2008 spudich & chiou_paper+anexos

7/27/2019 2008 Spudich & Chiou_paper+Anexos

1/116

Directivity in NGA Earthquake Ground

Motions: Analysis using Isochrone Theory

Paul Spudich, M.EERI, and Brian S.J. Chiou

Accepted,Earthquake Spectra

Likely publication date: June 2008

Corresponding (first) author: Paul Spudich

Mailing address: U.S.Geological Survey, MS977

345 Middlefield Road

Menlo Park, CA 94025

Phone: 1-650-329-5654

Fax: 1-650-329-5163

E-mail: [email protected]

Submission date for review copies: July 20, 2007

Date Accepted: August 21, 2007, November 15, 2007

Submission date for revised copies: November 9, 2007

Submission date for camera-ready copies: April 24, 2008

May 2, 2008: v19 - equation 2 denominator corrected

Abrahamson&Silva 2007 ref fixed

Appendix D added to list of appendices

Cell G57 fixed in example spreadsheet (Appendix A)

Ru, Rt, and Rri for hypo 2 fixed in example spreadsheet

Figure A12 fixed

Copyright (2008) Earthquake Engineering Research Institute. This article may bedownloaded for personal use only. Any other use requires prior permission of the Earthquake

Engineering Research Institute.


2/116



Paul Spudich,a)

M.EERI, and Brian S.J. Chioub)

v17 9 Nov 2007

We present correction factors that may be applied to the ground motion

prediction relations of Abrahamson and Silva, Boore and Atkinson, Campbell and

Bozorgnia, and Chiou and Youngs (all in this volume) to model the azimuthally

varying distribution of the GMRotI50 component of ground motion (commonly

called 'directivity') around earthquakes. Our correction factors may be used for

planar or nonplanar faults having any dip or slip rake (faulting mechanism). Our

correction factors predict directivity-induced variations of spectral acceleration

that are roughly half of the strike-slip variations predicted by Somerville et al.

(1997), and use of our factors reduces record-to-record sigma by about 2-20% at 5

sec or greater period.

INTRODUCTION

In a landmark paper Somerville et al. (1997) (henceforth 'SSGA') demonstrated the

correlated effects of rupture propagation, earthquake source radiation pattern, and particle

motion polarization on near-source ground motions. Their combined effect has subsequently

been referred to in the engineering literature as 'directivity,' although in the seismological

literature this term is reserved exclusively for rupture propagation effects. For use in

predicting ground motion amplification, duration, and polarizations SSGA introduced twopredictor variables, Xcos() for vertical strike-slip faults and Ycos() for dip slip faults

(see SSGA for definitions).

Despite the importance of SSGA's advance, use of their formulation has led to some

practical and conceptual difficulties. On the practical side, their formulation is a

discontinuous (step) function of magnitude, fault dip, fault rake, and rupture distance. Theirformulation does not predict ground motions in an excluded zone (called the neutral zone

below) around dipping faults, nor is its application to nonplanar faults clear. On the


3/116

conceptual side, there is only weak theoretical justification for their functional forms. For

long strike-slip faults, theirXfactor implies that ground motion increases progressively along

a rupture all the way to its end, a prediction clearly in conflict with the intensity map from the1906 San Francisco earthquake (Boatwright and Bundock, 2008). Abrahamson (2000) has

modified the SSGA model to avoid some of these problems by capping Xcos() at 0.4 and

by introducing magnitude- and distance-tapers to smooth discontinuities. Rowshandel (2006)

has cleverly generalized both the X and terms to smooth and extend the range of

applicability of the basic SSGA model, at the price, however, of requiring a surface integral

to be done over the fault for every receiver location.

This paper has the following goals. 1) We wish to develop physically-based predictor

variables by using isochrone theory (Bernard and Madariaga, 1984; Spudich and Frazer,

1984, 1987). We attempt to keep these predictors as simple and computationally rapid as

possible while retaining essential physics and limiting the domain of applicability as little aspossible. 2) Using this theory, we clarify the various factors that contribute to azimuthal

distribution of shaking around a source. 3) We develop directivity models with empirically

determined coefficients that can be used to calculate a 'directivity' correction to each NGA

developer's ground motion prediction model.

DEFINITION OF ISOCHRONE DIRECTIVITY PREDICTOR, IDP

Isochrone theory allows a simplification of an otherwise complicated formulation in

computational seismology. The theory simplifies the computation of synthetic seismograms

to an analytical expression, from which one can identify the main contributors to directivity

effects (or the azimuthal variation of near-fault ground motion). In the isochrone formulation

three main contributors to the azimuthal variation of ground motions are recognized. These

factors, which various formulations lump together under the term 'directivity,' include the slip

distribution, the radiation patterns, and true seismic directivity (in its guise here as isochrone

velocity). In the last few years we experimented with numerous candidate variables and

functional forms to search for a preferred representation of these three contributors for the

purpose of modeling directivity effects in a (even simpler) ground-motion prediction model.

Some of our earlier efforts are documented in Spudich et al. (2004) and Spudich and Chiou


4/116

(2006). In the following we present and justify this preferred predictor variable and compare

it to the predictor variables used by SSGA.

Our preferred predictor of directivity effects, IDP (the isochrone directivity predictor), is

a product of three terms

IDP = C S Rri (1)

C=min c ,2.45( ) 0.8

(2.45 0.8)(2)

S= ln min(75,max(s,h))[ ] (3)

All the above terms are evaluated at a site xs

in the geometry shown in Figure 1, in which s

is the along-strike distance in km from the hypocenter xh to the point xc on the fault closest

to the site, and h is the downdip distance in km from the top of the rupture to the hypocenter.Approximate isochrone velocity ratio c is defined below. C is a normalized form of c ,lying in the range [0,1]. Rri is a scalar radiation pattern amplitude defined below, ranging

from 0 to about 1, which we use for the GMRotI50 component of motion. More discussions

of the definitions ofSand Care given in the next section. In equation 1,S takes the role ofX,

andC

takes the role of cos() in

Xcos(

). The radiation pattern amplitude

Rri provides

the neutral region defined by SSGA for reverse events. Equation 1 differs from the

functional form recommended by Spudich et al. (2004). We will comment on this later.

Isochrone velocity ratio c is an approximation of the isochrone velocity defined inSpudich and Frazier (1984), which captures the seismic directivity amplification around a

fault. It has the advantages of being defined everywhere on the Earth's surface around

vertical and dipping faults using distance measures obtainable in typical practice. Spudich et

al. (2004) defined c to be proportional to the distance D (Figure 1) between the hypocenterand the closest point, divided by the difference in arrival times of S waves from these two

points. The physical meaning is simple; all the energy radiated between the hypocenter and

the closest point arrives in a time interval, and if that time interval is very short energies aretime-compressed, a directivity pulse is formed, and the spectral amplitude is amplified.

c is derived in Electronic Appendix A, and is given by


5/116

c :=

vr

RHYP RRUP( )

D

1

, D > 0

= vr

, D = 0 (4)

where rupture velocity is vr

and is the shear wave speed in the source region. In this work

we assume

vr

= 0.8,

which, on average, is a good approximation for most earthquakes. Note thatc depends only

on the locations of the hypocenter, the site, and the point on the fault closest to the site. clies in the range

vr

c

vr

1

1

which, for vr =

0.8 is the range from 0.8 to 4. For a fault having bilateral rupture

cachieves its maximum value when the rupture is traveling directly toward the site, and it

achieves the above minimum value when the rupture direction is exactlyperpendicularto the

direction to the site. Spudich et al. (2004)'s main results (their equations 9a and 9b, which we

use here) assumed that the earthquake's hypocenter is not on the edge of the fault. Their

special case of a hypocenter exactly on the edge of a fault (their equation 10) is not used here.The D = 0 limit of c (Equation 4) is multivalued when the hypocenter is on the edge of therupture area, and consequently we recommend that hypocenters not be placed on the edge of

rupture areas. Guidelines for sensible placement of hypocenters can be found in Mai et al.

(2005). For multisegment faults, we generalizes andD as shown in Electronic Appendix A.

Finally, scalar radiation pattern Rri is

Rri = max Ru2+ Rt

2,

,

where Rt and Ru are the strike-normal (transverse) and strike-parallel hypocentral radiation

patterns (Electronic Appendix A), with a water level = 0.2 filling the nodes. We

approximate the finite fault radiation pattern by a single point source radiation pattern.

Electronic Appendix A describes a generalization for use with multi-segment ruptures and it

gives a computed example.


