of layer 4 of Case 1 with 3% data errors. (a) to (j) are described

in the text.

h/h 0

(a) ∆c (%)

0.1

0.1

0.2

0.2

0.3 0.4 0.5

1

2

h/h 0

(b) ||∂β|| (m/s)

50 100

150

200 250

1

2

h/h 0

(c) R

0.20.4

0.40.4

0.6

0.6 0.6

0.8

0.8 0.8

1

11

1.21

2

h/h 0

(d) S

0.6103

0.6103

0.610310.61031

0.610320.61033 0.610340.61035

1

2

h/h 0

(e) log10

(δR) (m/s)

1.2

1.21.2 1.2

1.4

1.4 1.4

1.6

1.61.6

1.8 22.2

1

2

h/h 0

(f) δS (m/s)23.88623.886523.88723.887523.888

23.8881

2

h/h 0

(g) R11

0.99998

0.99998

0.999980.99998

0.99998 0.999980.99998 0.99998

1

2

h/h 0

(h) R22

0.780.78

0.790.79

0.80.8

0.810.81

0.82 0.83 0.84

1

2

β/β0

h/h 0

(i) log10

(R33

)

−4.8 −4.6 −4.6−4.4 −4.4

−4.2

−4.2−4

−3.8

0.5 1 1.5 2

1

2

β/β0

h/h 0

(j) log10

(R44

)

−7

−6.5

−6.5

−6.5

−6.5

−6−5.5−5

0.5 1 1.5 2

1

2

Figure 7.3: Resolution of layer 4 of Case 1 with realistic data errors. (a) to (j) are described

in the text.


h/h 0

(a) ∆c (%)

1

12 23

34

4

51

2

h/h 0

(b) ||∂β|| (m/s)

95 95

100

105

110

115

120

1

2h/

h 0

(c) R

0.51 1.5

2

2

2.5

2.5

3

3

31

2

h/h 0

(d) S0.10.15

0.20.250.3

1

2

h/h 0

(e) log10

(δR) (m/s)

0.5

11.5

1.5

22

2

2.5

333

1

2

h/h 0

(f) δS (m/s)

20

3040 50 60 70

80

1

2

h/h 0

(g) R11

0.15

0.2

0.2

0.25

0.30.35

0.4 0.45

1

2

h/h 0

(h) R22

0.1

0.20.3

0.4

0.5

0.60.7

0.80.91

2

β/β0

h/h 0

(i) log10

(R33

)

−4−3.5 −3 −3

−2.5

−2.5

−2−2

−1.5−1.5

0.5 1 1.5 2

1

2

β/β0

h/h 0

(j) log10

(R44

)

−10

−10−9−8

−8−7−6−5−4

−4

0.5 1 1.5 2

1

2


in the text.

h/h 0

(a) ∆c (%)

1 12 23 3

4

4

4

5

1

2

h/h 0

(b) ||∂β|| (m/s)

95 95

100

105

110

115

120 1

2

h/h 0

(c) R

0.20.4

0.6

0.6

0.8

1

1

2

h/h 0

(d) S

0.1

0.2

0.30.40.5

1

2

h/h 0

(e) log10

(δR) (m/s)

1

1

1

1.5222 2.53

3

33.5

4

1

2

h/h 0

(f) δS (m/s)

20

304050

60 70

70 1

2

h/h 0

(g) R11

0.750.80.85

0.9

0.9

0.95

0.95

1

2

h/h 0

(h) R22

0.10.20.3

0.4

0.4

0.5

0.5

0.6

0.6

0.70.70.

70.7 0.80.9

1

2

β/β0

h/h 0

(i) log10

(R33

)

−4

−4

−3−3

−2

−2

−1−1

−1

0.5 1 1.5 2

1

2

β/β0

h/h 0

(j) log10

(R44

)

−14

−12

−10−10

−8

−8

−6

−6

0.5 1 1.5 2

1

2


in the text.


h/h 0

(a) ∆c (%)

2

2

4

4 4

44

4

4

6

6

66

8101214

16

1

2

h/h 0

(b) ||∂β|| (m/s)

130

140 150 160 170 180 190

200

210 1

2

h/h 0

(c) R

0.5 1

1

1.5

2

22

2

2 22

2

2.5

2.52.5

3

3

3

3

3.5

3.5

3.5

3.5

4

444

4

4

4.5

4.5

1

2

h/h 0

(d) S

0.5

0.5 1 2

1

2

h/h 0

(e) log10

(δR) (m/s)

0

0

00

0

00

0

0.51

11.5

2

1

2

h/h 0

(f) δS (m/s)

6

668

88

8

8

10101012 12 12

12

12

12

14

1414

1416

16

1

2

h/h 0

(g) R11

0.88

0.90.92 0.

940.

96

0.98

1

2

h/h 0

(h) R22

0.10.20.3

0.40.50.60.70.8

1

2

β/β0

h/h 0

(i) log10

(R33

)

−3−2.5−2 −2−1.5 −1.5

−1

−1−0.5−0.5

0.5 1 1.5 2

1

2

β/β0

h/h 0

(j) log10

(R44

)−6

−5−4

−3

−3 −3

−3

0.5 1 1.5 2

1

2


in the text.

h/h 0

(a) ∆c (%)

1 1

2 2

3

45

1

2

h/h 0

(b) ||∂β|| (m/s)

130

140

150

160 170

180

190

200

210 1

2

h/h 0

(c) R

0.2 0.

4

0.4

0.4

0.6

0.80.8

1

1

11

1

11.2

1.41

2

h/h 0

(d) S0.5

0.55 0.6

1

2

h/h 0

(e) log10

(δR) (m/s)

1.51.5

22.53

3.5

4

1

2

h/h 0

(f) δS (m/s)

28

30

323436

38

38

40

1

2

h/h 0

(g) R11

1

2

h/h 0

(h) R22

0.1

0.2

0.30.4

0.50.60.7

0.80.9 1

2

β/β0

h/h 0

(i) log10

(R33

)

−6

−4 −4−4

−2

−2 −2

0.5 1 1.5 2

1

2

β/β0

h/h 0

(j) log10

(R44

)

−10

−8

−6

−6−6

−6

0.5 1 1.5 2

1

2


in the text.


h/h 0

(a) ∆c (%)

1

22

22

3

3

34

4

4

4

5

5

66

7

1

2

h/h 0

(b) ||∂β|| (m/s)

90

100

110

120

130

140

150

160

1

2h/

h 0

(c) R

1

1

12

22

22

3

33

3

33

3 3 3

3

4

4

4

4

4

5 5

51

2

h/h 0

(d) S

0.25 1

2

h/h 0

(e) log10

(δR) (m/s)

0

0

0

0

00

00

00.2

0.2

0.2

0.2

0.2

0.20.4

0.4

0.40.6

0.6

0.81

1

2

h/h 0

(f) δS (m/s)

1414 14

16

1618

18

20

1

2

h/h 0

(g) R11

0.993

0.9930.99

3

0.9930.994

0.994

0.995

0.995 0.996 0.996

1

2

h/h 0

(h) R22

0.02

0.020.02

0.04

0.040.06 0.06

0.08 0.080.10.12

1

2

β/β0

h/h 0

(i) log10

(R33

)

−3

−2.5−2−1.5−1

−1

−0.5

0.5 1 1.5 2

1

2

β/β0

h/h 0

(j) log10

(R44

)

−4−3.5

−3.5

−3.5

−3

−3−3

−3−2.5

−2.5

0.5 1 1.5 2

1

2


in the text.

h/h 0

(a) ∆c (%)

0.20.2

0.4

0.4

0.4

0.4

0.40.4

0.6

0.6

0.6

0.6

0.6

0.8

0.8

1

1

1.2

1

2

h/h 0

(b) ||∂β|| (m/s)

90100

11012

0

130

140

150

160

1

2

h/h 0

(c) R

0.20.2 0.4

0.4

0.4

0.4

0.4

0.4

0.4 0.6

0.6

0.6

0.6

0.6 0.60.6

0.8

0.8

0.81 1

11.2

1.21.4

1

2

h/h 0

(d) S

0.587

0.5875

0.5875

0.58

8

1

2

h/h 0

(e) log10

(δR) (m/s)

1.2 1.21.2

1.4

1.4

1.4

1.4

1.4

1.4

1.6

1.61.6

1.6

1.6

1.6

1.61.8

1.8

1.81.8

2

22

2

2

2.2

2.4

1

2

h/h 0

(f) δS (m/s)

30.54

30.5630.56 30.56

30.56 30.5830.6 1

2

h/h 0

(g) R11

0.99998

0.99998

0.99998

1

2

h/h 0

(h) R22

0.12 0.14

0.14 0.14

0.14

0.160.18

1

2

β/β0

h/h 0

(i) log10

(R33

)

−5

−4

−4

−3

−3

−3−2

−2

−2−2−1 −1

0.5 1 1.5 2

1

2

β/β0

h/h 0

(j) log10

(R44

)

−6.5−6

−6

−6

−6−5.5

−5.5

−5.5−5.5

−5.5

−5 −5

−5

−5

0.5 1 1.5 2

1

2


in the text.


Case 2 LVL (layer 2) The Case 2 LVL showed quite good shear velocity resolution in

the initial inversion tests with various layering assumptions. Figure 7.4(a) shows this as

the narrow width of the 1% misfit contour in the β/β0 direction. However, h is not as well

resolved, where the misfit minima appears to show a δh/δβ tradeoff. Thus, an initial rule

for model appraisal may be adopted: The degree of over- or under-estimation in LVL β

is most likely proportional to the error in h. In Figures 7.4(e) and (f), overall resolution

and sensitivity become approximately equal when the LVL is both thicker and of lower

velocity. However, the good sensitivity at very low velocities is offset by poor resolution,

and good resolution at higher velocity / thinner layer is offset by poor sensitivity. Of the

individual resolution kernel peaks, one point of note is the rapid improvement in the layer

1 resolution (Figure 7.4(g)) at the point where the LVL shear velocity becomes normally

dispersive. The generally good resolution of the LVL at low shear velocity is again evident

in Figure 7.4(h). For layer 3, it is naturally better resolved when the overlying LVL is

thinner (Figure 7.4(i)). This is partially due to the fact that since the half-space depth of

14 m is fixed, layer 3 becomes thicker in proportion to the thinning LVL.

When realistic data errors are used (Figure 7.5), although the LVL velocity is still

relatively constrained, its thickness is not resolvable as the misfit minima valley is open-

ended in the h/h0 direction. The appraisal rule noted above for 3% data error now appears

invalid. The general monoticity of the misfit images and presence of only one minimum is

further merit for the use of linear inversion in dispersion from a LVL structure, even though

the data are not continuous. Again, the overall resolution and sensitivity are better at a

slightly thicker and lower velocity LVL than the starting model (Figures 7.5(e) and (f)).

The marked changes between employing realistic data errors as opposed to a 3% level is

in individual layer resolution kernels, where all of the layer 1, 2 and 3 shear velocities are

better resolved (Figures 7.5(g) to (i)), in particular layer 1. Layer 2 is better resolved over

the transition from LVL to HVL and layer 3 better resolved especially when the overlying

LVL is thinner. Layer 4 (Figure 7.5(j)), however, appears much less resolvable, by several

orders of magnitude, due to the much larger error at low frequency.

Case 3 HVL (layer 2) The degree of nonlinearity can be seen in Figure 7.6(a), where

at high β/β0 ratios, there is no smooth convergence towards the true solution, however,

at lower velocity, the layer shear velocity converges steeply. Compared to the Case 2 LVL,

shear velocity appears better constrained, and the thickness at low h/h0 is well constrained.

This suggests that (at the true shear velocity) the HVL is unlikely to be interpreted as

too thin. However, similar to the Case 2 LVL, thickness in increasing h/h0 direction

is poorly constrained, where even twice the true thickness is within the valley of misfit

minima. In the area of model space where overall resolution and sensitivity are best

(Figures 7.6(e) and (f)) is where there is the largest degree of nonlinearity. However,

from Chapter 5, where convergence was generally better from higher velocity starting


models, it can now be seen that this is a consequence of better sensitivity in this area,

dominating the nonlinearity effects, which can be overcome by the ‘jumping’ nature of

the optimisation. As for the resolution of individual layers, unlike the Case 2 LVL, the

Case 3 HVL is poorly resolved around and above its true shear velocity (Figure 7.6(h)).

