Full Waveform Inversion Using Oriented Time Migration Method

Thesis by

Zhen-dong Zhang

In Partial Fulfillment of the Requirements

For the Degree of

Masters of Science

King Abdullah University of Science and Technology, Thuwal,

Kingdom of Saudi Arabia

Copyright © April 2016

Zhen-dong Zhang

All Rights Reserved

2

The thesis of Zhen-dong Zhang is approved by the examination committee

Committee Chairperson: Tariq Alkhalifah

Committee Member: Yike Liu

Committee Member: Ibrahim Hoteit

Committee Member: Ying Wu

3

ABSTRACT

Full Waveform Inversion Using Oriented Time Migration

Method

Zhen-dong Zhang

Full waveform inversion (FWI) for reflection events is limited by its linearized up-

date requirements given by a process equivalent to migration. Unless the background

velocity model is reasonably accurate the resulting gradient can have an inaccurate

update direction leading the inversion to converge into what we refer to as local min-

ima of the objective function. In this thesis, I first look into the subject of full model

wavenumber to analysis the root of local minima and suggest the possible ways to

avoid this problem. And then I analysis the possibility of recovering the correspond-

ing wavenumber components through the existing inversion and migration algorithms.

Migration can be taken as a generalized inversion method which mainly retrieves the

high wavenumber part of the model. Conventional impedance inversion method gives

a mapping relationship between the migration image (high wavenumber) and model

parameters (full wavenumber) and thus provides a possible cascade inversion strat-

egy to retrieve the full wavenumber components from seismic data. In the proposed

approach, consider a mild lateral variation in the model, I find an analytical Frechet

derivation corresponding to the new objective function. In the proposed approach, the

gradient is given by the oriented time-domain imaging method. This is independent

of the background velocity. Specifically, I apply the oriented time-domain imaging

4

(which depends on the reflection slope instead of a background velocity) on the data

residual to obtain the geometrical features of the velocity perturbation. Assuming

that density is constant, the conventional 1D impedance inversion method is also ap-

plicable for 2D or 3D velocity inversion within the process of FWI. This method is not

only capable of inverting for velocity, but it is also capable of retrieving anisotropic pa-

rameters relying on linearized representations of the reflection response. To eliminate

the cross-talk artifacts between different parameters, I utilize what I consider being

an optimal parameterization. To do so, I extend the prestack time-domain migration

image in incident angle dimension to incorporate angular dependence needed by the

multiparameter inversion. For simple models, this approach provides an efficient and

stable way to do full waveform inversion or modified seismic inversion and makes the

anisotropic inversion more practical. Results based on synthetic data of isotropic and

anisotropic case examples illustrate the benefits and limitations of this method.

5

ACKNOWLEDGEMENTS

Sincere thanks to my supervisor, Professor Tariq Alkhalifah and also my co-supervisor,

Professor Yike Liu at Institute of Geology and Geophysics, Chinese Academy of Sci-

ences. Professor Alkhalifah has consistently given me great care and concern especially

when I first arrived at KAUST. His profound knowledge, energy and enthusiasm for

Geophysics and excellent instructions help me a lot. Professor Yike Liu was my first

teacher in Geophysics. His patience and foresight in Geophysics help me to get in-

volved in research quickly.

Thanks to the members of my thesis examination committee, Professor Ibrahim Hoteit,

and Professor Ying Wu, for their valuable comments and precious time.

Thanks to KAUST for its support and specifically the seismic wave analysis group

(SWAG) members for their valuable insights.

Thanks to my parents and my friends for their support, patience and help.

6

TABLE OF CONTENTS

Copyright 1

Examination Committee Approval 2

Abstract 3

Acknowledgements 5

List of Figures 8

1 Introduction 10

2 Full Model Wavenumber Inversion 14

2.1 Velocity Model Building . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 Initial Velocity Model . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.2 Enhance the Resolution . . . . . . . . . . . . . . . . . . . . . 17

2.2 Migration and Inversion . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Oriented Time Migration . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Inversion Algorithm and Numerical Examples 24

3.1 Derivation of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.1 Impedance Inversion and FWI . . . . . . . . . . . . . . . . . . 24

3.1.2 Multi-parameter Inversion for VTI Media . . . . . . . . . . . . 26

3.1.3 Mono-parameter Inversion for Isotropic Media . . . . . . . . . 28

3.2 Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 The Workflow for This Method . . . . . . . . . . . . . . . . . . . . . 31

3.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.1 Monoparameter Case . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.2 Multiparameter Case . . . . . . . . . . . . . . . . . . . . . . . 35

7

4 Concluding Remarks 38

4.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Future Research Work . . . . . . . . . . . . . . . . . . . . . . . . . . 40

References 42

8

LIST OF FIGURES

2.1 The objective function corresponding to a model with the residual miss-

ing the wavenumber given by the horizontal axis (Courtesy of Alkhal-

ifah, 2015). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 The wavenumber issue. Notice that the intermediate wavenumber com-

ponents of the model are insensitive to the data (Adapted from Imaging

the Earth Interior). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Description of geometry parameters (Courtesy of Clayton, 1977). . . . 20

2.4 Velocity model used in the example. . . . . . . . . . . . . . . . . . . . 22

2.5 Conventional time migration image. Notice that time migration cannot

image the structures which has large dipping angles. . . . . . . . . . . 23

2.6 Extended time migration image. The third axis is the incident angle. 23

3.1 Radiation pattern for parameters Vp0, δ and ε. The red line indicates

Vp0; the green line indicates ε, and the blue line indicates δ. . . . . . . 30

3.2 Radiation pattern for parameters Vn, η and δ. The red line indicates

Vn; the green line indicates η, and the blue line indicates δ. . . . . . . 31

3.3 Monoparameter inversion results: (a) true velocity model, (b) initial

velocity model, and (c) inverted velocity model. . . . . . . . . . . . . 33

3.4 Comparison of velocity models. The red line indicates true velocity; the

dashed green line indicates initial velocity, and the black line indicates

inverted velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5 Normalized data residual for five iterations at 4 Hz. The data residual

is decreasing with the iteration steps increasing. More iterations should

be implemented to get a better approximation to the true velocity in

one stage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6 Multiparameter inversion results: (a) true Vn model, (b) true η model,