6/116

DIRECTIVITY IN SYNTHETIC DATA

Despite having 3551 records from 173 earthquakes, there are very few earthquakes in the

NGA dataset that are recorded at 10 or more azimuthally well-distributed stations having

good data at long periods where the directivity signal is strongest. In addition, the azimuthal

distribution of ground motion around these events, particularly at rupture distances less than

40 km, is strongly correlated with the local slip distribution. In such cases it is difficult to

separate the effects of true directivity from the effects of proximity to a local slip maximum,

making the inference of directivity effects very problematic.Consequently, we turned to the rich data set of synthetic data calculated by the URS

Corporation to provide guidance on the search for preferred predictor variable and an

effective functional form for a directivity model. URS calculated synthetic strike-normal and

strike-parallel seismograms at about 200 station locations surrounding 10 strike-slip events

and 12 reverse-slip events, described in Abrahamson (2003) and Somerville et al. (2006). We

used a subset of the events (Table 1) with deeper hypocenters located 10%, 30%, and 50%

from a fault edge.

Table 1. Synthetic URS events used

Event

Name

Mag

W (km) L (km) Dip (dg)

Top of

Rupture (km)RB 6.5 18 18 45 0

RG 7.0 28 36 45 0

RK 7.5 28 113 45 0

SA 6.5 13 25 90 0

SD 7.0 15 67 90 0

SE 7.5 15 210 90 0

SH 7.8 15 420 90 0

Because synthetic data contain effects like magnitude scaling and geometric spreading in

addition to directivity, we had to remove from the data the non-directive part of the motions.

This is done by fitting simulated data from each event and each hypocenter location to a

simple non-directive model

ln(yi ) = k1 + k2 ln RRUPi + k3( )+ i

where yi was the GMRotI50 spectral acceleration at station i, RRUPi was the closest distance


7/116

to the fault from station i,i

is the residual, and kiare unknown coefficients determined by

regression analysis. We then used the residualsi

, referred to as the 'directive residual'

below and shown in Figure 2, to guide the development of our directivity model.

We examined the correlation of directive residuals with a variety of candidate predictors

motivated by the isochrone theory. Spudich et al. (2004) noted that the logarithm of the

ground motion should be proportional to the logarithm of isochrone velocity ratio c , andSpudich and Chiou (2006) proposed that the predictor should contain the product Dc~ln ,

where D isD normalized by the diagonal of the fault. In the current work we further tested

various products of log or linear c and D against the directive residuals. FollowingAbrahamson and Silva (2007) we also tested forms involving s and ln(s). Below is a

summary of our findings and decisions that ultimately lead to the definitions ofCand Sgiven

in equations 2 and 3.

We noted that the directive residuals correlated well with c up to a value of 2.45, whichprompted us to cap c at 2.45. We decided to normalize the capped c so the resultingvariable, Cof Equation2, is in the range [0, 1], same as the cos() and )cos( used by

SSGA. We also noted the correlation of directive residuals with ln(s) was more linear than

with eitherD ors. Based on the above two observations, we speculated that a predictor

involving the term )sln(C would work well for modeling the directivity effect in URSs

simulated motions. This is confirmed by the plots in Figure 2, which show the correlation of

directive residuals with Cln(s) for both the strike-slip and reverse events listed in Table 1.

The residuals are a linear function of Cln(s) between about 0 and 4. Note that within the

interval [0, 4] the slope of the residuals is about the same over the magnitude range 6.5 - 7.8

and for strike-slip and reverse events. Note also that residuals from hanging wall, foot wall,

and neutral zone stations (zones defined in the NGA database documentation) show the same

approximate slope with respect to Cln(s). Other tested predictors did not share these

characteristics. Some magnitude dependence is seen in the average level of the residuals, but

no magnitude dependence was seen in the real earthquake data.

From Figure 2 it was obvious that some modifications to )sln(C were needed in order to

model the directive residuals outside the interval [0, 4]. For negative predictor values a


8/116

horizontal tail of residuals indicates that a floor of 1 km should be placed unders. However,

with this floor value the form )sln(C produces no directivity directly up-dip from the

hypocenter of a reverse event (because ln(s) is 0), a behavior that is contradictory to SSGAand not supported by the (limited) data in both simulations and the NGA dataset. A proper

floor (larger than 1 km) is needed to allow up-dip directivity to come through. We picked h

(the downdip distance in km from the top of the rupture to the hypocenter; Figure 1) to be the

floor ofs. With this floor our updip prediction is improved, but still underestimates the data

by about 0.2 log units. See Appendix D for plots of updip residuals.

The strike-slip residuals decline for )sln(C value greater than 4, but we chose not to

include this decline in our model. These residuals correspond primarily to higher values ofs,

meaning that they are farther down the rupture. A decline in spectral acceleration with

distance along long strike-slip ruptures was seen, not only in the URS synthetics, but also in

synthetic ground motions produced by Pacific Engineering and Analysis (Somerville et al.(2006) and by ourselves (not shown). The decline does not seem to be caused by the

diminution of slip toward the end of the rupture. We chose not to include this decline in our

model because we do not yet understand the cause of the decline, and not understanding the

cause, we cannot be confident that such a variation in spectral acceleration seen in synthetic

seismograms would be found in real motions from long strike-slip earthquakes.

Consequently, in our model we caps at a value of 75 km, derived from our synthetic ground

motions, meaning that like Abrahamson (2000), our predictor does not continue to rise

inexorably with distance along the rupture. However, by capping rather than tapering to zero

for very large s, our predictor might overpredict directivity effect at the ends of very long

strike-slip ruptures.

EMPIRICAL DATA

We used the same record selection criteria as did each developer. Developer teams in this

volume, Abrahamson and Silva (2008), Boore and Atkinson (2008), Campbell and Bozorgnia

(2008), and Chiou and Youngs (2008) (AS, BA, CB, and CY in the following) provided us

with their predicted ground motions and event terms for their selected records, from which

we derived total residuals (observed ground motion minus developer's median prediction). If


9/116

the developer provided a predicted motion for an NGA record, we used the record.

Developers total residuals were the response data used in our regression analysis to

develop models for directivity effects. More discussions on the data will be given in the

following sections. In general, all developers' data sets included post-1995 large earthquakes

not in the SSGA data set, such as the 1999 Kocaeli and Dzce, Turkey, earthquakes and the

1999 Chi-Chi, Taiwan, earthquake. Several well-recorded Chi-Chi aftershocks were

included in the AS and CY data sets.

DEVELOPMENT OF THE DIRECTIVITY MODEL

Development of our model proceeded through various stages. The first stage was data

exploration, when we tried to get some general idea of what domain of the data could be fit

by various directivity predictors including our chosen IDP above. Some earthquakes'

residuals correlate well with the IDP, others correlate poorly, and some have strong anti-

correlations, as can be seen in Figure 3, in which events are ordered by magnitude. To

produce this figure a simple least-squares straight line was fit through developer AS's

residuals in the 0 - 40 km distance bin for each earthquake for each period. Normalized slope

is the slope divided by its standard deviation, which we use as an indicator of significance of

the slope owing to the highly variable number of data for each quake. Recall that some

events had only 4 recordings, while others had more than 100.

In general there is a positive correlation with IDP, shown by the predominance of open

circles in Figure 3, except for a few events. The M5.99 1987 Whittier Narrows earthquake

was particularly problematic, being very well recorded and showing a strong anti-correlation

withIDP. The NGA rupture model for this event is peculiar, rupturing downward from a

hypocenter exactly on the upper edge of the rupture. Small ground motions at the Lamont

stations that recorded the 1999 Dzce earthquake also produced a poor correlation withIDP.

Our general conclusion from the data exploration was that the IDP correlated with the

developers' residuals best (i.e. had a non-zero linear slope with IDP) for earthquakes having

M 6.0, dips > 65, and periods greater than 2 sec. For near-vertical ruptures the IDP

worked best for distances less than 40 km. For earthquakes with low dips, there was some

evidence that the IDPcorrelated with the residuals better at distances beyond 40 km than


10/116

shorter distances. However, for this paper we decided to concentrate on directivity in the 0 -

40 km range.

Based on the above observations, we formulate the directivity effect as a function of

moment magnitudeM, rupture distance RRUP , andIDPas follows

fD = fr RRUP( ) fM M( ) a+ b IDP( ) (5)

Coefficients a and b are unknown and will be determined by regression analysis of the

empirical data (developers total residuals). fr is a distance taper, and fM is a magnitude

taper, where

fr = max 0, 1max 0, RRUP 40( )

30

and

fM

= min 1,max 0, M 5.6( )

0.4

.

fr has value unity for 0 RRUP 40 and tapers linearly to its value of zero at RRUP 70 .

fM has value zero for0 M 5.6 and rises linearly to its value of unity at M 6.0.

ESTIMATION OF THE MODEL COEFFICIENTS

For each developer team we conducted regression analysis to estimate model coefficients

a and b for each spectral period in the list of 0.5, 0.75, 1, 1.5, 2, 3, 4, 5, 7.5, and 10 sec, the

spectral periods common to the NGA models. To properly weight each event in the

estimation of coefficient a, we used a mixed-effects model (Abrahamson and Youngs, 1992;

Joyner and Boore, 1993). We selected as data the developer's total residuals from all

earthquakes with M 6.0 and all stations in the 0 - 40 km rupture distance range.