However, the top layer is best resolved in this area of model space (Figure 7.6(g)). As

for the deeper layers, the underlying LVL is best resolved when the overlying HVL is thin

(Figure 7.6(i)), analogous to the Case 2, layer 3 resolution. Half-space resolution is again

poor (Figure 7.6(j)), improving when the HVL is both thinner and/or higher velocity.

When realistic data errors are incorporated, there is no constraint in the HVL thickness

(Figure 7.7(a)), where a broad valley runs from the extremities in the h/h0 direction.

Shear velocity, however, is still modestly constrained where an allowed 1% RMS error

provides about a ±20% error range in shear velocity. Overall, resolution and sensitivity

are poorer (Figures 7.7(e) and (f)), but there is an improvement in the layer 1 and 2

resolutions (Figures 7.7(g) and (h)). Again, the underlying LVL (layer 3) and half-space

shear velocity are poorly resolved (Figures 7.7(i) and (j)).

Case 3 LVL (layer 3) The Case 3 inversions in Chapter 5 suggested that the LVL

underlying the HVL is poorly resolved. Its shear velocity apparently has little effect on

the dispersion, where the dominant higher mode is most sensitive to the HVL depth and

velocity. When a 3% error is assumed, the misfit plane of Figure 7.8(a) shows a sharp, well

contained minimum valley as a function of both h/h0 and β/β0. While this plane does not

represent the entire solution space, it suggests that convergence to the global minimum

could only be achieved if thickness was also allowed to vary. As thickness is usually fixed,

unless the true thickness is used, the global minimum will not be achieved. Similar to the

overlying HVL, resolution is better at higher β/β0 ratios (Figure 7.8(e)) and although the

convergence gradient is lower, local minima are mostly absent. The increase in nonlinear

effects correlates with model roughness (Figures 7.6(b) and 7.8(b)). Again, the top layer

resolution (Figure 7.8(g)) is excellent but the layer 2 (HVL) resolution is uniformly poor

(Figure 7.8(h)), implying that even if the LVL is correctly interpreted, accuracy of the

overlying HVL cannot be guaranteed. The resolution of the LVL is best when it is of low

velocity (Figure 7.8(i)), which is contrary to the overall model resolution, which is best

when it is at higher velocity.

When realistic data errors are incorporated, the misfit pattern is similar (Figure 7.9(a))

but with a much broader minimum. Considering a 0.5% criteria for convergence, layer

parameters will only be well constrained when it is both thin and of high shear velocity.

It is likely that an interpreted LVL will be thicker and/or higher velocity than the true

values. In this area of model space, resolution is poor and both resolution and sensitivities

are at least twice that of when 3% data errors are used (Figures 7.9(e) and (f)). In spite of

this, layers 1 and 2 are better resolved when the larger data errors are used (Figures 7.9(g)


β (m/s)

Dep

th (

m)

(a) 0.0 < ∆c < 1.0%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(b) 0.0 < ∆c < 2.5%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(c) 0.0 < ∆c < 5.0%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(d) 0.0 < ∆c < 10.0%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(e)

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(f)

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(g)

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(h)

0 200 400 600

0

5

10

15

20

25

30

True Dist.

PDF0

0.5

1

Figure 7.10: Shear wave velocity PDF of Case 1 for various RMS misfits by the

PSV method, with normally distributed VS perturbations. (a) to (d) employ realistic

error envelopes and (e) to (h) employ constant 3% errors.

and (h)). However, layers 3 and 4 are not resolved, to at least one magnitude poorer than

when 3% data errors are used (Figures 7.9(i) and (j)). The resolution of deeper layers

is thus highly dependent on the low-frequency data, which is the band where dominant

higher modes persist for Case 3 and data error increases sharply.

7.3.2 Monte Carlo model resolution The procedure of Section 7.2.1 applied to

the normally dispersive Case 1 model with the PSV method is illustrated in Figure 7.10.

In this figure, and all subsequent figures, the top row (usually (a) to (d)) employs a realistic

noise model for dispersion error in the RMS misfit calculation (modelled in Section 3.5 of

Chapter 3 and the the lower row (usually (e) to (h)) employs a constant 3% dispersion

error. VS is usually perturbed by a normally distributed value with standard deviation of

3σβ′(h). The maximum allowable RMS misfits are chosen according to the usual ranges

arrived at in inversion tests. For the synthetic cases, these are 1%, 2.5%, 5% and 10%.

A total of 300 random perturbations was employed for each case and the starting (true)

model overlain on all plots as the solid, dashed curve.

In Figure 7.10 the normalised probability density function of a given shear velocity with

depth is overlain with the probability function of the Monte Carlo shear velocities, averaged

over the true layer thicknesses. The width of the averaged distribution of feasible shear

velocities increases with both depth and maximum misfit. At low RMS, the distribution

is bimodal, which is due to the smoothing effect between the 3rd and 4th layers, causing

these to be both over- and under-estimated respectively. At larger RMS misfit, while the


β (m/s)

Dep

th (

m)

(a) 0.0 < ∆c < 1.0%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(b) 0.0 < ∆c < 2.5%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(c) 0.0 < ∆c < 5.0%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(d) 0.0 < ∆c < 10.0%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(e)

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(f)

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(g)

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(h)

0 200 400 600

0

5

10

15

20

25

30

True Dist.

PDF0

0.5

1


FSW method.

0 200 400 600

0

5

10

15

20

25

30

(a) 0.0 < ∆c < 1.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(b) 0.0 < ∆c < 2.5%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(c) 0.0 < ∆c < 5.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(d) 0.0 < ∆c < 10.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(e)

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(f)

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(g)

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(h)

β (m/s)

Dep

th (

m)

100 %50 % 10 %

Figure 7.12: Shear wave velocity likelihood ranges for Case 1 for various RMS misfits by

the PSV method, with normally distributed VS perturbations.


β (m/s)

Dep

th (

m)

(a) 0.0 < ∆c < 1.0%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(b) 0.0 < ∆c < 2.5%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(c) 0.0 < ∆c < 5.0%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(d) 0.0 < ∆c < 10.0%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(e)

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(f)

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(g)

0 200 400 600

0

5

10

15

20

25

30

β (m/s)D

epth

(m

)

(h)

0 200 400 600

0

5

10

15

20

25

30

True Dist.

PDF0

0.5

1


PSV method, with equally distributed VS perturbations.

0 200 400 600

0

5

10

15

20

25

30

(a) 0.0 < ∆c < 1.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(b) 0.0 < ∆c < 2.5%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(c) 0.0 < ∆c < 5.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(d) 0.0 < ∆c < 10.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(e)

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(f)

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(g)

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(h)

β (m/s)

Dep

th (

m)

100 %50 % 10 %

Figure 7.14: Shear wave velocity likelihood ranges for Case 1 for various RMS misfits with

equally distributed VS perturbations.


β (m/s)

Dep

th (

m)

(a) 0.0 < ∆c < 1.0%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(b) 0.0 < ∆c < 2.5%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(c) 0.0 < ∆c < 5.0%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(d) 0.0 < ∆c < 10.0%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(e)

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(f)

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(g)

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(h)

0 200 400 600

0

5

10

15

20

25

30

True Dist.

PDF0

0.5

1


PSV method, with normally distributed VS perturbations.

0 200 400 600

0

5

10

15

20

25

30

(a) 0.0 < ∆c < 1.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(b) 0.0 < ∆c < 2.5%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(c) 0.0 < ∆c < 5.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(d) 0.0 < ∆c < 10.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(e)

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(f)

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(g)

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(h)

β (m/s)

Dep

th (

m)

100 %50 % 10 %




β (m/s)

Dep

th (

m)

(a) 0.0 < ∆c < 1.0%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(b) 0.0 < ∆c < 2.5%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(c) 0.0 < ∆c < 5.0%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(d) 0.0 < ∆c < 10.0%

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(e)

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(f)

0 200 400 600

0

5

10

15

20

25

30

β (m/s)

Dep

th (

m)

(g)

0 200 400 600

0

5

10

15

20

25

30

β (m/s)D

epth

(m

)

(h)

0 200 400 600

0

5

10

15

20

25

30

True Dist.

PDF0

0.5

1


PSV method, with normally distributed VS perturbations.

0 200 400 600

0

5

10

15

20

25

30

(a) 0.0 < ∆c < 1.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(b) 0.0 < ∆c < 2.5%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(c) 0.0 < ∆c < 5.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(d) 0.0 < ∆c < 10.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(e)

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(f)

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(g)

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

30

(h)

β (m/s)

Dep

th (

m)

100 %50 % 10 %




bimodal nature is not as evident, there is a definite skew towards higher VS velocities

between 6 m and 14 m depth and lower VS for the half-space. The scatter does no show

a standard distribution, suggesting that error propagation is nonlinear. A similar pattern

emerges when the FSW method is used for the forward calculation (Figure 7.11). However,

there are slightly fewer low shear velocity outliers in the deeper layers.

The results of Figure 7.12 are displayed as the percentage of models with allowed

likelihood in discrete ranges in Figure 7.12. Here, the 100% range comprises all models

and narrows with gradually better fitting models. A similar pattern emerges, with the

most likely models tending towards an overestimate for layer 3 and underestimate for

layer 4. When the VS perturbations are generated by an equally distributed random

variable (Figure 7.13), the distributions are less bell-shaped, but still skewed from the

true shear velocities. The shear velocity likelihood bands (Figure 7.14), more clearly show

the misinterpretation at depth of even the most successful models, where the half-space

shear wave velocity is always lower than true.

The same procedure applied to the LVL model Case 2 are shown in Figures 7.15 and

7.16. The similar smearing between layers 3 and 4 is evident, along with the underestimate

of half-space VS in the most successful models. However, the LVL itself is relatively well

resolved, with the most likely cases interpreting its shear velocity exactly. The chance of

non-detection of the LVL, shown by the skew towards higher VS, is less than 50%. This

was suggested in the ‘blind’ inversion of this case in Figure 5.46. One point to note is that

if a low RMS misfit is sought, realistic dispersion errors will allow only a slightly broader

distribution of successful models.

However, when the HVL model Case 3 is tested for Monte Carlo data likelihood (Fig-

ures 7.17 and 7.18), the range of successful models at low RMS misfit is highly dependent

on dispersion uncertainty. With 3% dispersion errors, only the true model is allowed with

a data likelihood up to 1% (Figure 7.17(e)) but with realistic errors, a model without a

HVL structure may result (Figure 7.17(a)). For 5% misfit and above, the top of the LVL

is interpreted as being 1-2 m too deep with either data weighting (Figures 7.18(c) and

7.18(g)). In addition, due to the poor resolution of the underlying LVL VS, it may not

be interpreted as a buried HVL. This is an important consideration if the thickness of a

buried stiff layer (such as calcarenite) was desired for foundation stiffness potential.

7.3.3 Monte Carlo parameter statistics Monte Carlo sampling also allows a

statistical analysis of the various model parameterisations used for the random analysis.

An example for Case 1 is shown in Figures 7.19 and 7.20, where both realistic and constant

3% dispersion errors are respectively employed. The figures comprise three rows, each

representing a certain model style:

Upper Constant thickness layers;


0 10 20 300

0.2

0.4

0.6

0.8

1

Max. layers

(a)

Typ

e 1

(41.

1%)

0 1 2 3 4Min. thick, h

1 (m)

(b)

0 10 20Max. depth, z

max (m)

(c)

1 2 3 4Thick factor (n) in h=n

(d)

0 5 100

0.2

0.4

0.6

0.8

1

True σ RMS error (%)

(e)

Typ

e 1

(41.

1%)

0 10 20 300

0.2

0.4

0.6

0.8

1

Max. layers

(f)

Typ

e 2

(27.

4%)


1 (m)

(g)


max (m)

(h)

0 10 20 30

Thick factor (n) in hi=h

i−1n

(i)

0 5 100

0.2

0.4

0.6

0.8

1


(j)

Typ

e 2

(27.

4%)

0 10 20 300

0.2

0.4

0.6

0.8

1

Max. layers

(k)

Typ

e 3

(31.

