station (RIN) was sited on Rincon Hill near the western abutment of the Bay. Bridge .... average S-wave velocity for the deposit of older Bay Mud underlying the.
Bulletin ofthe SeismologicalSocietyof America, Vol. 81, No. 5, pp. 1754-1782, October1991
A GENERAL INVERSION SCHEME FOR SOURCE, SITE, AND PROPAGATION CHARACTERISTICS U S I N G MULTIPLY RECORDED SETS OF MODERATE-SIZED E A R T H Q U A K E S BY JOHN BOATWRIGHT,
JON B. FLETCHER, AND THOMAS
E. FUMAL
ABSTRACT We derive an inversion scheme that fits the logarithms of seismic body-wave spectra to oJ2 source shapes conditioned by anelastic attenuation with a frequency-independent Q. The residuals from this fit are then projected onto the set of sources and sites, with the source terms damped, to estimate residual source spectra and site response spectra. This two-part inversion process is iterated until the square of the residuals, summed over frequency for all recordings, is minimized. The absolute amplitudes of the source and site spectra are determined by fitting the site response spectra to estimates of the site amplification derived from geotechnical data. We apply this inversion process to three sets of recordings of Loma Prieta aftershocks. The first data set contains S waves recorded both in the Marina District and at nearby hard-rock stations. We constrain the site response using the S-wave velocity measured in a nearby borehole. The absolute site amplifications for the Marina stations are 15 to 20 at 1 Hz and 10 at 2.5 Hz. The other two data sets comprise the P- and S-wave recordings from 28 aftershocks ranging in size from M D = 2.1 to 4.6, obtained at accelerograph sites near the epicentral area. The P- and S-wave attenuations determined from the inversions are Q = 414 and 380, while the average near-site attenuations are t * -- 0.012 and 0.026 sec, respectively. The average ratio of the S- to P-wave seismic moments is 1.1 +_ 0.1, the ratio of the P- to S-wave corner frequencies is 1.24 _+ 0.11, and the ratio of S- to P-wave radiated energy is 13.9 _+ 3.2. The Brune stress drops for these aftershocks range from 8 to 800 bars. The largest earthquakes exhibit the largest stress drops, but there is no apparent dependence of stress drop on seismic moment for earthquakes with 1019 < M o < 1021 dyne-cm.
INTRODUCTION Identifying and separating source and site characteristics remains the chief impediment to the spectral analysis of small and moderate-sized earthquakes. Seismologists have applied the method of spectral ratios in two different strategies, depending on their objectives, to isolate these characteristics, For source studies, the spectrum of a large earthquake is divided by the spectrum of a small earthquake recorded at the same stations; for site studies, the spectrum recorded at a sedimentary site is divided by the spectrum of the same earthquake recorded at a nearby hard-rock site. The difficulty with the method of spectral ratios is that the resulting ratios have two degrees of freedom and can only be interpreted conditionally, for example, by assuming that the lowfrequency source spectrum of the smaller earthquake is approximately constant or that the hard-rock site is approximately unamplified. Andrews (1986) recast the method of spectral ratios into a generalized inverse problem by decomposing the body-wave spectra uk(f) into source, site, and 1754
SOURCE, SITE, AND P R O P A G A T I O N CHARACTERISTICS
1755
propagation components as
u (r)
1 rk
Here #hi(f) is the moment rate spectrum for the j t h earthquake, si(f) is the site response spectrum for the ith site, and r k is the appropriate geometrical spreading factor. The simplicity of Andrews' decomposition is obtained through his neglect of the body-wave radiation patterns: the source mechanisms for the earthquakes are not incorporated into the spectra model. Both Andrews (1986) and Bonamassa and Mueller (1988), who also use this formulation, demonstrate that this decomposition only separates the source and site spectra up to an unresolved degree of freedom, that is, up to an undetermined function of frequency that can be multiplied onto each source spectrum and divided from each site spectrum. At first glance, then, Andrews' decomposition does not appear to be a significant improvement over the method of spectral ratios, as the desired source or site response spectra are still undetermined. We note, however, that there is only a single degree of freedom missing for each connected or linked set of recordings. Here the term linked describes a data set in