(c) true δ model, (d) inverted Vn model, (e) inverted η model and (f)

inverted δ model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

9

3.7 Initial models used in mulitiparameter inversion: (a) Vn, (b) η and (c)

δ. They are smoothed version of the true models. . . . . . . . . . . . 36

3.8 Comparison of the inverted models at x = 1.5km. (a) The red line

indicates true Vn; the green line indicates initial Vn, and the blue line

indicates the inverted Vn. (b) The red line indicates true η; the green

line indicates initial η, and the blue line indicates the inverted η. (c)

The red line indicates true δ; the green line indicates initial δ, and the

blue line indicates the inverted δ. . . . . . . . . . . . . . . . . . . . . 37

10

Chapter 1

Introduction

Full waveform inversion (FWI) is a nonlinear optimization procedure which aims to

better estimate subsurface parameters to reduce the misfit between predicted and

observed data. Given initial estimates of subsurface parameters, the predicted data

is calculated by numerically solving the relevant wave equations such as acoustic,

elastic and quasi-acoustic. Usually, we evaluate the data misfit in the 2-norm sense

and calculate the perturbations by solving a least-squares local optimization problem

[1]. This procedure is repeated in an iterative fashion until the data residual is

acceptable. For gradient-based optimization methods, gradients could be calculated

by the multiplication of Frechet derivations and data misfits or by the adjoint-state

method. Frechet derivations cannot be calculated numerically in practice due to

the unaffordable computational cost for large models. This makes the adjoint-state

method the more practical and widely used method since Lailly [2] and Tarantola [3]

have shown that it is nothing but a process of cross-correlating the forward-propagated

source wavefield and the back-propagated data residual.

Full waveform inversion has not been widely used in the industry mainly because

of its local minima problem and huge computational cost. Gradient-based local op-

timization methods require that the initial models should be close to true models,

which guarantees convergence to the global minima. To be more specific, the initial

models should be able to produce synthetic data that is within a half-cycle of the ob-

11

served data everywhere, and the physical temporal length of a half cycle depends on

the frequency used in inversion [4]. It is not easy to build such precise initial models

in practice. Because of this, various methods have been introduced to mitigate the

local minima problem, including multiscale inversion method by Bunks et al. [5], the

damping method by Shin et al. [6] and the unwrapped phase method by Choi and

Alkhalifah [7] and Alkhalifah [8]. Global optimization methods introduced by Sen

and Stoffa [9] are intended to find the global minima with arbitrary initial models,

but they are not widely used mainly because their computational costs are not af-

fordable. Even for the local optimization methods, the computational cost is huge.

Most of the computation time of full waveform inversion are occupied by extrapo-

lating the wavefield using numerical methods including finite-difference, spectral and

finite-element methods. To speed up the forward and backward propagation of the

wavefield the Graphic Process Unit (GPU) based method by Yang et al. [10] was

introduced to FWI. There are also many dimensionality reduction approaches used

to reduce the computational cost of modeling and inversion including source encod-

ing methods [11, 12, 13] and randomized source methods [14, 15, 16]. Although we

cannot numerically calculate Frechet derivations in practice, for simple models, we

can find its approximate analytical solutions [17, 18]. No backward extrapolations are

needed in each iteration, and thus, the computational cost and memory requirements

are reduced.

General seismic inversion methods, including the 1D impedance inversion [19, 20,

21, 22], provide a possible way to calculate the velocity perturbation from the image

of the data residual using perturbation theory. Under the assumption of a mild lat-

eral velocity variation, the 1D inversion method could easily be used for 2D or 3D

inversions. Calculating the image corresponding to velocity perturbations, migration

of the data residual is needed. Both wave equation and ray-theory based depth migra-

tion methods depend strongly on the migration velocity. Actually, the velocity model

12

is usually considerably inaccurate at the beginning of full waveform inversion. This

leads to inaccurate focusing and positioning of reflectors (perturbations). Such inac-

curacies for depth based imaging tend to be larger than those corresponding to time

domain imaging methods. For conventional velocity analysis time domain processing

methods are considered more robust [23, 24]. Here vertical time replaces depth as

the vertical axis. The oriented time-domain imaging method developed by Fomel [25]

is independent of the velocity model, and provides a better chance of guaranteeing

an accurate representation of the image. This is especially true at the beginning of

the inversion process. In other words, the energy of the data is transformed to the

migration image without leakage to extended images. This property is crucial for our

proposed objective function which evaluates the data misfit in the image domain. We

know that local slopes estimated from prestack reflection data contain all the infor-

mation of the reflection geometry [26]. Once they are calculated correctly, the seismic

velocities, incident angles, and some other wave geometry parameters become data

attributes. For multiparameter inversion, incident angles are needed to distinguish

the contributions to the waveform from different parameters.

Since the aim of FWI is to fit the full information content of observed and predicted

data, the simulation engine embedded in the FWI algorithm should honor all of

the physics of wave propagation as much as possible [1]. Multiparameter inversion

including velocity, density, attenuation and anisotropic parameters helps improve

the FWI results and is drawing more and more attention in full waveform inversion

analysis [27, 28, 29, 30]. I propose a new multiparameter inversion method for VTI

media based on the pseudo-acoustic wave equation introduced by Alkhalifah [31].

One of the difficulties in multiparameter FWI is to eliminate the crosstalk artifacts

between different parameters. One remedy is to select the optimal parameterization,

which not only can adequately describe the subsurface model but also can be inverted

from the observed data [27, 29, 32]. When considering the parameters as virtual

13

sources in the model space, I can calculate their radiation patterns, which indicate

their contributions to the waveform recorded at the surface. Their contributions

depend on the scattering angles in the model space or roughly the offsets in the data

space. When back projecting the data residual in the data domain to the parameter

perturbation in the model domain, the radiation patterns act like filters. Due to the

limited separation of the acquisition geometry, some parameters cannot be retrieved if

their unique contributions are not identified in the observed data. Another difficulty

is the variable influence of different parameters on the waveform. The hierarchical

strategies [33, 28, 34] can retrieve different parameter classes successively and partly

mitigate the poor positioning of FWI. Since the incident angles help to distinguish the

contributions of the parameters, the extended images in the incident angle dimension

are needed. The extended images can be calculated easily because the incident angles

are also data attributes.