5%)


1 (m)

(l)


max (m)

(m)

1 2 3 4 5Thick factor (n) in h=z/n

(n)

0 5 100

0.2

0.4

0.6

0.8

1


(o)

Typ

e 3

(31.

5%)

Figure 7.19: Model parameter statistics of Monte Carlo perturbation of the Case 1 with

realistic dispersion errors and allowed data misfit up to 10%.

0 10 20 300

0.2

0.4

0.6

0.8

1

Max. layers

(a)

Typ

e 1

(40.

5%)


1 (m)

(b)


max (m)

(c)


(d)

0 5 100

0.2

0.4

0.6

0.8

1

3% σ RMS error (%)

(e)

Typ

e 1

(40.

5%)

0 10 20 300

0.2

0.4

0.6

0.8

1

Max. layers

(f)

Typ

e 2

(29.

0%)


1 (m)

(g)


max (m)

(h)

0 10 20 30


i−1n

(i)

0 5 100

0.2

0.4

0.6

0.8

1

3% σ RMS error (%)

(j)

Typ

e 2

(29.

0%)

0 10 20 300

0.2

0.4

0.6

0.8

1

Max. layers

(k)

Typ

e 3

(30.

5%)


1 (m)

(l)


max (m)

(m)


(n)

0 5 100

0.2

0.4

0.6

0.8

1

3% σ RMS error (%)

(o)

Typ

e 3

(30.

5%)


3% dispersion errors and allowed data misfit up to 10%.


0 10 20 300

0.2

0.4

0.6

0.8

1

Max. layers

(a)

Typ

e 1

(37.

6%)


1 (m)

(b)


max (m)

(c)


(d)

0 5 100

0.2

0.4

0.6

0.8

1


(e)

Typ

e 1

(37.

6%)

0 10 20 300

0.2

0.4

0.6

0.8

1

Max. layers

(f)

Typ

e 2

(32.

6%)


1 (m)

(g)


max (m)

(h)

0 10 20 30


i−1n

(i)

0 5 100

0.2

0.4

0.6

0.8

1


(j)

Typ

e 2

(32.

6%)

0 10 20 300

0.2

0.4

0.6

0.8

1

Max. layers

(k)

Typ

e 3

(29.

9%)


1 (m)

(l)


max (m)

(m)


(n)

0 5 100

0.2

0.4

0.6

0.8

1


(o)

Typ

e 3

(29.

9%)



0 10 20 300

0.2

0.4

0.6

0.8

1

Max. layers

(a)

Typ

e 1

(39.

1%)


1 (m)

(b)


max (m)

(c)


(d)

0 5 100

0.2

0.4

0.6

0.8

1


(e)

Typ

e 1

(39.

1%)

0 10 20 300

0.2

0.4

0.6

0.8

1

Max. layers

(f)

Typ

e 2

(31.

2%)


1 (m)

(g)


max (m)

(h)

0 10 20 30


i−1n

(i)

0 5 100

0.2

0.4

0.6

0.8

1


(j)

Typ

e 2

(31.

2%)

0 10 20 300

0.2

0.4

0.6

0.8

1

Max. layers

(k)

Typ

e 3

(29.

6%)


1 (m)

(l)


max (m)

(m)


(n)

0 5 100

0.2

0.4

0.6

0.8

1


(o)

Typ

e 3

(29.

6%)




Middle Nonlinear geometric increasing thicknesses; and

Lower Linear geometric increasing thicknesses.

For each style, several statistics are analysed and presented as normalised distributions.

From left to right, these are (eg. for the upper row):

(a) Maximum number of layers;

(b) Minimum thickness;

(c) Maximum depth (top of half-space);

(d) Thickness factor; and

(e) RMS data misfit.

The percentages indicated on the vertical axes are the relative proportion of successful

models for that style which generate RMS data misfits up to 10%. The normalised distri-

butions can thus be converted to amounts relative to the 300 Monte Carlo perturbations

by multiplication of this percentage.

In Figure 7.19, the largest proportion of successful models are those with constant

thickness layers and the most likely data misfit about 2.5%. For all model types, however,

the maximum number of layers is usually limited to less than about 20 and the most

likely layer thickness is 1 m. The poor resolution of half-space depth is illustrated by

the broad range of maximum depths. Nonlinear and linearly increasing thicknesses are

less common, but show the optimum minimum thicknesses around 1.5-2.5 m. Linearly

increasing thicknesses also offer a slightly higher probability of achieving a data misfit of

less than 1%. In Figure 7.20, with constant 3% dispersion errors, the statistics are mostly

unchanged except for RMS misfit. In that case, almost all successful models converge to

less than 6% data error.

The irregularly dispersive models Case 2 (LVL) and Case 3 (HVL) statistics for realistic

dispersion errors are in Figures 7.21 and 7.22. Again, constant thickness layers comprise

the largest proportion of successful models with data misfit up to 10%. In Case 2, there

is a minor shift towards thinner layers being preferable, and below a the 5% misfit level,

linearly increasing thicknesses have a high probability (Figure 7.21). With the Case 3

model, the probability with data misfit is more broad and centred around 3.5-4.5%. This

was expected since the dominant higher mode associated with the HVL is in the region of

low-frequency effects and has overall poor resolution.


7.4 Telfer gold mine

7.4.1 High velocity layer The final model of Line 9 (mill site, Figure 6.25) re-

vealed a stiff layer at about 1.5-2 m depth, interpreted as unconsolidated sand over sand-

stone and supported by an unrippable horizon identified in nearby furrows. This model

suggested a weak velocity reversal between about 4-8 m depth, possibly due to secondary

cementation of the uppermost portion of weathered sandstone. Monte Carlo tests showing

both VS probability and likelihood ranges are shown in Figures 7.23 and 7.24. Since those

inversions converged to <1% misfit, only small increments in data misfit are analysed.

Both the PDF images and model likelihood ranges show that the best fitting models do

show a LVL trend, albeit with a broad distribution, even at 0.5% data misfit. However,

if the amount of shear velocity perturbation is randomised from layer to layer (unlike in

Section 7.3.3 where it was constant for all layers at each Monte Carlo iteration), the pat-

tern is much broader and the deeper LVL cannot be interpreted with confidence (Figures

7.23 and 7.24). Nevertheless, the depth and shear velocity of the shallow HVL are very

well resolved with data misfit to 1% and the sandstone best estimated at a constant 700

m/s with depth.

7.4.2 Low velocity layer The final model of Line 11 (waste dump, Figure 6.30)

showed a revealed a stiff, surficial layer about 1-2 m thick, underlain by a thick soft zone

extending to about 7 m depth, below which a gradual increase to 350 m/s attained by 9

m. Monte Carlo tests about this model are shown in Figures 7.27 and 7.28. Similar to

the synthetic cases, the LVL resolution is good, and changes little with maximum allowed

data misfit.

When only the shallow zone is considered (Figure 6.32) the Monte Carlo resolution test

results are shown in Figures 7.29 and 7.30. In this case, the allowed standard deviation of

perturbation for each layer shear velocity is equally distributed within the linear window

of 5% of VS1 and 20% of VSN , where VS1 and VSN are the shear velocities of the shallowest

and deepest (half-space) respectively. The inversion converged to less than 1% RMS error,

and within this allowed misfit, the LVL is well resolved. However, for over 2.5% misfit, the

LVL is less well resolved, with a definite favouring of lower shear velocities, especially at

depth. The estimated half-space depth 6 m was too shallow, which did not allow resolution

of the increasing shear velocity gradient to 12 m depth as inverted in Figure 6.30.

7.4.3 Waveform matching

Tunnel pits The shot gathers at 0.5 m geophone spacing over the tunnel pit 1 (Line

1) were very high quality and amenable to a qualitative waveform matching. It is well

known this problem is highly nonlinear and no attempt is made to optimise the model

with respect to wavefield perturbations. The final models of this site were used to generate

48-channel synthetic shot gathers at 0.5 m geophone spacing and 5 m near offset. A sample


β (m/s)

Dep

th (

m)

(a) 0.0 < ∆c < 0.5%

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

β (m/s)

Dep

th (

m)

(b) 0.0 < ∆c < 1.0%

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

β (m/s)

Dep

th (

m)

(c) 0.0 < ∆c < 2.5%

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

β (m/s)

Dep

th (

m)

(d) 0.0 < ∆c < 10.0%

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

β (m/s)

Dep

th (

m)

(e)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

β (m/s)

Dep

th (

m)

(f)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

β (m/s)

Dep

th (

m)

(g)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

β (m/s)

Dep

th (

m)

(h)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

True Dist.

PDF0

0.5

1

Figure 7.23: Shear wave velocity PDF of Telfer gold mine mill site Line 9 for various

RMS misfits by the PSV method, with normally distributed VS perturbations. (a) to (d)

employ realistic error envelopes and (e) to (h) employ constant 3% errors.

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(a) 0.0 < ∆c < 0.5%

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(b) 0.0 < ∆c < 1.0%

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(c) 0.0 < ∆c < 2.5%

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(d) 0.0 < ∆c < 10.0%

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(e)

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(f)

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(g)

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(h)

β (m/s)

Dep

th (

m)

100 %50 % 10 %

Figure 7.24: Shear wave velocity likelihood ranges for Telfer gold mine mill site Line 9 for

various RMS misfits by the PSV method, with normally distributed VS perturbations.


β (m/s)

Dep

th (

m)

(a) 0.0 < ∆c < 0.5%

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

β (m/s)

Dep

th (

m)

(b) 0.0 < ∆c < 1.0%

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

β (m/s)

Dep

th (

m)

(c) 0.0 < ∆c < 2.5%

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

β (m/s)

Dep

th (

m)

(d) 0.0 < ∆c < 10.0%

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

β (m/s)

Dep

th (

m)

(e)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

β (m/s)

Dep

th (

m)

(f)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

β (m/s)

Dep

th (

m)

(g)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

β (m/s)

Dep

th (

m)

(h)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

True Dist.

PDF0

0.5

1

Figure 7.25: Shear wave velocity PDF of Telfer gold mine mill site Line 9 for various

RMS misfits by the PSV method, with normally distributed VS perturbations, which vary

randomly from layer to layer. (a) to (d) employ realistic error envelopes and (e) to (h)

employ constant 3% errors.

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(a) 0.0 < ∆c < 0.5%

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(b) 0.0 < ∆c < 1.0%

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(c) 0.0 < ∆c < 2.5%

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(d) 0.0 < ∆c < 10.0%

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(e)

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(f)

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(g)

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

16

18

(h)

β (m/s)

Dep

th (

m)

100 %50 % 10 %

Figure 7.26: Shear wave velocity likelihood ranges for Telfer gold mine mill site Line 9

for various RMS misfits by the PSV method, with normally distributed VS perturbations,

which vary randomly from layer to layer.


β (m/s)

Dep

th (

m)

(a) 0.0 < ∆c < 0.5%

0 200 400 600

0

5

10

15

20

25

β (m/s)

Dep

th (

m)

(b) 0.0 < ∆c < 1.0%

0 200 400 600

0

5

10

15

20

25

β (m/s)

Dep

th (

m)

(c) 0.0 < ∆c < 2.5%

0 200 400 600

0

5

10

15

20

25

β (m/s)

Dep

th (

m)

(d) 0.0 < ∆c < 10.0%

0 200 400 600

0

5

10

15

20

25

β (m/s)

Dep

th (

m)

(e)

0 200 400 600

0

5

10

15

20

25

β (m/s)

Dep

th (

m)

(f)

0 200 400 600

0

5

10

15

20

25

β (m/s)

Dep

th (

m)

(g)

0 200 400 600

0

5

10

15

20

25

β (m/s)

Dep

th (

m)

(h)

0 200 400 600

0

5

10

15

20

25

True Dist.

PDF0

0.5

1

Figure 7.27: Shear wave velocity PDF of Telfer gold mine waste dump Line 11 for various



0 200 400 600

0

5

10

15

20

25

(a) 0.0 < ∆c < 0.5%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

(b) 0.0 < ∆c < 1.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

(c) 0.0 < ∆c < 2.5%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

(d) 0.0 < ∆c < 10.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

(e)

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

(f)

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

(g)

β (m/s)

Dep

th (

m)

0 200 400 600

0

5

10

15

20

25

(h)

β (m/s)

Dep

th (

m)

100 %50 % 10 %

Figure 7.28: Shear wave velocity likelihood ranges for Telfer gold mine waste dump Line 11

for various RMS misfits by the PSV method, with normally distributed VS perturbations.