which each station can be linked to every other station through a mutually recorded event, or through a set of other stations and mutually recorded events. If a method can be devised to restrict or resolve the missing degree of freedom, then all the source and site spectra can be determined simultaneously. Clearly, the accuracy of the derived spectra depends on how well the method exploits the spectral information contained in the body waves. In this article, we resolve the missing degree of freedom through two assumptions: first, the undetermined function of frequency is restricted to a single undetermined factor by assuming that the source spectra are approximately fit by ¢o2 spectral models (Brune, 1970); and second, the undetermined factor is estimated by re-scaling the site response spectra to fit the geotechnical information available for the recording sites, that is, using estimates of the average wave velocity and density as a function of depth below one or more of the stations. Neither of these assumptions is particularly novel. The assumption of Brune's (1970) spectral model underpins almost all of the spectral analyses of near-source and regional earthquake recordings which have been performed in the last 20 years. Moreover, the effect of the near-surface velocity structure on body-wave amplitudes has been documented by many researchers since Haskell's original papers in 1953 and 1960. The utility of the proposed inversion derives from the simultaneous inversion for source parameters and site response. The interdependence of these parameters has generally precluded their objective, or unconditioned, determination. For example, Andrews (1986) averages acceleration and displacement spectra for frequency bands above and below the corner frequency, respectively, to estimate an average site response. Because these initial estimates of corner frequency are contaminated by site attenuation (Dibona and Rovelli, 1988), however, the corner frequencies shift significantly when he corrects for the average site response. To ensure that final corner frequency estimates are unbiased, the procedure of picking corner frequencies and estimating site
1756
J. BOATWRIGHT, J. B. FLETCHER, AND T. E. FUMAL
response should be iterated until the corner frequency estimates are stable. The inversion scheme derived in this article incorporates a similar iteration between source parameters and site response. The inversion has a wide range of application: it can be used to analyze the body-wave spectra from a suite of earthquakes recorded by a single station or from a set of recordings of a single earthquake. It is most effectively applied, however, to multiply recorded sets of earthquakes whose corner frequencies span the recorded frequency band. Obviously, the more recordings analyzed for each earthquake, the better the estimates of the source spectra and the consequent source parameters. Similarly, the greater the range of distances, the better the estimates of the propagation characteristics. Unfortunately, the model for the body-wave propagation used in this analysis is inadequate for analyzing body waves recorded at significantly different distances, specifically, at both near-field and regional distances. A companion article, Fletcher and Boatwright (1991), modifies the inversion presented in this article to discern distance- and frequency-dependent propagation characteristics. INVERSION FOR SOURCE, SITE, AND PROPAGATION CHARACTERISTICS
The inversion procedure has been broken into two interdependent parts, or subinversions, to reduce the number of parameters in the largest inversion. The first subinversion uses the entire frequency band to solve for source and attenuation parameters, while the second solves for source residuals and site response independently at each frequency. Combining these two parts into a single inversion would require that the residuals be solved for all frequencies simultaneously, greatly increasing the size of the inversion. Although the two subinversions are not exactly conjugate, they complement each other so that an iterative process that solves each subinversion in turn converges satisfactorily. The first subinversion combines a simple spectral shape for the seismic source, that is, the e2 spectral shape, with a simple model for the geometrical and anelastic attenuation that incorporates both a distance-dependent term and a near-receiver term. This part of the inversion represents an extension to multiple recordings of the inversion originally devised by Boatwright (1978), which has been recently revised by Lindley and Archuleta (1990) using a linear programming technique. Because the attenuation and the rate of spectral fall-off above the corner frequency cannot be independently determined from body-wave recordings, we assume that the displacement spectra fall off as ¢0-2 above the corner frequency, explicitly following the suggestion of Hasegawa (1974) and Anderson and Hough (1984), who assume that acceleration spectra are flat for frequencies above the corner frequency and that the anelastic attenuation is appropriately modeled using an exponential function of frequency. The source spectral shapes are modeled using Boatwright's (1978) approximation for the Brune (1970) shear-wave spectrum: =
2Trf~ o (1 + ( f l f c ) 4 ) 1/2
-
2~f~ o
B4112(flfc)
(1)
The notation B 4 ( f / f o ) = 1 + (f//fc) 4 is introduced to simplify the subsequent algebra. Each source spectrum is then determined by two parameters, the
S O U R C E , SITE, A N D P R O P A G A T I O N C H A R A C T E R I S T I C S
1757
low-frequency spectral level, Eo, and the corner frequency, f~. The dependence on the corner frequency is nonlinear: the appropriate inversion procedure is necessarily iterative. The attenuation of the body waves is modeled using the function
g( R, T, f) = RE ~f(t*+T/Q),
(2)
where R is the geometrical spreading factor, which can be calculated explicitly for a known velocity structure and source-receiver geometry; T is the travel time; T~ Q is the distance dependent attentuation; and t* is the near-receiver attenuation, averaged over all the recordings. This two parameter model for the attenuation represents a practical simplification of Hough and Anderson's (1988) fit to the attenuation observed at a set of stations near Anza, California. This attenuation function is readily linearized with respect to t* and 1/Q by taking logarithms. Although this formalism can be extended to parametrize the geometrical spreading as a function of distance, as performed by Frankel et al. (1990), the present article only seeks to demonstrate the method's ability to resolve source spectra and site response. We approximate the geometrical spreading as the hypocentral distance divided by two to model both the spreading and the amplification at the free surface. A companion article (Fletcher and Boatwright, 1991) addresses the more involved problem of discerning frequency-dependent geometrical spreading from recordings of an extended distribution of earthquakes on a linear array of stations. This first subinversion then solves for 2 J + 2 parameters, where J is the number of earthquakes, by fitting the logarithms of the instrument-corrected velocity spectra In uk(f~). In the following summations, the subscript k identifies the record written at the ith station by the j t h earthquake: The subscripts i and j are implicit functions of k. If the record spectra are sampled at the N frequencies f~, the appropriate error function is x 2=
E ]lnit~(f.)-ln%(f~)+lng(R~,Tk,f.)12/ak2(f~).
(3a)
k,n
The data variances ak2(fn) are derived from noise samples (see Appendix). Writing the various terms of this summation out explicitly yields
X2 =
In k,n
h k ( f ~ ) - l n ( 2 7 r f n ) + - l1n B 4 ( f n / f ~ j ) + l n R 2
k 2
- In _Uoj
2 Af~JB41(fcj/f~ ) + vf~t* + vfnTk/Q /ak2(f,,). fcj
(3b)
The dependence on the corner frequency (or the change in the corner frequency) has been linearized by expanding ln(1 + (fifty) 4) in a Taylor Series around f~j. Writing the set of model parameters as the vector x = (t*, 1/Q, ln :ol, Afc:,ln Eo2, Aft2 . . . . )
(4a)
1758
J. BOATWRIGHT, J. B. FLETCHER, AND T. E. FUMAL
and their respective coefficients as the frequency-dependent vector
= (- "lrfn,
-
2 (~jl B 1( 2 ~J2 "l~fnTk, ~jl, - 4 fcl/fn),Sj2,---=--B4
-
fcl
to2
1"
"
- (fc2/f~) . . . . .
/
!
(4b) where 6jq is the Kronecker delta function, condenses equation (3b) to
x
(3c)
E i
k,n
in which the scalar bk(f~) contains the terms in the first line of equation (3b). Setting the partial derivatives of x 2 with respect to the parameter vector x equal to zero yields the matrix equation A x = d,
(5)
in which the components of the matrix A are E ak(fn) ® ak( fn)/Z~2(fn), and the 2 components of the data vector d are E b k ( f n ) a k ( f n ) / a ~ (fn). Note that the cross terms between source parameters for different earthquakes are identically zero (that is, each record k corresponds to a single earthquake j). To minimize equation (3), we solve equation (5) iteratively using a singular value decomposition. For the data sets analyzed in this article, we begin the inversion assuming fcj = 5.0 and 2.5 Hz for P and S waves, respectively. The choice of the initial corner frequency has no effect on the inversion results. The second subinversion projects the residuals from the first subinversion onto each source and site. We assume that the residual for each record fit in r ~ ( f ) = In hA(f) - In bj(f) + In g ( R h, Th, f )
