Bulletin of the Seismological Society of America, 90, 4, pp. 977–992, August 2000

Inversion of Source Parameters and Site Effects from Strong Ground
Motion Records using Genetic Algorithms
by Aaron Moya, Jorge Aguirre, and Kojiro Irikura

Abstract The low-frequency flat levels and corner frequencies of the aftershocksof the Kobe earthquake, together with the site effects of the recording stations, areinverted by using genetic algorithms. All records were corrected by the quality factor,
Q, of propagation-path, and for each aftershock, a different omega-squared model
was assumed such that the low-frequency flat levels and corner frequencies would
minimize the standard deviation of the site effects at all stations. It was observed thatsite effects for rock sites showed significant amplification at high frequencies. Thisamplification was found to correlate with spectral ratios of records located at 70-m
depth and at the surface for one of the stations. After estimating the seismic moment
and the stress drop, it was observed that the relation of the seismic moment, M0,
versus the corner frequency, fo, followed the scaling law M0 ␣ foⳮ3. Large valuesof stress drop were found around asperities on the fault zone and at depth.

Introduction
Strong Ground Motion Records and Geologyof the Area

Although the calculation of source parameters such as
the seismic moment and stress drop is a simple procedure,
it remains not a trivial task for at least two significant reasons. The first one is the contamination of the signal by thepath and site effect. Conventionally, site effects are estimated relative to that for a rock site station, but the use of
relative estimates introduces some bias at the time of choosing the reference site. In addition to that, recent investigations (Archuleta and Steidl, 1998; Steidl et al., 1996) haveshown that rock sites are not completely void of amplification at the surface. The second reason is the reading of thelow-frequency flat levels and corner frequencies from thedisplacement spectra. In most cases, these readings are doneby eye, which makes the result subjective. A small variationin the value of the corner frequency would strongly affectthe result for the stress drop.
We have used genetic algorithms (GA) as an alternativemeans to invert the source spectra of the earthquakes andthe site effects of the stations. The inversion starts by choosing a Brune’s (1970) source model for each aftershock. Eachsource model is then used to compute the site effect of thestation that has recorded the earthquake. Assuming that thesite effect of a station should be the same regardless of theevent, GA minimizes the standard deviation of the site effects
by searching for new source models. The inversion thenguarantees that the flat levels and corner frequencies obtained at the end of the inversion be the ones that would
correspond to the most stable site effect estimation.

This study concentrates on the aftershock sequence ofthe Kobe, Japan, earthquake (Fig. 1). The Mw 6.9 earthquake
occurred in 1995 in the Osaka area and it ruptured a total
length of approximately 45 km along the fault strike (Sekiguchi et al., 1996). The mainshock triggered a series of aftershocks that were recorded by 11 strong-motion instruments.
Table 1 lists the location parameters of the aftershocks considered in this study. The instruments belong to the Committee for Earthquake Observation and Research for theKansai Area (CEORKA), and they are all velocity sensors
with a 16-bit digital recording system. The sampling rate is
100 samples/sec and instrument response is flat from 0.025
to 30.0 Hz in velocity (Toki et al., 1995).
Most of the instruments were located on soft sediments
inside the basin area (Fig. 2) because evaluation of theground-motion characteristics of the basin was the prime
goal of CEORKA group. Only two of them were placed on
rock: CHY and KBU. Studies from well-logging data andseismic profiles of the Osaka basin area (Kagawa et al.,1993) indicate that the basin has roughly an oval shape thatreaches a maximum depth of approximately 2.4 km toward
the south (Fig. 2). The basement rock topography becomes
more irregular toward the northeastern part, right below
Osaka city. The western flank, location of Kobe city, is very
steep, probably due to the presence of the Rokko fault system, a series of active faults that run parallel to that edge.
977

978

A. Moya, J. Aguirre, and K. Irikura

Figure 1.

Location of the 1995 Kobe earthquake, aftershocks (red dots), andCEORKA stations in the Osaka area. The fault trace of the mainshock is taken from

Sekiguchi et al. (1996).

