Modification of Empirical Strong Ground Motion Attenuation Relations to Include the Amplitude and Duration Effects of Rupture Directivity

Abstract

Rupture directivity effects cause spatial variations in ground motion amplitude and duration around faults and cause differences
between the strike-normal and strike-parallel components of horizontal ground motion amplitudes, which also have spatial variation
around the fault. These variations become significant at a period of 0.6 second and generally grow in size with increasing
period. We have developed modifications to empirical strong ground motion attenuation relations to account for the effects
of rupture directivity on strong motion amplitudes and durations. The modifications are based on an empirical analysis of
near-fault data. The ground motion parameters that are modified include the average horizontal response spectral acceleration,
the duration of the acceleration time history, and the ratio of strike-normal to strike-parallel spectral acceleration. The
parameters upon which the adjustments to average horizontal amplitude and duration depend are the fraction of the fault rupture
that occurs on the part of the fault that lies between the hypocenter and the site, and the angle between the fault plane
and the path from the hypocenter to the site. Since both of these parameters can be derived from the hypocenter location and
the fault geometry, the model of rupture directivity effects on ground motions that we have developed can be directly included
in probabilistic seismic hazard calculations. The spectral acceleration is larger for periods longer than 0.6 second, and
the duration is smaller, when rupture propagates toward a site. For sites located close to faults, the strike-normal spectral
acceleration is larger than the strike-parallel spectral acceleration at periods longer than 0.6 second in a manner that depends
on magnitude, distance, and angle. To facilitate the selection of time histories that represent near-fault ground motion conditions
in an appropriate manner, we provide a list of near-fault records indicating the rupture directivity parameters that each
contains.

Full text

EFFECTS OF RUPTURE DIRECTIVITY ON
PROBABILISTIC SEISMIC HAZARD ANALYSIS
Norman A. Abrahamson
1
ABSTRACT
For long period structures such as bridges that are near faults with high activity rates,
it can be important to explicitly include directivity effects in the attenuation relations for
either probabilistic or deterministic analyses. There are two rupture directivity effects. The
first effect is a change in strength of shaking of the average horizontal component of motion,
and the second effect is the systematic differences in the strength of shaking on the two
horizontal components oriented perpendicular and parallel to the strike of the fault. The new
San Francisco Oakland Bay Bridge is used as an example to show the effects of including
rupture directivity in probabilistic seismic hazard analysis. For this example, including both
effects results in about a 30% increase in the T=3 second spectral acceleration for a 1500
year return period as compared to traditional analyses.
Introduction
Rupture directivity effects can lead to large long period pulses in the ground motion.
Recently, models have been developed to quantify the directivity effect (e.g. Somerville et
al, 1997). With these models of the rupture directivity effect, directivity can be included in
either deterministic or probabilistic seismic hazard analyses. This paper demonstrates the
effect of rupture directivity on probabilistic seismic hazard analyses.
Attenuation Relations and Rupture Directivity
For design of long-period structures such as bridges, characterization of long-period
motion is essential. Attenuation relations commonly used in California do not explicitly
include rupture directivity effects but they can adjusted to account for near-fault directivity
effects using the Somerville et al. (1997) fault-rupture directivity model. The Somerville et
al. (1997) model comprises two period-dependent scaling factors that may be applied to
horizontal attenuation relationship. One of the factors accounts for the change in shaking
intensity in the average horizontal component of motion due to near-fault rupture directivity
effects (higher ground motions for rupture toward the site and lower ground motions for
rupture away from the site). The second factor reflects the directional nature of the shaking
intensity using two ratios: fault normal (FN) and fault parallel (FP) versus the average (FA)
1
Pacific Gas and Electric Company, 245 Market Street, Mail Code N4C, San Francisco, CA 94105
Proceedings of the Sixth International Conference on Seismic Zonation (6ICSZ)
EERI November 12-15, 2000 Palm Springs CA