Regression analysis was performed using the NLME package in the statistical software S-

Plus.

Our decision to focus on the directivity effects inside the domain of M 6.0 and RRUP

40 km required an adjustment to the NGA model residuals to account for NGA model misfitsin this domain and the differences in data distribution between the developers' total data set

and the data subset used in this analysis, which could upset the original event terms and


11/116

hence the constant term in the NGA model. This adjustment ensured proper centering of the

total residuals data and hence allows a reasonable and stable estimate of coefficient a as a

function of period. We did the following to make the adjustment. We fitted a mixed-effects

model with a single constant term ao

to developers total residuals

i)i(qoi at ++= (6)

where ti is developer's total residual for the ith record, and q is the earthquake index for

record i. Random variables and are random errors with zero mean, the former being the

record-to-record errors (the intra-event residual) and the latter being event terms. Random

errors and have standard deviations o and o, respectively. The estimated ao and o are

listed in Table 2. We then subtracted the estimated ao

from developers total residuals. This

adjustment was done independently for each NGA model and for each of the 10 spectral

periods.

Using the adjusted total residuals, we estimated coefficients a and b using Equation 7,

( ) i)i(qioi IDPbaat +++= (7)

where ao

is the correction explained above, and are random variables with zero mean and

standard deviations and , respectively. There is no need for the tapers fr and fM in

Equation 7 given that we used data only having M 6 and RRUP 40 km. Equation 6 can be

considered as the null model of Equation 7 and the differences in between equations 7 and

6 can be viewed as a measure of the significance of directivity in the selected dataset.

To ensure a smooth directivity effect, we smoothed the estimated coefficients a and b

over periods in two steps. Coefficient b was smoothed first. We required the smoothed value

to be non-negative because negative b is anti-directivity, which our model does not predict

and we do not understand. In the second step we developed a revised estimate of a using

Equation 7 again, but with b fixed at its smoothed value from step 1. The resulting a

estimates were smoothed over periods with the constraint that a equaled zero where b

equaled zero, because a non-zero a in that circumstance causes a constant bias to all predicted

motions. The final, smoothed coefficients a and b for each of the 10 periods for each

developer's model are given in Table 2, along with and from the 2nd step of smoothing.


12/116

Plots of the original estimated values and the smoothed curves for coefficients a and b are

shown in Electronic Appendix B. The four resulting directivity models are called AS6, BA6,

CB6, and CY6. Figure 4 shows the predicted directivity effect fD

as a function of period for

each of the models. They will be discussed later.

Figures 5, 6, and 7 show examples of the data fit for 3, 5, and 10 sec, respectively; the

fitted lines have a slope given by the smoothed b and smoothed intercept a (Table 2).

Electronic Appendix B shows figures like Figure 5 - 7 for all periods and developers.

Although the residual data in Figure 7 could be fit more closely using a bilinear function thatis flat below IDP = 2 and rises linearly above that IDP, we decided not to use the bilinear

form for the following reasons. The high residuals in Figure 7 around IDP = 3 at 7.5 and 10

sec are dominated by Chi-Chi stations close to the slip maximum at the northwest end of the

rupture, and thus are biased high by the particular slip distribution. In addition, drop-out of a

number of low non-Chi-Chi residuals at IDP ~ 0.5 going from 5 to 10 sec helps transform a

linear trend into a bilinear trend. (This is more clearly seen in Electronic Appendix C residual

plots for abscissa

fD .) Finally, we are not aware of a physical reason to justify a transition

from a linear directivity at T 5 sec to a bilinear directivity at T 7.5 sec.

DIRECTIVITY MODEL RESIDUALS

We have plotted our intra-event residual from the 2nd

step of smoothing for each

developer against several independent variables, specifically 1) vr

, the ratio of rupture

velocity to shear velocity for each earthquake, 2) distance D between the hypocenter and the

closest point, 3) fault dip angle, 4) magnitude, 5) predicted directivity effect fD , 6) station

categorization (footwall, hanging wall, neutral zone, and other), 7) earthquake slip rake, 8)station Vs30, 9) Joyner-Boore distance, 10) closest distance to fault (RRUP), 11) along-strike

distances, and 12) down-dip hypocentral distance h.All these plots for all directivity models

are shown in Electronic Appendix C. The data set for each plot consists of all records used by

each developer for earthquakes having M 6.0 andRRUP 40 km.

The two most noticeable trends in residuals are summarized in Figure 8 for periods 3 sec

and 7.5 sec. First, long-period motions (T= 7.5 sec) of hanging wall stations are on average

underpredicted after we corrected for directivity effects. We have not yet determined how


13/116


14/116

correlations withs andD are probably related to the 'U' shape in the residuals noted above,

caused by domination of Chi-Chi at long periods.

Finally, use of our directivity model reduces the record-to-record standard deviation by

about 16% at 10 sec, compared to a null-directivity model (Table 2). As a result of modeling

directivity, the corrected NGA model's intra-event standard deviations should become

smaller, but it is difficult for us to estimate the amount of reduction. The correct approach is

to re-estimate the standard deviations (for both inter-event and intra-event residuals) with a

directivity term in the NGA equation. However, if an interim solution for intra-event standarddeviation is needed immediately, one could consult the fractions of reduction listed in Table

2, after they are smoothed over periods.


15/116

Table 2. Directivity model coefficents and statistics.

Period

(sec)

No. of

data

No.

of

EQ

a b

ao o ( ) 0 o

AS6

0.5 573 40 0.0000 0.0000 0.5414 0.2247 4.210E-02 0.5414 0

0.75 572 40 -0.0447 0.0298 0.5586 0.2460 -2.137E-02 0.5596 0.002

1 572 40 -0.0765 0.0510 0.5553 0.3138 2.305E-02 0.5598 0.008

1.5 562 38 -0.1213 0.0809 0.5174 0.3260 3.380E-02 0.5225 0.01

2 538 38 -0.1531 0.1020 0.5240 0.3711 4.692E-02 0.5341 0.019

3 465 35 -0.1979 0.1319 0.5178 0.3595 3.191E-02 0.5393 0.04

4 436 34 -0.2296 0.1530 0.5294 0.3922 6.307E-02 0.5506 0.039

5 328 30 -0.2542 0.1695 0.5285 0.4192 8.158E-02 0.5510 0.041

7.5 276 28 -0.3636 0.2411 0.5147 0.4332 -5.407E-02 0.5560 0.074

10 158 17 -0.5755 0.3489 0.5566 0.3656 -1.517E-01 0.6626 0.16

BA6

0.5 419 27 0.0000 0.0000 0.5212 0.1945 -8.870E-03 0.5212 0

0.75 418 27 -0.0532 0.0355 0.5387 0.2618 -3.463E-03 0.5394 0.001

1 418 27 -0.0910 0.0607 0.5278 0.3068 5.466E-02 0.5327 0.009

1.5 412 26 -0.1443 0.0962 0.5052 0.3214 8.360E-02 0.5091 0.008

2 390 25 -0.1821 0.1214 0.5191 0.3815 9.181E-02 0.5301 0.021

3 371 25 -0.2353 0.1569 0.5197 0.3985 3.557E-02 0.5484 0.052

4 363 25 -0.2731 0.1821 0.5247 0.3637 3.217E-02 0.5559 0.056

5 263 20 -0.3021 0.2015 0.5513 0.3801 2.285E-02 0.5973 0.077

7.5 234 20 -0.4627 0.2727 0.5340 0.4514 4.121E-03 0.6005 0.111

10 129 12 -0.8285 0.4141 0.5171 0.3387 -1.210E-01 0.6503 0.205

CB6

0.75 438 36 0.0000 0.0000 0.5298 0.2247 2.525E-02 0.5298 0

1 438 36 -0.0329 0.0220 0.5234 0.2666 5.890E-02 0.5243 0.002

1.5 431 34 -0.0795 0.0530 0.4889 0.2699 8.493E-02 0.4899 0.002

2 409 34 -0.1125 0.0750 0.4921 0.2507 7.915E-02 0.4972 0.01

3 387 31 -0.1590 0.1060 0.4964 0.2556 5.891E-02 0.5129 0.032

4 379 31 -0.1921 0.1280 0.5075 0.2323 8.163E-03 0.5250 0.033

5 276 27 -0.2172 0.1450 0.5206 0.2677 -4.461E-02 0.5481 0.05

7.5 248 27 -0.3227 0.2147 0.5151 0.3580 -8.447E-02 0.5613 0.08210 129 16 -0.6419 0.3522 0.5365 0.4071 -2.105E-01 0.6497 0.174

CY6

0.75 570 40 0.0000 0.0000 0.5428 0.3615 8.438E-02 0.5428 0

1 570 40 -0.0260 0.0200 0.5393 0.4042 8.660E-02 0.5404 0.002

1.5 560 38 -0.0627 0.0482 0.5097 0.3998 9.405E-02 0.5113 0.003

2 536 38 -0.0887 0.0682 0.5307 0.4044 1.002E-01 0.5349 0.008

3 462 35 -0.1254 0.0965 0.5311 0.4335 1.064E-01 0.5431 0.022

4 432 34 -0.1514 0.1165 0.5503 0.4274 1.389E-01 0.5626 0.0225 324 30 -0.1715 0.1320 0.5527 0.4895 5.773E-02 0.5655 0.023

7.5 272 28 -0.2797 0.1865 0.5476 0.4693 5.731E-02 0.5713 0.041

10 154 17 -0.4847 0.2933 0.5819 0.3077 1.719E-02 0.6454 0.098


16/116

CONCLUSIONS

In this paper we introduce a new and physically-based isochrone directivity predictor

(Equation 1) and models (Equation 5 and Table 2) of directivity effects based on this

predictor. Our models AS6, BA6, CB6, and CY6 almost always predict about half the

directivity amplification or deamplification at every period compared to the model of SSGA,

although our forward directivity is comparable to that of Abrahamson (2000), as shown in

Figure 4. Capping of the directivity predictor (s by us,XCos() by Abrhamson (2000)) partly

contributes to the discrepancy noted at the high predictor value. Watson-Lamprey (2008)

shows that the reduced scaling of directivity effects inferred from the NGA data set is caused

by variations in the data set, compared to SSGA's, rather than differences of

parameterization.