β (m/s)

Dep

th (

m)

(a) 0.0 < ∆c < 0.5%

0 200 400 600

0

2

4

6

8

10

12

β (m/s)

Dep

th (

m)

(b) 0.0 < ∆c < 1.0%

0 200 400 600

0

2

4

6

8

10

12

β (m/s)

Dep

th (

m)

(c) 0.0 < ∆c < 2.5%

0 200 400 600

0

2

4

6

8

10

12

β (m/s)

Dep

th (

m)

(d) 0.0 < ∆c < 10.0%

0 200 400 600

0

2

4

6

8

10

12

β (m/s)

Dep

th (

m)

(e)

0 200 400 600

0

2

4

6

8

10

12

β (m/s)

Dep

th (

m)

(f)

0 200 400 600

0

2

4

6

8

10

12

β (m/s)

Dep

th (

m)

(g)

0 200 400 600

0

2

4

6

8

10

12

β (m/s)

Dep

th (

m)

(h)

0 200 400 600

0

2

4

6

8

10

12

True Dist.

PDF0

0.5

1

Figure 7.29: Shear wave velocity PDF of Telfer gold mine waste dump Line 11 shallow

layers for various RMS misfits by the PSV method, with normally distributed VS per-

turbations. (a) to (d) employ realistic error envelopes and (e) to (h) employ constant

3% errors.

0 200 400 600

0

2

4

6

8

10

12

(a) 0.0 < ∆c < 0.5%

β (m/s)

Dep

th (

m)

0 200 400 600

0

2

4

6

8

10

12

(b) 0.0 < ∆c < 1.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

2

4

6

8

10

12

(c) 0.0 < ∆c < 2.5%

β (m/s)

Dep

th (

m)

0 200 400 600

0

2

4

6

8

10

12

(d) 0.0 < ∆c < 10.0%

β (m/s)

Dep

th (

m)

0 200 400 600

0

2

4

6

8

10

12

(e)

β (m/s)

Dep

th (

m)

0 200 400 600

0

2

4

6

8

10

12

(f)

β (m/s)

Dep

th (

m)

0 200 400 600

0

2

4

6

8

10

12

(g)

β (m/s)

Dep

th (

m)

0 200 400 600

0

2

4

6

8

10

12

(h)

β (m/s)

Dep

th (

m)

100 %50 % 10 %

Figure 7.30: Shear wave velocity likelihood ranges for Telfer gold mine waste dump Line

11 shallow layers for various RMS misfits by the PSV method, with normally distributed

VS perturbations.


5 10 15 20 25

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(a)

5 10 15 20 25

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(b)

5 10 15 20 25

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(c)

5 10 15 20 25

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(d)

Figure 7.31: Comparison of (a) Observed; and (b)-(d) Synthetic shot gathers (see text)

from the Telfer gold mine tunnel pit Line 1.

0 0.05 0.1−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(a) 5.0 m

0 0.05 0.1−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(b) 10.5 m

0 0.05 0.1 0.15 0.2 0.25−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(c) 16.5 m

0 0.05 0.1 0.15 0.2 0.25−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(d) 28.5 m

Observed Synthetic

Figure 7.32: Comparison of observed and modelled seismic traces at the Telfer gold mine

tunnel pit Line 1 at various source offsets.


0 0.05 0.1−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(a) 5.0 m

0 0.05 0.1−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(b) 10.0 m

0 0.05 0.1 0.15 0.2 0.25−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(c) 16.0 m

0 0.1 0.2 0.3 0.4 0.5−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(d) 28.0 m

Observed Synthetic


mill site Line 9 at various source offsets.

0 0.05 0.1−10

−5

0

5

10

Time (s)

Am

plitu

de

(a) 5.0 m

0 0.05 0.1−10

−5

0

5

10

Time (s)

Am

plitu

de

(b) 10.0 m

0 0.05 0.1 0.15 0.2 0.25−10

−5

0

5

10

Time (s)

Am

plitu

de

(c) 16.0 m

0 0.05 0.1 0.15 0.2 0.25−10

−5

0

5

10

Time (s)

Am

plitu

de

(d) 28.0 m

Observed Synthetic


waste dump Line 11 at various source offsets.


interval of 0.5 ms for 512 points is set, with cosine rolloff filters from 0-8 Hz and 500-1000

Hz. While methods exist for estimating a minimum phase, pre-stack wavelet from on-land

impact [270] or explosive [372] sources in elastic media, no attempt was made to optimise

for the source function. Merely, the same 45 Hz causal Berlage wavelet [11] from a vertical

impact at the surface as used in the PSV inversion is employed.

Figure 7.31(a) shows the raw data and Figures 7.31(b) to (d) the synthetic shot gathers

for the tunnel pit models of models of Section 6.3.4, respectively Figures 6.20 (forward),

6.21 (reverse) and 6.23 (forward with fixed 4 m deep half-space). Gathers have a 16 ms

AGC window applied. The moveout of the ground-roll is similar in all cases, as is the

similar velocity and shingling of the guided waves. This is supportive of the assumed

Poisson’s ratio (0.4) being correct and the guided wavefield has been identified by others

as a possible candidate for waveform inversion due to the high frequency content and

pronounced tuning [272].

The best match at later times between the observed and synthetic shot gathers is

probably for Figure 7.31(c), which is the final model of Figure 6.21. Individual trace com-

parisons between the observed data and synthetic shot gather for this model at various

offsets are shown in Figure 7.32. The phase match between the main ground-roll wave-

trains is excellent, with mismatch increasing with offset. However, the surface wave group

velocity envelope and first arrival times are predicted well.

Mill and waste dump sites The results of Figures 6.29 and 6.30 were forward modelled

and waveform matching of individual traces shown in Figures 7.33 and 7.34. The only

estimated synthetic acquisition parameter was source pulse centre frequency, where for the

mill site it was assumed 75 Hz and waste dump site 50 Hz, based on the average spectra

of the observed near offset traces.

In Line 9 (shallow sandstone, Figure 7.33), only the very near offset shows a reasonable

surface waveform match, with increasing phase differences with offset. However, the high

frequency ringing at far offsets is reproduced in the synthetic seismograms and the group

velocity envelope matches the observed ground roll well. In addition, the first arrivals

coincide well at all offsets. In Line 11 (hard cap, Figure 7.34), the waveform match is

good except for the end of the spread, possibly due to scattering or mode conversions,

the lateral discontinuity identified in the observed waveforms and dispersion inversion

of Section 6.3.4. The synthetic waveforms show larger and/or earlier first breaks than

identified in the field data.



At this site, nearby borehole data allowed the surface wave models to be empirically

appraised. However, within the limitations of comparing invasive and surface measure-

ments, one feature of the surface wave inversion is the stiff silty sand layer at 20 m depth

not being interpreted. This was even the case for the inversion of a synthetic dispersion

curve based on the borehole model (Section 6.4.4). However, whether this limitation was

physical (due to the inherent resolution decrease with depth) or experimental (due to

insufficient spread length or data uncertainty) remained uncertain.

7.5.1 Waveform matching Both the ‘borehole’ and ‘blind’ inversion results of the

3 m spacing field data were used to generate synthetic 24-channel shot gathers, also at 3

m geophone spacing. A 35 Hz causal wavelet (Berlage with -90◦ phase) from a vertical

surface impact at 3 m near offset is used. However, a 70 ms delay is added, due to pre-

trigger samples recorded in the field. Otherwise, all parameters were the same as used

during the PSV inversion.

Synthetic traces of final models Figures 6.41 (borehole) and 6.44 (blind) are shown

in Figures 7.35 and 7.36 respectively. The offsets chosen correlate with the first and last

observed traces of shot 1 (3 m and 24 m) and the last traces of shots 2 and 3 (48 m and

72 m), each field shot at successive walkaway near offsets. The waveform fit is best at 3

m near offsets, but the 24 m offset shows a large phase difference between the observed

and modelled waveforms, albeit being from the same shot. However, the group velocity

envelopes coincide, suggesting that the inverted shear velocity models accurate slow in

the shallow zone, the elastic parameters of which dictate the fundamental mode group

velocity. At 48 m offset, there is an apparent good fit with the pulse phases and the

degree of dispersion (pulse spreading is similar), moreso with the borehole inverted model

(Figure 7.35(a) and (c)). However, the group velocity envelope of the modelled traces are

about 50 ms later in time than for the observed data, with a similar lag at 72 m offset.

This suggests too low shear velocities at depth, the elastic parameters which dictate the

longer propagation path dispersion. The inverted models Figures 6.41 and 6.44 showed

lower shear velocities than recorded with a nearby seismic cone penetrometer, thus far

offset surface waveforms may be more indicative of the true stiffness structure at depth.

7.5.2 Hard layer resolution The physical limitation of detecting this layer can be

approached through deterministic modelling. The borehole model of Figure 4.23 is simpli-

fied to a four-layer case and the resolution of the hard layer will be tested at four different

depths (to its top), namely 5, 10, 15 and 20 m, shown in Figure 7.37. A grid search over

perturbations of the thickness and shear velocity about the starting values (5 m and 349

m/s respectively) and measuring data likelihood will indicate the nonuniqueness of the

solution subspace. The thickness of the hard layer is perturbed from 40-200% at 10% in-


0.05 0.1 0.15 0.2 0.25

−5

0

5

Time (s)

Am

plitu

de

(a) 3.0 m

0.1 0.2 0.3 0.4

−5

0

5

Time (s)

Am

plitu

de

(b) 24.0 m

0.2 0.3 0.4 0.5

−5

0

5

Time (s)

Am

plitu

de

(c) 48.0 m

0.2 0.4 0.6 0.8

−5

0

5

Time (s)

Am

plitu

de

(d) 72.0 m

Observed Synthetic

Figure 7.35: Comparison between observed and borehole modelled seismic traces from the

Perth Convention Centre 3 m geophone spacing data at various source offsets.

0.05 0.1 0.15 0.2 0.25

−5

0

5

Time (s)

Am

plitu

de

(a) 3.0 m

0.1 0.2 0.3 0.4

−5

0

5

Time (s)

Am

plitu

de

(b) 24.0 m

0.2 0.3 0.4 0.5

−5

0

5

Time (s)

Am

plitu

de

(c) 48.0 m

0.2 0.4 0.6 0.8

−5

0

5

Time (s)

Am

plitu

de

(d) 72.0 m

Observed Synthetic

Figure 7.36: Comparison between observed and blind modelled seismic traces from the

Perth Convention Centre 3 m geophone spacing data at various source offsets.


0 200 400

0

5

10

15

20

25

30

35

40


Dep

th (

m)

(a)

0 200 400Shear velocity (m/s)

(b)

0 200 400Shear velocity (m/s)

(c)

0 200 400

0

5

10

15

20

25

30

35

40


Dep

th (

m)

(d)

Figure 7.37: Simple models based on the Perth Convention Centre seismic cone penetrom-

eter log SC2, used to start a deterministic grid search for resolution of the hard layer 2.

β/β0

h/h 0

(a)

1

2

2

2

2

2

2

3

3

3

3

3

3

3

3

3

4

4

4

4

4

4

4

4

55

5

6

0.5 1 1.5

0.5

1

1.5

2

β/β0

h/h 0

(e)

1

1

2

2

2

2

2

2

3

3

3

3

3

3

4

4

4

5

0.5 1 1.5

0.5

1

1.5

2

β/β0

(b)

0.5

0.5

1

1

11

1

1.5

2

0.5 1 1.5

β/β0

(f)

0.5

0.5

1

1

1

1.5

1.5

1.5

1.5

2

2

2

2

2.53

0.5 1 1.5

β/β0

(c)

0.1

0.2

0.2

0.20.3

0.3

0.3

0.4

0.4

0.5

0.5 1 1.5

β/β0

(g)

0.5

0.5

1

1

1

1.52

0.5 1 1.5

β/β0

h/h 0

(d)

0.05

0.1

0.1

0.15

0.15

0.2

0.2

0.2

0.25

0.250.