component ratios. The fault normal component is taken as the major principal axis resulting
in an FN/FA ratio larger than 1 and the fault parallel component is taken as the minor
principal axis with an FP/FA ratio smaller than 1. The two scaling factors depend on
whether fault rupture is in the forward or backward direction, and also the length of fault
rupturing toward the site.
There are several aspects of the empirical model for the average horizontal
component scale factors developed by Somerville et al that needed to be modified to make
the model applicable to a probabilistic hazard analysis. As published, the directivity model
is independent of distance. The data set used in the analysis includes recordings at distances
of 0 to 50 km. A distance dependent taper function was applied to the model that reduces
the effect to zero for distances greater than 60km.
T
d
(r) = 1 for r < 30 km
1 - (r-30)/30 for 30 km < r < 60 km (1)
0 for r > 60 km
As published, the model is applicable to magnitudes greater than 6.5. A magnitude taper
was applied that reduces the effect to zero for magnitudes less than 6.0.
T
m
(m) = 1 for m = 6.5
1 - (m-6.5)/0.5 for 6 = m < 6.5 (2)
0 for m < 6
The empirical model uses two directivity parameters: x and θ where x is defined as
the fraction of the fault length that ruptures toward the site and θ is the angle between the
fault strike and epicentral azimuth (see Somerville et al for details). The worst case is x=1
and θ=0. The empirical model uses a form that increases a constant rate as x increases from
0 to 1. There is little empirical data with x cos(θ) values greater than 0.6, and the
extrapolation of the model to larger x cos(θ) values is not well constrained. Based on an
evaluation of empirical recordings and numerical simulations, the form of the directivity
function is modified to reach a maximum at x cos(θ) =0.4. The slope is greater than the
Somerville model, but it flattens out at a lower level. The T=3 second value is used to guide
the adjustment of the model at all periods. The resulting model is given by
y
Dir
(x,θ,Τ) = C
1
(T) + 1.88 C
2
(T) x cos(θ) for x cos(θ) = 0.4 (3)
C
1
(T) + 0.75 C
2
(T)
for
x cos(θ) > 0.4
where C
1
(T) + C
2
(T) are from Somerville et al. (1997), and are listed in Table 1.

Table 1. Model Coefficients for the Modified Somerville et al. (1997) Directivity
Effects for the Average Horizontal Component
Period (sec) C1 C2
0.60 0.000 0.000
0.75 -0.084 0.185
1.00 -0.192 0.423
1.50 -0.344 0.759
2.00 -0.452 0.998
3.00 -0.605 1.333
4.00 -0.713 1.571
5.00 -0.797 1.757
Finally, including the directivity effect should results in a reduction of the standard
deviation of the attenuation relation. Based on an evaluation of the empirical data, at T=3
seconds, there is a reduction of the standard deviation of about 0.05 natural log units due to
adding the directivity term into the ground motion model. The period dependence of the
reduction was approximated by the period dependence of the slope of the directivity effect.
To account for the reduction in the standard deviation due to including the directivity effect
as part of the model, the standard deviations for the published attenuation relations were
modified for use in the hazard analysis using the following relation:
σ
dir
(M,T) = σ (M,T) - 0.05 C
2
(T)/1.333 (4)
where C
2
(T) is given in Table 1 and σ(M,,T) is the standard deviation from the published
attenuation relation (without directivity effects). The final modified Somerville model for
the average horizontal component for strike-slip faults is given by
ln Sa
dir
(M,r,x,θ,Τ) = ln Sa(M,r) + y
Dir
(x,θ,Τ) T
d
(r) T
m
(m) (5)
where Sa(M,r) is an empirical attenuation relation without directivity. This modified model
is shown in Figure 1a. The FN/Avg ratios are shown in Figure 1b for the modified
Somerville et al. (1997) model.
Incorporating Rupture Directivity Into Probabilistic Seismic Hazard Analysis
It is straightforward to include rupture directivity into a probabilistic hazard analysis.
The main change is that the location of the hypocenter on the rupture area needs to be
included as an additional source of (aleatory) variability (Figure 2).
In a standard hazard calculation, the hazard is given by:
ν
i
(A > z) = N
i
(M
min
)
Ey
1
∫
Ex= 0
1
∫
RA =0
∞
∫
f
m
i
(m) f
W
i
(m,W) f
RA
i
(m,RA)f
Ex
i
(x)f
Ey
i
(m,x)
m=M
min
M
max
i
∫
W= 0
∞
∫
P(A > z | m,r
i
(x,y,RA,W))dWdRAdxdydm
(6)