In addition to the difference in amplitude, maps of the predicted directivity effects (Figure

10) also reveal important spatial differences. To prepare these maps we computed directivity

effects by applying the AS6 model and the SSGA model to a grid of 2601 points at a spacing

of 4 km. These calculations were done for a period of 5 sec. For the vertical strike-slip fault

(the geometry of event SD, Table 1), the isochrone directivity in general resembles the

predictions of SSGA but predicts much narrower zones of amplification in the forward(south) direction and a small deamplification in the backward (north) direction. The maps for

reverse event RG show that the isochrone directivity also resembles the pattern predicted by

SSGA, but has a more gradual and natural transition going from the footwall or hanging wall

zones to the neutral zones.


17/116

ACKNOWLEDGEMENTS

This project was sponsored by the National Earthquake Hazards Reduction Program and by the

Pacific Earthquake Engineering Research Center's Program of Applied Earthquake Engineering

Research of Lifeline Systems supported by the California Energy Commission, California Departmentof Transportation, and the Pacific Gas & Electric Company. The financial support of the PEARL

sponsor organizations including the Pacific Gas & Electric company, the California Energy

Commission, and the California Department of Transportation is acknowledged. This work made use

of Earthquake Engineering Research Centers Shared Facilities supported by the National Science

Foundation under Award Number EEC-9701568. We thank D.M. Boore, K.C. Campbell, an

anonymous reviewer, and all the NGA developers and stakeholders for helpful reviews and

suggestions.

LIST OF ELECTRONIC APPENDICES

Electronic Appendix A. Isochrone Theory, Generalized Geometry, and Examples

Electronic Appendix B. Fits to Developer Residuals

Electronic Appendix C. Directivity Residuals vs. Various Quantities

Electronic Appendix D. Updip Directivity Residuals

REFERENCES

Abrahamson, N., 2003. Draft plan for 1-D rock motion simulations, unpublished manuscript for Next

Generation Attenuation Project, dated July 11, 2003.

Abrahamson, N. 2000. Effects of rupture directivity on probabilistic seismic hazard analysis, Proc.

6th Int. Conf. on Seismic Zonation, Palm Springs, CA.

Abrahamson, N. and Silva, W., 2007. Abrahamson & Silva NGA ground motion relations for the

geometric mean horizontal component of peak and spectral ground motion parameters, PEERReport, Pac. Earthq. Eng. Res. Center, Berkeley, CA, 378 pp.

Abrahamson, N. and Silva, W., 2008. Summary of the Abrahamson & Silva NGA ground motion

relations, Earthq. Spectra, (this volume).

Abrahamson, N. and Youngs, W., 1992. A stable algorithm for regression analysis using the random

effects model, Bull. Seismol. Soc. Am. 82, 505-510.

Bernard, P., Madariaga, R., 1984. A new asymptotic method for the modelling of near field

accelerograms, Bull. Seismol. Soc. Am. 74, 539-558.

Boatwright, J, and Bundock, H., 2008. The distribution of modified Mercalli intensity in the April 18,

1906, San Francisco, earthquake, Bull. Seismol. Soc. Am., submitted.

Boore, D.M., and Atkinson, G.A., 2008. Ground motion prediction equations for the average

horizontal component of PGA, PGV, and 5%-damped PSA at spectral periods between 0.01 s and

10.0 s, Earthq. Spectra, (this volume).

Campbell, K.C., and Bozorgnia, Y., 2008. Campbell-Bozorgnia NGA horizontal ground motion

model for PGA, PGV, PGD, and 5% damped linear elastic response spectra, Earthq. Spectra, (this

volume).

Chiou, B., and Youngs, R., 2008. Chiou-Youngs NGA Ground motion relations for the geometric


18/116

mean horizontal component of peak and spectral ground motion parameters, Earthq. Spectra,

(this volume).

Joyner, W.B., and Boore, D.M., 1993. Methods for regression analysis of strong-motion data, Bull.

Seismol. Soc. Am. 83, 469-487.

Mai, P.M., Spudich, P., and Boatwright, J., 2005. Hypocenter locations in finite-source rupture

models, Bull. Seismol. Soc. Am. 74, 965-980.

Rowshandel, B., 2006. Incorporating source rupture characteristics into ground-motion hazard

analysis models, Seismol. Res. Let. 77, 708-722.

Rowshandel, B., 2008. Directivity in NGA ground motions based on four NGA relations, Earthq.

Spectra, submitted.

Somerville, P. G., Collins, N., Graves, R., Pitarka, A., Silva, W., and Zeng, Y., 2006. Simulation of

ground motion scaling characteristics for the NGA-E Project, Proceedings of the 8th NationalConference on Earthquake Engineering, San Francisco, Calif.

Somerville, P.G., Smith, N.F., Graves, R.W., and Abrahamson, N.A., 1997. Modification of empirical

strong ground motion attenuation relations to include the amplitude and duration effects of

rupture directivity: Seismol. Res. Let. 68, 199-222

Spudich, P., and Chiou, B. S-J., 2006. Directivity in preliminary NGA residuals, Final Project Report

for PEER Lifelines Program Task 1M01,

http://quake.usgs.gov/~spudich/pdfs_for_web_page/Spudich&Chiou1M01_FinalReport_v6.pdf

49 pp.Spudich, P., Chiou, B. S-J., Graves, R., Collins, N., and Somerville, P. G., 2004. A formulation of

directivity for earthquake sources using isochrone theory, U.S. Geological Survey Open File

Report 2004-1268, http://pubs.usgs.gov/of/2004/1268/.

Spudich, P., and Frazer, L.N., 1984. Use of ray theory to calculate high frequency radiation from

earthquake sources having spatially variable rupture velocity and stress drop: Bulletin of the

Seismological Society of America, v. 74, 2061-2082

Spudich, P., and Frazer, L.N., 1987. Errata: Bulletin of the Seismological Society of America, v. 77,

2245.

Watson-Lamprey, Jennie, 2008. Modification of ground motion prediction equations for the effects

of rupture directivity: Earthq. Spectra, submitted.


19/116

Figure 1. Rupture and site geometry.


20/116

Figure 2. URS directive residuals for 5 sec period, as a function ofCln(s) for reverse events RB,

RG, RK, and strike-slip events SA, SD, SE, and SH (Table 1). Symbols: () footwall stations, (+)

hanging wall stations, () neutral zone stations, and () other stations (strike-slip faulting stations).


21/116

Figure 3. Circles show normalized slope of correlation ofIDP with each earthquake's AS totalresiduals at various periods. Earthquakes are arranged in order of magnitude, indicated after

earthquake name. Only data within rupture distance of 40 km are used. White/black circle indicates

positive/negative slope. Circle radius proportional to slope, with slope of 5 indicated in key.

indicates a bin having fewer than 4 data.


22/116

Figure 4. (black lines) Directivity effects predicted by the models forIDP= 0, 1, 2, 3, and 4. Solid

blue lines are predictions for strike-slip events at Xcos() = 0 and 1 using SSGA; green lines are

predictions for reverse events at the same values of Ycos(). Dashed red lines are the predictions

from strike-slip events from Abrahamson (2000).


23/116

Figure 5. Comparison of our corrected total residuals t-aowithIDPfor all developers for 3 sec period

in the 0 - 40 km distance and M 6.0 bin. Symbols: () footwall stations, (+) hanging wall stations,

() neutral zone stations, and () other stations (strike-slip stations), sloping line is a + b IDP( ),and dots with error bars are means and standard deviations in adjacent IDP bins.


