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Bulletin of the Seismological Society of America. Vol. 54, No. 2, pp. 627-679. April, 1964

S U R F A C E W A V E S I N M U L T I L A Y E R E D E L A S T I C M E D I A

I. R A Y L E I G H A N D L O V E W A V E S F R O M B U R I E D S O U R C E S I N A

M U L T I L A Y E R E D E L A S T I C H A L F - S P A C E

BY DAVID G. HARKRIDER

ABSTRACT

A matrix formulation is used to derive integral expressions for the time transformed displacement fields produced by simple sources at any depth in a multilayered elastic isotropic solid half-space. The integrals are evaluated for their residue contribution to obtain surface wave displacements in the frequency domain. The solutions are then generalized to include the effect of a surface liquid layer. The theory includes the effect of layering and source depth for the following: (1) Rayleigh waves from an explosive source, (2) Rayleigh waves from a vertical point force, (3) Rayleigh and Love waves from a vertical strike slip fault model. The latter source also includes the effect of fault dimensions and rupture velocity. From these results we are able to show certain reciprocity relations for surface waves which had been previously proved for the total displacement field. The theory presented here lays the ground work for later papers in which theoretical seismograms are compared with observations in both the time and frequency domain.

INTRODUCTION

Several years ago Dorman, Ewing, and Oliver (1960) successfully utilized the Thomson-Haskell matrix formulation (Haskel], 1953) in the calculation of surface wave dispersion on multilayered elastic media using a high speed computer. Since that time surface wave dispersion has been used extensively in the interpretation of the earth's structure. Calculation of dispersion had previously been limited to simple earth models consisting of at most three layers.

The success of surface wave dispersion in yielding additional knowledge on the earth's upper mantle structure and on the earthquake source mechanism has given hope to seismologists that the amplitude spectra of surface waves may provide further information concerning the mechanism of seismic sources. This is especially true for source depth which has no influence on dispersion. In order to determine the effect of source location on surface wave amplitudes it is necessary to include a source at depth in the multilayered fornmlation. Also without a specific source one is unable to determine the relative excitation between modes as a function of frequency.

There are two methods of attacking the source problenl for an n-layered medium. The classical technique uses the determinants that result from Cramer's rule for solving a set of inhomogeneous linear equations. One expresses the source as an integration of homogeneous solutions to which have been added homogeneous layer solutions with arbitrary coefficients so as to be able to satisfy the boundary conditions at each interface. In this way one arrives at a formal integral solution with an integrand given in terms of the ratios of two determinants of order (4n - 2) for Rayleigh waves and (2n - 2) for Love waves.

This method, although easy to formulate, is extremely cumbersonle if not dan- gerous to evaluate numerically on a computer. The danger involved results from the fact that determinants in general are not slowly varying functions of their elements.

627

628 B U L L E T I N OF T H E SEISMOLOGICAL SOCIETY OF AMERICA

The determinant solution was obtained by Jardetzky (1953) and Kellis-Borok (1953). Besides the numerical difficulties inherent in solving large order determinants there is the practical difficulty of reordering or simplifying the deter~ minan£s into a form which provides insight into the individual effects of receiver depth, source depth, and layering on the spectral amplitude.

The second method is to use a matrix formulation. Previously this has been done in two ways. Using the Thomson-Haske]l matrices to obtain the reflection and transmission coefficients for plane waves in multilayered media (Thomson, 1950), and an integral representation of a point source in terms of plane waves, Gilbert (1956) obtained a formal integral solution for the compressional point source, but made no effort to evaluate the integral for the surface wave contribution.

Gilbert and MacDonald (1961) applied the Thomson-Haskell matrix method to a layered sphere using the solutions of the equations of motion for an elastic shell. They obtained the solution to the source-at-depth problem for the sphere by operating on the source vector equation with a matrix product of the shell matrices. The source vector equation was obtained by evaluating the source at posi£ions infinitesimally above and below the source depth.

This paper derives in detail an integral solution for the time transformed displacements for certain elementary sources at depth in a multilayered isotropic half- space. The integrands are expressed in terms of elements from the matrix product of the Thomson-Haskell layer matrices in the layered array. These integrands are obtained by a technique similar to that used by Gilbert and MacDonald, namely, by a matrix operation on a general source vector equation. The elements of the vector equation depend on the integrand of the particular type of source under in- vestigation. The transformed sources considered are as follows: (1) An explosive or spherical pressure source, (2) A horizontal and vertical point force, (3) A model of vertical strike slip fault sources formed by integration of a time lagged horizontal singlet or doublet point force over the fault surface.

The vertical and horizontal point forces are not as restrictive as one might surmise. Their generality was shown by Kellis-Borok (1953) who pointed out that the field due to a point force, F, of arbitrary direction in a multilayered media can be obtained by the superposition of the fields due to a vertical point force, F sin A, and a horizontal point force, F cos A, where A is the vertical angle between F and the horizontal. Furthermore, displacement fields for multipo]e sources can be determined by spatial differentiation.

