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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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