2008 spudich & chiou_paper+anexos
TRANSCRIPT
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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.
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Directivity in NGA Earthquake Ground
Motions: Analysis using Isochrone Theory
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
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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
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(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
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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.
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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
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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
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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
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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
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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
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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.
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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
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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.
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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
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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.
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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
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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.
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Figure 1. Rupture and site geometry.
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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).
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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.
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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).
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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.
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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.
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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.
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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.
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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).
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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.
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Appendix A. Isochrone Theory, Generalized
Geometry, and Examples
to accompany
Directivity in NGA Earthquake Ground
Motions: Analysis using Isochrone Theory
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
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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
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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
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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
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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
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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
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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.
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and the algorithm that we sketch out below is appropriate for such a parameterization but will
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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
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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.
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m=2, c=2
m=2, c=2
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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.
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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.
Figure A9. Gray-scale plot of strike-normal generalized radiation pattern Rt
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.
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Figure A10. Gray-scale plot of strike-normal generalized radiation pattern Rt
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.
Figure A11. Gray-scale plot of strike-normal generalized radiation pattern Rt
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.
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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.
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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.
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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
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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
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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
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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
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to accompany
Directivity in NGA Earthquake Ground
Motions: Analysis using Isochrone Theory
Paul Spudich,a)
and Brian S.J. Chioub)
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."
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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
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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
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-2
-1
0
1
2
0 1 2 3 4
IDP
Correcte
dTotalResidualfromB
ooreandAtkinson
T0.750S ; Boore and Atkinson, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Correcte
dTotalResidualfromB
ooreandAtkinson
T1.000S ; Boore and Atkinson, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Correcte
dTotalResidualfromB
ooreandAtkinson
T1.500S ; Boore and Atkinson, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Correcte
dTotalResidualfromB
ooreandAtkinson
T2.000S ; Boore and Atkinson, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Correcte
dTotalResidualfromB
ooreandAtkinson
T3.000S ; Boore and Atkinson, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Correcte
dTotalResidualfromB
ooreandAtkinson
T4.000S ; Boore and Atkinson, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Correcte
dTotalResidualfromB
oorea
ndAtkinson
T5.000S ; Boore and Atkinson, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Correcte
dTotalResidualfromB
oorea
ndAtkinson
T7.500S ; Boore and Atkinson, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Correcte
dTotalResidualfromB
oorea
ndAtkinson
T10.000S ; Boore and Atkinson, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Correcte
dTotalResidualfromB
oorea
ndAtkinson
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Period (sec)
a
0.5 1.0 5.0
-1.
0
-0.
8
-0.
6
-0.
4
-0.
2
0.0
0.
5
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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
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Period (sec)
Dir
ectivityEffect
-1
0
-0.8
-0.4
0
0.4
0.8
0.5 1 5
AS-NGASSGA, SS
Abrahamson 2000, SSSSGA, DS
T0.500S ; Abrahamson and Silva, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Corrected
TotalResidualfromA
braham
sonandSilva
T0.750S ; Abrahamson and Silva, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4IDP
Corrected
TotalResidualfromA
braham
sonandSilva
T1.000S ; Abrahamson and Silva, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Corrected
TotalResidualfromA
braham
sonandSilva
T1.500S ; Abrahamson and Silva, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Corrected
TotalResidualfromA
braham
sonandSilva
T2.000S ; Abrahamson and Silva, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Corrected
TotalResidua
lfromA
braham
sonandSilva
T3.000S ; Abrahamson and Silva, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Corrected
TotalResidua
lfromA
braham
sonandSilva
T4.000S ; Abrahamson and Silva, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Corrected
TotalResidua
lfromA
braham
sonandSilva
T5.000S ; Abrahamson and Silva, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Corrected
TotalResidua
lfromA
braham
sonandSilva
T7.500S ; Abrahamson and Silva, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
CorrectedTotalResidua
lfromA
braham
sonandSilva
a
T10.000S ; Abrahamson and Silva, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
CorrectedTotalResidua
lfromA
braham
sonandSilva
0.0
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Period (sec)
a
0.5 1.0 5.0
-1.
0
-0.
8
-0.6
-0.
4
-0.
2
0
0.
5
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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
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Period (sec)
DirectivityEffect
-1
0
-1
0
0.5 1 5
CB-NGASSGA, SS
Abrahamson 2000, SSSSGA, DS
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
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-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
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-2
-1
0
1
2
0 1 2 3 4
IDP
Correc
tedTotalResid
ualfromC
hiou
andYoun
ngs
T1.000S ; Chiou and Youngs, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Correc
tedTotalResid
ualfromC
hiou
andYoun
ngs
T1.500S ; Chiou and Youngs, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Correc
tedTotalResid
ualfromC
hiou
andYoun
2n
gs
T2.000S ; Chiou and Youngs, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Correc
tedTotalResid
ualfromC
hiou
andYoun
2ngs
T3.000S ; Chiou and Youngs, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
Correc
tedTotalResid
ualfromC
hiou
andYou
2ungs
T4.000S ; Chiou and Youngs, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
CorrectedTotalResid
ualfromC
hiou
andYou
2ungs
T5.000S ; Chiou and Youngs, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
CorrectedTotalResid
ualfromC
hiou
andYou
2ungs
T7.500S ; Chiou and Youngs, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
CorrectedTotalResid
ualfromC
hiou
andYou
2ungs
T10.000S ; Chiou and Youngs, Corrected Total Residuals; Mag >= 6 ; Mag
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-2
-1
0
1
2
0 1 2 3 4
IDP
CorrectedTotalResid
ualfromC
hiou
andYou
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0.
4
0.
5
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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
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Motions: Analysis using Isochrone Theory
Paul Spudich,a)
and Brian S.J. Chioub)
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
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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