A composite source model for computing realistic synthetic strong ground motions

Abstract

A composite source model is presented for convolution with synthetic Green's functions, in order to synthesize strong ground motions due to a complex rupture process of a large earthquake. Subevents with a power-law distribution of sizes are located randomly on the fault. Each subevent radiates a displacement pulse with the shape of a Brune's pulse in the far field, at a time determined by a constant rupture velocity propagating from the hypocenter. Thus, all the input parameters have a physical basis. We simulate strong ground motions for event - station pairs that correspond to records obtained in Mexico by the Guerrero accelerograph network. The synthetic accelerations, velocities, and displacements have realistic amplitudes, durations, and Fourier spectra.

Full text

GEOPHYSICAL RESEARCH LETTERS, VOL. 21, NO. 8, PAGES 725-728, APRIL 15, 1994
A composite source model for computing realistic synthetic strong
ground motions
Yuehua Zeng, John G. Anderson, and Guang Yu
Seismological Laboratory and Department of Geological Sciences
Mackay School of Mines, University of Nevada, Reno
Abstract. A composite source model is presented for
convolution with synthetic Green's functions, in order to
synthesize strong ground motions due to a complex rupture
process of a large earthquake. Subevents with a power-law
distribution of sizes are located randomly on the fault. Each
subevent radiates a displacement pulse with the shape of
a Brune's pulse in the far field, at a time determined by a
constant rupture velocity propagating from the hypocenter.
Thus, all the input parameters have a physical basis. We
simulate strong ground motions for event - station pairs
that correspond to records obtained in Mexico by the
Guerrero accelerograph network. The synthetic acceler-
ations, velocities, and displacements have realistic ampli-
tudes, durations, and Fourier spectra.
Introduction
One goal of strong motion seismology is to develop a
capability to estimate strong ground motions from an
arbitrary future event, with sufficient accuracy that the
synthetic seismograms are useful for engineering appli-
cations. If realistic seismograms can be computed, then
any derived parameters of engineering interest, such as
peak acceleration or velocity, duration of shaking, or
response spectral values, can be obtained easily. Several
techniques are available to compute synthetic seismograms
for strong motion applications, but all of these have some
limitations. This paper presents a new approach that
overcomes some of the problems, and seems to effortlessly
yield synthetic strong motion records that have a very
realistic appearance. Two sample applications demon-
strate the realism of the results.
One approach for generating synthetic time series is a
stochastic simulation. Following Boore (1983), accelera-
tion time histories are generated by shaping an initially
random time series so that it has an appropriate duration,
and then filtering in the Fourier transform domain so that
it has the appropriate spectral shape. Limitations that have
not been addressed include that this approach only gen-
erates an S-wave pulse, and that it does not naturally
generate three-component seismograms with physically
expected coherency. Phases of arrivals, such as dispersed
surface waves, cannot be simply included. More infor-
mation about wave propagation can be included by the use
of empirical Green's functions (e.g. Hartzell, 1978;
Hutchings and Wu, 1990) in simulations. However, the
empirical Green's functions may be generated by earth-
Copyright 1994 by the American Geophysical Union.
Paper number 94GL00367
0094-8534/94/94GL-00367503.00
quakes with differing focal mechanisms from the desired
main shock, are often not available for the desired
source-station pair, and may not have a sufficient signal to
noise ratio at low frequencies. Finally, models for wave
propagation can be included by representing the ground
motion as a convolution of a slip function on the fault with
a synthetic Green's function (e.g. Aki and Richards, 1980).
Methods for computing synthetic Green's functions are
rapidly improving (see Anderson, 1991 for references). A
difficulty with this method has been to develop an appro-
priate source description. For example, in a recent
application Somerville et al. (1991) simply used an
empirical source function derived from smaller earth-
quakes. This paper proposes a synthetic composite source
model for use in these applications.
Method
Composite source time function
We hypothesize that the source slip function can be
simulated, in a kinematic sense, by randomly distributed
subevents on the fault plane. The size distribution of
subevents is based on a self-similar model proposed by
Frankel (1991). In this model, an earthquake is made up
of a hierarchical set of smaller earthquakes. The number
of circular subevents with radius R is specified by
dN -v
d(•n•)
where D is the fractal dimension, N is the number of
subevents, andp is a constant of proportionality. Frankel
predicted that if the static stress drop of the sub-events is
independent of their size, and if the sum of the areas of all
the sub-events equals the area of the main shock, the high
frequency roll-off of the displacement spectrum will be
proportional to co -(•-v/z). The condition on the area is
removed in our procedure, so this prediction may not
strictly hold. Integrating Equation (1), the number of
subevents with radii larger than R is
P -D -D (2)
m(•)-- 3(• - •m•x)
In (2), R max is the largest subevent allowed. We consider
it to be approximately the largest subevent that will fit inside
the fault plane. We use the power law distribution in
Equation (2) to define the relationship between the
number of sub-events and their radius.
After Keilis Borok (1959), the stress drop of a subevent
is related to its radius, R, and seismic moment, M o, by
Mo(R ) = ICR3Ao (3)
725