From the residue contribution of the integral solutions, we obtain the Rayleigh and Love wave displacements for various source types. If we had stopped here this problem would have been merely an extension of the matrix technique of Gilbert and MacDonald to Rayleigh and Love waves on a multilayered media for different types of sources. Extensive programming and numerical analysis would have been needed to compute amplitude spectra or theoretical seismograms. However by obtaining a simple form of the inverse of the product matrix in terms of the elements of the product matrix itself and simplifying the residue numerator we are able to separate the solution into factors representing source depth, receiver depth, layering, and path of propagation. The necessary simplification of the numerator is accomplished by using relations determined by setting the integrand denominator

S U R F A C E W A V E S I N M U L T I L A Y E R E D E L A S T I C M E D I A 629

to zero. The factors representing source and receiver depth are shown to be simple functions of quantities calculated in the plane wave problem. Using the numerical techniques to be described in Part II the excitation function for the layered medium can be calculated analytically by simple modifications of computer programs which are currently used in seismology to calculate dispersion for Rayleigh and Love surface waves.

THE EQUATIONS OF MOTION AND THE MATRIX RELATIONS FOR LAYERS

I~OT CONTAINING A SOURCE

First we consider a semi-infinite elastic medium made up of n parallel, solid, homogeneous, isotropic layers (figure 1). We number the array such that the layer at the free surface is layer 1 and the half-space is layer n. Placing the origin of a cylindrical coordinate system (r, 0, z) at the free surface, the layer interfaces are defined by z constant and any given layer m is bounded by z~_~ and z~ with z,~ > Zm--1 •

The Fourier time transformed vector equation of motion for an isotropic elastic solid m is

(X,, + 2v~) grad div S~ - tt.murl (curl ~m) = -o~p~S~2 ~ (1)

where the following notation has been used:

S m = ( ~ , ~ , ~m); the displacement vector with (r, O, z) components

co; angular frequency

pm; density

Xm, ~m ; Lame's constants

= -- • ; compressional wave velocity p~ /

= ~.~ \P,~l ; shear wave velocity

k ~ - ~0 ," compressional wave number O/m

50 kom ~m' shear wave number

Defining the potentials ,~, ~}, and 2 implicitly as

630 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

~.~(r, O, z) - 1 a~.~ _~ 1 a2m,7. O~., (2) r 00 r OzO0 Or

~ ( r , O, z ) - Oz r & \ r - - " q - - - Or / r 2 002

which represent the transformed radial, azimuthal and vertical displacements

F ["

2

3

Z i

Z2

t l 1 s + t

SI

S 2

7 s - I

D=z

Zs=

n - I Zn-I

FIG. 1. Direction of axes, numbering of layers, and the depth of interfaces and source.

respectively, and substituting into equation (1) we see that

div S~ V ~" ~m

and that equation (1) is satisfied if ~m, ~m and ~m are the solutions of

la(a,, 6 v~,~==-r~ 7 r o r ~ + - - -

1 a 2 2

r 2 002 q _ O~ m_ ~ _ O z 2 ]c o, ,,~ ~

(3)

2 - 2 = V X~ = --]c~mXm

Using (3) with (2) and the definitions of stress in terms of displacement, we obtain

SURFACE WAVES IN MULTILAYERED ELASTIC MEDIA 631

the normal, azimuthal tangenUal, and the radial tangential stress to the z plane or /5=,/50~ ' a n d / 5 respectively

~l~(r, O, z) Ocp,~ 02~ 1 02m = ~-r + o~0~0~ + J o~

10cpm 1 02~m 02.~ Ore(r, O, z) -- + r O0 r OzO0 Or

W~(r, O, z) 0~,.~ 02~ 2 - = Oz + ~ + k e ~ . ~

P~..,(r, O, z) = 2u~ ~ - - + X~ div N,~

= F ] k Oz~ + o Z + k~ ~ - ~k~e~

Po.~(r, O, z) = .~ \r O0 + Oz/

0 ~.m 2 03~ k~ x,~ r oTgo r

P .... (r,O,z) = . ~ \ Or + Oz /

r O~O0

(4)

We now define

~[m(r, O, z) = I qm(r, O, z; k) dk,

Om(r,O,z) = i vm(r,O,z;k) dk,

t,¢,o

~,~(r, o, z) = Jo w~(r, 0, z; k) &,

a o

(~,~(r, o, z) = fo ~m(r, o, z; k) &,

P ~ ( r , o, z) = fo ~° P=.,(r, o, z; k) dk,

Po~(r, O, z) = fo ~ Po~,~(r, O, z; k) dk,

P~.~(r, O, z) = i ~ P ~ ( r , O, z; k) dk,

~m(r, 0, z) = i v era(r, 0, z; k) dk,

(5)

In passing, we note that if 2.~ is independent of r and if ~m, @~ and 2m are independent of 0, equations (4) reduce to the usual po*ential expressions for cylindrical displacements involving azimuthal symmetry (Ewing, Jardetzky, and Press, 1957).