where N
i
(M
min
) is the rate of earthquakes with magnitude greater than M
min
from the i
th
source; m is magnitude; M
max
i
is the maximum magnitude (for the i
th
source); f
m
(m),
f
W
(m,W), f
RA
(m,RA), f
Ex
, and f
Ey
are probability density functions for the earthquake
magnitude, rupture width, rupture area, location of the center of rupture along strike and
location of the center of rupture down dip, respectively; and P(A>z|m,r,ε) is the probability
that the ground motion exceeds the test level z for a given magnitude and distance.
Including directivity effects for strike-slip faults, the hazard is given by
where r, X, and θ are computed from the rupture location and the hypocenter location. In
this calculation, we need to define and additional probability density function for the
location of the hypocenter on the rupture, f
hx
(h
x
). Here, I have assumed uniform
distributions for the hypocenter location (e.g. no preferred locations for the nucleation of the
rupture).
It could be argued that the effect on directivity on the average horizontal component
is already included in the standard deviation of the ground motion of the published
attenuation relations. There are two reasons why the published standard deviations do not
adequately account for directivity effects. First, the standard deviation of most attenuation
relations is averaged over all distances. As a result, the standard deviation of the near fault
ground motion is underestimated by the average standard deviation over all distances.
Second, the size of the directivity effect can vary significantly for different locations that are
the same distance away from the fault. That is, a particular site-fault geometry may be more
likely to experience forward directivity effects than other sites.
Example Calculation
As an example, the probabilistic seismic hazard is computed with and without
directivity. As discussed above, we can consider the effects of directivity on the average
horizontal component or on the individual components oriented parallel and perpendicular
to the strike of the fault. The base case is using the attenuation relations as published.
These are for the geometric mean of the two horizontal components. This case is label as
“without directivity” in Figure 3. Including just the effect of directivity on the average
horizontal component increases the hazard at return periods greater than 500 years. The
hazard with directivity can be deaggregated in terms of the rupture directivity parameter,
Xcos(θ). The deaggregation at the 1500 year return period is shown in Figure 4. This figure
shows that most of the long period ground motion hazard is from forward directivity cases.
ν
i
(A > z) = N
i
(M
min
)
hx= 0
1
∫
Ey
1
∫
Ex= 0
1
∫
RA =0
∞
∫
f
m
i
(m) f
W
i
(m,W) f
RA
i
(m,RA)f
Ex
i
(x) f
Ey
i
(m,x)
m=M
min
M
max
i
∫
W= 0
∞
∫
f
h
x
(h
x
)
P(A > z | m,r,X,
θ
)dW dRAdx dydh
x
dm
(7)

Deaggregation on the directivity can guide the selection of time histories in terms of
selecting forward directicity time histories vs neutral or backward directivity time histories.
Conclusions
Rupture directivity has not been included in most probabilistic seismic hazard
calculations. To accurately estimate the hazard from long period ground motions, directivity
should be directly included in the hazard analysis as one of the important sources of
variability of the long period ground motion. I believe that including rupture directivity will
soon become the standard of practice for computing the hazard for long period ground
motions near active faults.
References
Somerville, P. G., N. F. Smith, R. W. Graves. and N. A. Abrahamson (1997). Modification
of empirical strong ground motion attenuation relations to include the amplitude and
duration effects of rupture directivity, Seism. Res. Let, Vol. 68, 199-222.
QGa
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Directivity Model
Xcos(Theta)
Modified Somerville et al.
(1997) directivity model
0
0.05
0.1
0.15
0.2
0.25
0.3
0 102030405060708090
Ln(FN/Avg)
Theta
Somerville et
al. (1997)
FN/Average
Horizontal
Model
1a.
1b.
Figure 1. Modified Somerville et al. (1997) rupture directivity model.

QGa
15
10
5
0
0 102030405060708090100
Depth (km)
Distance (km)
Figure 2. Example of the variability in hypocenter location over the rupture surface for a single rupture
location. The heavy line defines the rupture area of an earthquake scenario, the circle indicates the center of
the rupture, and the stars indicate the range of hypocenters.
QGa
0.0001
0.001
0.01
0.1
1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Annual Probability of Exceedance
Spectral Acceleration (g)
Ave Horizontal without Directivity
Fault Parallel Component
Average Horizontal Component
Fault Normal Component
Figure 3. Effects of directivity on the hazard for T=3 seconds spectral acceleration for the new San
Francisco Bay Bridge.
Figure 4. Deaggregation of the hazard for a spectral period of 3.0 seconds at the 1500 year return period.

