Empirical Ground Motion Prediction Equations

Empirical Ground Motion Prediction Equations

Introduction to Strong Motion SeismologyDay 2, Lecture 3

Nay Pyi Taw, MyanmarTuesday, 29 January 2013

Empirical Specification of Ground MotionsEmpirical Specification of Ground Motions

• Ground-motion prediction equations • Collections of ground-motion values (and time series)

for magnitude and distance bins

2USGS - DAVID BOORE

Overview

• Empirical ground-motion prediction equations (GMPEs)– What they are and how they are used– The database

• Processing of data

• Combining horizontal components

– Expected dependency on magnitude and distance– What functional form to use– What to use for the explanatory variables– Example: PEER NGA-West2 GMPEs

3USGS - DAVID BOORE

The Engineering Approach

• Use physically based empirical relationships

• Take a large suite of recorded earthquake motions and perform regression analysis to obtain models that may be used for the estimation of the distribution of future ground-motions

• Simple models that only require knowledge of a few parameters

• Significant variability associated with the estimates of these equations– Partly reflects the simplicity of the models– Partly reflects the inherent variability of earthquake ground-

motions

• The treatment of this variability is crucial for hazard analysis

P. Stafford4USGS - DAVID BOORE

Ground-Motion Prediction EquationsGround-Motion Prediction Equations

Gives mean and standard deviation of response-spectrum ordinate (at a particular frequency) as a function of magnitude distance, site conditions, and perhaps other variables.

10-1 1 101 102

0.01

0.1

1.0

Shortest Horiz. Dist. to Map View of Rupture Surface (km)

La

rge

rH

ori

zon

talP

ea

kA

cce

l(g

)

1992 Landers, M = 7.31994 NR, M=6.7 (reduced by RS-->SS factor)Boore et al., Strike Slip, M = 7.3, NEHRP Class D_+

File

:D

:\m

etu

_0

3\r

eg

res

s\B

JF

LN

DN

R.d

raw

;Da

te:2

00

5-0

4-2

0;T

ime

:2

0:2

5:2

6

5USGS - DAVID BOORE

Call them “Ground-Motion Prediction Equations” (GMPEs)

• “Attenuation EquationsAttenuation Equations” is a poor term

– The equations describe the INCREASE of amplitude with magnitude at a given distance

– The equations describe the CHANGE of amplitude with distance for a given magnitude (usually, but not necessarily, a DECREASE of amplitude with increasing distance).

6USGS - DAVID BOORE

Deriving the Equations

• Regression analysis of observed data if have adequate observations (rare for most of the world).

• Regression analysis of simulated data for regions with inadequate data (making use of motions from smaller events if available to constrain distance dependence of motions).

• Hybrid methods, capturing complex source effects from observed data and modifying for regional differences.

7USGS - DAVID BOORE

1 10 100 1000

5

6

7

8

Mo

me

nt

Ma

gn

itud

e

Used by BJF93 for pga

Western North America

1 10 100 1000

5

6

7

8

Distance (km)

Mo

me

nt

Ma

gn

itud

e

AccelerographsSeismographic Stations

Eastern North America

File

:C:\m

etu_

03\re

gres

s\m

_d_w

na_

ena

_pg

a.d

raw

;D

ate

:20

03-0

9-05

;Ti

me:

14:4

0:01

Observed data adequate for regression exceptclose to large ‘quakes

Observed data not adequate for regression, use simulated data

8USGS - DAVID BOORE

What Measure of Seismic Intensity to Use?

• PSA (derived from SD)• SD? Need for displacement-based design

– Can be derived from PSA, so GMPEs in terms of PSA also give equations for SD

– If use SA, then need separate GMPEs for PSA and SD

• Usually horizontal component

9USGS - DAVID BOORE

How to Use Two Horizontal Components

• Use both independently• Use larger component as recorded• Use larger component after rotation to find maximum• Use vector sum• Use geometric mean as recorded• Use orientation-independent geometric mean (gmroti50)• Use orientation-independent, non-geometric mean (rotd50)

10USGS - DAVID BOORE

What to use for the Predictor Variables?

• Moment magnitude• Some distance measure that helps account for the

extended fault rupture surface (remember that the functional form is motivated by a point source, yet the equations are used for non-point sources)– Distance must be estimated for future event (leaves out distance to

energy center, hypocentral distance)

• A measure of local site geology

11USGS - DAVID BOORE

Moment Magnitude

• Best single measure of overall size of an earthquake• Can be determined from ground deformation or

seismic waves• Can be estimated from paleoseismological studies• Can be related to slip rates on faults

12USGS - DAVID BOORE

Many distance measures are used

• There is no standard, although the closest distance to the rupture surface is probably the distance most commonly used

• The distance measure must be something that can be estimated for a future earthquake

13USGS - DAVID BOORE

Most Commonly Used:

Rrup

RJB

14USGS - DAVID BOORE

How does the motion depend on magnitude?