726 ZENG ET AL.: COMPOSITE SOURCE MODEL
For this application, we take A o to be independent of the
subevent radius. To define the total moment, M E of a o,
collection of subevents with a distribution given by (2), we
note that
dN (4)
n(R) pR
dR
and that R max (5)
M •= / n(R)M (R)clR
0 0
Rmln
This constraint leads to the value for the constant of
proportionality, p, of:
œ
7,•o 3-o D =• 3
p = (6a)
max- rain
p- 16AOl.(Rmax/Rmln ) (6b)
R mi n is intended to be a purely numerical parameter
defined by computational constraints, and for D • 2 it
generally does not affect the value ofp.
To realize the size distribution (2) numerically, we
generate N random real numbers, N ,, which are uniformly
distributed from 0 to N. The size of the corresponding
subevent is:
(7)
R, P max
The actual seismic moment for this realization of the
probability distribution is, from (3):
~ ] 6 ~
I=l I=l
Thus in our numerical simulations wc adjust A o as nec-
essary to achieve M • = M E These adjustments arc gen-
erally less than 10%, determined by the sizes of the few
sub-events with the largest radius.
The source time function for each subevent is deter-
mined from its size. Wc assumed that the radiation from
each subevent takes the shape of the Brunc (1970) pulse.
Then:
,
•o = (2r[]•) M'oxexp(-2r[]•x)H(x )
wheref • is the corner frequency, q: is time after the subevent
is triggered, and H (•) is the Heavyside step function. The
function M o(t) is the net seismic moment at any instan-
taneous time during the rupture, and at sufficiently large
time it equals the momentM o. Its derivative is• o ( t ). We
relate the corner frequency/'• to the source radius of the
i th event following Brune (1970):
2.34[3 (12)
2e,
where r3 is shear velocity.
The subevents are distributed randomly on the fault
plane, and overlap of subevents is allowed. This is a major
difference from the model visualized in Frankel (1991), in
which boundaries of subevents do not intersect. Another
difference is that the total area of subevents exceeds the
area of the main event; the necessity for this is pointed out
by Tumarkin ½t al. (1994). Allowing overlap, subevents are
particularly easy to assign to the fault. Subevents are not
allowed to extend beyond the limits of the main fault, so
they are distributed uniformly over the area where overlap
of the main fault boundary will not occur. As an example,
Figure 1 shows the locations of the first 10% of subevents
generated for one realization. We then assumed a hypo-
center and a rupture velocity. The origin time of radiation
from each subevent is the time the rupture, propagating at
a uniform rupture velocity, reaches the center of the
subevent. Because the number of subevents is very large,
we do not compute the Green's function for all of them.
Rather, we divide the fault into a grid of approximately
square sub-faults, and sum the time functions for each
source in one grid element to obtain an effective time
function. This sum is performed adding in a delay for each
sub-event controlled by the rupture time and the geo-
metrical phase delay appropriate for the azimuth to the
station.
Synthetic seismogram
The Green's functions for this simulation are computed
using our code (see Zeng and Anderson, 1994) imple-
menting the generalized reflection coefficient method of
Luco and Apse1 (1983). The synthetic seismograms due to
the complex fault are obtained by convolving the Green's
function with the composite source time function gener-
ated above.
Example
We demonstrate this approach by generating synthetic
seismograms for a source - station geometry that has
previously produced strong motion accelerograms. In
particular, we chose to apply the method to two subduction
zone earthquakes recorded on the Guerrero, Mexico,
accelerograph network shown in Figure 2 (Anderson et al.,
1987, 1991). The simulated events are the M =6.9 earth-
quake on April 25, 1989 and the M =8.1 earthquake on
September 19, 1985.
For the M=6.9 event, we choose to simulate the
accelerograms at the station at La Venta, since Humphrey
and Anderson (1992) have demonstrated that there is a
nearly flat site response at that station. The source
geometry we used is as given by Anderson et al., (1991).
The hypocentral depth is 19 km, and the focal mechanism
is a low angle thrust, consistent with subduction, with dip
13 and strike 301. The seismic moment is
8.7 x 10 2s dyne - cm (Humphrey and Anderson,
unpublished data, 1993). Singh (personal communication)
found the aftershock zone to be roughly circular with radius
Figure 1. Spatial distribution of 10% of the subevents on
the fault for one simulation.