This thesis is divided into four chapters. After the introduction, I analyze the

wavenumber components of gradients used in full waveform inversion. In the third

chapter the algorithm and numerical examples of the proposed method are delivered.

Finally, I draw the conclusion from the developed theory and numerical examples and

suggest future work.

14

Chapter 2

Full Model Wavenumber Inversion

The velocity model can be separated into low and high wavenumber components [35].

The ultimate goal of FWI is to recover all the wavenumber components available in

recorded data from low to high [36]. To avoid the cycle-skipping problem the initial

velocity model should provide correct kinematic information (low wavenumber) for

the simulated data. High wavenumber components mainly provide the dynamic infor-

mation such as the reflection coefficient of the model. In practice, the low wavenumber

information is recovered by tomography and the high wavenumber part is provided by

migration. In the proposed method, I assume the initial model has correct kinematic

information and utilize time migration method to reveal the dynamic information

(high wavenumber). We evaluate the data misfit by the difference between the ob-

served and calculated data so that the forward modeling engine will describe the

propagation of the P wave as precisely as possible. This means that the calculated

data is acquired by solving the acoustic wave equation.

2.1 Velocity Model Building

Velocity information is the vital component in seismic data processing and it is one

of the most reliable ways to understand the subsurface structures. For migration

purpose a smooth RMS velocity model (usually used as the initial velocity model for

15

FWI) is good enough, however, with the help of full waveform inversion methods we

can get a higher resolution velocity model and thus a better understanding of the

subsurface properties. This is especially relevant for some complex velocity models

where tomography complicates a solution. In such cases the FWI method can provide

an accurate solution [37].

2.1.1 Initial Velocity Model

Limited by the sinusoid nature of seismic data, the initial velocity model should be

close to the true velocity model to avoid getting stuck in local minima. Specifi-

cally, the initial velocity model might have low resolution, but it should be kineti-

cally close to the true velocity model. Alkhalifah [36] demonstrates the importance

of low-wavenumber components in full waveform inversion as shown in Figure 2.1.

Low-wavenumber components which are mainly built by the initial model missing

could lead to large data residual. However high-wavenumber components which could

provide a high-resolution model cannot generate large data residual when they are

missing in the inverted model.

Figure 2.1: The objective function corresponding to a model with the residual missingthe wavenumber given by the horizontal axis (Courtesy of Alkhalifah, 2015).

For a successful application of full waveform inversion, it is critical to generate a

critical initial velocity model which guarantees the inversion is converging to the global

16

minimum. Conventional velocity estimation methods have been well developed in the

past years and provide kinematic accurate velocity models as the initial model for

full waveform inversion. These methods include first-arrival traveltime tomography,

normal move-out (NMO) velocity scanning in data domain, and migration velocity

analysis (MVA) or differential semblance optimization (DSO) in the image domain.

Through implements on the BP salt dome model, Chauris et al. [38] indicate that in

the simplest zone corresponding to smooth velocity models, the tomographic models

are good enough for waveform inversion with realistic frequency contents. In the

complex part going through a salt body, one need very low frequencies or a further

refinement of the tomographic model. Weibull and Nilsen [39] suggest using wave-

equation-migration velocity analysis (WEMVA) to build reliable initial models for

full waveform inversion.

The migration velocity analysis (MVA) approach has a long history and has plenty

of variations in application, but the underlying principle has not been changed. This

approach is usually measuring the focus of energy in the image domain; However,

Van Leeuwen and Mulder [40] implemented MVA in the data domain to recover the

background velocity model which could be used as the initial velocity model for full

waveform inversion. This method could be incorporated with the proposed inversion

method; here we give a short introduction to this method. Different from wave-

form inversion, this method focuses on fitting the curvatures of the calculated data

and observed data by a correlation method. The objective function for temporary

correlation is defined below,

Jt =1

St

∫ ∫ ∆tmax

−∆tmax

W (∆t)(Ct[p, q](∆t, h))2d∆tdh, (2.1)

and

17

Ct[p, q](∆t, h) =

∫p(t, h)q(t+ ∆t, h)dt, (2.2)

St =1

St

∫ ∫ ∆tmax

−∆tmax

(Ct[p, q](∆t, h))2d∆tdh. (2.3)

p(t, h) and q(t, h) are calculated and observed seismograms, where t and h denote

time and offset respectively. Ct is the temporal correlation of these two seismograms.

∆t denotes the maximal shift of the correlation which could reduce the influence of

crosstalk. W (∆t) is a weighting function measuring the amount of the energy away

from zero shift or around zero shift.

The weighting function used to measure the energy away from the center is,

W1(∆t) = (∆t

∆tmax)2. (2.4)

This weighting function penalizes peaks at non-zero shift and thus measures the

amount of energy away from the center.

A weighting that measures energy around zero shift is a Gaussian function,

W2(∆t) = −e−α∆t2 . (2.5)

The minus sign ensures that minimizing the functional will maximize the amount of

energy around the center, and α controls the bandwidth of the function.

The gradient for the optimization could be derived using the adjoint-state tech-

nique. See the paper [40] for more details.

2.1.2 Enhance the Resolution

The initial velocity model which is kinematically accurate with respect to the true

model could fulfill the imaging task. However, the refined velocity model which has

18

higher resolution (up to one-half of the propagated wavelength) could be used for di-

rect interpretation [37]. For short-offset acquisition the seismic wavefield recorded at

the receivers is insensitive to intermediate wavenumber components which generate a

gap in wavenumber components between the initial velocity model and the final-high

resolution velocity model as shown in Figure 2.2. To narrow the wavenumber gap,

it requires that the initial velocity should be sufficiently accurate [1], otherwise, the

initial velocity might be not very accurate. Because of this the data set should con-

tain low-frequency large-offset information [41]. The full waveform inversion method

utilizes the waveform residual to update the initial velocity model and the updates

are corresponding to the high wavenumber part of the velocity model.