• Source scaling theory predicts a general increase with magnitude for a fixed distance, with more sensitivity to magnitude for long periods and possible nonlinear dependence on magnitude

15USGS - DAVID BOORE

16USGS - DAVID BOORE

How does the motion depend on distance?

• Generally, it will decrease (attenuate) with distance• But wave propagation in a layered earth predicts

more complicated behavior (e.g., increase at some distances due to critical angle reflections (“Moho-bounce”)

• Equations assume average over various crustal structures

17USGS - DAVID BOORE

M and R dependence shown by data

18USGS - DAVID BOORE

• The scatter remaining after removing magnitude and distance dependence is large

• Can it be reduced by introducing other factors?• The most obvious additional factor tries to capture

the effect of site response

19USGS - DAVID BOORE

Uncertainty after Mag & Dist Correction

0.1

1

10

4503001500

Observation Number (no particular order)

95-percent within factor of 3.5 of average

(sigma = 0.63 (natural log))

(E. Field)

Simplest: Rock vs Soil

0.1

1

10

4503001500

Sorted: Soil to left - Rock to right

motion is ~1.5 times greater than Soil Rock

(E. Field)

0 ( )

SZ z

S

zV

dV

0

1 1( )

z

SZ SSZ

S S dV z

Time-averaged shear-wave velocity to depth z:

Average shear-wave slowness to depth z:

22USGS - DAVID BOORE

Site Classifications for Use WithGround-Motion Prediction Equations

• Rock = less than 5m soil over “granite”, “limestone”, etc.• Soil= everything else

2. NEHRP Site Classes (based on VS30)

3. Continuous Variable (VS30)

1. Rock/Soil

620 m/s = typical rock

310 m/s = typical soil

23USGS - DAVID BOORE

200 1000 20000.40.5

1

2

345

V30, m/sec

Am

plifi

catio

n2003-06-08 23:12:15

T=0.10 s

slope = bv, where Y (V30)bV

200 1000 20000.40.5

1

2

345

V30, m/sec

T=2.00 s

VS30 as continuous variable

Note period dependence of site response24USGS - DAVID BOORE

Why VS30?

• Most data available when the idea of using average Vs was developed were from 30 m holes, the average depth that could be drilled in one day

• Better would be Vsz, where z corresponds to a quarter-wavelength for the period of interest, but

• Few observations of Vs are available for greater depths

• Vsz correlates quite well with Vs30 for a wide range of z greater than 30 m (see next slide, from Boore et al., BSSA, 2011, pp. 3046—3059)

25USGS - DAVID BOORE

26USGS - DAVID BOORE

Effect of different site characterizationsfor a small subset of data for which V30values are available

• No site characterization• Rock/soil• NEHRP class• V30 (continuous variable)

27USGS - DAVID BOORE

-0.5

0

0.5

1

1.5

Obs

-B

JF(M

,R)

RockSoil

T = 2.0 secResiduals computed as obs-bjf (with no site correction,which is the same as assuming BJF class A or V30 = VA).

200 300 400 1000-1

-0.5

0

0.5

1

V30 (m/sec)

Obs

-B

JF(M

,R)

-S

ite(r

ock,

soil)

RockSoil

T = 2.0 secrock & soil site classification

File

:C:\p

sv\D

EV

EL

OP

\RS

DL

2A

_B

_p

pt.d

raw

;D

ate

:20

03

-06

-13

;Tim

e:

19

:07

:42

σ=0.25

σ=0.25

28USGS - DAVID BOORE

-1

-0.5

0

0.5

1

Obs

-B

JF(M

,R)

-S

ite(B

,C,D

)

RockSoil

T = 2.0 secNEHRP B,C,D site classification

200 300 400 1000-1

-0.5

0

0.5

1

V30 (m/sec)

Obs

-B

JF(M

,R)

-S

ite(V

30) Rock

Soil

T = 2.0 sec

V30 site classification (continuous)

File

:C:\p

sv\D

EV

EL

OP

\RS

DL

2C

_D

_p

pt.d

raw

;D

ate

:20

03

-06

-13

;Tim

e:

19

:08

:04

σ=0.21

σ=0.20

29USGS - DAVID BOORE

Nonlinear Soil Response

• Clearly seen in studies of pairs of records• Not as large as found in lab experiments• Account for in regression equations by looking for

systematic variation of soil amplitudes relative to predicted rock amplitudes at given distance and magnitude

30USGS - DAVID BOORE

2-stage Regression

• Use 2 stage regression• Use weighted least-squares (random effects model)

31USGS - DAVID BOORE

Stage 1 Regression: Determine the distance function and event offsets

32USGS - DAVID BOORE

Stage 2 Regression: Determine the magnitude scaling

33USGS - DAVID BOORE

Quantifying the Uncertainty

• The GMPEs predict the distribution of motions for a given set of predictor variables

• The dispersion about the median motions can be crucial for low annual-frequency-of-exceedance hazard estimates (rare occurrences for highly critical sites, such as nuclear power plants, nuclear waste repositories)

• Must be clear on type of uncertainty• The scatter is very large; can it be reduced?