3

0.35

0.4

0.5 1 1.5

0.5

1

1.5

2

β/β0

h/h 0

(h)

0.2

0.2

0.4

0.4

0.4

0.6

0.6

0.6

0.8

1

1.21.41.6

0.5 1 1.5

0.5

1

1.5

2

Figure 7.38: Dispersion RMS relative error for a directed grid search of perturbations of

layer 2 in the models of Figure 7.37. (a) to (d) are for the hard layer at 5, 10, 15 and 20

m respectively, with the RMS error weighted by a true errors and (e) to (f) are the same

but weighted by constant 3% errors in dispersion.


h/h 0

(a) ∆c (%)

11

122

2

345

1

2

h/h 0

(b) ||∂β|| (m/s)

100

150

200

250 1

2

h/h 0

(c) R

0.1

0.10.1

0.15

0.15

0.150.15

0.2 0.2

0.20.2 0.2

0.2

0.2

0.25

0.25

0.250.25

0.250.250.250.25

0.3

0.30.3

0.3

0.3 0.3

0.3

0.30.3

0.351

2

h/h 0

(d) S

0.385

0.385

0.390.395

0.4 0.405 0.41

1

2

h/h 0

(e) δR (m/s)

5

55

55

55

5 55

5

55

10

10

101010

10

15

15 15

20

20 202525 30

1

2

h/h 0

(f) δS (m/s)

3.3 3.

353.

4

3.4

3.4

3.45

3.45

3.45

3.5

3.5

3.5

3.55

3.55

3.55

1

2

h/h 0

(g) log10

(R11

)

−0.001 −0.0008−0.0006

−0.0004

1

2

h/h 0

(h) log10

(R22

)

−5−5

−4.5

−4.5

−4

−4

−3.5−31

2

β/β0

h/h 0

(i) log10

(R33

)

−5−4.5−4.5−4 −4

−3.5 −3.5

−3 −3

−3

−3

0.5 1 1.5

1

2

β/β0

h/h 0

(j) log10

(R44

)−4

.8

−4.6 −4.6

−4.

4

−4.4

−4.

2

−4.2

−4.2

−4

−4

−4

−4

−3.8 −3.8

−3.8

−3.6−

3.4

0.5 1 1.5

1

2

Figure 7.39: Resolution of layer 2 of Figure 7.37(d) with constant 1% data errors. (a) to

(j) are described in the text of Section 7.3.1.

h/h 0

(a) ∆c (%)

0.5

0.5

0.51

1

1

1.5

1.5

22.53

1

2

h/h 0

(b) ||∂β|| (m/s)

100

150

200 250

1

2

h/h 0

(c) R0.10.10.10.2

0.20.3

0.3

0.3 0.3

0.30.3

0.4

0.4

0.40.4

0.50.5

0.50.6

0.6

0.6

0.70.7

0.7

0.81

2

h/h 0

(d) S

0.38

0.38

0.385

0.39

1

2

h/h 0

(e) δR (m/s)

5

5

5

5 5

55

10 1010

1010

15 151520

1

2

h/h 0

(f) δS (m/s)

3.323.333.34

3.35 3.35

3.36

1

2

h/h 0

(g) log10

(R11

)

−0.0013−0.0013

−0.0012

−0.00111

2

h/h 0

(h) log10

(R22

)

−5

−5

−5

−4.5

−4−3.5

−3−2.5−2

1

2

β/β0

h/h 0

(i) log10

(R33

)

−4.5−4−3.5

−3.5

−3

−3−2.5

−2.5−2

0.5 1 1.5

1

2

β/β0

h/h 0

(j) log10

(R44

)

−3.3−3.2

5

−3.2 −3.2

−3.2 −3.15−3.15−3.15

−3.1−3.1

−3.05−3.05

−3 −3

0.5 1 1.5

1

2

Figure 7.40: Same as Figure 7.39 but using the FSW method for dispersion modelling.


tervals of the starting value and shear velocity scanned from 20-150% at 10% intervals of

the starting value of 349 m/s.

To incorporate acquisition dependencies in detecting a HVL at various depths, disper-

sion is generated by the PSV method, under the same conditions as the 56-channel shot

gathers. The relative weighted RMS error is calculated between the perturbed and start-

ing dispersion curves, employing both realistic and 3% data standard deviations. These

are shown in Figure 7.38, where the upper row (a) to (d) employs the ‘true’ dispersion

standard deviations measured from the field walkaway shooting and the lower row (e) to

(h) assumes a constant 3% dispersion uncertainty. The strong undulation in the solu-

tion subspace when the hard layer is at 5 m depth (Figure 7.38(a) and (e)) and in the

higher shear velocity region when at 10 m depth (Figure 7.38(a) and (e)) indicate that

large variations in dispersion are occurring due to the shape of the dominant higher mode

structure. This is not noticeable when the layer is deeper as dominant higher modes are

not expected. In these zones however, if a misfit of 1% is desired, the parameters of the

hard layer at convergence will obviously not be resolved , even if a constant 3% disper-

sion error is applied (Figure 7.38(g) and (h)). With a more realistic dispersion standard

deviation (Figure 7.38(c) and (d)) the possible solution subspace is even wider.

When the case of a hard layer at 20 m is considered, other sensitivity and resolution

measures such as those of Figures 7.2 to 7.9 are shown in Figure 7.39. Moreover, a very

conservative estimate of 1% dispersion error over the entire frequency range measured

(1.5-50 Hz) has been applied, thus no decrease in data weight at low frequency. While

overall resolution and sensitivity of the model is good (Figure 7.39(e) and (f)), resolution

of the layer of investigation is very small, even lower than that of the underlying layers at

some thickness-shear velocity combinations. However, this still incorporates acquisition

dependent resolution (spread length, source bandwidth, plane-wave transform padding,

etc).

A measure of whether the poor resolution of the hard layer is at 20 m is a physi-

cal or experimental limitation can be accomplished through dispersion calculated by the

FSW method, which removes any acquisition parameter-independent effects, such as low-

frequency resolution and plane wave transform limitations. With the layer at this depth,

dominant higher modes are not generated, even at very high thickness-shear velocity,

thus plane wave dispersion remains valid. Again, with an underestimated dispersion er-

ror estimate of 1%, the resolution measures are shown in Figure 7.40. Overall the con-

tours show similarity to the acquisition-dependent values of Figure 7.39. The data misfit

(Figure 7.40(a)) is only slightly steeper than the equivalent PSV solution space, but the

2% data likelihood range covers the entire range of allowable thickness-shear velocity com-

binations for a HVL. The layer 2 resolution (Figure 7.40(h)) is of similar low amplitude

to the PSV resolution (Figure 7.39(h)).


Since inversion of the field dispersion converged to 4-5% with realistic errors, the

range of h/h0 and β/β0 covered in this deterministic grid search shows a subset of the

expected nonuniqueness of the hard layer during an inversion, with either the PSV or

FSW methods. That is, the layer is not resolvable due to the physical limitation of loss of

sensitivity with depth and no surface wave dispersion technique, measured at the surface,

could interpret this layer. However, a waveform inversion may be able to resolve this layer,

albeit with a high degree of nonlinearity in the solution space.


7.6 Road cutting

7.6.1 Granitic overburden The inversion of Line 1 in Section 6.5.3 revealed a

strong dependence on the assumed layer interfaces. Monte Carlo inference around the

final model of Figure 6.50 are shown in Figures 7.41 and 7.42. Even though the inversion

converged to less than 1% misfit, the scatter in VS probability is very broad. In addition,

the best fitting models vary greatly from the final solution where the roughness indicates

a large degree of nonuniqueness. This data cannot be interpreted with confidence below

about 3-4 m depth.

7.6.2 Laterite caprock The inversion of Line 5, located over relatively thick caprock,

revealed little dependence on subdivisions of the uppermost layer, converging to less than

1% misfit. Monte Carlo inference around the final model of Figure 6.53 are shown in Fig-

ures 7.43 and 7.44. A smaller range of allowed data misfits is chosen to show the effect of

the assumed dispersion errors. Up to 0.5% misfit, the assumption of a 3% phase velocity

standard deviation (Figure 7.43(e) to (f)) only permits the final model as an acceptable

solution. Conversely, when realistic dispersion errors are assumed, the full range of feasible

models is achieved with 0.5% data misfit. Although the overall inversely dispersive trend

is maintained in Figure 7.43(a) to (d), even at 0.5% data likelihood, there are instances of

nonuniqueness, where the best fitting models are much rougher than that of the inversion

and invariably of lower shear velocity. However, the very shallow zone shows best fitting

models at higher velocity.

Figure 7.45(a) and (b) shows two, 48-channel walkaway shot gathers, each formed

by concatenation of two 24-channel gathers shot at 1 m and 25 m near offset, in the

forward (western shot) and reverse (eastern shot) directions respectively. The synthetic

shot gathers (Figure 7.45(c) and (d)) are modelled from the inversion solutions of Figures

6.54 and 6.55 respectively. The former was a ‘borehole’ inversion, with the nearby laterite

thickness subdivided into 3 layers. The latter was a ‘blind’ inversion, where a stack

of 15 layers, all 0.5 m thick was used. Note that those inversions were made of the

dispersion from 24-channel gathers at 5 m near offset. Nevertheless, the modelled surface

wave velocity is very similar to that observed in the field. The solution of Figure 6.55

was essentially a smoothed average of that in Figure 6.54 and the nearly imperceptible

difference between Figure 7.45 suggests a nearly linear relationship of waveform changes

to small model perturbations in this structure.

Comparisons of the individual traces over the 5-28 m offsets for the observed data

(western shot) and modelled data with a subdivided laterite horizon (Figure 6.54) are

shown in Figure 7.46, with some similarity, better at the far offsets. Closer to the shot,

observed ringing is not modelled by the synthetics (Figure 7.46(a)), possibly due to un-

consolidated gravels near the shot. This is unlike the stiff surficial layer at Telfer Line 11,

where it was thinner and of lower shear velocity and apparently more laterally homoge-


β (m/s)

Dep

th (

m)

(a) 0.0 < ∆c < 0.5%

0 500 1000 1500

0

2

4

6

8

10

12

14

β (m/s)

Dep

th (

m)

(b) 0.0 < ∆c < 1.0%

0 500 1000 1500

0

2

4

6

8

10

12

14

β (m/s)

Dep

th (

m)

(c) 0.0 < ∆c < 2.5%

0 500 1000 1500

0

2

4

6

8

10

12

14

β (m/s)

Dep

th (

m)

(d) 0.0 < ∆c < 10.0%

0 500 1000 1500

0

2

4

6

8

10

12

14

β (m/s)

Dep

th (

m)

(e)

0 500 1000 1500

0

2

4

6

8

10

12

14

β (m/s)

Dep

th (

m)

(f)

0 500 1000 1500

0

2

4

6

8

10

12

14

β (m/s)

Dep

th (

m)

(g)

0 500 1000 1500

0

2

4

6

8

10

12

14

β (m/s)

Dep

th (

m)

(h)

0 500 1000 1500

0

2

4

6

8

10

12

14

True Dist.

PDF0

0.5

1

Figure 7.41: Shear wave velocity PDF of the road cutting site Line 1 model for various



0 500 1000 1500

0

2

4

6

8

10

12

14

(a) 0.0 < ∆c < 0.5%

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

(b) 0.0 < ∆c < 1.0%

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

(c) 0.0 < ∆c < 2.5%

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

(d) 0.0 < ∆c < 10.0%

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

(e)

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

(f)

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

(g)

β (m/s)

Dep

th (

m)

0 500 1000 1500

0

2

4

6

8

10

12

14

(h)

β (m/s)

Dep

th (

m)

100 %50 % 10 %

Figure 7.42: Shear wave velocity likelihood ranges for of the road cutting site Line 1 model



β (m/s)

Dep

th (

m)

(a) 0.0 < ∆c < 0.1%

0 500 1000

0

2

4

6

8

10

12

14

16

β (m/s)

Dep

th (

m)

(b) 0.0 < ∆c < 0.5%

0 500 1000

0

2

4

6

8

10

12

14

16

β (m/s)

Dep

th (

m)

(c) 0.0 < ∆c < 1.0%

0 500 1000

0

2

4

6

8

10

12

14

16

β (m/s)

Dep

th (

m)

(d) 0.0 < ∆c < 2.5%

0 500 1000

0

2

4

6

8

10

12

14

16

0 500 1000

0

2

4

6

8

10

12

14

16

(e)

β (m/s)

Dep

th (

m)

0 500 1000

0

2

4

6

8

10

12

14

16

(f)

β (m/s)

Dep

th (

m)

β (m/s)

Dep

th (

m)

(g)

0 500 1000

0

2

4

6

8

10

12

14

16

β (m/s)

Dep

th (

m)

(h)

0 500 1000

0

2

4

6

8

10

12

14

16

True Dist.