The eastern part of the basin dips more gently toward thewest, and it contains the Ikoma fault, which is reverse typewith its fault plane dipping toward the east (Research Groupfor Active Faults of Japan, 1991).
The superficial geology of the CEORKA sites is summarized in Table 2. Also, geotechnical borehole results forthe first 30 m of sediments below the stations are shown in
Figure 3. This indicates that the sediments below FKS,
AMG, and YAE stations are very soft compared to the rest.

plitude of the NS and EW components. The lowest frequency
in the spectrum is 0.33 Hz. The antialias filter for the
CEORKA instruments has a cutoff frequency of 30 Hz (Tokiet al., 1995).
As mentioned before, only two instruments were placedon rock: KBU and CHY. In the following section, we examine the amplification at KBU site by studying the spectral
ratios of two earthquakes.

Amplification at KBU Site from Spectral Ratios
Fourier Spectra of the Strong-Motion Records
To estimate the Fourier spectra of the aftershocks, therecords of every station were first corrected to zero baseline.
We took a 3-sec time window beginning from the S-wave
arrival time from the north–south (NS) and the east–west(EW) components. A cosine taper was then applied to data
0.5 sec before and after that window to ensure that the taper
applied would not disturb the amplitude of the S wave. The
Fourier amplitude spectrum was calculated using an FFTroutine applied to the NS and EW components. Each component’s spectrum was then divided by 2pf , (f being thefrequency) in order to get the displacement spectra. The resultant horizontal displacement spectrum was then estimatedasO( f ) ⳱ 冪(NS( f ))2 Ⳮ (EW( f ))2(1)

where O(f ) is the resultant of the Fourier displacement am-

KBU station is located in the middle of the ruptured
area of the mainshock (Fig. 1). The instruments are locatedin a 10-m-deep tunnel in weathered granite at Kobe University campus.
Figure 4 shows the displacement spectra for selected
events. It is clear that the uncorrected displacement spectra
at KBU are characterized by a steep decay for frequencies
higher than 4 Hz while the low-frequency spectra are mostly
flat. The high-frequency decay for the event 95JAN17-13:05
is affected by amplification from 3 to 6 Hz approximately,
and this can be easily recognized because that earthquake
has a corner frequency at around 0.9 Hz. Figure 5 shows thedisplacement and acceleration spectra for another earthquake. In this particular case, it can be seen that there issignificant amplification in the frequency range from 1.5 to
5 Hz observed from the acceleration spectrum. The tendency
for all events to decay from 4 Hz could be caused by the site
effect. This amplification becomes a real problem when try-

Inversion of Source Parameters and Site Effects from Strong Ground Motion Records using Genetic Algorithms

Table 1
Location Parameters for the Aftershocks
Event

Local Date–TimeLong. (East)

Lat. (North)

Depth (km)

MJMA

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48

95JAN17–06:29
95JAN17–06:40
95JAN17–06:42
95JAN17–06:54
95JAN17–07:00
95JAN17–07:38
95JAN17–08:15
95JAN17–08:21
95JAN17–08:30
95JAN17–08:48
95JAN17–08:58
95JAN17–09:58
95JAN17–12:32
95JAN17–13:05
95JAN17–18:01
95JAN17–22:18
95JAN18–00:51
95JAN18–05:25
95JAN18–06:50
95JAN18–13:34
95JAN19–01:00
95JAN19–10:36
95JAN19–18:59
95JAN21–21:12
95JAN23–00:33
95JAN23–06:02
95JAN23–21:44
95JAN25–23:15
95JAN29–16:02
95FEB02–16:04
95FEB02–16:19
95FEB03–04:36
95FEB03–20:36
95FEB06–13:00
95FEB18–21:37
95MAR30–14:24
95APR06–10:50
95MAY04–05:53
95MAY04–17:42
95MAY19–20:35
95JUN16–07:55
95JUN23–22:18
95JUL24–03:59
95SEP12–06:30
95OCT14–02:03
96JAN25–20:17
96MAY11–14:46
96MAY19–14:06

134.913135.188
135.430134.938
134.860
135.437135.158
134.953135.007135.048135.007134.567
135.188
135.172
135.177134.863135.172
135.182
135.170
135.175
135.328135.278134.902134.995
134.952134.907135.318135.313135.180135.040135.150135.187
135.267
135.325134.807135.295
135.323135.182134.900
135.025135.292
135.288
135.538
135.197135.107135.343
134.962135.433

034.523034.693034.778034.550034.473034.782034.695
034.953034.578034.600
034.587034.510034.708034.688034.690034.475034.677034.697034.683034.688034.797034.722034.537034.565034.555034.530034.793034.790034.685034.582034.695
034.697034.730034.793034.445034.753034.787034.700
034.537034.598034.763034.760
034.920034.692034.617034.770034.523034.780