632 B U L L E T I N O F T H E S E I S M O L O G I C A L S O C I E T Y O F A M E R I C A

and ¢o

5~m(r, O, z) = fo xm(r, O, z; k) dk.

Assuming the following separable radial and azimuthal dependence for the potential integrands

~,~(r, O, z; k) = ~m(z)J,(t~r) cos 1O

¢ . , ( r , e, z; k) = ¢m(z)J,(kr) COS le (6)

x,~(r, O, z; k) = xm(z)J~(kr) sin lO

we substitute relations (5) and (6) into equations (4). Equating integrands, we obtain

qm(r, O, z; £) = ~,~(z) -t- dz _1 dr

(1 itR~(z) dJt(tcr) \ k c dkr

i OL~(z) Jz(kr)~ k c ~ j cos le

[ J,(kr) v,~(r,o,z;!~)= - ~m(z)+ dz _1 r x , ~ ( z ) ~ . ) s id l e

= { 1 aR~(z) J,(kr) q_i ~L~(z) dJz(kr)~ k c kr k c ~

sin 10

Fd(~(z) d~C~m(z) w~(r,O,z;k) = k dz A- dz ~ cos lO

i wRm(z) Jt(kr) cos lO k c

rd2m(z) da¢m(z) P,,~(r, O,z; k) = 2tt,~ k dz2 + d z ~ +k~.o d¢.~(z)Jdz - X~ k =. ~ ~=(z)}

• Jt(k~,) cos lo - ~ ( z ) & @ r ) cos lo

I d+m(z) Po,,~(r, O, z; k) = --,t,~ 2 ~ z - - + 2 d~¢~(z)

dz 2 )1 J~(k,,) r

-I-- dx~(z) dJ , (kr ) \ { Jz(kr) dz ~- .j sin lO - --ir~,,(z) kr r~,dz) d.Iz&r,~( ~ sin lO

&r j

I d~,~(z) P ~ ( r , O, z; k) = ~ 2 ~ + dz 2 2 1 dJz(kr) - - -+- 1~ ¢~m(z) &r

dx,~(z) Jz!kr)} t -+- dz - - cos lO =- irR,~(Z) dJ~(kr) Jz(Icr)\ dk~ - ~ q- r~m(z) ~ r ) cos 10

(7)


where we have used the following relations:

d2'~'~(Z)dz 2 - (It ~ - k.~)~m(z)2 = --k2r,~..,pm(z)2

- - ~ ) ~ ( z ) - - k r ~ ~ ( z ) dz 2

dz 2 (k2 2

- - - k r ~ o , x m ( z ) ] ~ ) x ~ ( z ) =- ~

( 8 )

50 C - -

k

the first three relations (8) being a consequence of substituting equations (6) and (5) into equations (3).

Since there are no boundaries at r or 0 constant, the solution to the multilayered halLspace problem will have the same r and 0 dependence as the source integrands. The r and 0 dependence of equation (6) were chosen for this reason. For the horizontal point force their dependence is given by / = 1 and for the azimuthal independent sources by I = 0.

At the welded interface between two layers, we impose the boundary condition of continuity of displacement and stress. Since ( J z ( k r ) ) / k r and ( d J d k r ) ) / d k r are linearly independenb, we must impose continuity on their individual coefficients in the integrand expressions (7) in order that continuity of displacement and stress be satisfied for all r. From equations (7) this continuity at ~he z = z~_~ interface between layers m and m - 1 can be expressed in vector form as follows.

7UR m ( Zm--1 ) C

?J)R m (zm-1) C

O'Rm ( Zm--1 ) C

_TRm ( Zm--1)

c

~J)R m -1 ( Zm--1 ) C

O'Rm --1 ( Zm--1) c

TRm_l (Zm--1)

(9a)

and

~- -~)Lm--l (cZm~l) 1

TZ,~_,(Zm--1)..J (9b)

From (7) we see that these z dependent coefficients of the displacement and stress integrands for layer m are given by

634 B U L L E T I N OF T H E SEISMOLOGICAL SOCIETY OF AMERICA

and

I d¢.~(z)l ~,.(Z)c - ~ ~,.(z) + dz

wn,~(z) [ - + - - +

c L cz dz ~

z ~ . ( z ) = 2tL~ k dz2 + dz - - ~ - - ~" dz J

V d~m(z) -~ . ( z ) = - i k , . | 2

l_ 2 d~¢'(z) k ~

d z ~ + ~.~(z) J

( loa )

- ~ x~(z) C

dz

(lOb)

For layers not containing a source, we use for ~m(z) and ¢~(z) the general solutions of equations (8) with arbitrary coefiicients.