(6)
can be decomposed into a site response spectrum and a residual source spectrum as In rh( f) = In si( f) + In Ej( f ) ,
(7a)
where si(f) is the site response spectrum for the ith station and c j ( f ) is the residual source spectrum for the j t h earthquake, that is, the difference between the best spectral model and the fitted ~2 spectrum by(f). This part of the inversion is similar to the inversion performed by Andrews (1986) except that we are projecting spectral residuals rather than body-wave spectra. Note that this projection onto the sources is not essential for estimating the site response spectra: the residuals could be projected directly onto the sites without considering the variation of the source spectra from the fitted ¢02 models. Andrews (1986) determines his source and site spectra by solving the system of K equations in equation (7a) (where K is the number of recordings) for the I + J unknown spectra (the number of sites I plus the number of earthquakes J ) at each frequency fn and constraining the solutions. A more rapid solution of
S O U R C E , SITE, AND P R O P A G A T I O N C H A R A C T E R I S T I C S
1759
equation (7a) can be obtained by minimizing the error function x 2 ( f ) = E Iln r k ( f ) -
In
k
si(f)- ln¢j(f) 121vk2(f)
(7b)
at each frequency f = fn' The variances , 2 ( f ) are derived in the Appendix by combining the data variances ak2(f) with the variances of the spectral models fit in the first subinversion. Setting the partial derivatives with respect to the parameter vector y ( f ) = (ln
s~(f),...,in
s s ( f ) , l n e l ( f ) , . . . . In c a ( f ) )
(s)
equal to zero yields the matrix equations G(f)y(f)
= h(f).
(9a)
The diagonal terms of G ( f ) Gpp(f) or Gqq(f)
: E (Sip or Sjq)/vk2( f)
(9b)
k
contain summations either over all the earthquakes recorded at site p or over all the sites which recorded earthquake q. The off-diagonal terms
Gin(f) = ~ 5ip Sjq/V2(f)
(9c)
k
contain only the variance for the earthquake q recorded at the site p. The matrices G ( f ) are relatively sparse, because there are no cross terms between different earthquakes or between different sites and because K is generally about I. J/2 for aftershock data sets. The components of the data vector h ( f ) h p ( f ) or
hq(f)
-- ~ ( ~ , or
~jq)lnrk(f)lpk2(f)
(9d)
k
contain summations of the spectral residuals which are similar to the summations for the diagonal terms of G(fn). Multiplying the residual source spectrum onto the fitted two-parameter source spectrum yields the source spectral model for the j t h earthquake
2 ~f,~ojC~(f)
oAr) Ar) : (1 + (rlrcffl/ ,)
0o)
which is specified by N + 2 parameters. The residual source spectra allow the spectral models to vary from the co2 spectral shape, to model spectral fall-offs greater than ~-2, two corner frequencies separated by a frequency band with an intermediate spectral falloff, spectral modulation resulting from an episodic moment release rate, and contamination from low-frequency recorder noise. Collating these residual source spectra allows us to test if the variations from the ~2 spectral shape are systematic.
1760
J. BOATWRIGHT, J. B. FLETCHER, AND T. E. FUMAL
The residual source spectral components can trade off with the corner frequency and the low-frequency spectral level. We resolve this trade-off by adding the damping term
(7c)
x~(f) = ),E Ilned(f) 12/Pk2(f) k
to the error functions in equation (7b). The Gqq(f) terms of the resulting matrix are simply obtained by multiplying the terms given in equation (9b) by (1 + k). Setting X = 0.5 yields residual source spectra that are relatively small, that is, 0.5 _ In ~ for all of the stations and most of the frequencies used in the inversion. The estimate In ~"~ rain {ln i,n
s*(fn)}
can then be used to re-scale the source and site response spectra. It is possible t h a t a strong resonance could produce a narrow de-amplification, however, so this constraint should be applied with some caution. Including one or more hard-rock sites in the data set would clearly improve this approximate method of rescaling. A SITE RESPONSE STUDY IN THE MARINA DISTRICT As a first test of this inversion technique, we analyze the recordings from 13 aftershocks of the 17 October 1989 Loma Prieta earthquake obtained at four stations in San Francisco. Two of these station (NPT and BEA) were sited