11.3
9.1
15.6
12.0
11.0
11.0
13.1
14.2
10.4
14.0
18.8
13.3
17.6
11.7
14.7
11.5
18.0
14.6
13.9
15.2
13.6
9.9
15.3
14.7
8.7
15.4
16.0
16.7
13.3
11.5
17.9
14.9
11.5
11.6
12.6
12.1
12.9
13.0
16.2
20.8
11.9
11.8
14.8
15.7
16.8
13.8
10.6
14.2

4.8
3.9
4.2
4.4
4.4
4.9
4.0
4.1
4.5
4.2
4.7
4.3
4.2
4.7
3.8
4.5
4.5
4.5
4.5
4.0
4.0
4.1
4.2
4.3
4.1
4.5
4.3
4.7
3.6
4.1
4.2
3.7
3.4
3.5
4.9
3.6
4.1
3.7
4.3
4.1
3.8
3.8
4.2
3.9
4.8
3.4
4.0
3.3

ing to estimate source parameters from small earthquakes,
because the site amplification tends to “hide” the aftershock’s source-originated corner frequencies.
In 1997 the Electric Company Group installed two morerecorders at KBU site. One of them is 70 m deep and theother is on the surface. The new instruments had recorded
only two earthquakes by the time this article was being written. Unfortunately, neither earthquake was recorded at the
instrument 10-m deep in the tunnel. Figure 6 shows the spectral ratio of the signals between the record at the surface and979

that at 70-m depth. According to this result, the spectrum at
KBU is flat up to 1.5 Hz approximately. For frequencies
higher than 1.5 Hz there is amplification. This result holds
even when considering the effect of the destructive interference caused by the reflective waves from the surface. The
frequency at which this interference takes place is given byf⳱

2n ⳮ 1
Vs
4H(2)

where n is an integer, Vs is the S-wave velocity, and H isthe depth. Assuming an average Vs ⳱ 663 m/sec (Table 3)
and H ⳱ 70 m, then f ⳱ 2.3 Hz. There is no trough at 2.3
Hz from the spectral ratios shown in Figure 6.
GA Inversion for Source Spectra and Site EffectTo estimate the source spectra, observation needs to becorrected by propagation path and site effects. Ground motion of the ith earthquake observed at the jth receiver, O(f )ij,
can be represented as the multiplication of three physicalparameters (Iwata and Irikura, 1988):
O( f )ij ⳱ S( f )iP( f )ijG( f )j,(3)

where S(f )i is the seismic source effect of the ith earthquake,P(f )ij is the term of the propagation path from the ith source
to the jth receiver, and G(f )j is the site effect of the jth receiver. The propagation-path effect, P(f )ij, has the following
form assuming radiation of S waves from a point source:P( f )ij ⳱ Rijⳮ1 eⳮpRij f / Q( f )Vs,

(4)

where Rij is the hypocentral distance from the ith source tothe jth receiver, Q(f ) is the quality factor, and Vs is the Swave velocity of the medium. Substituting (4) into (3) andsolving for the source term, we obtainS( f )i ⳱O( f )corrij,
G( f )j(5)

whereO( f )corr
⳱ RijepRf / Q( f )Vs O( f )ij ,ij(6)Here, O(f )ij is the observed spectrum for the ith earthquakeat the jth station corrected by the propagation-path effect.
As can be seen, in order to recover the source spectrum,
observation needs to be corrected for the propagation-pathand site effect. Site effect is usually estimated by dividing
the observed spectra at the sediment sites by those at therock sites (Borcherdt, 1970). This technique works under
the assumption that the site effect of rock sites is flat in thefrequency range of interest. However, as we showed in theprevious section, that assumption does not hold for KBU

980

A. Moya, J. Aguirre, and K. Irikura

Figure 2. Osaka basin basement-rock topography and CEORKA stations. Contour
lines indicate depth of the sediments in the basin in meters. The sediment cover below
TDO, TOY, and SKI is thinner than that below the rest of the stations. KBU and CHY
are located on granite.