~ ( z ) = ~ J e -~k~° + ~ " e ~ ° (11)

~ ( z ) = ~. 'e -~k'~ + ~ " e ' ~

And from (8), the exponents of (11) are given by

(k,-~°) ~ ~ o ~ - k ~ -- 1~ ~ - 1

(k~.~°) ~ = k~o - k~ = k~ [ ~y2~ 1 -

where k ~ = ~/a ,~ , k~,~ = ¢o/~m and am and tim are the eompressional and shear velocities respectively of layer m. We use the following sign criteria for r ~ and rs~ as given in Haskell (1953):

r~. = -- 1 for c > c~

r ~ = - - i 1 - - ~ _ ] f o r e < c~

(12)

r~m = -- 1 for c > ~m

[ c2 71/2 r ~ = -- i 1 - - ~ j f o r c < fl,~


Def in ing

m = e -/Xm~

^ ¢t _ _ k 2 l C ~ 2 1 X ikr~mz,n ~v ~

f i k 3 - - i k r ~ m Z m i . ! O)m -~- - - e - ~Om

( 1 3 )

a n d

i k 3

3%

a n d subs t i tu¢ ing equa t ions ( 11 ) in equa t ions ( lOa) , we o b t a i n

- [ ( £ J + £,~") cos ~r .~(z - z.~_~) c

-- i(~.~' - - £m") sirL kr,~m(z -- z~- l ) ]

- ~,, ,r~,~[(~' - ~m") cos k r ~ m ( z - z m - 1 )

- i(~,~' + ~m") sin ]cr~,~(z - z~_l)]

- ~ r . ~ [ - - i ( ~ m ' + /~m") sin kr~m(z - - zm-1) C

£ , + ( .~ - ~ J ' ) cos k~.~,o(z - z~_~)]

-+- "rm[--~(o~.~ -- & ~ ) sin kr~,~(z - - z ~ - l )

+ ( ~ . / + ~%/') cos b ~ , ~ ( z - zm_~)]

-p , .~ ,~ (~,,~-~)[(~,~ + ~ ) cos

- i ( £ . ' - L J ) s i n Ic~-,~,~(z - z,~_~)]

2 2 r [ ~ , ' -- pmc ~',~ r~.~tio~m -- c~,J) cos kro~(z - - z~_~)

- - ~(w,~ + Co./') sin k r ~ ( z - - z~_~)]

r . m ( z ) = p , ~ a m 2 " Y ~ r . , ~ [ - - i ( ~ ' + ~,~") sin kr~m(z - - zm-1)

+ ( £ J - £ , . " ) cos kro,~(z - z.~_~)]

2 . ! - - pmc "y~('/m - - 1)[--~(~m - &~") sin k r ~ ( z - - z.~-l)

(14)

-'}- ((-~m ! --}- (.~m tt ) COS k r f l m ( Z - - Z m _ l ) ]

636 B U L L E T I N OF THE SEISMOLOGICAL SOCIETY OF AMERICA

where we have made use of the following relations

k2 k 2c 2 am - - - - ]2C2 Pm

a~ 2 X~ + 2tL.~

k2 lc2c 2 k 2c 2 P~ ~rn -- -- - -

fl, 2 g . ,

and

Evalua t ing equat ions (14) at z = z . and at z = z~_~, we can then el iminate the coefficients (Am + 7~") , (A~ -- A~"), ( ~ -- ~ " ) , and (w,, + ~") in the same manner as Haskell (1953) and obta in his matr ix result.

-~(z~) c

~m(z~) C

Gl~ra(Zm)

r/~m (gin)

-~.~(z~_~) C

= (~Rm C

GRm(Zm--1)

TRm ( Zm--1)

The lements of the layer matr ix a~,. are given by

((~.~)n = ((~.m)44 = "rm cos P~ -- ( ' ~ -- 1) cos Q~

((~..,)i, = (~.,.)~4 = i [('/.~ - 1) sinP'r.. ~r" sin Q~ 1

((~..)1, = ((~..)24 = - ( p . c 2 ) - ' [cos P . -- cos Q.]

((~R~)i~ " 2 -i sin P~ L ra~

( a . ~ ) , i = (a. . , )43 = - i F v . , r . , ~ s i n P , ~ q- (V,~ - 1 ) s i n Q~] L re~ J

(a.~)22 = (Ct.~)33 = - - ( v ~ -- 1) cos P~ -t- ~'~ cos Q~

(6:.,~)23 = i (p. ,c2) -1 Fr.. sin P.~ q- sirL Q_ ~] k re~ j

( a ~ , , J ~ = (a.%0~2 = p,~J ' r , . ( '7 ,~ - 1 ) [ c o s P ~ - cos q,~]

2 [ sin P____~ ~ J ((~m)~2 = io~c (v~ - 1) 2 A- 7~ r ~ sin Q~

(~Rm)41 ='SPin C2 ['~m 2 ram s in Pm ~- ('Ira - - 1) 2 sin Qm]rom J

(15)

(16)


with

P m = rk~md~, Q,~ = kr~d~ and d~ = z~ - z~_l

In addition, the coefficients are related to the z dependent quantities at z = z~_~

by

v%~(z,._O-

V ' ' £"1 A~ - -?X" I E - 1 (vRm ) (17)

where

E - l = / 0 c2(~ 2 - - veto ram ) 1)/(o~,~ r . . ) o (p,. ~ - ~ Rm [ 2 --1 ('),,~ -- 1) /(~r~m) 0 --(pmc~/~r~) 0

C 'Ym) O 1 0 (Pm 2 --i

(18)

Equation (17) was used in obtaining (15) and it will be used again for the half- space boundary condition.