1763
S O U R C E , SITE, A N D P R O P A G A T I O N C H A R A C T E R I S T I C S
within the Marina District, which suffered significant damage during the Loma Prieta mainshock. A third station (MAS) was sited 1 km to the east of the Marina on an outcrop of Franciscan Sandstone at Fort Mason. The fourth station (RIN) was sited on Rincon Hill near the western a b u t m e n t of the Bay Bridge, on another outcrop of Franciscan Sandstone. These station locations are plotted in Figure 1. The earthquake times, local magnitudes, and hypocentral distances from station MAS are compiled in Table 1; the average hypocentral distance for the Loma Prieta aftershocks is about 100 km. There are two earthquakes t h a t are much closer, however: event 3010835 occurred near Daly City, while event 3080716 occurred near San Leandro on the Hayward Fault. Even with the range in epicentral distance t h a t these two earthquakes provide, however, the distance-dependent attenuation (T/Q) could not be distinguished from the average near-surface attenuation beneath the receivers (t*); all of the attenuation is then forced into the near-surface terms. Note t h a t these four stations are relatively close to each other even for the nearer earthquakes; the assumption of an isotropic source is appropriate to this geometry. The inversion reduced the overall error in equation (11) to 0.23% of the initial power in the unfit spectra. Figure 2 shows the residual source spectra for the largest and the smallest of the set of earthquakes, plotted as shaded confidence intervals. The inversion was performed using the frequency band from 0.3 to 30 Hz. The source parame-
I
i San
o
0
BEA
o
oo
Francisco
Bay
1
o
NPT
o
2
I
I
/
o
km
FIG. 1. Map of San Francisco showing the station locations for the site study in the Marina District. Stations NPT and BEA are located in the Marina, while stations MAS and RIN are sited on outcrops of Franciscan sandstone.
1764
J. B O A T W R I G H T , J. B. F L E T C H E R , A N D T. E. F U M A L TABLE 1 AFTERSHOCKS RECORDED IN THE MARINA Event
2921014 2930018 2930813 2940049 2951424 2980127 2990901 3010835 3021311 3031117 3040835 3060550 3080716
Magnitude
Distance
5.0 3.9 3.6 4.3 3.7 4.5 3.6 2.5 2.9 3.6 3.7 4.5 3.6
109 93 77 97 108 97 99 19 97 100 100 101 25
NPT
BEA
MAS
L B T R
L B T R N O F D J B I C
Q
Q B R N O F D J
RIN
O F D C
I
I C
S-Waves, Event 29909010
,
i
i
~ll~
i
J
i. '
'
' '
i0.
i
1
1.
10.
',I
S - W a v e s , E v e n t 2921014Q o
gl @ r~
i.
I0. Hz
1.
10. Hz
FIa. 2. Residual source spectra for four of the events analyzed in the Marina site study. The 2921014Q event was the largest earthquake studied, while the 3010835F event was the smallest. The fitted low-frequency levels are given in units of lnlo~.
ters In ~o and fc are written on the plots. The residual source spectrum for event 2921014 indicates that the spectral model for the earthquake exhibits a more extended shoulder t h a n the fitted source spectrum, falling off gradually between 0.6 and 2.0 Hz. The residual spectrum for the M D = 2.5 event 3010835 is contaminated by the low-frequency noise from one sensor and one GEOS recorder (Borcherdt et al., 1985).
SOURCE,
SITE, AND PROPAGATION
1765
CHARACTERISTICS
The site response spectra have been rescaled by fitting the geotechnical constraints determined from the near-surface velocity and density. The geotechnical estimates for the two Marina stations are obtained from a borehole drilled between the two sites (Kayen et al., 1990); the S-wave velocity as a function of depth in the hole has been reduced to averages over 15 and 30 m depth. The average S-wave velocity and density for the hard-rock stations were determined from boreholes drilled in similar rock (Fumal, 1991). These geotechnical constraints are summarized in Table 2. The corrected and rescaled site response spectra are plotted in Figure 3; the site specific attenuations t* are written on the plots while the geotechnical
TABLE 2 G E O T E C H N I C A L C O N S T R A I N T S FOR T H E M A R I N A S T A T I O N S
Rock T y p e
Density (g/cc)
S-Wave Velocity (m/sec)
Impedance
2.4 1.7 1.5
830 200 150
2.2 5.3 6.5
Franciscan Sandstone Marina Bay mud
Weight
Depth (m)
Quarter-Wave Frequency (Hz)
0.2 0.4 0.2
30 30 15
6.9 1.7 2.5
r _
I
S - W a v e s , S t a t i o n MAS t* = 0.047_+ 0 . 0 0 5 s
S - W a v e s , Statio
~--~-
/
30 m
1 I
1.