Table 2
CEORKA Stations: Location and Soil Conditions
Name

InitialsLongitude (East)

Latitude (North)

Altitude (m)Abeno
Amagasaki
Chihaya
Fukushima
Kobe University
Morikawashi
Motoyama-Kobe
Sakai
TadaokaToyonaka
Yae

ABN
AMG
CHY
FKS
KBU
MKT
MOT
SKI
TDO
TOY
YAE

135.519135.408
135.659
135.474
135.240135.572
135.281135.462135.408135.501135.612034.636034.718034.439034.687034.725034.680034.725034.564034.480034.801
034.680

12
0
280
0
110
1
25
2
12
55
3

Superficial Geology

Late Pleistocene deposits
Thick Holocene deposits
Mesozoic granite
Thick Holocene deposits
Mesozoic granite
Thick Holocene deposits
Late Pleistocene fan deposits
Thin Holocene deposits
Thin Holocene deposits
Pliocene deposits
Thick Holocene deposits

From Toki et al. (1995).

site. In addition, relative estimate would introduce serious
bias by the choice of reference station.
Tsurugi et al. (1997) introduced a new method to estimate the site effect. After correcting the observed spectra at
the rock site by the propagation path, they assumed anomega-squared model. The site effect at the jth station, G(f )j,is then calculated by solving (5) for the site-effect termG( f )j ⳱O( f )corr
ij,
S( f )ir(7)

where S(f )ir is the following omega-squared model (Brune,1970) for the ith event obtained at the reference site, r. Thisis determined by reading a low-frequency flatlevel, Xr, anda corner frequency, fo, from the displacement spectrum re-Inversion of Source Parameters and Site Effects from Strong Ground Motion Records using Genetic Algorithms

Figure 3.

Blow count at the sediment site stations. A large N value corresponds toa hard soil type, small N to soft soil. Data for MKT site are missing.

981

982

A. Moya, J. Aguirre, and K. Irikura

Figure 4.

The “hiding” of the corner frequency by the site effects. Note how all of
the four events presented here seem to have a corner frequency at around 4 Hz. Thisis caused by the site effect because the source-originated corner frequency is lower inevery case.

Figure 5.

The (a) displacement and (b) acceleration spectrum for the 95JAN25-23:15 aftershock recorded at KBU. By looking at both
spectra, it can be easily recognized that there
is amplification of the signal from 1.5 to 5 Hz
approximately. The corner frequency (fo) is not
at 4 Hz as it would seem from the displacement
spectrum.

Inversion of Source Parameters and Site Effects from Strong Ground Motion Records using Genetic Algorithms

Figure 6.

Spectral ratios at KBU for the 14 May
1997 earthquake (magnitude 3.2, depth 10 km) (blackline) and the 12 June 1997 earthquake (magnitude 3.9,
depth 15 km) (gray line) between spectra at the surface and at 70-m depth at the borehole. The spectral
ratios are roughly flat for frequencies lower to 1.5 Hz.

Table 3
Geophysical Parameter Information at KBU Borehole
Thickness(m)

Density(g/cm3)

P-wave velocity(m/sec)S-wave velocity(m/sec)

6.0
4.0
1.0
4.5
5.0
12.0
7.5
12.0
18.0

1.30
1.53
1.96
1.97
2.25
2.08
2.06
2.18
2.12

640
400
490
700
1700
2050
2250
2850
3100

320
200
240
340
590
850
960
1120
1350

Xr /2
f 2
1Ⳮ
fo

冤冥

range of fields. This searching algorithm is not constrained
to solve problems following deterministic formulas. The algorithm is based on the biological principle of natural selection: only the strongest or fittest organisms survive to a given
environment. In GA, data are treated as a population of individuals that experience a process of evolution throughmany generations in search of solutions. Each unknown parameter is a gene. In our case, we are interested in finding
the low-frequency flat level of the corrected displacement
spectrum at the surface, X0, and its corresponding cornerfrequency, fo. Several genes make up individuals, and several individuals make up a population. Just as in nature, mutation and crossover affect the selection process. Mutationmeans that an entirely alien structure can be introduced tothe population at any given time. In many cases, mutation
helps the algorithm overcome a local minimum (solution).
Crossover means that, in the same population, individuals
are allowed to exchange certain characteristics, which might
help them improve the performance of the species in the longrun.
The data needed to be prepared for the inversion by GA.
First of all, we assumed a frequency dependent Q(f ) ⳱ 33f(Moya and Irikura, 1998) and Vs ⳱ 3.2 km/sec to correct
the observed spectra for the propagation path given by (6)at the CEORKA stations. After this correction, we proposeda point-source model in a similar way to (8) to estimate thesite effect, G(f )j,G( f )j ⳱O( f )corr
ij
X0 /2
f m
1 Ⳮ
fo