24/116

Figure 6. Comparison of our corrected total residuals t-aowithIDPfor all developers for 5 sec period

in the 0 - 40 km distance and M 6.0 bin. Symbols: () footwall stations, (+) hanging wall stations,

() neutral zone stations, and () other stations (strike-slip stations), sloping line is a + b IDP( ),and dots with error bars are means and standard deviations in adjacentIDPbins.


25/116

Figure 7. Comparison of our corrected total residuals t-ao with IDP for all developers for 10 sec

period in the 0 - 40 km distance and M 6.0 bin. Symbols: () footwall stations, (+) hanging wall

stations, () neutral zone stations, and ( ) other stations (strike-slip stations), sloping line is

a + b IDP( ), and dots with error bars are means and standard deviations in adjacentIDPbins.


26/116

Figure 8. Comparison of directivity residuals with rupture distance and station classification. Data

are from all events M 6.0, rupture distance < 40 km. Left column: 3 sec period. Right column: 7.5

sec period. Rows from top to bottom are AS6, BA6, CB6, and CY6 directivity models. Symbols: ()

footwall stations, (+) hanging wall stations, () neutral zone stations, and () other stations (strike-

slip stations). Within each box the stations are plotted according to rupture distance along a cyclic

rupture distance scale. Black horizontal bars are mean values of the directivity residual, very short

black vertical bars are standard errors of the mean.


27/116

Figure 9. Comparison of directivity residuals with

fD , the predicted directivity effect. Data are fromall events with M 6.0, rupture distance < 40 km. Left column: 3 sec period. Right column: 7.5 sec

period. Rows from top to bottom are AS6, BA6, CB6, and CY6 directivity models. Symbols: ()

footwall stations, (+) hanging wall stations, () neutral zone stations, and () other stations (strike-

slip stations).


28/116

Figure 10. Comparison of maps of predicted directivity effect

fD from AS6 (left column) and the

predicted effects from SSGA (right column), both for 5 sec period. White lines show vertical

projection of rupture boundaries. White or black dots indicate epicenter. Dashed line is the top edge

of reverse fault. a) and b) Reverse event RG (Table 1) for AS6 and SSGA, respectively. c) and d)

strike-slip event SD for AS6 and SSGA, respectively.


29/116

Appendix A. Isochrone Theory, Generalized

Geometry, and Examples

to accompany



Paul Spudich,a)

and Brian S.J. Chioub)

v17 9 Nov 2007

INTRODUCTION

In this appendix we

derive the isochrone velocity ratio, comment on the IDP, present the equations necessary to calculate the hypocentral radiation patterns Rt

(transverse, or strike-normal) and Ru (strike-parallel),

give a computed example of radiation pattern and generalized geometry calculation, and describe an algorithm for calculatings andD for multisegment faults.

We start by developing expressions suitable for a rupture which occurs on a single plane,

and then we generalize to a multi-planar rupture geometry.

THEORETICAL DEVELOPMENT

Because most of the theory has already been presented in great detail by Spudich et al.

(2004) and Spudich and Chiou (2006), we briefly summarize the theory here. We also

comment on some properties of theIDP.

a) U.S. Geological Survey, 345 Middlefield Road, Menlo Park, CA 94025b) California Department of Transportation, 5900 Folsom Blvd., Sacramento, CA 95819


30/116

Isochrone velocity ratio c is a simple approximation of seismic directivity amplificationaround a fault. It has the advantages of being defined everywhere on the Earth's surface

around vertical and dipping faults. Let xc

be a vector indicating the point on the rupture

closest to a station at xs, so that rupture distance RRUP = xc xs . Let

xh be the hypocenter

so that the hypocentral distance RHYP = xh xs . Let distance D be the distance along the

fault between the hypocenter and the closest point, D = xc xh (for a planar fault; it will be

generalized later for a multisegment fault).

Directivity is strong when all the S wave energy radiated from a long stretch of a rupture

arrives at a site in a short time window. Using the above distances typically calculated in

engineering practice, we can develop a simple parameter that encapsulates this idea. We take

as our "long stretch of the rupture" the part of the rupture between the hypocenter and the

closest point to the site, length D. The "short time window" is the difference between the

arrival times of the hypocentral S wave and the S wave radiated from the closest point on the

rupture. Let tax,x

s( ) be the arrival time at xs of an S wave radiated from the rupturing of

point x. Its arrival time is the sum of the time point x ruptures tr(x) and the S-wave travel

time tS,

ta x,xs( ) = tr x( )+ tS x,xs( ).

We assume that the rupture propagates at uniform rupture velocityvr 0

=vr

, D= 0 .

Note that the latter equation, which results from the fact that RHYP RRUP goes to zero asD

goes to zero, is only appropriate for hypocenters not on the edge of the rupture area. The

D = 0 limit of c is multivalued when the hypocenter is on the edge of the rupture area, andconsequently we recommend that hypocenters not be placed on the edge of rupture areas. It

should be noted that xc

is not always the place on the rupture from which directivity is

strongest, so maps of c on the ground surface, particularly around dipping faults, can beadversely affected by the use of x

cas one end of the "short stretch of rupture."

The ln(s) factor in equation 3 (main body of paper) is a somewhat ad hoc factor that

loosely simulates two physical effects. Comparisons with synthetic seismograms, shown in

the main body of the paper, support its use. First, this factor tapers the IDP to zero for

receivers in the 'backward' direction when the hypocenter approaches the edge of the rupture.

Second, the ln(s) factor approximately models the fact that slip tends to grow with distance

away from the hypocenter. In this view, both ln(s) and Abrahamson's capping ofX are

recognitions that hypocenters tend to occur about 1/3 of the way from the end of a rupture

(Mai et al., 2005), and that fault slip tends to be biggest in the middle of a rupture.

Scalar radiation pattern Rri = max Ru2+ Rt

2,

, where Rt and Ru are the generalized

strike-normal (transverse) and strike-parallel hypocentral radiation patterns defined below,

with a constant water level = 0.2 filling the nodes. We approximate the finite fault

radiation pattern by a single point source radiation pattern. This differs from the approachadvocated by Spudich and Chiou (2006), who used a radiation pattern that was a sum of the

hypocentral pattern and a floating optimal point. The testing we have done indicates that the

single generalized hypocentral radiation pattern fits the NGA data as well as the two-source

pattern, and is simpler to code. To handle the case of non-planar ruptures, we have developed

a generalized coordinate system that warps itself parallel to the rupture. This has the effect of

warping the radiation pattern to follow the contortions of the multi-segment fault. The

suitability of the single generalized hypocentral radiation pattern might break down for the


32/116

largest faults, if they have a complicated geometry. Examples of the generalized strike-

normal hypocentral radiation pattern for the Landers and Chi-Chi earthquakes are shown

below, and an example calculation is given.

RADIATION PATTERN DEFINITION

Consider a buried, dipping, rectangular rupture (Figure A1). Define a coordinate system

u,t,z( ) with unit vectors u parallel to the fault strike, t transverse to the strike, and z

increasing downward. Note that in this coordinate system z = 0 at the elevation of the

station, which might be above sea level. The top of the fault is parallel to u at z = ZTOP, as

in Figure A1. The hypocenter is at xh = uh , th ,ZHYP( ) and the station is at xs = us ,ts ,0( ) .

Then the S wave radiation pattern at xs

for a source at xh , for the component of motion in

the p direction can be written (from SCGCS),

Rp xs ,xh( ) = (p b0 ) (n r)(s b) + (n b)(s r)[ ] + (p c0 ) (n r)( s c) + (n c)(s r)[ ] (2)

Expressions for all the dot products in Equation 2 are given in the following equations,

where we use the notation sand c to mean sin() and cos() , respectively. Terms in

Equation 2 not defined here are defined in SCGCS and illustrated in Figure A1, but are not

necessary for evaluation of Equation 2. We will evaluate the radiation pattern for the strike-

parallel (

p = u) and the strike-normal (

p = t ) components of S-wave motion, where p is the

unit vector in the desired direction of horizontal polarization. Note that dip and rake are

known parameters, so terms like sand c , (i.e. sin() and cos() ), and

sand c sin() and cos()( ) are directly evaluated. Angles and f are shown in Figure

A1, and their sines and cosines must be calculated using the simple algebraic expressionsbelow (Equation 3).

ForRu

, p = u (u b0 ) = c, (u c0 ) = s

ForRt, p = t (t b0) = s, (t c0) = c

Note that the radiation patterns for any desired horizontal polarization direction can be

obtained from a vector combination ofRu and Rt, and that the radiation pattern that we are

applying to the GMROTI50 component of motion is


33/116

Rri = max Ru2+ Rt

2,

, = 0.2.

From SCGCS

(n r) = sf ss+ cf c

(n b) = cf ss sf c

(n c) =cs

(s r) = csf c+ scsfs sscf

(s b) = ccf c+ sccfs+ sssf

(s c) =c s+ scc

and

sf =R / rh ; cf =ZHYP/ rh ; s= t / R; c= u / R , (3)

where the different sign of cf here, compared to SCGCS, results from the differing

directions of positivez.