34USGS - DAVID BOORE

35USGS - DAVID BOORE

Two types of Aleatory Uncertainty

• Within event (intraevent) (φ) (event-to-event variation has been removed)

• Event-to-event (interevent) (τ): generally smaller than φ

• Total (σ): 2 2 σ φ τ

36USGS - DAVID BOORE

What functional form to use?

• Motivated by waves propagating from a point source• Add more terms to capture effects not included in

simple functional form

37USGS - DAVID BOORE

h

d

r

File

:C:\

me

tu_

03

\re

gre

ss\p

oin

t_so

urc

e.d

raw

;Da

te:2

00

3-0

9-0

5;T

ime

:11

:47

:14

38USGS - DAVID BOORE

39USGS - DAVID BOORE

40USGS - DAVID BOORE

Modeling Considerations

• Not too simple• Not too complex (over paramerized)• Reasonable extrapolations to data-poor but

engineering-important regions of predictor variable space (e.g., close to M~8 earthquakes)

41USGS - DAVID BOORE

Represent M-scaling by joined quadratic and linear or simple quadratic?

42USGS - DAVID BOORE

43USGS - DAVID BOORE

Pacific Earthquake Engineering Research Center (PEER) NGA Projects:

NGA (2008)NGA-West2 (2013)

• Abrahamson & Silva• Boore, Stewart, Seyhan, and Atkinson• Campbell & Bozorgnia• Chiou & Youngs• Idriss

44USGS - DAVID BOORE

NGA Project Details

• All developers used subsets of data chosen from a common database

– Metadata (e.g., magnitude, distance, etc.)

– Uniformly processed strong-motion recordings

– U.S. and foreign earthquakes

– Active tectonic regions

• The database development was a major time-consuming effort

45USGS - DAVID BOORE

• 1264 (173) worldwide shallow crustal events from active tectonic regions

• 19409 (3551) recordings (mostly 3-components each) uniformly processed strong motion stations

• M 3.0 (4.2) to 7.9 (7.9)

PEER NGA-West2 Strong-Motion Database

Blue = Previous NGA

46USGS - DAVID BOORE

BSSA13 GMPEs

Dave Boore, Jon Stewart, Emel Seyhan, Gail Atkinson

48

Our equation for predicting ground motions is:

1 1 2 2ln ln lnY w Y w Y (1)

where iw are weights that sum to unity. We use an average of two

predicted motions as a way of dealing with possible uncertainties

in the functional forms and the derived coefficients.

Boore, Stewart, Seyhan, and Atkinson (BSSA13) Ground-Motion Prediction Equations (GMPEs)

USGS - DAVID BOORE

49

In equation (1) ln iY is given by

30ln ( ) ( ) ( , ) ( )i i i ii D JB S S JBY F F R F V R M M , M , M M , (2)

In this equation, iFM , iDF , and i

SF represent the magnitude scaling,

distance function, and site amplification, respectively, for the ith set of coefficients (we drop the superscript in subsequent

equations). M is moment magnitude, JBR is the Joyner-Boore

distance (defined as the closest distance to the surface projection of

the fault), and 30SV is the time-averaged shear-wave velocity over

the top 30 m of the site. The predictive variables are M, JBR and

30SV ; the fault type is an optional predictive variable that enters

into the magnitude scaling term as shown in equation (6) below.

USGS - DAVID BOORE

50

is the fractional number of standard deviations of a single

predicted value of lnY away from the mean value of lnY (e.g.,

1.5 would be 1.5 standard deviations smaller than the mean

value). All terms, including the coefficient , are period

dependent. is computed using the equation:

2 2(M) (M) , (3)

where is the within-event (intraevent) aleatory uncertainty and is the event-to-event (interevent) aleatory uncertainty.

USGS - DAVID BOORE

The Distance Function

51

The distance function is given by:

1 2 3( , ) [ ( )]ln( / ) ( )D JB ref ref refF R c c R R c R R M M M , (4)

where

2 2JBR R h (5)

and 1c , 2c , 3c , refM , refR , and h are the coefficients to be

determined in the analysis.

USGS - DAVID BOORE

52

The magnitude scaling is given by:

a) hM M

20 1 2 3 4 5( ) ( ) ( )M h hF e U e SS e NS e RS e e M M M M M ,

b) > hM M

0 1 2 3 6( ) ( )M hF e U e SS e NS e RS e M M M ,

where U, SS, NS, and RS are dummy variables used to specify

unspecified, strikeslip, normal slip, and reverse slip fault type.