KCampbell@eqecat.com
To "Gilles Bureau" <gbureau@geiconsultants.com>
03/03/200510:12
AM
ee
john_barneich@geopentech.com,
kcampbell@absconsulting.com,
Nancy
_ Collins@urscorp.com,
bee
Subject Re: Correction for fault normal
I fault parallel effects
Gilles,
Note
that
there
is
an
error
in
the
strike-slip
equation
that
Norm
gives
in
his
paper
(personall
corom.,
2003).
The
equation
(in
different
format)
should
be:
(Embedded
image
moved
to
file:
pic03902.jpg)
Ken
Kenneth
W.
Campbell,
PhD
Vice
President
EQECAT
Inc
(503)
533-4359
(503)
533-4360
(fax)
kcampbell@eqecat.com
http://www.eqecat.com
"Gilles
Bureau"
<gbureau@geiconsu
Itants.com>
To
<kcampbell@absconsulting.com>,
03/03/2005
08:55
<KCampbell@eqecat.com>,
AM
<john_barneich@geopentech.com>,
<Phalkun_Tan@geopentech.com>,
<yoshi_moriwaki@geopentech.com>,
<Nancy_Collins@URSCorp.com>
<Paul_Somerville@URSCorp.com>
Subject
Re:
Correction
for
fault
normal
/
fault
parallel
effects
To
all:
Thank
you
for
the
paper
and
for
clarifying
this
situation.
I
have
deeply
appreciated
the
fast
response
from
Paul
and
Ken,
after
learning
one
was
on
Australia
and
the
other
in
Italy.
cc

Best
regards,
Gilles
Gilles
Bureau,
P.E.,
G.E.
Senior
Consultant
GEl
Consultants,
Inc.
2201
Broadway,
Suite
321
Oakland,
CA
94612-3017
510-835-9838
x
105
510-835-9842
(fax)
gbureau@geiconsultants.com
»>
<Nancy_Collins@URSCorp.com>
3/1/2005
5:07:18
PM
»>
Gentlemen:
Attached
is
the
Abrahamson
paper
of
which
Paul
speaks
below
(See
attached
file:
Abrahamson_6lCSZ.pdf)
Nancy
F.
Collins,
Project
Scientist
URS
Corporation
566
EI
Dorado
Street,
Suite
200
Pasadena,
CA
91101
(626)
449-7650;
FAX
(626)
449-3536
Forwarded
by
Nancy
Collins/Pasadena/URSCorp
on
03/01/2005
04:52
PM
Paul
Somerville
To:
KCampbell@eqecat.com,
robert_graves@urscorp.com,
03/01/2005
yoshi_moriwaki@geopentech.com,
PM
john_barneich@geopentech.com
<gbureau@geiconsultants.com>,
01:54
nancy_collins@urscorp.com,
cc:
"Gilles
Bureau"
kcampbell@absconsulting.com,
naa2@pge.com,
paul_somerville@urscorp.com
fault
normal/fault
parallel
Subject:
effects
Re:
Correction
for
Gilles
et
al.
Ken
has
correctly
described
the
situation.
My
assistant
Nancy
Collins
is
going
to
fax
you
the
paper
by
Norm
Abrahamson
that
describes
the
tapering
of
the
model
for
small
magnitude
and
large
distance.
Paul