For a rupture consisting of a single planar segment:

rh = RHYP, the hypocentral distance, R = R

EPI, the epicentral distance, u = us uh , and t = ts th .For a rupture consisting of multiple segments, for which the generalized geometry is

used, the station is at xs = US,TS,0( ), the hypocenter is at xh = UH,TH,ZHYP( ), and

u =USUH (4) t = TSTH R = US UH( )2 + TS TH( )2 , and (5) rh = ZHYP2 + R2 .

Generalized coordinates US, TS, UH , and TH are given in the next section.

In the special case ofR = 0, f = 0 and we have

Ru = cc, Rt = c2s.

GENERALIZED GEOMETRY


34/116

GENERALIZED GEOMETRY

To handle the case of non-planar faults, we have developed a generalized coordinate

system that warps itself parallel to the fault. This has the effect of warping the radiation

pattern to follow the contortions of the multi-segment fault.

Sine and cosine of the angle , defined as the azimuth of the epicenter-to-station

direction measured clockwise from the fault strike direction (Figure A1), are commonly used

in the calculation of radiation coefficient and other seismological parameters (such as the

angle used by SSGA). In this short note we describe the generalized coordinate system we

use to extend the definition of to the multi-segment case. Special attention is given to the

spatial smoothness of.

First a heuristic of the generalized coordinate is provided to help reader understand the

basic idea, and then the computation algorithm is presented. We give two examples to

demonstrate this algorithms utility in yielding a smooth distribution of near a multi-

segment fault.

HEURISTIC

We start with configuration #1 of Figure A2 in which the fault is vertical and the

epicenter-to-station azimuth is 0. In configuration #2, segment (P2-P3) and station S are

rotated to the left from their original positions in configuration #1. The azimuth to point S in

configuration #2 could be either1 or2, depending on which segment is selected. Our goal

is to devise a formulation ofso that its value is uniquely defined. One could use the

azimuth with respect to the closest segment. This approach is simple but might produce a

discontinuity in . One could use the weighted average of, but it requires tedious tracking

of each angle and the associated weight, and the outcome may be sensitive to the details of

the segmentation. Here we propose

= tan1 t2S

L1 + u2S( ) uH(6)

Use of Equation 6 amounts to flattening the fault trace, and L1 + u2S, t2S( ) is the (strike-

parallel, strike-normal) coordinate of point S in the 2-D curvilinear system defined by the

fault trace.

E ti (6) k ll f fi ti #2 b t it i t bl i th d tt d


35/116

Equation (6) works well for configuration #2, but it runs into problems in the dotted areas

where a discontinuity in may emerge. In the following section we describe a modified

version of L1 + u2S, t2S( ) which will yield a reasonably smooth distribution of . The

modification also help extends the algorithm to faults with more than 2 segments.

ALGORITHM

We define the vertical projection of the top edges of a multi-segment fault to be thefault

trace. The top edges of all segments must be horizontal and at the same depth. The fault

trace consists of n connected linear segments defined by (n + 1) end points

{P1 ,P2,KPn ,Pn+1}. The end points and the fault segments are numbered consecutively

along the fault strike direction (Figure A3). Each fault segment has its own local Cartesian

coordinate system ui ,ti, z, where ui is the unit vector along Pi+1 Pi ,

ti is perpendicular to

ui , and zpoints down into the Earth, with ui ti = zi , and the origin is at Pi (Figure A3). Li

is the length of the i-th segment. We require that each segment is either vertical or dips in the

direction ofti .

For a given station, let be the horizontal distance to the closest point on the fault trace

and c be the index of the segment closest to the station. If the station is equidistant from

more than one segment, choose c to be the lowest segment number of the equidistant

segments. Note that when the station is within the two ends of the fault trace, tc, the t-

coordinate of the station (not the t-coordinate of the point on segment c closest to the station)

is not necessarily . Point D in Figure A3 is an example of such a station.We define the

generalized coordinate U,T( )

T= sign(tc ) , if u1 0 and un Ln (i.e. station is within the two ends of the fault)

= tc

, otherwise.

U= u1 , n = 1, (7)

= min(u1 ,L1 )+ max(u2 ,0 ) , n = 2,

= min(u1,L1 ) + min max(ui ,0),Li( )2

n1 + max(un ,0 ) , n > 2.

Using this algorithm the generalized coordinates of hypocenter U T( ) and the station


36/116

Using this algorithm the generalized coordinates of hypocenter UH, TH( ) and the station

US, TS( ) can be determined. TH is 0 for a vertical fault and non-negative for a dipping fault.

These coordinates are inserted into equations 4 and 5, the results of which are then inserted

into Equation 3. Using the generalized coordinates we can determine

= tan1 TTH

UUH

(8)

where UH,TH( ) is the generalized coordinate of the hypocenter. TH is 0 for a vertical fault

and non-negative for a dipping fault. Note that we do not recommend use of equations 8 or 6

to determine for use in the radiation pattern, because we cannot guarantee that Equation 8

will alway produce the proper sign of s and s. Equation 3 should be used instead.

LIMITATIONS

There are limitations on the fault complexity this algorithm can handle. One important

limitation is that every fault segment should be dipping in the same general direction, which

is equivalent to the requirement that the along-strike direction ui should always point from

the lower to the higher segment number. For example, this algorithm will work for Chi-Chi

earthquake because all segments are dipping to the east, but not for Kobe earthquake because

it consists of two segments, one dipping to the west and the other to the east. In addition,

users should note that for sites off the ends of faults, the transverse coordinate is controlledby the end segment of the fault trace, so short terminal fault segments rotated strongly from

the main strike of the fault should be avoided. This algorithm works best when the strike

change between any two segments is less than 60. For complicated fault geometries, the user

should make maps of the Uand Tcoordinates to confirm that the algorithm is producing a

sensible coordinate system.

EXAMPLES

Figures A4 and A5 show contours of generalized coordinates Uand T, , and plots of the

strike-normal direction for two vertical faults, one a straight fault and one a two-segment

fault. Contours of generalized coordinates U and T, and plots of the strike-normal

direction are shown for a vertical five-segment fault in Figures A6 and A7.


37/116

and the algorithm that we sketch out below is appropriate for such a parameterization but will


38/116

and the algorithm that we sketch out below is appropriate for such a parameterization but will

not work for a rupture described by a set of contiguous triangles. Hence, the user may have to

develop his or her own algorithm for computings andD.

Because D in equation 1 represents an actual physical distance that the rupture front

travels in going from the hypocenter to the closest point, regardless of the fault surface

parameterization used the algorithm should strive to approximate this physical distance.

Inaccurate D will lead to c outside the range from 0.8 to 4 (for vr

= 0.8). In the case of

single segment, D,RRUP and RHYP form a triangle, therefore 0 (RHYP-RRUP)/D 1. In the

case of a multiple segment fault,D no longer is the length of any side of the triangle formed

by the site, hypocenter, and closest point, and the adopted algorithm should not violate the

above inequality. In particular, an excessively smallD can cause singular or negative values

of c ( when (RHYP-RRUP)/D equals or exceeds 1.25, respectively). s is a measure of distancealong the fault surface at hypocentral depth from the hypocenter to the closest point. Because

ourIDPis proportional to the logarithm ofs, theIDPis less sensitive to errors ins.

Figure A13 shows a plotted example of the determination ofs andD. The fault trace in

Figure A13 is identical to that of Figure A12, but the down-dip extension is different. We

briefly explain the algorithm we used to develop the downdip extension, as this extension

was developed using the same algorithm that we used to define the downdip fault surfaces for

the NGA faults. However, as explained above, this extension algorithm can be replaced by

users own algorithm; the user is not required to use the same approach described in the rest

of this paragraph. Given a fault trace and the dips of each fault segment, our algorithm

extends a fault segment in the downdip direction to a given depth. One dipping rectangle is

created for each fault trace segment. Each rectangle extends to the same depth. For non-

vertical fault, there are two problems with the resulting rectangles: 1) two adjacent rectangles

may have a gap (open space) between them; 2) two adjacent rectangles may penetrate each

other. These two problems are solved by moving the (bottom) corners of the rectangles to

the intersection points of the bottom edges. This procedure yields quadrilaterals rather than

rectangles after moving the bottom corners. Dashed lines in Figure A13 show the downdip

segment boundaries of the adjacent quadrilaterals.

To calculate s andD we identify a line (green in the figure) that follows the fault strike at


39/116

y (g g )

the hypocenter depth. For a particular site we need to identify xc, the closest point on the

fault. We then find point P, the intersection point of the green line with the vector originating

from xcin the down-dip direction. Distances is the distance along the green line from the

hypocenter to point P. Distance dis the distance from xc

to P. D is measured along the red

line, which is a continuous segmented line lying in the fault surface. This line intersects the

boundary between segments i and i+1 a distance di, measured along the segment boundary,

up- or downdip from the green line. Ifsi is the distance along the green line from the

hypocenter to the boundary between segments i and i+1 , then di is chosen to satisfy

di d= si s.