PDF0

0.5

1

Figure 7.43: Shear wave velocity PDF of the road cutting site Line 5 model for various



0 500 1000

0

2

4

6

8

10

12

14

16

(a) 0.0 < ∆c < 0.1%

β (m/s)

Dep

th (

m)

0 500 1000

0

2

4

6

8

10

12

14

16

(b) 0.0 < ∆c < 0.5%

β (m/s)

Dep

th (

m)

0 500 1000

0

2

4

6

8

10

12

14

16

(c) 0.0 < ∆c < 1.0%

β (m/s)

Dep

th (

m)

0 500 1000

0

2

4

6

8

10

12

14

16

(d) 0.0 < ∆c < 2.5%

β (m/s)

Dep

th (

m)

0 500 1000

0

2

4

6

8

10

12

14

16

(e)

β (m/s)

Dep

th (

m)

0 500 1000

0

2

4

6

8

10

12

14

16

(f)

β (m/s)

Dep

th (

m)

0 500 1000

0

2

4

6

8

10

12

14

16

(g)

β (m/s)

Dep

th (

m)

0 500 1000

0

2

4

6

8

10

12

14

16

(h)

β (m/s)

Dep

th (

m)

100 %50 % 10 %

Figure 7.44: Shear wave velocity likelihood ranges for the road cutting site Line 5 for

various RMS misfits by the PSV method, with normally distributed VS perturbations.


0 10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(a)

0 10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(b)

0 10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(c)

0 10 20 30 40 50

0

0.05

0.1

0.15

0.2

0.25

Offset (m)

Tim

e (s

)

(d)

Figure 7.45: Comparison between observed (a) forward and (b) reverse (flipped) shot

gathers with synthetic shot gathers (c) and (d), modelled from two different inversion

solutions (see text) from the road cutting site Line 5.

0 0.05 0.1−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(a) 5.0 m

0 0.05 0.1−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(b) 12.0 m

0 0.05 0.1 0.15 0.2 0.25−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(c) 20.0 m

0 0.05 0.1 0.15 0.2 0.25−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(d) 28.0 m

Observed Synthetic

Figure 7.46: Comparison between observed and modelled seismic traces from the road

cutting site Line 5 at various source offsets.


nous. The laterite here is of much higher velocity which, combined with strong lateral

heterogeneity, makes waveform matching difficult.



7.7.1 Common shot data The inversion results of the single shot vertical and

inline component dispersion (Figures Figure 6.64 and Figure 6.65) converged to weighted

RMS misfits less than 0.5%, thus the Monte Carlo inference around the final model are

shown over low ranges of allowed data misfits to investigate if models with even lower data

likelihood exist in the vicinity. For the vertical component (Figures 7.47 and 7.48), there

are no better fitting models at 0.2% misfit and at 0.5% misfit, the scatter is still small

for realistic dispersion uncertainty (Figure 7.47(b)). The indication that the final inverted

model is indeed the best fitting model (global minimum) is supported by the shear velocity

ranges at higher misfit (Figure 7.48(d)). With 3% dispersion errors, the model space is

even less restricted, not providing a realistic scatter of possible models (Figures 7.47 and

7.48 (e) to (h)).

When the model from the inline component dispersion is analysed, the possible shear

velocity ranges are even more restricted (Figures 7.49 and 7.50). Only at realistic data

misfits above 1% (and 2% misfit for 3% dispersion errors) does a scatter of models emerge.

There is an anomaly at 1% misfit where the best-fitting model is quite different from the

final model (Figure 7.50(c)), but this may be due to the limited range of model space

scanned (300 Monte Carlo iterations). In general, the best-fitting models follow the trend

of the final model, with discrepancies only with the deeper layer shear velocities and thin,

high velocity layers (‘rough’ layers).

The final inverted models from both vertical and inline component dispersion of the

test array (Figures 6.64 and 6.65) were used to generate synthetic shot gathers with a 35

Hz causal wavelet from an explosion at 0.3 m the source. The same 4.5-44.5 m offset range

as used in the inversion of field data was modelled and comparisons between observed and

theoretical waveforms at various offsets along the spread are shown in Figures 7.51 and 7.52

for the vertical and inline components respectively. A similar waveform shape is observed

for the fundamental mode of both components up to near offsets of 30.5 m. Further

offsets and/or higher surface wave modes, along with scattered wavefield influence, are not

properly modelled. The slight phase lag in the modelled vertical component waveforms

(Figure 7.51) suggests that the shallow shear velocity of the inverted model is too low,

which for the uppermost layer (0.78 m thick) shear velocity was 25 m/s slower than that

interpreted from the inline component data (254 m/s as opposed to 279 m/s). However,

because the lag does not increase with offset, it is inferred to be due to an early trigger

error in the raw data. This problem is not evident in the inline component (Figure 7.52)

which show a modest fundamental mode surface waveform fit up to 17.5 m (albeit clipping

of the observed waveforms at the very close offsets) which is an effective appraisal of the

inverted models.


0 200 400 600 800

0

5

10

15

20

25

30

35

40

(a) 0.0 < ∆c < 0.2%

β (m/s)

Dep

th (

m)

β (m/s)

Dep

th (

m)

(b) 0.0 < ∆c < 0.5%

0 200 400 600 800

0

5

10

15

20

25

30

35

40

β (m/s)

Dep

th (

m)

(c) 0.0 < ∆c < 1.0%

0 200 400 600 800

0

5

10

15

20

25

30

35

40

β (m/s)

Dep

th (

m)

(d) 0.0 < ∆c < 2.0%

0 200 400 600 800

0

5

10

15

20

25

30

35

40

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(e)

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(f)

β (m/s)

Dep

th (

m)

β (m/s)

Dep

th (

m)

(g)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

β (m/s)

Dep

th (

m)

(h)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

True Dist.

PDF0

0.5

1

Figure 7.47: Shear wave velocity PDF of Hyden fault scarp vertical component for various



0 200 400 600 800

0

5

10

15

20

25

30

35

40

(a) 0.0 < ∆c < 0.2%

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(b) 0.0 < ∆c < 0.5%

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(c) 0.0 < ∆c < 1.0%

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(d) 0.0 < ∆c < 2.0%

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(e)

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(f)

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(g)

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(h)

β (m/s)

Dep

th (

m)

100 %50 % 10 %

Figure 7.48: Shear wave velocity likelihood ranges for Hyden fault scarp vertical compo-

nent for various RMS misfits by the PSV method, with normally distributed VS pertur-

bations.


0 200 400 600 800

0

5

10

15

20

25

30

35

40

(a) 0.0 < ∆c < 0.2%

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(b) 0.0 < ∆c < 0.5%

β (m/s)

Dep

th (

m)

β (m/s)

Dep

th (

m)

(c) 0.0 < ∆c < 1.0%

0 200 400 600 800

0

5

10

15

20

25

30

35

40

β (m/s)

Dep

th (

m)

(d) 0.0 < ∆c < 2.0%

0 200 400 600 800

0

5

10

15

20

25

30

35

40

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(e)

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(f)

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(g)

β (m/s)

Dep

th (

m)

β (m/s)

Dep

th (

m)

(h)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

True Dist.

PDF0

0.5

1

Figure 7.49: Shear wave velocity PDF of Hyden fault scarp inline component for various



0 200 400 600 800

0

5

10

15

20

25

30

35

40

(a) 0.0 < ∆c < 0.2%

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(b) 0.0 < ∆c < 0.5%

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(c) 0.0 < ∆c < 1.0%

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(d) 0.0 < ∆c < 2.0%

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(e)

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(f)

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(g)

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(h)

β (m/s)

Dep

th (

m)

100 %50 % 10 %

Figure 7.50: Shear wave velocity likelihood ranges for Hyden fault scarp inline component



0 0.05 0.1−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(a) 4.5 m

0 0.05 0.1−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(b) 17.5 m

0 0.05 0.1 0.15 0.2 0.25−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(c) 30.5 m

0 0.05 0.1 0.15 0.2 0.25−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(d) 44.5 m

Observed Synthetic

Figure 7.51: Comparison between observed and modelled seismic traces from the Hyden

fault scarp vertical component data at various source offsets.

0 0.05 0.1−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(a) 4.5 m

0 0.05 0.1−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(b) 17.5 m

0 0.05 0.1 0.15 0.2 0.25−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(c) 30.5 m

0 0.05 0.1 0.15 0.2 0.25−6

−4

−2

0

2

4

6

Time (s)

Am

plitu

de

(d) 44.5 m

Observed Synthetic

Figure 7.52: Comparison between observed and modelled seismic traces from the Hyden

fault scarp inline component data at various source offsets.


0 200 400 600 800

0

5

10

15

20

25

30

35

40

(a) 0.0 < ∆c < 0.2%

β (m/s)

Dep

th (

m)

β (m/s)

Dep

th (

m)

(b) 0.0 < ∆c < 0.5%

0 200 400 600 800

0

5

10

15

20

25

30

35

40

β (m/s)

Dep

th (

m)

(c) 0.0 < ∆c < 1.0%

0 200 400 600 800

0

5

10

15

20

25

30

35

40

β (m/s)

Dep

th (

m)

(d) 0.0 < ∆c < 2.0%

0 200 400 600 800

0

5

10

15

20

25

30

35

40

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(e)

β (m/s)

Dep

th (

m)

β (m/s)

Dep

th (

m)

(f)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

β (m/s)

Dep

th (

m)

(g)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

β (m/s)

Dep

th (

m)

(h)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

True Dist.

PDF0

0.5

1

Figure 7.53: Shear wave velocity PDF of Hyden fault scarp stacked dispersion over SP1011

to SP1060 for various RMS misfits by the PSV method, with normally distributed VS per-

turbations. (a) to (d) employ realistic error envelopes and (e) to (h) employ constant

3% errors.

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(a) 0.0 < ∆c < 0.2%

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(b) 0.0 < ∆c < 0.5%

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(c) 0.0 < ∆c < 1.0%

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(d) 0.0 < ∆c < 2.0%

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(e)

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(f)

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(g)

β (m/s)

Dep

th (

m)

0 200 400 600 800

0

5

10

15

20

25

30

35

40

(h)

β (m/s)

Dep

th (

m)

100 %50 % 10 %

Figure 7.54: Shear wave velocity likelihood ranges for Hyden fault scarp stacked dispersion

over SP1011 to SP1060 for various RMS misfits by the PSV method, with normally

distributed VS perturbations.


1001 1011 1021 1031 1041 1051 1061 1071 1081 1091

0

0.05

0.1

0.15

0.2

0.25

Shotpoint (m)

Tim

e (s

)

(a)

1001 1011 1021 1031 1041 1051 1061 1071 1081 1091

0

0.05

0.1

0.15

0.2

0.25

Shotpoint (m)

Tim

e (s

)

(b)

Figure 7.55: Comparison between shot gathers of the Hyden fault scarp rollalong P -wave

reflection data and stacked dispersion inversion models: (a) Observed 96-channel data from

SP1066; and (b) Synthetic shot gather of model from 1D inversion of stacked dispersion

over 50-channel window SP1011 to SP1060.

7.7.2 Rollalong data The inversion of stacked dispersion (Figure 6.66) converged

to a weighted RMS misfit of 0.25%, thus the Monte Carlo inference around the final model

are shown over a low range of allowed data misfits (Figures 7.53 and 7.54). However, at

0.2% misfit, after 150 iterations no random models pass the data likelihood test, both for

for realistic (Figure 7.53(a)) and 3% (Figure 7.53(e)) dispersion errors. This suggests that

the model is in a well-bounded global minimum, but a much larger span of model space

may reveal otherwise. With small increases of allowed data misfit, the scatter in possible

models broadens rapidly, with distributions showing two or more maxima, averaged over

depths between the input model interfaces. In general, there is a preference to lower shear

velocities (Figure 7.54), but at all data misfits, the depth and trend of the buried HVL is

repeated. In general, the best fitting models predict a thinner HVL.