As pointed out by Haskell (1962), the inverse of the layer matrix is given by

~-z l~ j+k/ ( t ( .~)J~=(- ~ ~ . ~ (19)

Combining (19) with the relations between elements given in (16), we can write the inverse matrix as

I (a .m) . - (a .~)34 (a .~ )~ - (a .~ ) l~ 1 (~-1 - ((~.~)4a ((~.~)3a -- ((~.~)2a (a.,.)13 .o = (20)

-(a~m)~ (a~,~)~ - ) a ~ ) ~ (a~,.)l~

In a similar manner, we formx~(z) ~ o m t h e general solutions of (8) for layers without a source

xm(z) = ~m'e -~kr~m~ + ~m"e ~k'~" (21)

and substitute into equations (9b). Defining


i " em" ' and ~e-~kr~ .... l A ?! ~e¢kr~ .... 1 ~ ,, (22) 6m

we o b t a i n

f f . a f f .

~)Lm(Z) - - (~m ! + ~m )?~]~ COS ]~r~m(Z - - Zm--1) -~- (6m t - - em )]~ SIR kff~m(Z - - Zm-1) C

~ ( z ) = - ( ~ ' + ~")~u~r~ sin kr~.~(z - zm_~) (23)

- E~ ) ~ k ~ r ~ cos ~r~,~(z - zm_~)

E v a l u a t i n g equa t ions (23) a t z = zm and a t z = z~_~ we can e l imina te the co-

efficients and o b t a i n the fol lowing:

~(z~) (~Lm

-~)Lm ( Zm--1)

C

TLm(Zm--1)

(24)

where the e lements of (~L~ are

( a ~ ) ~ l = ( a ~ ) 2 2 = cos Q~

i (aLm)l~ - sin Q~

~ m ~ m (25)

( a~.~)21 = i u m r ~ sin Q.~

Here the coefficients are re la ted to these z d e p e n d e n t quan t i t i e s a t z = zm-1 b y

. , , l i _IL j ^ ! A t! ~ Lm

Em - - •m TLm[Zm_I"~

(26)

where

B : l o = - (*kum r~,o

(27)

The inverse of the l aye r ma t r i x (~Lm is g iven b y

= ~ (a~.)~2 -(a~Dl~ 1 a~i~ L-(a~o)~ (a~o)1~ J (28)


Comparing equation (28) with (20) we see that the elements of the inverse of both (~R,~ and aL,~ can be written as

-- xJ+k/a" (a-l) ~k (--1] ( )z, (29)

w h e r e l = n + 1 - k , p = n + 1 - j a n d a i s a n X n m a t r i x . ItT~s easy to show ~hat if

c = ab

and the inverse of matrices a and b are given by relation (29) then the inverse of the product matrix c will also be given by relation (29). This is an important matrix relation which will be used later to simplify the form of our solution.

~/[ATRIX RELATIONS FOR THE SOURCE LAYER

The point sources considered in this paper can be represented in two equivalent forms. The first form is the source potential. For the source layer s defined by the planes z, and z~_l, we use a general point source located at D, (z~ < D < z,_~) such that the z-dependent integrands of the source potentials are solutions of equations (8) everywhere in s and continuous with continuous derivatives except at D or

~0(z) = ~01eC± - ik~l , - , i

and X~o( z ) = ,Jo3eZ± --ikr~slz~'DI

: i : a s z ~ D (30)

where S0~1, S$2 and S0% are spatially independent constants which depend on the source type and the elastic constants of the layer containing the source.

The second form is a discontinuity in the z dependent coefficients of the displacement and stress integrands at the source plane. The equivalence of the two forms will become obvious as we obtain the necessary matrix relations for the source layer.

Combining equations (30) with the solutions of equations (8) with arbitrary coefficients, the general potentials for the source layer are

~ o s ( Z ) = ~oieZ~: - - i k r ~ l ~ - - D J - ~ 7 X z l e - i k r " . ~ .~_ ~ s t l e i k r ~ z

• ~ f - - i k r~sz ~ If i k r~sz t / z ) = Z0~2e -~kr~l~-'j + ~ e + ~ e

c'~=]= - - i k r ~ [ z - - D ] ~ ! - - i k r~sz ~ t2 ikr~sz

(31)

or rewriting

640 B U L L E T I N O F T H E S E I S M O L O G I C A L S O C I E T Y O F A M E R I C A

~ ( z ) = (S+o~e ~ ° ~ + L ' ) e -~r°°~ + 7,~" e ~ ~

/ ~ - ~ ikr~sD ~ t \ - - ikr~s z G ( z ) = t~o2e + o~ )e + G"e ~ k ~ z~ = z > D

( e'~q- ikrflaD . t - - i k r~s z 2 t l p i k r#s z

~,~(z) = ~s'e -i~"~ q- ( ~ " Jr S~oe-i~'~O)e~"~

G(z) = G'e -i~a~ + (G" q- S o ~ e - ~ ' ~ ) e ~

(32)