Figure 2.2: The wavenumber issue. Notice that the intermediate wavenumber com-ponents of the model are insensitive to the data (Adapted from Imaging the EarthInterior).

For gradient-based optimization methods, gradients are added to initial models

after properly scaling. Thus, the wavelength components of the gradients (model

updates) decide the resolution of the inverted model. Conventional full waveform

inversion methods start from a quite low frequency or frequency band. With more

frequency information more details of the substructure are resolvable and add resolu-

tion to the initial velocity model. The wavenumber formula for the gradient is given

by

19

Kδv = Ks + Kr =ω

v0

cosθ

2n. (2.6)

Where v0 is the background velocity, ω is the angular frequency and θ is the

scattering angle. n is the unit vector which is defined as n = Ks+Kr

|Ks+Kr| . The updated

model has the best resolution in n direction which is perpendicular to the wavepath.

A wavenumber filter could be designed to select the appreciate wavenumber com-

ponents added to the background velocity model [36]. A specially designed filter can

filter out the unwanted part of the gradients, and it provides a possible way to apply

the proposed method to more complex models, which will be my future work.

2.2 Migration and Inversion

Seismic migration and inversion share the same objectives and underlying physical

principles; Migration can be taken as an inversion process which inverts for the re-

flectivity from seismic data [42]. Reflectivity can be turned into impedance or other

subsurface parameters by a linear transform provided by convolution model. Thus it

is a migration process that is the essential part embedded in waveform inversion.

Both wave equation and ray theory based depth migration methods depend criti-

cally on the migration velocity. Actually, at the beginning of a full waveform inversion,

the velocity model is usually considerably inaccurate, which leads to inaccurate focus-

ing and positioning of reflectors (perturbations). Such inaccuracies for depth-based

imaging tend to be larger than those corresponding to time domain imaging methods.

For conventional velocity analysis, time domain processing methods are more robust.

2.2.1 Oriented Time Migration

The oriented time-domain imaging method developed by Fomel [25] largely applies a

prestack time migration utilizing velocity information extracted from the local event

20

Figure 2.3: Description of geometry parameters (Courtesy of Clayton, 1977).

slopes in the reflection data. Thus, this method is independent of the velocity model,

which could help us avoid putting the reflectors (or perturbations) in the wrong

position due to the erroneous initial velocity model. This method is applicable for

VTI media under the assumption that the moveout is non-hyperbolic.

The basic idea of this migration method is to find a mapping relationship between

images and data attributes. Equations 2.7, 2.8 and 2.9 are geometrical descriptions

of Figure 2.3, and the description is valid for both isotropic and anisotropic media.

t =2h cosα

v sin β, (2.7)

ph =∂t

∂h=

2 cosα sin β

v, (2.8)

py =∂t

∂y=

2 sinα cos β

v. (2.9)

Where h denotes the half-offset, y denotes the midpoint in the image domain. ph

and py are estimated slopes which will be explained later.

21

Sava and Fomel [43] provide a description for data attributes shown in equations

2.10 and 2.11.

τ = tcos2 α− sin2 β

cosα cos β, (2.10)

y − x = hsinα cosα

sin θ cos θ. (2.11)

Here x denotes true reflection point on surface.

The time-migrated image domain (τ, x) is mappable from the prestack domain

(t, h, y) using the following formulas,

τ 2 =tph[(t− hph)2 − h2p2

y]2

(t− hph)2[tph + h(p2y − p2

h)], (2.12)

x = y − htpytph + h(p2

y − p2h). (2.13)

The event slopes in recorded data, given by ph = ∂t/∂h and py = ∂t/∂y, can be

estimated using the plane-wave destruction algorithm of Fomel [44]. Different from

the isotropic case, we need to calculate the incident angle using equation 2.14 besides

the time τ and midpoint x in the image domain for the anisotropic case. Since the

incident angle is one of the attributes of the data, the extended images can be easily

calculated by sweeping over all the possible incident angles and reorganize them in

the incident angle dimension.

sin2θ =hpht. (2.14)

2.2.2 Numerical Example

Time migration plays an essential part in the proposed inversion algorithm. In this

part, we verify the extended migration algorithm by part of Marmousi-II model as

shown in Figure 2.4. There are 50 shots evenly distributed on the surface. Each shot

22

Figure 2.4: Velocity model used in the example.

has a Ricker wavelet with a peak frequency of 5 Hz as the source. The migration

image is shown in Figure 2.5 and it is a conventional image without incident angles.

Notice the fact that time migration cannot image the large-dipping portion of the

model. The extended image is shown in Figure 2.6. The third dimension is incident

angles, and the images look discontinuous mainly because the sampling is not frequent

enough in the estimated slopes. However, after stacking all the images along incident

angles the image should coincide with Figure 2.5.

23

Figure 2.5: Conventional time migration image. Notice that time migration cannotimage the structures which has large dipping angles.

Figure 2.6: Extended time migration image. The third axis is the incident angle.

24

Chapter 3

Inversion Algorithm and

Numerical Examples

3.1 Derivation of the Algorithm

In this section, I derive an analytical solution to approximate Frechet derivations.

The analytical solution is fit for a new objective function which is equivalent to the

conventional objective function. The analytical approximations of Frechet derivation

help reduce the computational time and memory requirement in every iteration.

3.1.1 Impedance Inversion and FWI

Impedance inversion methods provide a relationship between impedance and image

in time domain [19, 20]. Under the assumption that the density is constant, and the

lateral velocity variation is negligible, the 1D inversion method can be extended to a

2D or 3D case. After converting the seismic data residual to images, we can combine

the full waveform inversion method with the acoustic impedance inversion method

and derive an efficient way to calculate the parameter’s perturbations. In this section,

I propose a new objective function and derive the gradients for different parameters.