(9)

冤冥

corded at a rock site at the surface. The source spectrum isgiven byS( f )ir ⳱

983

(8)

where the flat level, Xr, in (8) is divided by 2 to account for
the free surface effect, fo is the corner frequency, and f is
the frequency. Hand reading of flat level and corner frequency from observed spectra is not objective, and in somecases it could be prone to human error as the shape of thespectra becomes complex. The method applied in this study
adopts the idea proposed by Tsurugi et al. (1997) with thevariation that we performed a nonlinear inversion using GA
to obtain the flat levels and corner frequencies.
GA Implementation
Ever since the pioneering work of Holland (1975), GA
have proven to be a powerful tool for scientists in a wide

where m is 2 to meet the omega-squared model (Aki, 1967;
Brune, 1970). The source term, S(f )i, in (9) is very similar
to that of (8), but this time the flat level, X0, and the cornerfrequency, fo, are not obtained from hand readings from thedisplacement spectrum. Instead, we make use of a range inwhich GA is to search for the values of X0 and fo.
The upper and lower limits for the flat level are defined
as follows. Considering that the low-frequency band in therock site, KBU, is little affected by the site effect, the asymptote of the corrected amplitude level of the displacement
spectrum at that station is taken as an upper limit for X0.
The minimum limit is taken as a fraction of the upper limit
which is X0 /2 or X0 /5 depending on the earthquake. In thecase of the limits for the corner frequency, the lower limit
is 0.3 and the upper limit is 5.0 Hz for all earthquakes.
Figure 7 illustrates the inversion process. The main idea
is that the site effect at a given station should be the same
regardless of the earthquake. GA select values for the X0 and
fo from the ranges already specified to make a different
source model for each earthquake. Then, the model proposedby GA divides the corresponding corrected amplitude spectrum of the event, and a site effect is calculated using (9).
If, for example, we had only three earthquakes and three984

A. Moya, J. Aguirre, and K. Irikura

Figure 7.

Outline of the GA approach to estimate the source spectrum and the site effect.

stations, then GA select three values for X0 and three values
for fo. With this information we can calculate three site effects for the same station (three spectral ratios). However,the site effect should be the same regardless of the earthquake. The average and the normalized standard deviation
are then calculated. The standard deviation is the value weare interested in minimizing, so the inversion continues byselecting different values of X0 and fo until a certain numberof iterations (generations) is reached and the standard deviation of the site effects is minimum in every station.
In order to give equal weights to the low- and highfrequency bands, the displacement spectrum was sampled atregular intervals in the logarithmic frequency domain. This
prevents the algorithm from fitting the model very well in
the high frequency and poorly in the low frequency. Also,
the inversion was performed using only 7 out of the 48 aftershocks listed in Table 1, because only 7 events were re-corded by at least 10 stations. If we included more earthquakes not recorded by all stations, then the stations that
contained a larger number of earthquakes would be better
resolved than the rest.

Inversion Results
We performed the inversion for a total of 200 generations and a population size equal to three times the numberof genes. The residual of the normalized standard deviation
is estimated as
11

residual ⳱p兺 兺 dlk(10)

k⳱1 l⳱1

where p is the total number of sampled frequencies, and dlk
is the normalized standard deviation of the site effect of theInversion of Source Parameters and Site Effects from Strong Ground Motion Records using Genetic Algorithms

985

events used in the GA inversion is different from the oneused in the linear inversion. However, the main features are
the same, such as the deamplification in the high frequency
for the sediment site stations and the peaks observed at FKS,
MKT, and YAE stations.
Table 4 shows the flat levels and the corner frequencies
for the seven events used during the inversion. The cornerfrequencies are smaller than the ones obtained by Tsurugi etal. (1997). We corrected the observed spectrum for eachstation by removing the propagation-path and site effects.
Then we took the average and compared it to the source
model as given by (8) using the flat levels and corner frequencies from Table 4. The result is shown in Figure 11.
The fitting of the average source spectra to that of the one
found by GA is very good considering that we are assuming
a simple shape for the source in the inversion process.
Figure 8.

