REPRESENTATION OF AN EXTENDED SEISMIC SOURCE IN A ...

Bulletin of the Seismological Society of America, Vol. 77, No. 1, pp. 14-27, February 1987

REPRESENTATION OF AN EXTENDED SEISMIC SOURCE IN A PROPAGATOR-BASED FORMALISM BY BERNARD CHOUET ABSTRACT

A method is presented for the computation of synthetic seismograms for an arbitrary seismic source embedded in a stratified medium. The technique is applied to a source extending in three space dimensions for which the radiation has been decoupled into P, SV, and SH motions. A matrix formulation is given in which the displacements and stresses due to individual plane wave components of the P.SV and SH radiations from the source are separated into contributions at the upper and lower interfaces of the source layer where they are subtracted from the displacement-stross vector due to the total wave field. With this method, the propagator in the source layer connects only the reverberated components of the waves at the upper and lower layer interfaces, and there are no discontinuities in the displacements and stresses at the source depth. An application of Dunkin's formulation of the Thompson-Haskell algorithm is given for the solution of the P-SV problem, along with a brief summary of the well-known propagator matrix formulation for the associated SH problem. The formulation is extended to the case of a shallow source penetrating two layers, and examples are presented for both shear and tensile sources. INTRODUCTION

An extensive body of literature is available concerning the calculation of the ground response to seismic sources embedded in various types of earth models. A common approach to this problem involves solutions in laterally homogeneous media, which offers the advantages of cylindrical symmetry and translational invariance [see Kennett (1983) for a summary]. Among the various techniques currently in use, the theory for the computation of synthetic seismograms is most fully developed for stratified media made of uniform plane layers bounded by discontinuities [see Harvey (1981) and Chin et al. (1984) for a review of various methods]. While a cylindrical coordinate system is the natural one for point seismic sources in such media, a Cartesian coordinate system can be quite useful for studies of the ground motion over large regions when extended sources are considered. Although this approach has been used to model the ground response in the near and intermediate fields of extended sources of various types embedded in homogeneous or vertically heterogeneous structures (Bouchon, 1979a, b, 1980a, b, c; Chouet, 1981, 1982, 1983), the representation of such sources in a propagator-based formalism has not, to this author's knowledge been fully described in the literature to date. The purpose of this paper is to fill this gap. Unlike the methods discussed in Kennett (1983), which are aimed at solutions to point sources, the method used by Bouchon (1979a) and Chouet (1981) to represent an extended source does not produce a discontinuity of the displacement-stress vector at an interface; rather, the radiation from the source is separated into contributions at the upper and lower interfaces of the source layer where the displacement-stress vectors they generate are subtracted from the displacementstress vectors due to the total wave field to give the "reverberated" wave field. The propagator in the source layer connects only the reverberated components of the 14

SEISMIC S O U R C E IN A P R O P A G A T O R - B A S E D

FORMALISM

15

waves at the upper and lower layer interfaces. The source contributions are obtained from the integration of a point force system over an extended source, which is easily performed analytically for simple dislocation sources (e.g., Bouchon 1979a; Chouet 1982) or numerically for more complex source space-time functions (e.g., Chouet 1981; Bouchon 1982). The first step in propagating the solution through the structure is to decouple the shear wave displacement associated with the source into S V and SH motions. In the formulation of the source potentials used by Bouchon (1979a), this is achieved by exploiting the cylindrical symmetry of the problem and decomposing the rotational components of the source elastic field into the corresponding S V displacement potential and SH displacement. The compressional and S V displacement potentials and SH displacement for each plane wave component of the source elastic field are then propagated through the horizontally layered structure using the appropriate propagators for P-SV and SH radiations. The P - S V problem is treated using Dunkin's formulation of the Thompson-Haskell method (Dunkin, 1965). Dunkin's method involves a decomposition of the propagator matrix into second order subdeterminants and avoids the computation of unnecessary matrix components, which limit the resolution of the wave field to low frequencies. The Cartesian components of the free-surface displacement vector are reconstructed by rotating the displacements obtained in the cylindrical coordinates back to the Cartesian coordinates. Each plane wave component of the resulting wave field is integrated over the horizontal wavenumber space to provide the total wave field solution for a given spectral component of the source. The individual spectral components of the ground response are then synthesized into a continuous spectrum, which can be transformed into the time domain by using a Fourier transform. Of the various methods available for a numerical calculation of the ground response to an extended seismic source, one involves the discrete wavenumber synthetic method (Aki and Larner, 1970; Bouchon and Aki, 1977; Bouchon, 1979a). This method relies on the exact discretization of the elastic wave field resulting from a periodic disposition of sources. The individual plane wave components of the source wave field decomposed in this manner naturally lend themselves to a solution based on the standard propagator formalism as described above. Other approaches can be found in Kennett (1983). We shall start with a brief presentation of the decoupling equations expressing the three-dimensional problem in terms of the P - S V and SH radiations and follow with the application of Dunkin's decomposition to the propagation of plane P and S V waves generated by a buried source. To complete our description of the problem, we shall briefly retrace the treatment of the associated SH source wave field using the well-known SH propagator matrix. We shall conclude with an example of an extended source penetrating through a layer interface and give examples of the displacement-stress vectors for the simple cases of a right-lateral strike-slip fault and circular vertical tensile crack. FREE SURFACE D I S P L A C E M E N T S Let us consider a horizontally layered, elastic half-space and assume a Cartesian system of coordinates (x, y, z) with the z axis taken positive downward (Figure 1). The displacement vector u can be written in the form u = V ¢ + V x ~I,.