If we take the full waveform inversion as a least-squares local optimization problem

25

then the conventional misfit function is given by

J =1

2‖ L(m)− dobs ‖2, (3.1)

where m represents the model parameters and here in an acoustic, constant density,

case, it is given by the anisotropic parameters and velocities. L denotes the for-

ward modeling operator and dobs is the observed seismic data. The operator ‖ . ‖2

corresponds to the l2 norm.

Instead of evaluating the data misfit directly, I propose a new misfit function

which is given by the image of data residual,

J =1

2‖ I(τ,x) ‖2, (3.2)

where I(τ,x) is the prestack time migration image of the data residual. Thanks to the

velocity-independent property of the dip-oriented migration method, the images are

always focused at zero time lag and zero subsurface offset. In other words, equations

3.1 and 3.2 are equivalent to each other because there is no energy leakage.

To minimize the new misfit function, gradient-based optimization methods are

needed [1]. Here, I use the steepest descent method which is given by

mnew = mold − α∂I(τ,x)

∂mI(τ,x), (3.3)

wherem denotes the model parameters, which could be velocity, density and anisotropic

parameters according to the assumed approximations of the subsurface model, and α

is the step length which can be calculated using a line search method [45]. The Frechet

derivation, ∂I∂m

, has an explicit form with the help of the widely used convolutional

model [20] in conventional impedance inversion method.

26

The general form of the convolutional model is given by

I(τ,x) = R(τ,x) ∗ s(τ). (3.4)

Here, I(τ,x) denotes the image in the time domain, R(τ,x) denotes the reflection

coefficient and s(τ) denotes the source wavelet used for forward modeling.

The parameters, including the velocity, density, and anisotropic parameters, are

hidden in the specific form of R(τ,x) which depends on the assumptions made of the

subsurface. In the next section, I derive the specific form of equation 3.3 for VTI

media. As a special case, the isotropic form is also given when there is no anisotropic,

and thus, no incident angle dependence.

3.1.2 Multi-parameter Inversion for VTI Media

For VTI media, the reflection coefficient given by [46] is shown below,

RV TIP (θ) =

1

2

∆Z

Z+

1

2(∆Vp0Vp0

− (2Vs0Vp0

)2 ∆G

G+ ∆δ)sin2θ +

1

2(∆Vp0Vp0

+ ∆ε)sin2θtan2θ,

(3.5)

where θ denotes the incident angle, which is important to suppress the crosstalk

artifacts between different parameters, Z = ρVp0 is the vertical P-wave impedance,

and G = ρVs0 denotes the vertical shear modulus, which is assumed to be zero in the

pseudo-acoustic approximation. The perturbations ∆δ and ∆ε are the differences in

anisotropy between two points. Here I set the density to equal 1.

Using the approximation introduced by [20] in conventional impedance inversion,

1

2

∆Z(τ, x)

Z(τ, x)=

1

2

∂ln(Z(τ, x))

∂τ, (3.6)

27

Inserting equations 3.6 and 3.5 into equation 3.4 and get,

IV TIP (θ) =1

2[lnVp0(1 + sin2θ + sin2θtan2θ)] ∗ s+

1

2[δsin2θ + εsin2θtan2θ] ∗ s. (3.7)

If each of the parameters, Vp0, δ, and ε, is perturbed successively, the image with

respect to the perturbations of each parameter is given by,

(I + ∆I)V TIvp0(θ) = 1

2[ln(Vp0 + ∆Vp0) ∗ (1 + sin2θ + sin2θtan2θ)] ∗ s (3.8)

+12[δsin2θ + εsin2θtan2θ] ∗ s,

(I + ∆I)V TIδ (θ) = 12[lnVp0 ∗ (1 + sin2θ + sin2θtan2θ)] ∗ s

+12[(δ + ∆δ)sin2θ + εsin2θtan2θ] ∗ s,

(I + ∆I)V TIε (θ) = 12[lnVp0 ∗ (1 + sin2θ + sin2θtan2θ)] ∗ s

+12[δsin2θ + (ε+ ∆ε)sin2θtan2θ] ∗ s.

Subtracting equation 3.7 from equation 3.8 and using Taylor expansion for ln(1+x) ≈

x, when x is small, I get the relationships between perturbed parameters and the

image of data residual, given by

∆IV TIvp0(θ) ≈ 1

2[ δVp0Vp0

(1 + sin2θ + sin2θtan2θ)] ∗ s, (3.9)

∆IV TIδ (θ) ≈ 12[∆δsin2θ] ∗ s,

∆IV TIε (θ) ≈ 12[∆εsin2θtan2θ] ∗ s.

By virtue of the associativity law of convolution and integral over the reflection

28

angles, the gradients are given by

∇vp0 ≈ 12[∑θ I(τ,x,θ)(1+sin2θ+sin2θtan2θ)

Vp0] ∗ s, (3.10)

∇δ ≈ 12[∑

θ I(τ,x, θ)sin2θ] ∗ s,

∇ε ≈ 12[∑

θ I(τ,x, θ)sin2θtan2θ] ∗ s,

where I(τ,x, θ) is the extended image along the incident angle axis. The incident

angle exposes the varying contributions to the waveform from the parameters, Vp0, δ

and ε, which is used to mitigate the crosstalk artifacts between them in the inversion

step. In addition to utilizing the incident angle directly, I also want to choose an

optimal parameter set which has fewer overlaps as a function of incident angle. More

details about choosing the optimal parameterization for the proposed method are

shown in the next section.

3.1.3 Mono-parameter Inversion for Isotropic Media

The derivation for monoparameter case is straightforward. Noting the fact that equa-

tion 3.5 is a good approximation for the reflection coefficient of vertical-incident P-

wave by setting the shear velocity, anisotropic parameters and incident angle equal

zero. Thus, the reduced form of equation 3.10 is the gradient for the monoparameter

inversion such that the vertical P-wave velocity inversion is given by,

∇vp0 ≈1

2[I(τ,x)

Vp0] ∗ s. (3.11)

29

No incident angle information is needed since I only invert for a monoparameter, ver-

tical P-wave velocity. The steepest descent method could be implemented to minimize

the proposed objective function.

