Use of Small Earthquake Records to Determine the Source Time ...

Bulletin of the SeismologicalSociety of America, Vol. 85, No. 4, pp. 1249-1256, August 1995

Use of Small Earthquake Records to Determine the Source Time Functions of Larger Earthquakes: an Alternative Method and an Application by Aldo Zollo, Paolo Capuano, and Shri Krishna Singh

Abstract

Small earthquake records are increasingly being used as empirical Green's functions to estimate the source time functions (STFs) o f l a r g e e a r t h -

quakes. This is generally accomplished in the frequency domain by computing the ratio of the large to the small event spectrum and then transforming it back to the time domain. When the data quality is poor, the resulting STFs often show unrealistic long-period signal and short-period oscillation. In this study we propose an alternative method in which a large-event STF is approximated by a series of pseudotriangular pulses whose parameters are determined by a nonlinear frequency-domain inversion, involving the spectrum of the large and the small events. The method allows a "positivity" constraint to be imposed on the STF. The misfit between the observed and computed large-event seismograms is measured in the frequency domain over the range of frequency in which the data are reliable. We present tests of the method on synthetic data and on earthquake records from Mexico, both of which show the robustness of the method. The quality of the solution is dependent on the quality of the input data. As the input data become more bandlimited and more noisy, the reliability and usefulness of the solution will decrease.

Introduction Since the pioneering work of Hartzell (1978), the use of recordings of small earthquakes as empirical Green's functions (EGFs) has gained considerable popularity, both i n the synthesis of ground motion (e.g., Irikura, 1983; Joyner and Boore, 1986; Wennerberg, 1990; Kanamori et al., 1993) as well as in the estimation of source time functions of large earthquakes (e.g., Mueller, 1985; Frankel et al., 1986; Mori and Frankel, 1990; Kanamori et al., 1992; Li and ToksSz, 1993) The great advantage of using EGF is that propagation, attenuation, and site effects are naturally included in such records. These path effects are poorly known at high frequencies in most regions. The source time function of the large shock is estimated by deconvolution of the records by EGF. This is generally accomplished by spectral division (e.g., Mueller, 1985; Frankel et aI., 1986). In this method the holes in the amplitude spectra of EGF are smoothed in order to avoid large amplitudes in the spectral ratios. An examination of the derived source time functions, however, shows unrealistic long-period trends and short-period oscillations although a rough estimation of source characteristic is still possible. In this article we propose an alternative method for the retrieval of the source time function of the larger earthquake using EGF. In this technique the source time function is approximated by a sequence of pseudotriangular pulses whose

amplitudes are obtained from a nonlinear frequency-domain inversion. The method has the advantage that it permits a "positivity" constraint on the source time function and allows specification of a frequency band, permitted by the quality of the larger and smaller earthquakes' data, to be used in the inversion. The misfit between the observed and the computed seismograms is measured in the frequency domain over the specified frequency band. Numerical tests and an application to earthquake data from the Mexican subduction zone show that the method is robust and yields reliable STF.

The M e t h o d Let us consider a small event S occurring in the source region of a large event M. We assume that the minimum distance to the receiver from the rupture area of the large event is much greater than the fault size. Let Us(t) and UM(t) be displacement records of small and large events, respectively. We may write

1249

Us(t) = Ss(t)*G(t),

(1)

Uu(t) = Su(t)*G(t),

(2)

1250

Short Notes

where Ss(t) and Su(t) are the source time functions of the small and large event, respectively, and G(t) represents path, site, and instrument response. In the frequency domain, equations (1) and (2) can be written as W~(o)) = Ss(oO)'G(o)),

(3)

U~(o) = SM(co).G(o)).

(4)

From equations (3) and (4) we obtain U,Ao2) = &,(oo)W~(oo)lSs(o)).

(5)

In the EGF approximation the source time function of the small event, Ss(t), is a delta function and, therefore, Ss(og) is a constant. Since even small events have a finite duration, Ss(CO) is a bandlimited function whose cutoff frequency, related to the comer frequency of the event, is "a priori ~" unknown. This function acts as a low-pass filter on the mainshock source function. In the proposed method the mainshock source time function is approximated by a sum of N functions

SM(t) = ~ A,f(t

-

"r,, L,),

i

(6)

with i = 0 . . . . . N - 1 and r0 = 0. In equation (6)f(t, L~) is a Hanning window function of amplitude A~ and duration L,, whose Fourier transform is given by f/(co, L~) = 0.5W0(o9) + 0.25W0(co + n/Li) + 0.25W0(fo - niLe), with sin (mL~) Wo(CO) = 2 L , coL~ The Hanning window in equation (6) was chosen because it may reasonably approximate the far-field radiation from subfaults of the large event. Substituting equation (6) in equation (2) and taking the Fourier transform yields UM(co) = ~ A,f(co, L,) exp( - cor~)G(co).

(7)

i

Using the EGF approximation we can replace G(co) in equation (7) by Us(co)/C, where C is a constant that equals Ss(o9 = 0) and is proportional to the seismic moment of the small event. Equation (7) is used in a nonlinear inversion scheme to determine the parameters of the mainshock source time function in equation (6). This is performed by an exhaustive search for the minimum of a misfit function defined by

=

~

Ou~(oJ)l- lu,f,(~)O ~,

¢Omin

8o

F

°

-... . .. . ... ..

_8o

"'~ ~''" "t I ! ~ ""-' . . . . . T

A

o

.a

"

_

.

1