= , / e ~k~o~ + ( ~ . + xo(z) - -

Decomposing layer s into two layers with for z~ >-- z -> D and layer sl for D -> z >- zs-1, and defining new constants

- ! ry-b ikra D -- f -- f! ~ j ! A~2 = ~ole ~ ~ - A s , A s 2 =

ws2 = ~302e q - ws ~ ws2 = ws

~? ~- ikrf lsD ~ ! J ! ~ f! es2 ~ ~ 0 3 e ~ - es~ es2 ~ es

~1 = As, A~t = As + Sole ~kr~,.o (33)

(.Osl ---- COs ~ (,Osl ~ (.Os ~ t,~O2e

~t ~ ! . t ! - t! ~ - - - - i k r~sD

we can write equation (32) as

w~(z)

Gffz ) zs ~_> z => D

x ~ ( z )

~a(z ) (34)

Gl (z ) D => z => z~-i

x~,(z)

Comparing equations (34) with equations (11) ~nd (21) and the relations derived from (11) and (21), we have the following matrix relations for the source layer

D > z => z~-i

~ -- - - i k r ~ s D \ i k r ~ z oze ) e

the same elastic constants; layer s2

"%! - - i k rasz ~ •

~ f - - i k r ~ z ~ f! - - i k r~sz O~s2e ~ - ¢os2e

~! - - i k r~sz ~it ikrf lsz = es2e ~ - e s 2 e

~ ! - - i k ra s z ~ ? ! ikrasZ = / ~ s l e q - L ~ s l e

f - - i k r~sz ~ f ! ~kr~sz ~ - 0 2 s l e - ~ o~sle

_ t - - i k r~sz j ! ikr~s z esle ~- e s l e


-~(Z~)c 1

~o~(z~)

- ~ ( D )

c

w~. (D) C

z R . ( D )

_ ~o~ ( D )

(~Rs 2

(~Rs 1

C

[_~-.o~(D)

C i

C

~.o~ (z ._~)

(35)

where the elements (~,~ and aR~ are identical with the exception that

d~2= z ~ - D and d ~ l = D - z ~ - I

Furthermore it can be shown that their matrix product yields

~ R s ~ (~Rs 2 (~Rsl

where a.~ is the layer matrix for layer s if no source is present. In addition we have the vector equation

- ~ ( D ) C

C

(YR~2 ( D )

- ~ ( D ) - C

v:R. (D) C

~ . (D)

_~,(D) _

where from equations (lOa) and (32)

~{TR s

8rR~ i

/~-. \ 8 ~ ) = k2[(S+1 - SOl)- ikr~.(S+2 + So2)]

8 ( - ~ ) = ik[-ikr..(S+l + S01)+ k2(S+~ - S~)]

2 2 " r +

2 2 + ~ = k cp~[-7~r.~(S0~ + S~) - ik(-y~ - 1)(S+2 - S~)]

(36)


Furthermore we obtain

and the vector equation

I ~ ( ~ L s 2

• - ~ L s l

(~Ls I

=

(38)

(39)

where by equations (lOb) and (32)

+

(40)

v--lkasR

Go(R) P°~ as

where R is the distance from the source, a~ is the cavity radius, P0~ is the Fourier time transformed pressure at the cavity walls,

and

Ose = tan -1 k,.a,

ei(ka,as--O~P) (41)

• 2 +

RAYLEIGH WAVES FROM AN EXPLOSIVE SOURCE AT DEPTH

For an explosive source, we use the Fourier time transformed spherical compressional potential for a pressure applied to the walls of a spherical cavity in medium s. It must be poiHted out that only the source term "sees" the spherical cavity• In other words, we do not impose on the multilayer problem the boundary condition that normal and tangential stress over the cavity walls vanish for the homogeneous terms; thus waves reflected and scattered by the cavity are not considered.

The potential for this type of explosive source has been given by many authors (Kawasmni and Yosiyama, 1935; Sharp, 1942; Fu, 1945; Mengel, 1951; Blake, 1952). Here we use the following form


Since the source is located at (0, O, D) in the cylindrical coordinate system (r, 0, z), we can write R as

R = [r ~ + (z - D)~] ~/~ (42)

By means of the Sommerfeld integral, we can rewrite equation (41) as

where

and as before

(D~o(r, z) = fo S°~e-~k~"~'z-~lJ°(kr) dk (43)

- 3 i(kasas--Osp) Sol = i po~ a~ e

E( 2 2 2 k2 ]c r ~ = / ~ --

Since the source is symmetric and the only boundary conditions are at the layer interfaces, z constant, the problem is axially symmetric. Therefore setting 1 = 0 in relations (6) and (7), the potential, displacement, and stress integrands for a layer m reduce to

~m(r, O, z; k) = ~.~(Z)Jo(kr)