ZENG ET AL.: COMPOSITE SOURCE MODEL 727
19.5
19.0
18.5
18.0
17.5
17.0 i
165
16O
MEXICO CIT •
.
O Accelerogroph site
I 0 ? i100 km
i i i
-103 -102 -101
I I
-100 -gg -98
Figure 2. Epicenter, fault size, and station locations for
the observed seismograms that are simulated in this paper.
about 10 kin, which we approximate with a rectangle. The
La Venta station is sited on a granitic outcrop on the
southern flank of the Sierra Madre mountains. The velocity
model used to compute the synthetic seismograms is given
by Anderson et al. (1987), but modified by addition of a
low velocity layer near the surface. Attenuation in this low
velocity layer has been set to assure that t* has a value about
equaling the spectral decay parameter, •:, estimated by
Humphrey and Anderson (1992).
Figure 3 shows observed (Anderson et al., 1991) and
computed acceleration, velocity, and displacement seis-
Acceleration (cm/sec '2 )
ao I- !'
60
40
ql , I i
0 to 20 30 40 50
Time (sec)
25=
20
16
I0
5
o
-5 o
Velocity (cm/sec)
i i ;0 i i
I o 20 40 60
Time(sec)
lO t
100
10-•
10 -2
Displacement (cm)
i i i i i
10 20 30 40 50
Time (sec)
Acceleration Spectra (cm/sec)
U E S
I 0 ø 10 ø I 0 ø
Frequency (Hz)
Figure 3. Observed (Anderson et al., 1991) and simulated
acceleration, velocity, displacement, and Fourier ampli-
tude spectra at the station La Venta. The observed time
series (left) and Fourier amplitude spectra (dashed) are
derived from accelerations recorded in the April 25, 1989
earthquake (Figure 2). The simulation uses D--2,
AO= 30 bars, and rupture velocity of 2.7 km/sec.
Component orientations are S: south; E: east; U: up.
Seismograms are bandpass fiRered between 0.03 and 10
Hz.
mograms and Fourier amplitude spectra of acceleration.
The initial impression is that the synthetic seismograms
have a realistic appearance and about the correct ampli-
tudes. Peak values are consistent to within a factor of two.
The phases do not match in detail, of course, since there
is no attempt to achieve this in the source function. We
conclude that the method described here has been very
successful in generating a synthetic time series that is
appropriate for this geometry.
The second application is for a source - station geometry
that was observed in the M =8.1 earthquake on Sept 19,
1985. The station selected is at Caleta de Campos, which
is almost directly above the hypocenter (Figure 2). The
hypocentral depth is 21 km, and the focal mechanism is
again a low angle thrust, consistent with subduction, with
dip 18 and strike 301. The seismic moment is nearly
1.1 x 10 28 dyna-cm (Anderson et al., 1986). The
hypocenter is at the downdip limit in the northwestern part
of the fault. The velocity model used to compute the
synthetic seismograms is a given by Anderson et al. (1987),
but modified by addition of a low velocity layer near the
surface, constrained to have the P- and S- velocities of a
short baseline refraction experiment (Anguiano Rojas,
1987). Attenuation in this low velocity layer has again been
set to assure that t* has a value about equaling the spectral
decay parameter, •0, estimated by Humphrey and
Anderson (1992), 0.04 sec in this case.
The results of this simulation are shown in Figure 4.
Once again, we are impressed by the realism of the syn-
thetics, especially in acceleration and velocity. All the
synthetics have about the correct duration and amplitude.
Predominant frequencies on acceleration and velocity are
consistent with the data. The displacement, which is
band-pass filtered, shows about the correct amplitudes, but
has two somewhat higher-frequency pulses than the
observations This is a result of the stochastic placement,
by chance, of smaller asperities near the station rather than
Acceleration (cm/sec 2 ) Velocity (cm/sec)
500[ ,t.,l•l• •[],LI ...... s
t .
200 E
,, ,;,,
I I i i i
0 20 40 60 80 100
Time (sec)
120
100
80
40
o
-2.0
0 20 40
I i I
60 80 l oo
Time (sec)
Displacement (cm)
40 E •
20
0 20 40 60 80 1 O0
Time(sec)
Acceleration Spectra (cm/sec)
U If'
10 •
lO 0
1,0 -• I I
I 0 ø I 0 ø
Frequency (Hz)
s
lO 0
Figure 4. Equivalent of Figure 4 for the September 19,
1985 earthquake recorded at Caleta de Campos. ^ccel-
erograms are described by Anderson et al. (1987). Seis-
mograms are bandpass filtered between 0.03 and 10 Hz.