(I)

16

BERNARD CHOUET

Zo=O ZI

~X

(I)

hI

al ,'/31 , P l

(2)

h2

a2,~2,P2

(n)

hn

.z2

z n'Zn

an ,,Bn , Pn x=L

Z $....

(8)

hs

Z=ZlO as,/~s,Ps

.... Z=Z20 Zs

Z

m--

(m)

am,Bin,Pro

Z FIG. 1. Cross-section of the layered elastic earth model. Material properties are c o n s t a n t within each layer. The source is represented by the rectangular fault in the x - z plane extending from x = 0 to x = L, and bounded by depths of zl ° and z2°.

¢ and q are the displacement potentials, and are solutions of the wave equations

V2~b-

1 02¢

a20t

2

1 02~

V2~ -/32

(2)

Ot 2 ,

where a and fl denote the compressional and shear velocities of the medium, and obeys the additional condition v

• ~, = 0.

(3)

Within a uniform layer, the solution for simple harmonic motion of frequency o~ may be written as plane waves of the form ¢(o~) = A e x p ( - i k x X - i k y y +_ i v z ) • (,~) = B e x p ( - i k x x

- i k y y +_ i~(z)

(4)

where the time factor is omitted and where the + and - signs correspond to upward and downward propagation, respectively. Here, v and 7 are the vertical wavenumbers, given by v ---- ( k . 2 -

kx 2 -

]~y2)1/2,

I m v _- z,. This represents the situation in which the source crosses the interface between layers with different material properties. We obtain the following relations

Sl(Zl) --~ Sl°(Zl) -Jr- G l [ S l ( z o ) ¢2-(z

) = E2(z

-

Zl)T2-1[S2(zI)

-

Sl°(Zo)] -

$2°(zl)]

+ E2(z - z2°)~2°-(z2°), z > z2°

(34)

S2(Zl) = SI(Z1) where

¢2-(z) = [0, 0, ¢2-(z), ¢2-(z)] (I)20-(Z20) = [0, 0, ~b2°-(Z2°), ¢2°-(z2°)],

(35)

and where E 2 ( z - Zl) and E 2 ( z - z2 °) are given by (13), in which hn is replaced by (z - zl) and (z - z2°), respectively. This yields

¢2-(Z)

= E2(z -

Zl)T2-1S1°(zl)

+ E2(z - zl)T2-1G, Sl(zo) - E2(z - zl)T2-1G, Sl°(zo) - Ee(z - zl)T2-'S2°(z,)

+ E 2 ( z - z2°)~2°-(z2°).

(36)

Using S , ( z o ) given in (14) and keeping only the first two rows of this matrix equation, we then derive

(U°) --R-~1[(R11R12)S1°(z°)+(P11P12)(S2°(zl)-S1°(zl))]wo

(37)

SEISMIC SOURCE IN A PROPAGATOR-BASED FORMALISM where R = T 2 - 1 G 1 rewritten as

()

and P = T2-1. Proceeding as shown earlier, this equation can be

°

,O(zo

rl '2,2 r ''2,x3 r l l ~ ] [ [2112~1 t2 2k,~k2 12~1 ~2 3k$:~k2 12~1 ~2 I 4k,~k2 12.1 \ I lkgk2 + / z2,~2~1 ~2 ,2.1 _[2 ~2.~ : 2 ~ 2 ~ / ( S 2 ° ( z 0 -

wo = r ~ 2

23

\~t,

I l k , ~ k l --t,

2k,~kl

3k,~kl ~ 1 ,

I 4k,~kl]

]

Sx°(z~))j

(38)

where the vectors S~°(Zo),Sa°(z~), and $2 ° (zl) are obtained from the source potentials through equation (9), and r l ~k ~ is given by

o~ I~.

(39)

Equation (38) shows that the source is separated into two components, one of which has a depth extent from zl ° to z~ in layer 1 and produces the displacement-stress vectors S~°(Zo)and S~°(z~), while the other extends between the depths of z~ and z2° and produces $2 ° (Zl). The SH radiation is obtained by an equation similar to (37) in which (Uo, Wo) is replaced by Vo; Sl°(zo), Sl°(zl), and $2°(zl) are derived from equation (29), and RI~, R~2, P~, and P~2 are given by

(RHR~) = I

1

2iu25'2

(PnP~2) =

(ig272COS(q(lhl) -- /.tl'ylsin('Ylhx) 2i~3,

i/x2~2 sin('yxhx) + cos('ylhl)) ]XI'~I

"

(40)

EXAMPLES OF SHEAR AND TENSILE SOURCES Using (9) and (10a), we derive

I~ O+(z,_l)\

!z) /,

-ik -i% \ _,.

\ 2~,v,k

-~,l,/

(~ I / ~-iit'k~ -ik i% kl /¢~°-(z,)~ Ss°(Z~)= T~ 4~O_(z~) = ~ g~l~ -2Us% \¢s°-(Zs)] \¢.°-(z.)/ for the

(41b)

-zA /

P-SV radiation, and using (29) and (30a) we get ~

for the

\-2.s~.k

(41a)

SH radiation.

~-iJ

= T.

= i#,% V~°+(z~-l)

(42a)

BERNARD CHOUET

24

Consider now a strike-slip fault buried in layer s. We assume a rectangular fault plane set in the plane y = 0 and extending between the depths z~° and z2° (z~ _->z2° > z~° = z~