¢~m(r, O, z; k) = ¢Jm(Z)Jo(kr)

x.~(r, o, z; k) - 0

[ d¢~(z)7 q~(r, o, z; k) -l~ _~m(z) + dz J J l ( k r )

v~(r, 0, z; k) --- 0

1 i t ~ ( z ) J l ( k r ) - - k C "

k dz - - + - - d~¢m(z)

dz 2 -l- k~¢m(z) 1 Jo@r)

t [d2,zm(z ) d3~bm(z) P ~ ( r , O, z; k) = 2u~ [. dz 2 + dz 3

Po~,.(r, O, z; k) =- 0

i w.,.(z) Jo@r) k c

-- ~ k ~ , ~ ( z Jo(kr) - ~ . A Z ) J o ( k r )

p~,o:(r,o,z; k) =-k~,mI2d~'°(z)~ + 2 a2'~°(~)d~- + k,,~2 d~,o(~)ls,(~r

(45)

(46)

- i r a , , ( z ) J l ( k r )

6 4 4 B U L L E T I N OF T H E SEISMOLOGICAL SOCIETY OF AMERICA_

For the source layer, s, the integrand of the source term is from equation (43)

¢~o(r, O, z; k) = ~o(Z)J~(kr) (47)

where

~.o( Z) = So~e -~k~°.''-ÈI (48)

Comparing equations (47) and (30) we see that S+~ = S o l = S01 and 5~2 = S0% = 0 and thus, by equations (37) and (40)

--- 2k r ~ S01

~0"R~ ~ 0

2 2 4 2

(49b)

It should be pointed out that equations (49b) are the reason that ~m(r, O, z) and 0~(r, 0, z) are equal to zero and not because of the 0 independence. As a counter example, we can have a ring torque source about the vertical axis which will pro- duce ~ ( r , 0, z) or a 0re(r, 0, z) displacement alone with no 0-dependence.

Using source relations (49a), the source vector equation (36) may be written as

-~,~: (D) c -] -~R~I (D) c

w..~(D) [ c = ~bR~l(D)c

_~(D) ~I(D)

Combining equations (9a) and (15), we have

-~_~(z~_~) c

w~_~(z~-l) C

~_~ (z~_~)

__ TR~_I ( Z ~ - - l )

= A ~

+ 0 7 0

~TR 8

(~o)

- ~ ( D ) c

C

o'n~ 2 (D)

(51)


-~[~Rs 1 ( D ) - c

w~l(D) C

CTRs 1 ( D )

_ T R , I ( D ) _

ARsl

- ~ ( o ) - - 7 -

~ o~)

~ (0)

~1 (0)

(52)

where A~ 2 - (~R._I "'" aR,= and A~, -- aR~, . . . aR,. At the free surface z = 0, we require that the stresses vanish. Thus by equation (46), equation (50) reduces

%~.(D)

?J)i%1(D)

c

[ ~,~. (D)

r~oi (D)

to

- AR llW o j (53)

where

~._~0 = ~R~(0) a n d WR0 _ ~2R~(O) C C C C

We now define W, X, Y and Z by the matrix operation

Rsl [;] c

wR~(D) c

(~R81 ( D )

(54)

- -1 Multiplying the vector source equation (50) by A~I and using equations (54) and (53) we have

[;7 I WR~ y j-- -~

L ° 0

_.~ A -1 Rsl

0

0

~TR s

(55)

646 ]~ULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

or

• [(~) 7 - - A - 1 - 1 Uzo _ W a ( . ~ )12 + ~tr.~(A.o~)14

C

• _ [ ( v ) . ~ ] ,Zl - -1 --1 Weo _ X a ( . . . )22 ~- a v . ~ ( A . ~ ) 2 , C

(56)

From relation (29) the inverse of AR~ is given by

-1 ARal

I (AR~,)44 --(AR~)34 (AR~,)24 -(A.~)14

L

(57)

Replacing the (A~1~1) elements in equation (56) by their AR~I equivalents, yields

~-- -[~ (~)(~,)~ + ~.~,~,~1 (5s)

Z-~-[~(~) (ARs l )31 Jf-~TRs(AR~l)11]

From equation (17), we have for the half-space or layer n

Am + n t -- ^ N

nt E~I l ~ Rn

- %tRn (Zn--1) C

~)Rn (Zn--1) C

O'Rn ( Zn--1) _ TRn(Zn--1)

(59)


As a boundary condition for the half-space, we require that the coefficients £~" and ~n" vanish. For c greater than either of the half space body velocities, this is equivalent to requiring that there be no radiation from infinity into the wave guide due to equation (14) and the sign criteria of r,~ and r~ . Similarly for c less than either of the body velocities, this is equivalent to requiring that displacements and stress remain finite as the depth becomes infinite. With this boundary condition, equation (59) reduces to

COn

L TRn (Zn--1)

Defining the matrix AR by the matrix product

AR A ~2 = R Az~I -- (~n~-i "'" (~R~(~R~I "'" (~1 = (~R._I "'" ( ~ "'" ~ 1

since it can be shown that (~. = aR~.~ a ~ z , and in turn defining J by

J = E~,~ Aa (61)

we obtain from equations (60), (51) and (54)