S - W a v e s , S t a t i o n RIN t* = 0.037_+ 0.005 s
30
~
i i illll
i
i
i i J lliJ
1.
10.
10.
/x~ S-Waves, t .................
S t a t i o n BEA
!
In
1
i
1.
10. Hz
i
I
i
ill
_ _ l
J
1.
J
i
i
i
i ii
10. Hz
FIG. 3. S - w a v e site response spectra for the four sites analyzed in the Marina site s t u d y . The short horizontal lines indicate the frequency-specific constraints used to determine the absolute scaling of the site response spectra; they are labeled with the depth in meters over which the velocities are averaged (see Table 2). The t* v a l u e s written on the plots are the site-specific estimates determined from equation (13).
1766
J. B O A T W R I G H T , J. B. F L E T C H E R , A N D T. E. F U M A L
constraints are plotted as short lines centered at the quarter-wave frequency and labeled with the appropriate depth. The site-specific attenuations vary from 0.037 sec for station RIN to 0.058 sec for both of the Marina stations. Note that these estimates of t* also contain the distance-dependent attenuation ( T / Q ) : Fletcher and Boatwright (1991) estimate T / Q ~ 0.036 sec for this source-receiver distance. The site-specific attenuation for station RIN may be somewhat underestimated owing to the trade-off between the attenuation and the amplification: The resulting site response implies a slight deamplification from 2 to 6 Hz and from 15 to 30 Hz. In contrast, the site resonse for station MAS is relatively flat, increasing gradually from 1 at 0.5 Hz to 2 at 10 Hz. This spectral behavior suggests that station MAS would make a good reference station for a site study of the Marina District, as it has already been used by Boatwright et al. (1991). The strong peak near 1 Hz in the site response spectra for the Marina stations NPT and BEA can be modeled, to first order, as the quarter-wave peak for a 80-m-deep layer with an average S-wave velocity of 300 m sec, the depth and average S-wave velocity for the deposit of older Bay Mud underlying the Marina District (Boatwright et al., 1991). Because equation (14) is not derived for such a peaked response, this depth is not used to constrain the response spectra for these stations. Similarly, the geotechnical constraints for the deeper (30 m) depth under the Marina stations are given smaller weights because of the proximity of their quarter-wave frequencies to this spectral peak. The site response of the Marina stations is substantial near 1 Hz, peaking at absolute amplifications of 14 and 20 for stations NPT and BEA, respectively. The amplifications of the sidelobes at 2.5 Hz are about 10 for both stations; this frequency band probably contributed strongly to the damage to the larger (four-story) apartment buildings in the Marina district. The strong attenuation at these stations masks the site amplification at frequencies greater than 5 Hz. A SOURCE STUDY IN THE EPICENTRAL AREA As a second test of the inversion technique, we analyze the P and S waves from 28 aftershocks of the Loma Prieta earthquake recorded at four stations relatively close to the aftershock zone. This set of stations is chosen to approximate the source-receiver geometry for aftershock studies of M L = 6 earthquakes, where the stations are characteristically sited near the rupture area of the mainshock. The earthquake epicenters and station locations are plotted in Figure 4. Although some of the source-receiver distances are relatively large (= 30 km) compared to the source depths (5 to 20 km), we continue to approximate the geometrical spreading using the hypocentral distance. This approximation may bias the site response spectra for the more generally distant stations. The earthquake times, duration magnitudes, and range in hypocentral distances are compiled in Table 3. The 28 earthquakes represent the Loma Prieta aftershocks that occurred in the first 10 days after the mainshock that were recorded by two or more of these four stations. They range in size from M D = 2.1 to M D ~-4.6; the P- and S-wave inversions reduced the error in equation (11) for this data set to 0.30% and 0.34% of the initial power of the unfit spectra, respectively. The frequency bands used for the inversion were 0.5 to 50 Hz for the p waves and 0.8 to 35 Hz for the S waves. Figure 5 shows the residual S-wave source spectra for the largest and the smallest earthquakes in
1767
S O U R C E , SITE, AND P R O P A G A T I O N C H A R A C T E R I S T I C S 122 o ,
,
SAR
r
,
501 i
,
,
,
,
,
,
,
[
40 t ,
""
,
,
,
,
,
,
,
,
301 ,
,
,
,
,
,
,
t
"
--
"2 KOI
37 o
,
•
-
\
GA2
\
F:c. 4. M a p of the epicentral area showing the locations of the 28 aftershocks and the four stations. The largest octogon indicates the epicenter of the main shock. The other octagons are scaled to the duration magnitudes of the earthquakes, which range from M D = 2.1 to M D = 4.6.