A waveform match of a synthetic shot gather calculated for the 1D model of Figure 6.66

is shown in Figure 7.55. Again, a 35 Hz explosive source at 0.3 m depth is employed. The

synthetic split-shot gather is actually is symmetrical about the shotpoint due to the 1D

restriction of P -SV reflectivity modelling. However, the trace normalised shot gathers

show matching moveouts of the fundamental ground-roll mode to about 30 m near offset,

albeit with less ‘shingling’ in the synthetic data. At further offsets the match is poorer,


the difference mainly due to lateral discontinuities over the large range of offsets causing

scattering and mode conversions to arrivals at earlier times. A more accurate comparison

would possibly be made by using the section of Figure 6.69 and a 2D finite-difference code

to properly model the lateral variation.



7.8.1 Synthetic models Homogenous half-space shear wave velocity is poorly re-

solved in all cases. Shear velocity of a LVL is well resolved, slightly better than for a HVL,

however, the maximum thickness for each is poorly resolved. A LVL at depth has a strong,

almost linear thickness-shear velocity tradeoff. Both the shallow HVL and underlying LVL

in Case 3 show nonlinear and multiple minima in the solution subspaces. Nonuniqueness

is exacerbated when realistic dispersion errors are introduced. Monte Carlo misfit tests in-

variably show best-fitting models with lower shear velocity than the true values for deeper

layers. Moreover, distributions are not Gaussian, when both realistic and 3% errors are

assumed in the weighted RMS misfit. A LVL at depth, however, shows preference towards

higher shear velocity. In general, with assumed layer interfaces, the shear velocities be-

tween the homogenous half-space and overlying layer become a smoothed average. The

best layering option for inverted shear velocity accuracy is for stacks of constant thickness

layers, instead of geometrically (linearly or nonlinearly) increasing thickness with depth.

7.8.2 Field models

Telfer gold mine The large contrast HVL at shallow depth (sandstone) is well resolved

for its depth and shear velocity. However, the LVL below this interface cannot be inter-

preted with confidence. A LVL beneath a thin, cemented surficial layer is better resolved,

but at depth the best-fitting models are generally at lower shear velocity than the inverted

model. Moreover, scatter is not Gaussian in either case, the distribution clearly bi-modal

in the LVL structure at depth. Waveform matching of the closely spaced shot gathers over

backfilled and compacted material is exceptional and conclusive of the inversion accuracy.

Over a shallow HVL, phase matching is better at near offsets, but the ringing at far off-

sets is duplicated and group velocity envelopes match well. On a cemented caprock over

inhomogenous rockfill, phase matching is more successful than the HVL case, but prone

to lateral cementation and underlying compaction variations.

Perth Convention Centre Waveform matching at near offsets of the 3 m spacing data

is good, with group velocity lags introduced at farther offsets. The 5 m thick stiff layer at

20 m depth cannot be interpreted with any surface wave dispersion method, due to loss

of sensitivity with depth.

Road cutting Soft overburden, as well as granitic basement is poorly resolved. How-

ever, laterite caprock and the softer substratum is better resolved, but solution space

broadens rapidly above optimum data likelihoods, especially for the soft homogenous half-

space. Waveform matching is best at far offsets.

Hyden fault scarp Single CSG inline component dispersion is inverted to a model with

narrower global minimum (better resolved) than the vertical component. Soft layers un-

der a buried laterite show large nonuniqueness, similar to the synthetic LVL at depth.

Waveform matching is also better with the inline component. Stacked vertical compo-


nent dispersion models show more even scatter with data likelihood, however, synthetic

seismograms at far offsets do not show higher modes due to observed wavefield splitting.

379

CHAPTER 8

Conclusions and recommendations

8.1 Conclusions from this work

The three major contributions of this work towards the use of surface wave dispersion

for shallow site investigation are:

1. Dominant higher mode simulation and inversion;

2. Realistic dispersion curve uncertainty; and

3. Model resolution and accuracy appraisal.

All features have been tested on synthetic models which generate dispersion with dom-

inant higher modes, that is, low- and high-velocity layers (LVL/HVL) typical of many

engineering sites. The same dispersion features have been observed and inverted in field

data from a variety of engineering site investigations in Western Australia.

8.1.1 Dominant higher mode simulation and inversion The kernel for forward

modelling is a full-waveform P -SV reflectivity method to generate synthetic shot gathers

from which a phase velocity dispersion curve is automatically observed using standard

multichannel f − k or f − p transform. Surface and body wavefields are generated and

dominant higher surface (Rayleigh) wave modes are properly modelled. By simulation of

the actual field test, incorporating acquisition and processing parameters, experimental-

dependent effects are also incorporated. The numerical accuracy is verified by comparison

to several other independent methods from the literature.

The use of dispersion curves over a different dataset, such as plane-wave spectra or

waveforms, has two advantages: (1) It permits experimental errors to be easily incorpo-

rated for better measures of final model resolution and accuracy; and (2) Allows rapid

analytic partial derivatives to be employed in a linearised inversion scheme.

Although the nonlinearity of the problem increases from a normally dispersive site

(nearly linear) to a LVL under caprock and a buried HVL (multiple solution minima), the

linearised inversion converges well, from poor starting models and parameter assumptions.

Some specific issues identified and addressed are:

1. Plane-wave matrix methods will not interpret a LVL and overestimate the depth,

thickness and shear velocity of a HVL and deeper layers;

2. In normally dispersive cases, Poisson’s ratio can be underestimated by 25% (VP un-

derestimate of 40%) for inverted shear velocity convergence to 10%;

3. A LVL case is less sensitive to Poisson’s ratio underestimates than a HVL case;

4. In general, Poisson’s ratio is best slightly overestimated in LVL/HVL cases, but

avoiding the fluid limit (close to 0.5);

380 Chapter 8. Conclusions and recommendations

5. Convergence to local minima in LVL/HVL cases with realistic dispersion error can be

avoided with a starting shear velocity overestimate, or, generated by the approximate

inversion method;

6. A stack of constant layer thicknesses should comprise layers a minimum of 15% the

depth to the homogenous half-space, if it’s depth is known a priori ; and

7. A large number of thin layers causes ‘smearing’ between inverted and true shear

velocity contrasts at depth, particularly at the half-space depth, along with ‘rough’

structure.

In field tests, although the longest measured wavelength exceeded the theoretical measur-

able limit (0.4L, see below), recovery of shear velocity structure down to over a half (or

more) of the spread length was attained. Some other points of note are:

1. Shallow HVL and LVL structures generate dominant higher modes, predicted by

theory and properly inverted;

2. Stiffness of homogenous fill is recovered well, less so its thickness;

3. Digging furrows to emplace the seismic array closer to a horizon of interest causes

scattering - more accurate results are obtained with the overburden is in situ;

4. A 5 m thick stiff layer at 20 m depth is not interpreted with a number of acquisition

layouts (56 x 1 m or 24 x 3 m geophone spacing spreads);

5. Laterite caprock thickness is interpreted based on a minimum stiffness criteria;

6. Vertical and inline component dispersion cause different interpretations; and

7. Dispersion spectra stacked with near offset provides smoother data in areas of lateral

heterogeneity, and inverted sections correlate with the expected structures, albeit the

averaging effect of dispersion observed over a linear array.

Some cases identified from field tests where pitfalls exist for employing the PSV method

are for inverting:

1. Large dispersion discontinuities generated by steep shear velocity gradients in the

very shallow zone;

2. Transition from an inversely dispersive Rayleigh mode to normally dispersive flexural

mode; and

3. Dispersion with multiple modal transitions between fundamental and dominant

higher modes.

In the last case, dispersion is best inverted as the fundamental mode alone if sufficient

signal to noise ratio allows its identification.

8.1. Conclusions from this work 381

8.1.2 Realistic dispersion curve uncertainty Parametric tests employing con-

stant wavenumber functions (dispersion curves) and full-waveform simulation of synthetic

models (with parametric and Monte Carlo acquisition and processing parameter/error

generation revealed that the repeatability envelope of a multichannel dispersion curve is

nonlinear, proportional to the inverse logarithm of frequency. This is based on the theo-

retical spread length (L)-dependent slowness resolution with frequency (f): ∆p = 1/(fL).

Some specific findings were:

1. Maximum measurable wavelength is 0.4L, that is, L should be at least 2.5 that of

the longest desired wavelength;

2. Trace padding should be at least to 128 traces;

3. The f − p method is generally preferred to the f − k method for dispersion;

4. The f−k method is more accurate in mapping inverse dispersion, assuming minimum

trace padding is applied;

5. ‘Near-field effects’ due to surface wavefield cylindrical spreading are slightly depen-

dent on shear velocity model dependent, but only extend to about 0.15λ;

6. Long spreads satisfying the L > 2.5λmax rule are not subject to near offset variation

effects;

7. ‘Low-frequency effects’ (from both a near-offset perspective and individual acquisi-

tion error effects) are not Gaussian distributed, where larger outliers better modelled

with a Lorentzian;

8. Higher frequency distribution (due to near-offset variations and/or individual acqui-

sition errors) are Gaussian distributed;

9. Dominant higher mode dispersion from a buried HVL structure is especially prone

to low-frequency effects;

10. Depth of explosive source is a primary consideration in dominant higher mode in-

version, even in normally dispersive structures;

11. Most influential individual sources of dispersion error are (from most to least) –

additive Gaussian noise, DC seismogram shifts, trace-to-trace static errors, geophone

placement error and propagating noise;

12. Least influential error sources are – geophone tilt error, source frequency, geophone

coupling and dead traces;

13. Incorporating near offset, spread length and channel density, dispersion at all fre-

quencies is Lorentzian distributed;


14. The ‘background’ standard deviation at high frequency is around 1%;

15. Around the frequencies of modal transitions, errors are several-fold larger;

16. Large, nonlinear increase in error at low frequency can exceed 30%, still within the

usable frequency range; and

17. A buried HVL dispersion is both contaminated by large error at low frequency,

over the dominant higher mode bandwidths, along with poor signal to noise due to

non-propagation of low-frequency source components.

The noise model of multichannel surface wave dispersion (Section 3.6.5) was verified by

repeated field testing. Other discoveries from experimental observations are:

1. Shorter spreads and/or near offsets preferred where possible - longer offsets only for

depth resolution as necessary;

2. CSG and CMP dispersion over coincident trace locations match well;

3. Walkaway shooting is a robust method to increase trace density and/or volume, with

minimal Gaussian-distributed error inclusion; and

4. Multiple geophone spacing and shot offsets (with redundant shooting) are recom-

mended.

Repeatability envelopes observed form repeated field tests are used for data weights in the

inversion.

8.1.3 Model resolution and accuracy appraisal From synthetic model appraisal

methods (deterministic grid search and Monte Carlo likelihood scans), many observations

are made from the inference of models which generate dominant higher modes:

1. Homogenous half-space shear wave velocity is poorly resolved in all cases;

2. Shear velocity of a LVL is well resolved, slightly better than for a HVL, however,

the maximum thickness for each is poorly resolved;

3. Both a shallow HVL and the underlying LVL show nonlinear and multiple minima

in the solution subspaces;

4. Nonuniqueness is exacerbated when realistic dispersion errors are introduced;

5. Best-fitting Monte Carlo models show lower shear velocity than the true values for

deeper layers and distribution of scatter is not Gaussian;

6. With realistic data errors, a LVL at depth shows preference towards higher shear

velocity than the true value;

8.1. Conclusions from this work 383

7. With assumed layer interfaces, the shear velocities between the homogenous half-

space and overlying layer become a smoothed average; and

8. The best layering option for inverted shear velocity accuracy is with stacks of con-

stant thickness layers.