40/116


41/116

m=2, c=2

m=2, c=2


42/116

seg1

seg3

seg5

u1A

u2A

u3A

m=3, c=1

UA=u1A+u2A+u3A

TA=t1A

1t1

u

UB=L1+u2BTB=t2B

u4C

u5C

m=5, c=4

UC=L1+L2+L3+u4C+u5CTC=t4C

B

A

C

seg2

t1At2A t3A

t4Ct5C

t1B

t2Bu1Bu2B

P1

P6

P3

P5

D

m=3, c=3

UD=L1+L2+L3TD=DP4

P4Hypocenter

seg4

seg1

seg3

seg5

u1A

u2A

u3A

m=3, c=1

UA=u1A+u2A+u3A

TA=t1A

1t1

u

UB=L1+u2BTB=t2B

u4C

u5C

m=5, c=4

UC=L1+L2+L3+u4C+u5CTC=t4C

B

A

C

seg2

t1At2A t3A

t4Ct5C

t1B

t2Bu1Bu2B

P1

P6

P3

P5

D

m=3, c=3

UD=L1+L2+L3TD=DP4

P4Hypocenter

seg4

Figure A3. The local and generalized (global) coordinate systems for a multi-segment fault. Fault is

dipping toward point A.Li is the length of the i-th segment.


43/116


44/116


45/116

Figure A8. Gray-scale plot of strike-normal generalized radiation pattern Rt

around the near-

vertical Imperial Valley earthquake. Red line is fault trace, dot is hypocenter. Light colored 'bird-

foot' patterns on either side of the fault are radiation pattern nodes.


around the dipping

Northridge earthquake. Red line is vertical projection of fault area, dot is hypocenter. Light colored

colored band is radiation pattern node. Two strike-normal radiation pattern maxima are seen.


46/116


around the near-

vertical multi-segment Landers earthquake. Red line is fault trace, dot is hypocenter. Light colored'bird-foot' patterns on either side of the fault are radiation pattern nodes. Radiation pattern maximum

bends along the fault, owing to the generalized geometry.


around the dipping

multi-segment Chi-Chi earthquake. Red line is fault trace, dot is hypocenter. Light colored patterns

on either side of the fault are radiation pattern nodes.


47/116

Figure A12. Map view of example multisegment rupture for calculation of generalized coordinates

and radiation pattern. Fault dips to the right. Red line is the fault trace, green quadrilaterals are the

vertical projection of the fault segments. See text for more explanation.


48/116

Figure A13. Map view of example multisegment rupture for calculation ofD and s. Blue line isvertical projection of the fault. Fault dips to the right. Dashed lines are intersections of numbered

planar fault segments. D is measured along the red line. Green line lies in the fault surface at the

hypocenter depth. P is point on green line closest to xc. s is measured along orange line from

hypocenter to P. See text for details.


49/116

Example calculation of generalized geometry and radiation

pattern to accompany "Directivity in NGA Earthquake Ground

Motions: Analysis using Isochrone Theory," by Paul Spudich

EXPLANATION OF WORKBOOK

This Excel workbook consists of 4 worksheets, this worksheet ("read me"), an "illustration"worksheet which shows the geometry of a computed example, the "example" worksheet showingthe calculated values and a "version histor " worksheet. See tabs below.

EXPLANATION OF EXAMPLE WORKSHEET

The 'example' worksheet (see tab below) gives an example calculation for the fault geometry andtwo possible hypocenters depicted on the "illustration" worksheet. That particular geometry wasnot chosen to be realistic. It was chosen to so that the exact analytic answer for many geometricquanitities could be determined. In particular, the geometry is somewhat unnatural in that theboundaries between fault segments alway run east-west, regardless of the strike of thesegments.

WARNING

This spreadsheet is not a calculational engine that implements the Spudich and Chiourelationship. It is simply a table of correct answers for a particular geometry for which manygeometric answers could be determined analytically. Some entries in the table have beencalculated outside this spreadsheet and pasted into the proper cell. Some entries are the exactanalytic answer (e.g. =sqrt(5)) when it could be determined. Some entries are derived fromother entries. DO NOT CHANGE ANY VALUES IN THIS EXAMPLE. THE RESULTS OF SUCHA CHANGE ARE UNPREDICTABLE AND UNRELIABLE.

SUGGESTED USE

The best use of this table is to check your calculation of generalized coordinates U and T, and tocheck your calculation of radiation pattern terms Ru and Rt derived from U and T (and Uh andTh). It is likely that you will parameterize your fault trace as connected line segments, as do we,so that exact agreement on the U and T coordinates might be expected. Regardless of yourparameterization, if you plug our U and T coordinates (and segment strikes and dips) into your

radiation pattern function, you should get our result for Ru and Rt.

v4, May 2, 2008: Please see version history worksheet


50/116

NOTE 1: Bold font is used to indicate input parameters; non-bold indicates derived parameters. Empty cells indicate values not needed in the calculation.

NOTE 2: Strictly speaking, to calculate the generalized coordinates and the radiation patterns, only the coordinates of the fault trace, the coordinates of the

hypocenter(s) and the dip and rake of the hypocenter(s) are needed We define the downdip extent of the fault segments only for visualization purposes


51/116

upper leftcorner

upper rightcorner

lower rightcorner

lower leftcorner

sin(dip) dip, dg

E 25 20 30 35

N 20 30 30 20

Z 0 0 10 10

E 20 20 30 30

N 30 40 40 30

Z 0 0 10 10

E 20 26 36 30

N 40 48 48 40

Z 0 0 10 10E 26 34 44 36

N 48 52 52 48

Z 0 0 10 10

E 34 37 47 44

N 52 61 61 52

Z 0 0 10 10

Hyp 1 Hyp2 Defining computational constantsE 27 35 name formula value

N 31 47 rtwo sqrt(2) 1.414213562

Up 7 9.75 rf sqrt(5) 2.236067977rake 37 143 rten sqrt(10) 3.16227766

dip 45 51.340192

UH 9.94427191 38.78297101

TH 6.708203932 4.91934955

51.34019175

Site

Segment 1

Segment 2

Segment 3

Segment 4

0.707106781

hypocenter(s), and the dip and rake of the hypocenter(s) are needed. We define the downdip extent of the fault segments only for visualization purposes.

45

Definition of rupture geometry

0.780868809

Segment 5

Definition of two hypocenters

Hypocenter

coords

Definition of Site Geometry and Related Geometrical Parameters

1 2 3 4 5 6 7 8 9 Seg len, L

E 20 15 20 26 31 60 26 39 33

N 15 50 54 58 73 45 32 32 26.5

Z 0 0 0 0 0 0 0 0 0

2 236067977 6 708203932 10 2859127 4 472135955 2 236067977o

Sitecoordinates


52/116

u -2.236067977 6.708203932 10.2859127 4.472135955 2.236067977

t -6.708203932 42.48529157 6.260990337 17.88854382 10.0623059

u 15 2 2

t 40 6 19

u 5 11.2 28 5

t -10 -8.4 29 20

u 4.472135955 29.06888371 4.472135955

t -8.94427191 17.88854382 20.1246118

u 3.16227766 18.97366596 1.58113883 -17.3925271

t -9.48683298 -9.486832981 26.87936011 11.06797181

c, clst fault

trace 1 3 3 4 5 5 2 1 1

rupturedistance r 7.071067812 10 8.485281374 8.94427191 13.41640786 20.1246118 4.242640687 13.33333333 7.5

7.071067812 10 8.485281374 8.94427191 13.41640786 26.87936011 6 17.88854382 10.0623059

U -2.236067977 26.18033989 31.18033989 38.8147535 59.09827776 37.23361467 12.2859127 15.94427191 2.236067977

T -6.708203932 -10 -8.48528137 -8.94427191 -9.486832981 26.87936011 6 17.88854382 10.0623059

water level 0.2

1 2 3 4 5 6 7 8 9

Ru -0.23826052 -0.23323648 -0.28397058 -0.32776607 -0.41311801 -0.26774148 0.22252261 0.45905748 0.28690694

Rt 0.00457177 0.79835215 0.86255951 0.86488706 0.8214969 -0.43081299 0.31343035 -0.69526032 -0.47470592

Rri 0.238304378 0.83172436 0.908101425 0.924910927 0.919523598 0.507233016 0.384388991 0.833139054 0.554672248

1 2 3 4 5 6 7 8 9

Ru 0.38690236 0.19834457 0.19485878 0.22322077 0.23545963 -0.76189424 0.58186756 0.39270164 0.61753616

Rt 0.86463954 0.81437836 0.7678403 0.50665724 -0.16702067 -0.56847625 0.5762423 -0.2029701 0.43504425

Rri 0.94725655 0.83818416 0.7921797 0.55365068 0.28868173 0.95060406 0.81891699 0.44205366 0.75539024

9.486832981

Segment 2

Segment 3

Segment 4

Segment 5

11.18033989

10

10

8.94427191

uandtcoordinateswithrespectto

eachsegmen

t

Segment 1

Site

Radiation pattern for hypocenter 2

Radiation pattern for hypocenter 1Site

Appendix B. Fits to Developer Residuals


53/116

to accompany



Paul Spudich,a)


v4 9 Nov 2007

DESCRIPTIONS OF THE FILES

In this Appendix are four files,

AppendixB_AS6.pdf AppendixB_BA6.pdf AppendixB_CB6.pdf AppendixB_CY6.pdf

Each file starts with an enlarged plot showing the directivity effects predicted by the

directivity model for each developer, i.e. enlarged versions of Figure 4. The caption for all

these figures should read, " Directivity effects predicted by the preliminary models for IDP =

0, 1, 2, 3, and 4. Blue lines are predictions for strike-slip events at Xcos() = 0, 0.25, 0.5,

0.75, and 1 using Somerville et al. (1997); green lines are predictions for reverse events at the

same values of Ycos(). Red lines are the predictions from strike-slip events from

Abrahamson (2000)."