3.2 Parameterization

For multiparameter inversion, the coupled effects of the parameters prevent the con-

vergence [34, 29]. Usually, different parameters have different sensitivities to different

scattering angles, which are also called radiation patterns. In this part, I test two

sets of parameters based on the previous analysis (i.e. equation 3.10) and choose the

optimal one based on the inversion purpose and inversion strategy. Figure 3.1 shows

the radiation patterns for the first set which includes Vp0, δ, and ε. It is obvious that

Vp0 is sensitive to all the scattering angles. On the other hand, δ and ε have over-

lapped scattering angles, which means that they have some crosstalk errors if I invert

these two parameters simultaneously. This set of parameters is reasonable if I keep

δ and ε fixed and invert Vp0 only. However, if I want to invert all three parameters

simultaneously, I need to change the parametrization. By virtue of the chain rule, I

represent the VTI media by the following parameters Vn, η and δ.

∣∣∣∣∣∣∣∣∣∣∂I∂Vn

∂I∂η

∂I∂δ

∣∣∣∣∣∣∣∣∣∣=

∣∣∣∣∣∣∣∣∣∣∂Vp0∂Vn

∂ε∂Vn

∂δ∂Vn

∂Vp0∂η

∂ε∂η

∂δ∂η

∂Vp0∂δ

∂ε∂δ

∂δ∂δ

∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣∂I∂Vp0

∂I∂ε

∂I∂δ

∣∣∣∣∣∣∣∣∣∣. (3.12)

The relationships between different anisotropic parameters are written as,

Vn = Vp0√

1 + 2δ

η = ε−δ1+2δ

=⇒

Vp0 = Vn√1+2δ

ε = (1 + 2δ)η + δ

δ = δ

. (3.13)

30

0.20.40.60.81

30

210

60

240

90 270

120

300

150

330

180

0

Vp0EpsilonDelta

Figure 3.1: Radiation pattern for parameters Vp0, δ and ε. The red line indicates Vp0;the green line indicates ε, and the blue line indicates δ.

Then I can get a new formula for the parameter perturbations and the image of

the data residual, given by

∣∣∣∣∣∣∣∣∣∣∆I

∆Vn

∆I∆η

∆I∆δ

∣∣∣∣∣∣∣∣∣∣≈

∣∣∣∣∣∣∣∣∣∣1√

1+2δ0 0

0 1 + 2δ 0

−Vn(1 + 2δ)−32 1 + 2η 1

∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣∆I

∆Vp0

∆I∆ε

∆I∆δ

∣∣∣∣∣∣∣∣∣∣. (3.14)

The gradients for Vn, η and δ are given by,

∣∣∣∣∣∣∣∣∣∣∇vn

∇η

∇δ

∣∣∣∣∣∣∣∣∣∣≈

∣∣∣∣∣∣∣∣∣∣1√

1+2δ0 0

0 1 + 2δ 0

−Vn(1 + 2δ)−32 1 + 2η 1

∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣∇vp0

∇ε

∇δ

∣∣∣∣∣∣∣∣∣∣. (3.15)

The radiation patterns for parameters Vn, η and δ are shown in Figure 3.2. For this

31

0.20.40.60.81

30

210

60

240

90 270

120

300

150

330

180

0VnmoEtaDelta

Figure 3.2: Radiation pattern for parameters Vn, η and δ. The red line indicates Vn;the green line indicates η, and the blue line indicates δ.

set of parameters, Vn is sensitive to all the scattering angles, δ is more sensitive to small

scattering angles and η is more sensitive to large scattering angles. Thus, I choose this

set of parameters as the optimal one since I intend to recover all the parameters of the

subsurface. The crosstalk effects between δ and η could be eliminated by adding the

scattering angle information in the inversion process. To reduce the variable effects

of Vn and the other two parameters, I invert Vn for the first several iteration and then

invert these three parameters jointly, which is referred to as the hierarchy method

[33, 28] and is fully described in the workflow.

3.3 The Workflow for This Method

In summary, the hierarchy-inversion strategy used for the multiparameter inversion

includes the following steps:

32

1. Establish initial parameter models in depth and time. The initially estimated

parameters should be good enough to avoid ending up in a local minimum.

2. Select a frequency band to invert within the data.

3. Using the depth version of the models, calculate the synthetic data by solving

the pseudo-acoustic wave equation for VTI media, which was first introduced

by [31].

4. Evaluate the data residual for each shot and organize the data into CMP gath-

ers.

5. Migrate the data residual using the oriented time-domain imaging method. Here

I calculate the extended images along the incident-angle axis.

6. Calculate the parameter’s perturbations.

7. Update Vn for the first several iterations and then parameters δ and η.

8. Repeat steps 3–7 until the data residual is acceptable.

9. Repeat steps 2–8 for the multi-scale approach until the data residual is accept-

able and the frequency range from low to high is covered.

For monoparameter inversion, the implementation is more straightforward and

simple, I need to solve the pure acoustic wave equation to generate the data in step 8,

and only P-wave velocity is inverted in step 7. The only significant computational cost

in this implementation is generating the observed data by solving the wave equation.

The gradient might not be perfect, but it tends to be stable. Besides, the gradient is

rarely perfect in FWI, as it depends on the background velocity model.

33

Velo

city (km

/s)Velo

city (km

/s)Velo

city (km

/s)

Figure 3.3: Monoparameter inversion results: (a) true velocity model, (b) initialvelocity model, and (c) inverted velocity model.

3.4 Numerical Example

3.4.1 Monoparameter Case

I use the multi-scale method introduced by Bunks et al. [5] for the inversion and start

with a peak frequency of 4 Hz. There is a total of four-frequency selection stages used

here ranging from 4 Hz to 10 Hz, and each stage includes five iterations. According

to the assumptions made in the approach I choose the left, laterally smooth, part of

the Marmousi velocity model to test the proposed method. I consider 50 shots with

100 geophones in each shot uniformly distributed in the horizontal direction. Each

shot is a point-source Ricker wavelet.