Similar tests to models inverted from field data revealed:

1. A large contrast HVL at shallow depth (sandstone) is well resolved for depth and

shear velocity;

2. A deeper LVL below a HVL cannot be interpreted with confidence, where the dis-

tribution of most likely models is bimodal;

3. An LVL beneath a thin, cemented surficial layer is well resolved;

4. A 5 m thick stiff silt-sand layer at 20 m depth, in a mostly soft, clay background can-

not be interpreted with any surface wave dispersion method, due to loss of sensitivity

with depth;

5. Laterite caprock and the softer substratum is well resolved for shear velocity struc-

ture;

6. Single CSG inline component dispersion is inverted to a better resolved model than

the vertical component; and

7. Soft layers under a buried laterite horizon show large nonuniqueness.

Qualitative waveform matching (between field and modelled seismograms) provided excel-

lent heuristic model appraisal, where some observations for better phase matching include:

1. Homogenous fill - excellent at all offsets;

2. Shallow HVL - better at near offsets, but ringing at far offsets and group velocity

envelopes match well;

3. Cemented caprock - at near offsets, prone to lateral cementation and underlying

compaction variations;

4. Thick soft material - at near offsets;

5. Thick stiff material - at far offsets;

6. Inline component - moreso than vertical component.


In summary, this new work has provided more accurate model estimates, with improved

understanding of model likelihood, in shallow surface wave inversion.

The most difficult parameters to accurately interpret (aside from deep, thin layers,

outside the theoretical resolution) with surface wave inversion will be layers below a buried

HVL, for example, sands under a cemented limestone or clays under a laterite horizon,

including the deeper, stiff homogenous half-space, if present.

8.2 Recommendations for further work

8.2.1 Lateral discontinuity effects In shallow surface wave field work, 2D effects

can be assumed from dispersion curves which vary with propagation direction, usually

from forward and reverse shots. This is an obvious problem with larger spreads, as phase

velocity dispersion is an average over the array and lateral discontinuities may be smeared

[241]. This problem has been addressed semi-quantitatively through 2D finite-difference

seismogram modelling and 1D analysis of the dispersion. Same analysis of field data over

obvious lateral discontinuities supports the errors observed in numerical 2D modelling

[313]. However, while 1D modelling continues to be employed and before a full 2D anal-

ysis becomes achievable, a thorough quantitative understanding is needed on the effects

2D earth has on 1D models. Dipping and undulating layers as well as abrupt discontinu-

ities and localised geological anomalies will all affect the wave propagation, primarily by

scattering and mode conversions.

A parametric approach to this problem is through numerical modelling and analysis of

a variety of geological and acquisition scenarios. A finite-difference synthetic seismogram

code would be an obvious contender for the forward modelling, however, a more rapid,

even if more approximate, method would be desirable time-wise. In any case, an accu-

rate modelling of surface waves would be essential, which excludes ray-based methods.

Knowing the ‘geological’ noise envelopes would enable limits on where 1D inversion be-

comes unsuitable to model the field data and recommendations on approximate detection

of lateral discontinuities could then be made. The noise envelopes revealed in the 1D

study of this thesis may be useful information. Analysis of dispersion would be an initial

methodology, but extension to waveforms or a suitable secondary observable would be a

logical step in this work.

A more rigorous approach to this problem may require tomographic imaging methods.

Alternatively, the perturbation theory work of [36, 24] may provide an analytic solution,

combined with a global search optimisation [277, 278] to interpret and appraise dipping

layered models.

8.2.2 Model appraisal Neglecting 2D effects, the geotechnical ‘accuracy’ of 1D

models in ideal scenarios still needs to be quantitatively understood and categorised. It

is often seen in borehole logs that traditional destructive geotechnical log data does not

8.2. Recommendations for further work 385

correlate perfectly with geophysical log data taken from the same borehole. An example

from this thesis is the Perth Convention Centre site, where the penetrometer hardness only

correlated with the shear wave velocity over broad zones. Indeed they are two different

parameters, a mechanical versus physical response respectively.

This problem could be approached from two directions: (a) Theoretically, employing

Bayesian inferences to statistically limit unfeasible results, or; (b) Experimentally, by ei-

ther a thorough physical modelling study or surveying a number of sites with good borehole

control. In particular, the effects of a water table would be a vital consideration as well

as velocity gradiations, such as the saprolite-crystalline basement weathering sequence. In

either approach, a neural network or fuzzy logic training algorithm may be required to

account for the immense geological/geophysical library which would be generated for var-

ious profiles and lithologies. The scatter in inverted models by various parameterisation

and optimisation techniques would be an optional inclusion for this work.

8.2.3 Higher mode and coupled inversion A primary candidate for a coupled

inversion would employ surface wave inversion and first arrival times (refractions). Since

refraction algorithms are well established, they could provide better constraints on the

layer thicknesses and Poisson’s ratios assumed in dispersion inversion, which in turn would

resolve ambiguities when velocity reversals are present. While this was used in [81], more

than one seismic component (vertical, radial or transverse) has not been reported.

Another coupling could be with Rayleigh and Love wave dispersion. Three component

recording and full-waveform methods would be preferred for the forward modelling. In-

line (radial) component dispersion may also be a possible dataset, as it does vary from the

vertical component dispersion in full-waveform observation and modelling. Alternatively,

a different non-seismic, non-wave parameter such as electromagnetic (EM) or geoelectrical

(VES) soundings could be used. While coupled VES soundings have been studied [211, 53]

high-frequency EM has not been investigated. From a Western Australian regolith point

of view, this would be useful for groundwater detection, even employing airborne EM

datasets.

8.2.4 Alternative optimisations Almost entirely, local, linear methods with var-

ious derivations (eg. Occam’s) have been employed in surface wave (dispersion) inversion.

Publications on the application of global inversion for shallow surface wave prospecting,

both SASW and MASW, have appeared since about 10 years ago and include simulated

annealing, genetic algorithms, and neural networks. In the course of this thesis, it was

occasionally noticed that intermediate solutions were often much better than the final solu-

tion. This is mostly due to the highly nonlinear nature of the solution space for irregularly

dispersive cases which generate dominant higher modes.

The application of the Neighbourhood Algorithm (NA) [280] remains untested in shal-

low surface wave inversion. A new datum, other than dispersion, may be viable as partial


derivatives are not required and final model appraisal would also be important.

8.2.5 Waveform inversion This has been researched and applied in the earth-

quake seismology field, both over single and multiple (tomographic) source-receiver paths

and arrays, but not rigorously in shallow surface wave inversion. The waveform fits of

Chapter 7 are supportive for research into this field. Initially, a 1D inversion employ-

ing the surface wave components of seismograms would be an interesting comparison to

standard 1D dispersion inversion, followed by multi-component data inversion.

A derivative-free global search would be preferred, possibly require the use of new

secondary observable as datum. The wavelet transform may provide a useful compression

of seismograms whose coefficients could be used in the optimisation. For a true wave-

form inversion, all the knowledge of noise and resolution envelopes, 2D effects and modal

appraisals for dispersion modelling would need to be reformulated and reassessed.

8.2.6 Signal processing In most surface wave inversion, the accuracy and resolu-

tion of the dispersion curve is vital. Methods to reduce smearing and undesired wavefield

interference such as time-frequency methods [193, 247] or Bayesian statistics [333]. Indeed,

such methods could be applied to filter the raw data first.

There is considerable opportunity to overlap the above suggestions for an overall im-

proved procedure, such as: Wavelet transforms of synthetic seismograms to invert desired

portions of the observed wavefield (surface/guided/body) by a NA optimisation with a

priori constraints. If this could be accomplished economically in 2D or 3D the method

would probably replace many existing geophysical and geotechnical methods.

8.3 Postscript

Shallow surface wave inversion has probably not been as widely employed in commercial

engineering applications as it has in academic earthquake seismological investigations. In

global applications, such the structure of the crust and upper mantle, detecting global free

oscillations and nuclear test ban treaty monitoring there are no immediate economic or

safety concerns for the practitioners, nor are there any ‘invasive’ methods by which the

results can be proven or disproven. In site investigations, however, critical decisions must

be made and are unlikely to be based on geophysical results alone, especially only one

survey method. Traditional invasive tests will probably always be required to support the

geophysics results for use in engineering design. Commercial shallow surface wave systems

do exist, however, care must be made if they do not both properly simulate dominant

higher modes in irregularly dispersive sites and give an indication of the accuracy of the

output of the inversion procedure.

A discussion in [48] revealed the strong application overlap, yet large communication

gap, between geophysics and civil engineering on surface wave inversion. This work has

8.3. Postscript 387

hopefully added to the cross-discipline knowledge and suggest further applications for the

method.


389

APPENDIX A

Glossary

A.1 Selected terminology

Airy phase A maximum or minimum on the group velocity dispersion curve due to

energy from a range of periods arriving simultaneously [297].

dispersion 1. A change in propagating pulse shape due to a frequency dependent char-

acteristic of the medium; 2. (Surface wave) The measured frequency-velocity relationship

due to vertical stiffness variations, eg. phase velocity dispersion curve.

dispersive 1. (Wave) Exhibiting dispersion; 2. (Medium) (a) Non-dispersive (constant

stiffness half-space), (b) Normally dispersive (stiffness increasing with depth), (c) Irreg-

ularly dispersive (low and high stiffness layers interbedded) and (d) Inversely dispersive

(layers overlying a softer half-space).

evanescent A wave whose amplitude decreases with distance travelled, due to vis-

coelastic losses and/or spherical spreading. Compare homogeneous (2).

FSW In this thesis, the method for calculation of a dispersion curve by plane wave

matrix methods, using the code of [290]. Compare PSV .

ground roll High amplitude and relatively low velocity and frequency signals seen on

land seismic shot records, comprised primarily of Rayleigh waves.

homogenous (1) (Medium) Of uniform properties throughout; (2) (Wave) Amplitude

does varying with distance from the source. Compare evanescent.

HVL, HVZ High Velocity Layer / Zone.

Love wave An elastic SH wave at the Earth-air interface, only generated in media

with vertical stiffness variation.

LVL, LVZ Low Velocity Layer / Zone.

MASW (Multichannel Analysis of Surface Waves) An abbreviation introduced by the

Kansas Geological Survey in the early 1990’s. See also SASW.

microseism Low amplitude seismic energy sourced from cultural and other natural

sources, such as distant wave and weather effects.

mode Of surface waves, a propagating wavetrain with discrete wavenumber function,

k(f). Theoretically, a surface wave comprises a fundamental mode with higher modes at

successively higher phase velocities. Dominant higher modes (or superposed modes) are

modes with higher energy causing abrupt transitions in the dispersion curve.

osculation A point where two modal dispersion curves become very close and appear

to touch. From field observations and full-waveform synthetics, these points occur at

frequencies where higher modes become dominant.

PSV In this thesis, the method for calculation of a dispersion curve by full-waveform

P -SV reflectivity synthetic seismograms and automatic picking of the plane wave trans-

form. Compare FSW .

390 Appendix A. Glossary

Q Quality factor, the inverse (Q−1) being proportional to attenuation.

Rayleigh wave An elastic P -SV wave at the Earth-air interface.

SASW (Spectral Analysis of Surface Waves) An abbreviation introduced by the Uni-

versity of Texas at Austin in the early 1980’s. See also MASW.

Scholte wave An elastic wave at the water-Earth interface, such as the seafloor.

source generated noise The collective surface and guided waves seen on land seismic

records.

Stoneley wave An elastic wave at an elastic-elastic interface.

A.2 Mathematical symbols

A summary of the common mathematical symbols used throughout this text:

x = distance (m), or horizontal (inline) component (A.1)

y = transverse horizontal (crossline) component (A.2)

z = depth (m), or vertical component (A.3)

t = time (s) (A.4)

f = frequency (Hz) (A.5)

c = phase velocity (m/s) (A.6)

U = group velocity (m/s) (A.7)

p = slowness (s/m) (A.8)

τ = time intercept (s) (A.9)

k = wavenumber (m−1) or bulk modulus (Pa) (A.10)

λ = wavelength (m) or Lame’s parameter (A.11)

µ or G = shear modulus (Pa) (A.12)

σ = Poisson’s ratio or standard deviation (A.13)

VP or α = compressional wave velocity (m/s) (A.14)

VS or β = shear wave velocity (m/s) (A.15)

ρ = density (g/cc) (A.16)

c will generally be implied to be the Rayleigh wave phase velocity (VR) neglecting the

need to write cR.

391

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