I,
Paul
Somerville
To: KCampbell@eqecat.com, robert-.9raves@urscorp.com,
:td::~t-
nancy-collins@urscorp.com, yoshi_moriwaki@geopentech.com.
-,
-',
03/01/200501 :54 PM
john_barneich@geopentech.com
cc: "Gilles Bureau" <gbureau@geiconsultants.com>,
kcampbell@absconsulting.com, naa2@pge.com,
pauLsomerville@urscorp.com
Subject: Re: Correction for fault normal
I fault parallel effects
I+/s.o
t/-v
-j>
-PM
I~
UlVI
J
CUA.I
Gilles et al.
1/i-j-
79b-
9/9/
Ken has correctly described the situation.
My
assistant Nancy Collins is going to fax you the paper by
Norm Abrahamson that describes the tapering
of
the model for
small magnitude and large distance.
Paul
Paul G. Somerville
URS Corporation
566
EI
Dorado Street
Pasadena,
CA
91101-2560
Voice: (626) 449-7650; FAX: (626) 449-3536
email: PauLSomerville@urscorp.com
-KCampbell@eqecat.com
wrote: -----
To: "Gilles Bureau" <gbureau@geiconsultants.com>
From: KCampbell@eqecat.com
Date:
03/0112005
11
:42AM
cc: kcampbell@absconsulting.com, naa2@pge.com, pauLsomerville@urscorp.com
Subject: Re: Correction for fault
normal/fault
parallel effects
Gilles,
I was in Italy and unable to respond to your question before now. I have
carefully reviewed the 1997 SRL paper and I believe that I have made a
mistake in representing the directivity model in
my
CRC Handbook (and two
other chapters that I have written). Even though they refer to the
FN
to
FP factor in many places throughout the text, including the equations
themselves, the authors state that the coefficients listed in the tables
are for the FN to
AV
(average) components. I believe that I was confused
by the ambiguity in their statements and I apologize for the error.
Therefore, I believe that
my
Equation (5.106) should not have the 1/2 in
the first two terms, Furthermore, the last term (0) should also apply when
M > 6, according to the authors. I also neglected to put that constraint
in as well. Furthermore, since the FN and FP relationships are not
numerically constrained at M > 6 and Rrup > 50, I would suggest that a
magnitude and distance taper similar
to
the one Norm applied to the spatial
directivity term
(my
f1) should also be applied to my f2 term, except that
the taper should be applied between M
= 5.5 and 6.5, centered on 6.0.
Paul and/or Norm, can you please confirm this and respond to my taper
suggestion?
Ken

cc
Kenneth
W.
Campbell, PhD
Vice President
EQECAT
Inc
(503)
5334359
(503)
5334360
(fax)
kcampbell@eqecat.com
http://www.egecat.com
"Gilles Bureau"
<gbureau@geiconsu
Itants.com>
To
<kcampbell@absconsulting.com>,
02/24/200512:52 <naa2@pge.com>,
PM <pauLsomerville@urscorp.com>
Subject
Correction
for
fault normal 1
fault parallel effects
Gentlemen:
Greetings.
I am facing an issue with the use
of
the correction factor for fault normal
(FN)
I fault parallel effects (FP)
for
ground motion estimates based on
your attenuation equations.
Paul and Norm, in their 1997, paper talk on page 214 (Seismological
Research Letters Vol. 68, No.1) about fitting the strike-normal to
strike-parallel ratio y
(y
being defined as the natural logarithm
of
the
strike-parallel to strike normal ratio
at
a given period), with an equation
of
the form:
y
=cos 2 ?
[ci
+ c2 In(Rrup + 1) + c3(Mw -
6)]
However, elsewhere in the text and in Table 6 and 7
of
the 1977 reference,
the c coefficients are defined for the strike-normal to average ratio.
In
Chapter 5
of
the
CRe
Handbook (2003), Ken uses a similar equation
for
FN
(and specifically refers to Norm's 2000 model) as follows:
f2(Rrup, Mw,
?) =% (cos 2 ?) [c3 +
c4ln(Rrup
+ 1) + c5
(Mw
- 6)] [5.106]
The
c3, c4 and
c5
are defined in Table 5.25 (Campbell, 2003) and are the

very same values as c1, c2 and c3 in the aforementioned Table
7.
Hence, a factor % (half the fault-normal effects) has been introduced in
Equation 5.106. Equation [5.106] also provides an equal in amplitude but
opposite sign correction
(-1/2 etc.) for FP effects when compared with the
average estimates.
My
question is:
How
where the c coefficients developed? Do they relate FN to FP,
or
FN to
average? Is a 0.5 factor correctly applied by Ken, or does it result from
the language ambiguity regarding y in the 1977 paper?
Thank you all very much for clarifying how to correctly apply the FN
correction. Your help will be greatly appreciated.
Best regards
to
all three
of
you,
Gilles.
Gilles Bureau, P.E., G.E.
Senior Consultant
GEl Consultants, Inc.
2201
Broadway, Suite
321
Oakland,
CA
94612-3017
510-835-9838 x 105
510-835-9842 (fax)
gbureau@geiconsultants.com