Evolution of the residual for 200 generations for the site effects. The residual never approaches zero, because it is the summation of the normalized standard deviations.

lth frequency for the kth station. In general, the convergenceof the algorithm is fast; the result does not change muchafter the 50th generation (Fig. 8). Since the residual is madeof the summation of the normalized standard deviation for
the 11 sites, it can never approach zero.
Figure 9 shows the site effects with their corresponding
standard deviation. For stations located inside the basin(FKS, MKT, YAE, and AMG), the amplification is verylarge and the maximum values are reached at low frequencies. YAE and FKS have the fundamental frequency at 1.2
Hz, MKT at 1.7 Hz, and AMG at 1.2 Hz and 2.5 Hz approximately. The amplification at ABN and TOY is constantfor the entire frequency range of interest.
As expected, the rock sites KBU and CHY are the stations that show the lowest level of amplification for frequencies below 1.5 Hz. However, for frequencies above 1.5
Hz, there is significant amplification. Both stations exhibit apeak at 4 Hz. After 4 Hz there is deamplification.
As can be seen, the amplitude of the low frequency ofthe site effect at KBU is two. This is theoretically acceptable
for a rock site in the low-frequency range. Also, the highfrequency amplification (2–5 Hz approximately) correlates
very well with the previous results from spectral ratios between signals in the borehole and on the surface (Fig. 6).
The shape of the site effect for KBU also agrees with theone obtained by Tsurugi et al. (1997) even though the lowfrequency amplification estimated by them is one at KBU.
We believe this is caused by differences in the definitions of
the site effects.
Normalizing the site effects in Figure 9 to KBU site, we
obtain relative site effects. Figure 10 compares the relativesite effects with the result from Moya and Irikura (1998).
They also obtained relative estimates by using a linear inversion for the relative site effects and the Q value. The
imperfect matching is expected because the number ofSource Parameters
Based on Brune’s (1970) model, the seismic moment,
M0, of an earthquake can be derived from the flat level ofthe displacement spectrum given by
M0 ⳱

4pqVs3X0
,
Rh␾

(11)

where q and Vs are the density and the S-wave velocity ofthe medium, respectively, X0 is the flat level of the sourcedisplacement spectrum at a unit distance form a source centroid, and Rhu is the radiation-pattern coefficient. We used adensity of 2.7 g/cm3, and an S-wave velocity of 3.2 km/sec,
and assumed a simple form of the radiation pattern for the
S waves equivalent to 冪2/5 (Andrews, 1986).
The stress drop can be obtained byDr ⳱q(2pfo)3X0,
2.34Rh␾(12)

2.34Vs
,
2pfo

(13)

and the source radius byradius ⳱Here, fo is the corner frequency and the parameters Vs, Rhu,
X0, and q are the same ones we have previously describedfor (11).
The GA inversion yielded the source-spectrum flat levels and corner frequencies for seven aftershocks. In order toestimate these values for the rest of the events listed in Table
1, we first calculated their source spectra by removing the
propagation-path and site effects from the observed spectra.
Then, for the same earthquake, the average source spectrum
was calculated.
The flat levels and corner frequencies for these eventswere estimated by using GA. In this case, GA were used tofit an omega-squared source model to the averaged source

986

A. Moya, J. Aguirre, and K. Irikura

Figure 9. Site effects obtained by GA inversion of seven aftershocks. Thick blackline corresponds to the average; gray lines correspond to the standard deviation. Notethe amplification at the rock sites KBU and CHY in the high-frequency band.
spectrum for each individual earthquake such that the differences between the model proposed by GA and the averaged source spectrum were minimal. In other words, wesearched for the best X0 and fo that minimized the summation of the error, ⑀,
p

e⳱

兺

k⳱1

(teok ⳮ obsk)2冪(teok)(obsk)

,

(14)

in which p corresponds to the total number of sampled frequencies in the spectrum, teok corresponds to the amplitudeof the theoretical source model (proposed by GA) at the kth
frequency, and obsk to the amplitude of the averaged source
spectra at the kth frequency. Figure 12 shows the evolutionof ⑀ for two selected earthquakes. Note that in this case thesolution approaches zero very quickly, so the inversion isset for 50 generations only. Figure 13 shows the averaged
and proposed spectrum for each earthquake. The fit is generally good, although care must be taken because the error
given by (14) is always positive. This causes the fitting to
be computationally correct (smallest error is found) evenwhen the matching from theory to observation would not be
perfect, such is the case of event 95JUN25-17:55 in Figure
13 for which the observed spectra is not as smooth as the
proposed model. These differences could also occur because
the shape of the observed spectra would not adjust very wellto the omega-squared model that we have assumed throughout this analysis. Events that showed these bad fittings were
not considered in the determination of source parameters.
After introducing the values for X0 and fo into (10) to(12), we estimated the seismic moment, stress drop, andInversion of Source Parameters and Site Effects from Strong Ground Motion Records using Genetic Algorithms

987

Figure 10. Comparison between the normalized site effects to KBU (thisstudy) (gray lines) with the relative site effects obtained by Moya and Irikura(1998) (black lines). The main features are the same even when there is adiscrepancy in the amplitude.