728 ZENG ET AL.' COMPOSITE SOURCE MODEL
a single large one that is inferred, from waveform inver-
sions, to have ruptured there (Campillo et al., 1989).
Campillo et al. proposed that the "ripples" that are visible
on the displacement and conspicuous on the velocity result
from acceleration and deceleration of the rupture from.
Figure 4 suggests that failure of randomly sized and places
asperities is an alternative hypothesis to explain this effect.
Discussion and Conclusions
This paper proposes the hypothesis that the composite
source model provides a kinematic description of the
earthquake source time function that, when combined with
a realistic Green's function, leads to the synthesis of real-
istic strong ground motions. The initial trials give synthetic
seismograms that appear to be quite realistic, with
appropriate amplitudes, durations, and frequency content.
These trials were selected to be cases where site effects are
believed to be minimal, and thus the Green's functions are
relatively simple and the source effect dominates the
ground motions. Considering the fundamental difficulties
of separating source and site effects, this simple situation
seemed most appropriate. This is not a fundamental
limitation, as the composite source model can be combined
equally well with synthetic Green's functions for a more
complicated 2-D or 3-D structure, or even with empirical
Green's functions if they are available.
As a kinematic description of an earthquake source, the
composite source model is easy to implement. All the
parameters involved are potentially constrained by physical
phenomena. Fault mechanism, dimension and slip, and
R max are constrained from geology. Rupture velocity can
be taken from the relatively narrow range of prior obser-
vations. The appropriate stress drop of the subevents needs
to be investigated more, but the value of 30 bars used here
is a typical stress drop for large earthquakes. Finally, fractal
dimension is, according to Frankel (1991), related to the
b-value of earthquakes.
One of the fundamental problems in seismology has
been the inverse problem of describing the seismic source
from observed strong motion accelerograms and other
data. The success in using this composite source model as
a source description for generating realistic ground
motions suggests the idea that there might be some kinship
between the actual earthquake source and a power-law
distribution of random-sized asperities. It will thus be
interesting to see if it is possible to perform an inverse
problem for location and size of main subevents, as a
supplement to the present procedure of estimating slip time
functions as a function of location on the fault.
Acknowledgements. We benefitted from helpful discus-
sions with J. Brune and helpful comments from anonymous
reviewers. This research was supported by the Southern
California Earthquake Center and National Science
Foundation Grant BCS 9120027.
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(Received: September 16, 1993; Revised: December 17,
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