Each file contains enlarged plots of the developer's residuals at 0.5, 0.75, 1, 1.5, 2, 3, 4, 5,

7.5, and 10 s vs IDP, like Figures 5 - 7 in the main paper. The caption for all these figures

should read, "Comparison of our corrected total residuals t'with IDP for the indicated

developer's directivity model, using data in the 0 - 40 km distance and M 6.0 bin. Red plus

signs are hanging wall stations, blue triangles are foot wall stations, green symbols are

neutral zone stations, black circles are other stations (typically for strike slip events), red line

is a + b IDP( ), and red dots with error bars are means and standard deviations in adjacent

IDP bins."


54/116

The final two pages of each file show unsmoothed and smoothed a and b values for the

developer. The caption for these pages should read, "Boxes are unsmoothed a orb values for

this developer at each period, with error bars. Blue line is the smoothed function."

1

Boore and Atkinson; S5 Floor = HypFY


55/116

Period (sec)

DirectivityEffect

-1

0

-0.8

-0.4

0

0.4

0.8

0.5 1 5

BA-NGASSGA, SS

Abrahamson 2000, SSSSGA, DS

T0.500S ; Boore and Atkinson, Corrected Total Residuals; Mag >= 6 ; Mag


56/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correcte

dTotalResidualfromB

ooreandAtkinson



57/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correcte

dTotalResidualfromB

ooreandAtkinson



58/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correcte

dTotalResidualfromB

ooreandAtkinson



59/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correcte

dTotalResidualfromB

ooreandAtkinson



60/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correcte

dTotalResidualfromB

ooreandAtkinson



61/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correcte

dTotalResidualfromB

ooreandAtkinson



62/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correcte

dTotalResidualfromB

oorea

ndAtkinson



63/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correcte

dTotalResidualfromB

oorea

ndAtkinson



64/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correcte

dTotalResidualfromB

oorea

ndAtkinson



65/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correcte

dTotalResidualfromB

oorea

ndAtkinson


66/116

Period (sec)

a

0.5 1.0 5.0

-1.

0

-0.

8

-0.

6

-0.

4

-0.

2

0.0

0.

5


67/116

Period (sec)

b

0.5 1.0 5.0

-0.

1

0.

0

0.

1

0.

2

0.

3

0.

4

1

Abrahamson and Silva; S5 Floor = HypF


68/116

Period (sec)

Dir

ectivityEffect

-1

0

-0.8

-0.4

0

0.4

0.8

0.5 1 5

AS-NGASSGA, SS


T0.500S ; Abrahamson and Silva, Corrected Total Residuals; Mag >= 6 ; Mag


69/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Corrected

TotalResidualfromA

braham

sonandSilva



70/116

-2

-1

0

1

2

0 1 2 3 4IDP

Corrected

TotalResidualfromA

braham

sonandSilva



71/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Corrected

TotalResidualfromA

braham

sonandSilva



72/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Corrected

TotalResidualfromA

braham

sonandSilva



73/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Corrected

TotalResidua

lfromA

braham

sonandSilva



74/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Corrected

TotalResidua

lfromA

braham

sonandSilva



75/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Corrected

TotalResidua

lfromA

braham

sonandSilva



76/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Corrected

TotalResidua

lfromA

braham

sonandSilva



77/116

-2

-1

0

1

2

0 1 2 3 4

IDP

CorrectedTotalResidua

lfromA

braham

sonandSilva

a



78/116

-2

-1

0

1

2

0 1 2 3 4

IDP

CorrectedTotalResidua

lfromA

braham

sonandSilva

0.0


79/116

Period (sec)

a

0.5 1.0 5.0

-1.

0

-0.

8

-0.6

-0.

4

-0.

2

0

0.

5


80/116

Period (sec)

b

0.5 1.0 5.0

-0.

1

0.

0

0.

1

0.

2

0.

3

0.

4

11

Campbell and Bozorgnia; S5 Floor = Hyp


81/116

Period (sec)

DirectivityEffect

-1

0

-1

0

0.5 1 5

CB-NGASSGA, SS


nia

T0.500S ; Campbell and Bozorgnia, Corrected Total Residuals; Mag >= 6 ; Mag = 6 ; Mag = 6 ; Mag = 6 ; Mag = 6 ; Mag = 6 ; Mag = 6 ; Mag = 6 ; Mag = 6 ; Mag = 6 ; Mag


95/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correc

tedTotalResid

ualfromC

hiou

andYoun

ngs

T0.750S ; Chiou and Youngs, Corrected Total Residuals; Mag >= 6 ; Mag


96/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correc

tedTotalResid

ualfromC

hiou

andYoun

ngs



97/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correc

tedTotalResid

ualfromC

hiou

andYoun

ngs



98/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correc

tedTotalResid

ualfromC

hiou

andYoun

2n

gs



99/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correc

tedTotalResid

ualfromC

hiou

andYoun

2ngs



100/116

-2

-1

0

1

2

0 1 2 3 4

IDP

Correc

tedTotalResid

ualfromC

hiou

andYou

2ungs



101/116

-2

-1

0

1

2

0 1 2 3 4

IDP

CorrectedTotalResid

ualfromC

hiou

andYou

2ungs



102/116

-2

-1

0

1

2

0 1 2 3 4

IDP

CorrectedTotalResid

ualfromC

hiou

andYou

2ungs



103/116

-2

-1

0

1

2

0 1 2 3 4

IDP

CorrectedTotalResid

ualfromC

hiou

andYou

2ungs



104/116

-2

-1

0

1

2

0 1 2 3 4

IDP

CorrectedTotalResid

ualfromC

hiou

andYou


105/116

0.

4

0.

5


106/116

Period (sec)

b

0.5 1.0 5.0

-0.

1

0.

0

0.

1

0.

2

0

.3

Appendix C. Directivity Residuals vs. Various

Quantities

to accompanyDirectivity in NGA Earthquake Ground

M ti A l i i I h Th


107/116


Paul Spudich,a)


v5, 9 Nov 2007

DESCRIPTIONS OF THE FILES

In this Appendix are four files,

as6_resids ba6_resids cb6_resids cy6_resids

Each folder contains enlarged plots of the directivity model residuals at 0.75 (if used), 1,

1.5, 2, 3, 4, 5, 7.5, and 10 s vs various quantities of interest, like Figure 9 in the main paper.

In all plots the residual is the directivity intraevent residual. File names indicate contents, and

have the form,

(directivity model)-IDPintraresid-(independent variable)-(bm)(number code).eps

for example, AS6-IDPintraresid-dip-bm268.eps,

where

(directivity model) = AS6, BA6, CB6, or CY6 (independent variable)=

VrOnBeta = average rupture velocity to shear velocity for the earthquake D = diagonal distanceD from the hypocenter to the closest point

dip = earthquake fault dip M = earthquake magnitude fD = directivity function fD fwhw = data are grouped by station location, i.e. footwall, hanging wall, neutral

zone, or other location

Vs30 = 1 / Vs30


108/116

rake = 'reflected' earthquake slip rake, explained below. rjb = Joyner-Boore distance rrup = rupture distance s =s, along-strike distance from the hypocenter to the closest point h = h, downdip distance from top of fault to hypocenter

(bm) = if 'bm' is present, it means that the mean of the directivity residuals grouped intobins has been plotted

(number code) - a meaningless (not to us) 3-digit number

The caption for all figures without 'bm' in the file name is, "Comparison of directivity

residuals with independent variable. Data are from all events used by the developer M 6.0,

rupture distance 40 km. Symbols: footwall stations (triangles), hanging wall stations (plus

signs), neutral zone stations (crosses), and other stations (typically strike-slip stations)

(circles). Symbols are colored by earthquake."

The caption for all figures with 'bm' i

2008 spudich & chiou_paper+anexos

Documents