the data set. Event 2980127 comprised part of the data set for the Marina study discussed previously. Because the average magnitude of the epicentrally recorded aftershocks was M D = 2.9, while the average magnitude of the aftershocks recorded in San Francisco was M D = 3.7, the instrument noise in more pronounced in the epicentral data set. Damping the source terms in the projection of the residuals through equation (7c) causes this noise to leak into the site response spectra. To minimize this leakage, we modify the damping coefficients in equation (7c) as h ~ h(ln ~oj, f ) =
In
Uk
4-2f
min(ln ~oj,ln ~ )
for f < 2 Hz. We use In ~ = - 2 . 0 and - 1 . 5 for the P- and S-wave inversion, respectively. This damping down-weights the spectra from the smaller earthquakes (ln ~oj < In ~ ) in constraining the low-frequency projections. The modification of the damping factor appears to work better for the S waves than for the P waves, however. The site response spectra plotted in Figure 5 rise systematically at low frequencies.
1768
J. BOATWRIGHT, J. B. FLETCHER, AND T. E. FUMAL TABLE 3 EPICENTRALLYRECORDEDAFTERSI-IOCKS Event 2940049 2940102 2941254 2942214 2950254 2950944 2951424 2951551 2960300 2961238 2961514 2961923 2962014 2971929 2980127 2980313 2980538 2981300 2982201 2990901 2991326 2991613 3001329 3002216 3012127 3021311 3022044 3022155
Magnitude Distance KOI 4.3 2.6 3.1 4.6 2.4 3.4 3.7 2.5 2.1 2.4 3.0 3.1 2.2 2.6 4.5 2.4 2.8 3.7 3.7 3.6 2.4 2.9 3.0 2.9 3.3 2.9 2.2 2.9
27-31 14-26 18-27 30-35 18-30 11-26 28-29 5-22 5-20 13-26 10-19 12-25 21-29 15-24 11-28 10-21 11-22 11-30 17-29 15-28 19-32 9-20 28-45 24-27 12-30 20-32 28-30 15-27
* P N * K T M M T H H R C I C O O Q J O I
R T P
DMD
SAR
P Q O T K A N A I I S P C J D O P R K P J R A R A F P
GAV GA2 P
A T N P H R P J
O R
S A Q T E P
G e o t e c h n i c a l e s t i m a t e s of t h e a v e r a g e P- a n d S - w a v e v e l o c i t y a n d d e n s i t y o v e r 30 m d e p t h s w e r e d e t e r m i n e d f r o m b o r e h o l e s d r i l l e d in s i m i l a r r o c k ( F u m a l , 1991). T h e s e c o n s t r a i n t s a r e s u m m a r i z e d in T a b l e 4. T h e s e n s o r s a t s t a t i o n G A V w e r e m o v e d six d a y s a f t e r t h e m a i n s h o c k i n s i d e a t w o - s t o r y b u i l d i n g . T h e s e c o n d s e n s o r l o c a t i o n is d e s i g n a t e d as s t a t i o n GA2. T h e w e i g h t s a s s u m e d for t h i s s i t e ' s c o n s t r a i n t s a r e l e s s t h a n t h o s e for s t a t i o n DMD, s i t e d on s i m i l a r a l l u v i u m , a n d s t a t i o n S A R , s i t e d on Q u a t e r n a r y g r a v e l s . No g e o t e c h n i cal c o n s t r a i n t s a r e u s e d for s t a t i o n K O I , w h i c h M c L a u g h l i n et al. (1988) m a p as s i t e d on a Q u a t e r n a r y l a n d s l i d e d e p o s i t . T h e P- a n d S - w a v e r e s u l t s for t h e n e a r - s i t e a t t e n u a t i o n c l e a r l y d e m o n s t r a t e t h a t t h e p h e n o m e n o n of fmax is d e r i v e d f r o m t h e s i t e r e s p o n s e r a t h e r t h a n f r o m t h e s e i s m i c source. T h e a v e r a g e d i s t a n t - i n d e p e n d e n t a t t e n u a t i o n (t*= 0.012 a n d 0.026 sec) is