Table 4
Flat Levels and Corner Frequencies for Seven Events Obtainedby GA in the First Inversion
Event

Flat Level

Corner Frequency

95JAN17–07:38
95JAN17–08:30
95JAN23–21:44
95JAN25–23:15
95FEB02–16:19
95FEB18–21:37
95OCT14–02:03

14.42
9.51
1.31
9.99
1.79
23.45
6.39

0.77
0.73
1.61
1.00
1.60
0.95
1.27

source radius, respectively. The seismic moment falls in therange of 1021 to 1024 dyne cm. We plotted the seismic moment versus the corner frequency (Fig. 14(a)) and found thatthe slope of the curve yielded a value of ⳮ2.7. In general,
we can say that this relation follows the scaling law M0 ␣
foⳮ3. We also found that the dependence of the seismic moment on the magnitude (Fig. 14(b)) followed the equation,
Log[M0] ⳱ 1.54[MJMA] Ⳮ 15.8,(15)which agreed with determinations obtained by other authors(Sato, 1979; Katsumata, 1996) for the MJMA scale.

988

A. Moya, J. Aguirre, and K. Irikura

Figure 11. The fitting of GA proposed source model(gray line) to the averaged source spectra (black line) for
the seven events used in the inversion.

Inversion of Source Parameters and Site Effects from Strong Ground Motion Records using Genetic Algorithms

989

Figure 12. Evolution of the residual for two selected events. The fitting of the source model is done
separately for each event.
We plotted the result for the stress drop against the seismic moment (Fig. 15(a)), the source radius (Fig. 15(b)), andthe depth (Fig. 15(c)). We found no apparent dependency ofthe stress drop on any of the earlier parameters. However,this result could have been influenced by the site effect of
the soft-sediment sites (ABN, AMG, FKS, MKT, MOT, and
YAE). Even though we have made the proper correction forthis factor, we should consider that the soft-sediment sites
could still have a strong influence on the determination of
the corner frequency and the flat levels of the aftershocks
due to their amplification in the low-frequency band. Moyaand Irikura (1998) noticed that the inversion of the relativesite effects and Q value yielded a smaller standard deviation
when they used only a limited number of stations, and thatthe deviation was large when they used all stations. They
presumed that a very local value of Q affected the softsediment sites of CEORKA stations inside the basin area. For
this reason, we performed another estimation of source parameters using only the hard-sediment site stations (TOY,
SKI, and TDO) as well as the rock sites (KBU and CHY).
These are the same stations that yielded an accurate Q value
in the work by Moya and Irikura (1998). From Figure 9 wecan also see that those stations are roughly flat in the lowfrequency band where the corner frequency is to be found.
Since we are using a smaller number of stations thistime, the number of aftershocks decreases. GA were usedagain search for the corner frequency and flat levels of theevents recorded by the selected stations. For this new result,we plotted the stress-drop values on the hypocenters of the

Figure 13.

Fitting of GA proposed source modelfor the two selected events. The residual from Figure
12 is low in both cases; however, sometimes the fitting of the theoretical model explains the observation
very well (95JAN17-13:05) and sometimes it does not(95JUN16-07:55).

aftershocks together with the moment-release distribution of
the mainshock (Sekiguchi et al., 2000) in Figure 16. According to Mendoza and Hartzell (1988), it was expectedthat the stress drop of earthquakes at the edges of the ruptured area be higher than the ones inside it. However, eventson the Awaji Island side as well as on the Kobe City side,which are outside the fault plane, show large and small values of stress drop. For those events that are inside the fault,
the distribution with depth reveals a more interesting pattern.
The reddish spots are associated with zones of large slip
during the mainshock, which correspond to asperities. Sekiguchi et al. (2000) recognize the existence of three of them.
The first one is located close to the starting point on the
Suma fault, the second one is below Kobe City on the Suwayama fault, and the third one is located on the Nojima
fault, between 2 and 10 km deep. The asperity on the Nojima
fault is also the shallowest. Aftershock activity on the Nojima fault also took place at shallower depths (Nemoto etal., 1996) as compared to the activity of the other faults
where deeper events were recorded. Figure 17 shows thestress drop versus depth for the events, which is very scattered, but a slight indication of dependency can be recognized from it, at least for a certain group of earthquakes. On