The initial velocity shown in Figure 3.3b is a smoothed version of the true ve-

locity model shown in Figure 3.3a. The smoothing operator is given by a triangle

34

filtering with a window size of 1/4 of the total grid points in depth and 1/10 of the

total grid points in the horizontal direction. Figure 3.3c shows the inverted velocity

model. To make the comparison more intuitive, I plot the velocities versus depth in

Figure 3.4. Since not all the wavenumber components can be recovered from the pro-

posed inversion, the true velocity here is wavenumber-filtered one. The corresponding

cutting wavenumber is calculated by equation 2.6. Figure 3.5 illustrates the normal-

ized data residual for the first five iterations in the first stage. The data residual

decreases as a function of iteration steps, which means that the simulated data fits

the observed one better and better per iteration. Comparing to the adjoint-state

method, the proposed method saves about two thirds of the total computational cost

per iteration. The most time-consuming part of FWI and the proposed method is

the extrapolation of wavefield. Conventional FWI needs two forward extrapolations

and one backward extrapolation per iteration; the proposed method requires only one

forward extrapolation per iteration.

3.4.2 Multiparameter Case

The next synthetic example is part of Marmousi II model, and the original model is

not anisotropic. I use the following equations ε = 0.25ρ − 0.3 and δ = 0.125ρ − 0.1

to generate the anisotropic models [47]. I also use the multi-scale method for the

inversion and start with a peak frequency of 2 Hz. There is a total of 4 frequency

selection stages used here ranging from 2 Hz to 5 Hz, and each stage contains twenty

iterations. Only the parameter Vn is updated in the first ten iterations in each stage,

then Vn, η and δ are inverted simultaneously. The water layer is set to be isotropic,

and the velocity is assumed to be already known. I consider 50 shots with 100

geophones in each shot uniformly distributed in the horizontal direction. Each shot

is a point-source Ricker wavelet.

The initial models shown in Figure 3.7 are smoothed versions of the true models

35

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Dep

th (k

m)

1.5 2 2.5 3Velocity (km/s)

Velocities at the Dashed Lines

Initial VelocityTrue VelocityInverted Velocity

Figure 3.4: Comparison of velocity models. The red line indicates true velocity;the dashed green line indicates initial velocity, and the black line indicates invertedvelocity.

1 1.5 2 2.5 3 3.5 4 4.5 50.7

0.75

0.8

0.85

0.9

0.95

1

Nor

mal

ized

Dat

aR

esid

ual

Iteration Steps

Normalized Data Residual for 5 Iterations at 4 Hz

Figure 3.5: Normalized data residual for five iterations at 4 Hz. The data residual isdecreasing with the iteration steps increasing. More iterations should be implementedto get a better approximation to the true velocity in one stage.

36

True Vnmo

0 1 2 3Distance (km)

0

1

2

3

Dep

th (k

m)

1.5

2

2.5

True Eta

0 1 2 3Distance (km)

0

1

2

3

Dep

th (k

m)

0.02

0.04

0.06

0.08

0.1

True Delta

0 1 2 3Distance (km)

0

1

2

3

Dep

th (k

m)

0.05

0.1

0.15

0.2

Inverted Vnmo

0 1 2 3Distance (km)

0

1

2

3

Dep

th (k

m)

1.5

2

2.5

Inverted Eta

0 1 2 3Distance (km)

0

1

2

3

Dep

th (k

m)

0.02

0.04

0.06

0.08

0.1

Inverted Delta

0 1 2 3Distance (km)

0

1

2

3

Dep

th (k

m)

0.05

0.1

0.15

0.2

Vel

ocity

(km

/s)

Vel

ocity

(km

/s)

a)

b)

c)

d)

e)

f)

Figure 3.6: Multiparameter inversion results: (a) true Vn model, (b) true η model, (c)true δ model, (d) inverted Vn model, (e) inverted η model and (f) inverted δ model.

Initial Vnmo

0 1 2 3Distance (km)

0

1

2

3

Dep

th (k

m)

1.5

2

2.5

Initial Eta

0 1 2 3Distance (km)

0

1

2

3

Dep

th (k

m)

0.02

0.04

0.06

0.08

0.1

Initial Delta

0 1 2 3Distance (km)

0

1

2

3

Dep

th (k

m)

0.05

0.1

0.15

0.2

Vel

ocity

(km

/s)

a)

b)

c)

Figure 3.7: Initial models used in mulitiparameter inversion: (a) Vn, (b) η and (c) δ.They are smoothed version of the true models.

37

shown in Figures 3.6a, 3.6b, and 3.6c. The smoothing operator is given by a triangle

filtering with a window size of 1/10 of the total grid points in depth and 1/10 of the

total grid points in the horizontal direction. Figures 3.6d, 3.6e, and 3.6f are inverted

velocity, η and δ at 5 Hz, respectively. Note the fact that the updating range of η

in Figure 3.6e decreases with increasing depth, which matches the previous analysis;

the parameter η mainly affects the traveltime of large scattering angle data. To make

the comparison more intuitive, I plot the parameters versus depth at x = 1.5km

in Figure 3.8. The Vn profile shown in Figure 3.8a can be well recovered since it

has an isotropic radiation pattern. This will influence the full offset range. I only

managed to invert the shallow part of η shown in Figure 3.8b and the inverted result

is acceptable. Since the maximum peak frequency used in the inversion is 5 Hz, the

resolution of the inverted parameter δ is not very high. More details of the models

are expected to be retrieved if I use higher peak frequencies in the inversion. In

multiparameter case, the proposed method saves more computational cost because

more wavefield extrapolations are needed for different parameters in conventional

FWI. However, the implementation of conventional multiparameter FWI is tricky,

for straightforward implementation of FWI, our proposed method can save up to six

sevenths of the total computational cost.