s i g n i f i c a n t l y d i f f e r e n t for t h e P a n d S w a v e s , w h i l e t h e v a r i a t i o n in t h e a t t e n u a t i o n specific to t h e s t a t i o n s a r e s t r o n g l y c o r r e l a t e d for t h e t w o w a v e t y p e s . T h e s i t e - s p e c i f i c P- a n d S - w a v e a t t e n u a t i o n s a r e t* = 0.00 a n d 0.014 sec for s t a t i o n s G A V a n d GA2, 0.008 a n d 0.025 sec for s t a t i o n DMD, 0.008 a n d 0.028 sec for s t a t i o n S A R , a n d f i n a l l y 0.035 a n d 0.048 sec for s t a t i o n KOI. A l t h o u g h such s t a t i o n - d e p e n d e n t v a r i a t i o n s have been pointed out by m a n y p r e v i o u s r e s e a r c h e r s , t h e v a r i a t i o n b e t w e e n t h e s e f o u r s t a t i o n s is r e m a r k a b l e . T h e c o r r e l a t i o n of fmax w i t h s t a t i o n s , r a t h e r t h a n sources, is o v e r w h e l m i n g in t h i s d a t a set.
SOURCE,
SITE, AND PROPAGATION
P - W a v e s , StaLion SAR t* = 0 . 0 0 8 _ + 0 . 0 0 2 s
P - W a v e s , S t a L i o n DMD t* = 0 . 0 0 8 - + 0 . 0 0 1 s o =
10
~
1
,
,
,,I
r
,
,
,
1.
,
,
i,I
10.
1.
10.
.
~
1
.
.
.
.
I
.
t* = 0 . 0 3 5 _ + 0 . 0 0 1
L* = 0.000_+ 0.003 s 1o
.
P-Waves, Station K0I
P-Waves, Station GAlg ~ o
1769
CHARACTERISTICS
s
10
,
1.
,
,
,
1.
i0. ttz
,
~
L,r
10. Hz
FIG. 5. P-wave site response spectra for the four sites analyzed in the epicentral study. The short lines indicate the frequency-specific constraints that determine the absolute scaling of the site response spectra; they are labeled with the depth in meters over which the velocities are averaged (see T a b l e 4). The site response spectra for stations GAV and GA2 are plotted together because the sensors were only moved within the building.
The estimates of distance-dependent attenuation obtained in these inversions are Q = 4 1 4 _ + 4 for P waves and 3 8 1 _ + 8 for S waves. The attenuation estimates appear somewhat overdetermined. However, when an ad hoc jackknife procedure was used to test the resolution of the S-wave attenuation by solving equation (11) with different earthquakes removed from the set of events, the standard deviation of the ensemble of Q estimates was less than 2, suggesting that the uncertainties obtained from the singular value decomposition are conservative. These attenuation estimates are only appropriate to the ray paths for these distances, that is, they are characteristic of the average wave propagation in the upper crust. S O U R C E P A R A M E T E R S FOR THE E P I C E N T R A L L Y R E C O R D E D A F T E R S H O C K S
Fitting the site response spectra to the geotechnical constraints fixes the source amplitude through the scaling factor ~. Then the seismic moment of the j t h earthquake is
Moj -
4 7rPoVo3 Fie ~Uoj'
(17a)
where Fie = exp(ln F e) are the logarithmically averaged radiation patterns equal to 0.33 for P waves and 0.55 for S waves (Boore and Boatwright, 1984).
1770
2
J. B O A T W R I G H T ,
J. B. F L E T C H E R ,
A N D T. E. F U M A L
S-Waves, Station DMD
I
S-Waves, Station SAR
t*
]
t*
=
0.024±
0.003
s
1
= 0.028± 0 . 0 0 5
s
t
o
i.
i0. S-Waves, Station t* = 0.014-+0.006
I. GAN s
t |
I0. S-Waves, Station t* = 0.048+-0.002
K0I s
m
a~
10 m
Q)
a