990

A. Moya, J. Aguirre, and K. Irikura

Figure 14. Result for seismic moment: (a) seismic
moment versus corner frequency, (b) seismic moment
versus magnitude.
the other hand, we observed no dependency of the stressdrop on the seismic moment or on the source radius throughthis new estimation.

Conclusions
An automated method for estimating simultaneously thesite effects of stations and the source parameters of earthquakes by using genetic algorithms has been introduced. The
site effects determined by GA show that stations located on
soft sediments such as FKS, AMG, MKT, and YAE experience deamplification in the high-frequency range. This isin good agreement with the site-effect estimations carriedout by Moya and Irikura (1998) from a linear inversion ofrelative site effects and Q value. Through the use of GA it
has also been possible to determine the site effect for KBU
site. This rock site is amplified for frequencies higher than
1.5 Hz. This result was similar to the one derived from spectral ratios of records located at the surface and at 70 m deepat that site.
We found that the majority of the aftershocks had stressdrop values lower than 50 bars, assuming the Brune’s (1970)

Figure 15. Result for the stress drop: (a) stressdrop versus seismic moment, (b) stress drop versus
source radius, and (c) stress drop versus depth.

model. It was observed that those events having stress drops
above that limit were usually located around asperities andat depth. We can support this conclusion by examining theoverall distribution of aftershocks on the fault plane and associating that distribution with the state of stress we obtained. Nemoto et al. (1996) worked on the relocation of theaftershocks that took place from 5:46 a.m. until 2:00 p.m.

Inversion of Source Parameters and Site Effects from Strong Ground Motion Records using Genetic Algorithms

991

Figure 16. Stress-drop distribution along the waveform inversion result of
Sekiguchi et al. (2000). The stress drop is systematically larger at depth and aroundasperities.
on 17 January. Their analysis indicates that most of the aftershocks took place between 8 and 18 km deep approximately. Only a limited number of them with magnitude

above 3.0 were located on the large Nojima asperity region.
Mendoza and Hartzell (1988) have linked the occurrence toaftershocks following large earthquakes to an increase of

992

A. Moya, J. Aguirre, and K. Irikura

Figure 17. Stress drop versus depth for the new
estimation using only five stations to calculate thecorner frequency and flat levels of the aftershocks.

stress in the periphery of the zone of large slip during themainshock. Our result for the stress drop of those events is
consistent with the idea that large stress drop should be expected on those areas not ruptured by the mainshock.
Even though asperities on the other fault segments are
not well constrained by the aftershock activity, we can see
the common pattern that high values of stress drop are prevalent for deeper events. This is especially significant for thegroup of aftershocks on the Suwayama fault, which extends
almost vertically from approximately 10 to 18 km and haslarge stress-drop values.

Acknowledgments
We thank Anshu Jin, Haruko Sekiguchi, and Tomotaka Iwata for theirhelpful discussions. We also wish to thank two journal referees for theircareful and critical review of the original manuscript. We are grateful to
the CEORKA group for providing the strong-motion data used in this study,
and to Takao Kagawa for providing the basement rock topography data of
the Osaka basin. This study was completed while one of the authors (A.
Moya) was a student at Kyoto University, and it was partially supported
by a Grant-In-Aid for Scientific Research, No. 08248111, from the Ministryof Education, Science, Sports, and Culture of Japan. The program used for
the GA inversion was GENESIS 5.0 by John J. Grefenstette.

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Laboratorio de Ingenierı́a Sı́smica
Instituto de Investigaciones en Ingenierı́a
Universidad de Costa Rica
San José, Costa Rica
(A. M.)
Instituto de Ingenierı́aCiudad Universitaria
Apdo. Postal 70-472
Coyoacán, C.P. 04510
México, D. F., Mexico(J. A.)
D.P.R.I., Kyoto University
Gokasho, Uji
Kyoto, 611-0011, Japan(K. I.)
Manuscript received 4 January 1999.