38

0

0.5

1

1.5

2

2.5

3

3.5

Dep

th(k

m)

1 2 3 4Velocity(km/s)

True VnmoInitial VnmoInverted Vnmo

0

0.5

1

1.5

2

2.5

3

3.5

Dep

th(k

m)

0 0.05 0.1

True EtaInitial EtaInverted Eta

0

0.5

1

1.5

2

2.5

3

3.5

Dep

th(k

m)

0 0.1 0.2

True DeltaInitial DeltaInverted Delta

a) b) c)

Figure 3.8: Comparison of the inverted models at x = 1.5km. (a) The red lineindicates true Vn; the green line indicates initial Vn, and the blue line indicates theinverted Vn. (b) The red line indicates true η; the green line indicates initial η, andthe blue line indicates the inverted η. (c) The red line indicates true δ; the green lineindicates initial δ, and the blue line indicates the inverted δ.

39

Chapter 4

Concluding Remarks

4.1 Discussion

Considering the inaccurate velocities we tend to work with at the beginning of the

FWI process; the gradient is usually far from perfect. Using an imperfect algorithm

to compute the gradient is, thus, justified considering its update linearized nature

and the imperfect physical assumptions we usually make. The proposed approach is

based on the assumptions that the density of the media is constant, and the lateral

velocity change is small. Subject to the prestack time-domain migration limitations,

the velocity model can not be too complicated. The cycle skipping problem is a

troublesome problem for local optimization methods and to avoid it we will need a

good initial velocity model. The numerical examples indicate that the performance

of this method depends on the accuracy of the initial velocity model to obtain proper

residual. The results are stable overall thanks to the time domain implementation.

Time-domain based imaging tends to be less sensitive to inaccurate velocities than

depth ones. The new formula for calculating the velocity perturbation is also derived

as a special case when there is no anisotropic and no angle dependence. From the

radiation patterns of the anisotropic parameters, we can find the data dependency on

different parameters. In the data domain, Vn has equal contributions to all scattering

angles, and thus, influences all offsets, which allows for a reasonable retrieval of it

40

from the data. On the other hand, δ mainly affects the amplitude of the near-offset

data, and thus, it promotes the waveform fitting at short offsets in FWI. Meanwhile,

η mainly affects the traveltime of far-offset data; thus, the FWI updating capability

of η reduces with depth.

The real value of this approach is in its ability to handle anisotropy in a more

robust matter, which is usually a challenge in complex media. The opportunity to

divide the update process to an imaging step and a parameter-inversion step allows

for better control of the anisotropy update. A featured seismic impedance inver-

sion method has been utilized for years. However, I utilize this at the update stage

maintaining the more elaborate (FWI) objective of fitting the modeled data to the

observed ones. This approach promises to deliver a practical orthorhombic anisotropy

inversion in a less complex setting. For that case, the number of parameters is much

larger (double at minimum), but so is the required azimuth coverage in the acquisition

to hopefully constrain these parameters.

4.2 Conclusion

Based on the analysis of model wavenumber components, I suggest a cascade inversion

strategy to reconstruct the full model wavenumber components. After time-domain

migration, I apply the analytical Frechet derivations to map the perturbations of

images to the perturbations of model parameters. And I propose a new objective

function in the time image domain. It is based on the fact that the energy of migration

image of the data residual should be zero when the predicted data and observed data

are the same. Because of the oriented time domain imaging method, the migration

image always focuses at zero subsurface offset. This means that no extended migration

images are needed to evaluate the data misfit in the image domain. In addition using

the oriented time-domain migration method allows us to avoid relying on the initial

41

velocity model for the gradient and provides more stable updates. Considering the

poor velocity models we tend to start the FWI method with, the velocity free gradient

calculation can help improve convergence as it has a more convex nature. Another

advantage of the proposed method is that the computational cost is about half of

the conventional full waveform inversion method. This method is also applicable

to VTI anisotropic media. By extending the oriented time-domain migration image

along incident angle axis, I can separate the data dependency on different anisotropic

parameters, which helps eliminate the crosstalk artifacts between them. I used the

optimal parametrization for the proposed VTI anisotropic inversion method, and

the numerical examples verify its effectiveness. The proposed method provides a

robust inversion strategy for anisotropic media. It reduces the extrapolation time

and memory requirements comparing to the conventional FWI algorithms in each

iteration. This method helps more when it is applied to the orthorhombic media.

4.3 Future Research Work

Since the migration algorithm and impedance inversions have been widely used in the

industry, the proposed method should also be applicable for field data. Some primary

tests have been done on a field data set. However, I need to try more parameter sets

to make the inversion results more reliable.

This method can partly handle complex models when applying wavenumber filter

to the data. To fulfill the assumptions made previously, I need to design a wavenumber

filter which preserves only horizontal parts of the subsurface structures.

This proposed method is also applicable to orthorhombic media. It provides a

practical method for orthorhombic inversion due to its reduced memory and compu-

tation requirements. I will extend the existing 2D oriented time migration method to

3D case and also find an optimal parametrization for orthorhombic multiparameter

42

inversion. For 3D case, the computational efficiency of the proposed method becomes

more important for its practical use. Thus, a set of high-level parallel codes needs to

be developed in the future.

43

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A Papers Submitted and Under

Preparation

• Rao, Y., Wang, Y., Zhang, Z.D., Ning, Y., Chen, X, and Li, J., “Reflection seismic

waveform tomography of physical modelling data“, Journal of Geophysics and Engi-

neering, 13, no. 2 (2016): 146.

• Zhang, Z.D. and Alkhalifah, T, “Full waveform inversion using oriented time-domain

imaging method for VTI medium“, Submitted to Journal of Geophysics International,

2016.

• Zhang, Z.D. and Alkhalifah, T., “Full Waveform Inversion using Oriented Time-

Domain Imaging Method“, SEG Technical Program Expanded Abstracts, 2015.

• Zhang, Z.D. and Alkhalifah, T., “Efficient Quasi-P Wavefield Extrapolation Using

an Isotropic Lowrank Approximation“, EAGE Expanded Abstracts, 2016.

• Zhang, Z.D. and Alkhalifah, T., “Full Waveform Inversion Using Oriented Time-

domain Imaging Method for VTI Medium“, EAGE Expanded Abstracts, 2016.