The Magnitude Function

USGS - DAVID BOORE

Mechanism U SS NS RSunspecified 1 0 0 0strikeslip 0 1 0 0normal 0 0 1 0reverse 0 0 0 1

53

The Magnitude Function

USGS - DAVID BOORE

ln lnS LIN NLF F F

The Site Amplification Function

30ln ln( / )LIN lin S refF b V V

Linear Amplification:

The coefficient blin depends on period. We use the Stewart and Seyhan (2013) model.

( 760 m/s)refV

54USGS - DAVID BOORE

ln lnS LIN NLF F F

The Site Amplification Function

760m/s 0.1ln ln

0.1NL nl

PGA gF b

g

Nonlinear Amplification:

The coefficient bnl depends on period and VS30. We use Stewart and Seyhan’s (2013) model.

55USGS - DAVID BOORE

56

Amplification using the Stewart & Seyhan (2013) Site Amplification Model

USGS - DAVID BOORE

Not Included• Directivity• Basin depth• Hanging wall (using RJB accounts for this to

some extent)• Other possible predictor variables (e.g., dip,

Ztor, etc.)

57USGS - DAVID BOORE

Determination of Coefficients (2-stage regression)

• Select data: – no basement or large structure records, etc;– RJB < 80 km– event class 1, as determined by CRJB=10 km

• Adjust observations to Vs30=760 m/s using SS13 site amps • Constrain c3 (anelastic term) and Mh to values from

special study• Regress for other coefficients, including pseudodepth h

58USGS - DAVID BOORE

Constrained Coefficients

59USGS - DAVID BOORE

60USGS - DAVID BOORE

61USGS - DAVID BOORE

62USGS - DAVID BOORE

63USGS - DAVID BOORE

64USGS - DAVID BOORE

65USGS - DAVID BOORE

66USGS - DAVID BOORE

67USGS - DAVID BOORE

68USGS - DAVID BOORE

69USGS - DAVID BOORE

70USGS - DAVID BOORE

I don’t want you to think that BSSA13 is the only GMPE available, or that it is superior to others. There are 100s of GMPEs worldwide.

The large epistemic variations in predicted motions are not decreasing with time

From Douglas (2010)

Factor ~ 20

Mw6 strike-slip earthquake at rjb = 20 km on a NEHRP C site

Figure 2. Predicted PGA and SA(1 s) (unfilled black circles) for a Mw6 strike-slip earthquake at 20 km on a NEHRP C site against publication date for over 250 models published in the literature. Filled red circles indicate models published in peer-reviewed journals and for which basic information on the used dataset is available. Also shown are the median PGA and SA(1 s) within five-year intervals (black line) and the median ±1 standard deviation (dashed black lines) based on averaging predictions. From Douglas (2010).

72USGS - DAVID BOORE

73

Within the PEER NGA project there are significant differences between the functions and data used. The BSSA13 GMPEs are probably the simplest, but there may be situations where they should be used with caution.

Here are some comparisons for the first set of GMPEs.

USGS - DAVID BOORE

Strike-Slip Fault Mechanism @ 10 kmMagnitude = 6.5, 7.5

NEHRP BC Site Conditions (VS30=760 m/s)

Slide from Yousef Bozorgnia74USGS - DAVID BOORE

45 degree dip to right, to 15 km depth 75USGS - DAVID BOORE

Differences after 10+ years

• SRL 1997

• Magnitude (~ 5-7.5)• Distance (~ 0-100km)• Period (~ 0-4/5s)• Site Class (1 Vs30)• Hanging Wall (simple)• Nonlinear site class (1 model)• Style-of-faulting• Geometric Mean• Magnitude Dependent Sigma

• NGA 2007, 2013

• Magnitude (~ 3-8+)• Distance (~ 0-200km)• Period (~ 0-10s)• All Vs30• Hanging Wall & Footwall

(complex)• All Nonlinear Site models• Depth to Top of Rupture• Sediment Depth• Style-of-faulting• Orientation independent GM• Magnitude Independent Sigma

P. Stafford76USGS - DAVID BOORE

Do the NGA models have it all?

• NGA models have advanced a lot in terms of consideration of new parameters and the capturing of effects that were previously not modelled by empirical equations

• So, have they captured everything? No.

• What remains to be done?

– Damping (in NGA-West2)

– Directivity (some accounting for this in NGA-West2)

– Fault-normal/Fault-parallel; SaMax

– Vertical component

– Correlations between periods

P. Stafford77USGS - DAVID BOORE

78USGS - DAVID BOORE

79USGS - DAVID BOORE