- J L j [j co,~ g

A ! wn Z

(62)

Eliminating ~ ' from the linear equations given by equation (62) yields

O = ( & l - J 2 ~ ) W + ( A 2 - & ~ ) X + ( & ~ - & ~ ) Y + ( J 1 4 - & 4 ) Z (63)

Similarly eliminating ~ ' yields

0 = ( . A ~ - A ~ ) W + ( & ~ - & 2 ) X + ( & ~ - A 3 ) Y + ( & ~ - J . ) Z (64)

Solving equations (63) and (64) for X and W we obtain

X = [GN -- L H J Y + [ R N -- S L ] Z (65) INK- LM]


W = [ N K - L M

Z (66)

where

G J~ -- J~ H J~ - - J4~ R J ~ -- J~ L J ~ - - J 2 ~ ' N J3l - - J ~ ' L J ~ - J ~

S _ J ~ 4 - J~4 K _ J 1 ~ - J~ aad M = J ~ - J~: N J~ - J4~' L J~ - J~ N J~ - J4~

(67)

Using the definition of J and the elements of E~: given in equation (18), write the following

r ~ (An)41

we can

Pn C 2

(A . )~ + (A.),~ pn C 2

Pn C 2 Pn C2

R = ~r .~(AR)I , + ( ~ -- 1)(AR)24 -- - - r.~ (AR)34 + (A.),~ Pn C 2 pn C 2

N = --(7~ -- I)(AR)~ + ~r~(A. )2~ + - pnc~ ~ (A.)~

(68)

M = - ( ~ / n - - 1 ) (AR)12 -~- 5 ~ n r ~ ( A , ) 2 2 - ~ - - pn C 2 p ~

r~ (A.)4~ P~ C ~ Pn C"

(A.)34 -k r~. pnC2 ~ (AR)44

From definitions (68) and the relations between AR elements, obtained by per- forming the operation A R A ~ ~ = I and replacing the A~ ~ elements by their AR equivalents, e.g.

( A R ) 1 3 ( A . ) 2 2 + ( A . ) 2 4 ( A R ) l l - ( A R ) 1 4 ( A R ) 2 ~ - ( A R ) 2 ~ ( A R ) I 2 = 0

it can be shown that

R N - S L = G M - H K (69)

S U R F A C E W A V E S IN M U L T I L A Y E R E D E L A S T I C M E D I A 649

Using equations (65) and (66), we have from equation (58)

and

where

@R0 _ N a ( 1 ) N R (a

c F~

• &T ( 3 ) A T (4) UR 0 -- ~¥R -LVR c FR

(7o)

(71)

FR ~-- [NK -- LM]

L H (;-)1 +

J~R(4) ~ 1 *-[- NR-- FR(3) [(~ ( ~ ) ( A R s ) 3 4 - ~ - ( ~ T R s ( A R s ) 1 4 ]

(72)

It is convenient at this point to examine equations (72) when FR is equal to zero. From the definition of FR in equations (72), we have

3I K N L

(73)

Combining equations (73) and (69) yields

R N - SL K GN - H L L

(74)

and thus for the case of FR = 0, equations (72) reduce to

NR (I) = [GN - L H ] I Y + K z 1

N (2) R = ] (75)

j~ (3) K _ (1)

650 ]BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

and

NR (4) = 1

Thus by equations (5), (46), (70) and (71), the displacements of the free surface are given by

'~ ?~T (1)~T (2) P ~o =-- ff)~(r,O,O) = i ] ~'" "'R Jo(lcr) dk (76)

F . 3o

and

oo RT (3) AT (4)

fo 1 lv~ lvR -Yi(lcr) die ~o =- ~ l ( r , 0, o) = - k F/~

(77)

Evaluating equations (76) and (77) for the residue contribution, we obtain for each j th mode or root, ~o fixed, at FR(~o, k.j) = 0

AT (I) :~T (2)

k,, (oFq

91" N (~) AT(4)

{qO}Ri = { ~R] (::RZi Ri HI(2)(~R:~')

\ oh ],, ,

(78)

or by equation (75)

N2i H0(2)(k.,

.K ~" N()~ Hl(2)(/%r)

k ok ]~,:

(79)

where (OF,/lc)~ j nr(1) N(R2~, N(3~ and N (4) , , ~, ~:, ~j are evaluated at (% k~i) such that F,(~o, kR) = 0.

FR(~o, kRj) = 0 is a form of the period equation for Rayleigh wave propagation in plane multilayered solids (Haskell, 1953; Dorman, M. Ewing, and Oliver, 1960; Dorman and Prentiss, 1960; Press, Harkrider and Seafeldt, 1961; Dorman, 1962; Harkrider and Anderson, 1962). For all real (% k) or (c, k) the elements in the arm matrix are either always real or always imaginary according to the following criteria (Haskell, 1953).

S U R F A C E W A V E S IN M U L T I L A Y E R E D E L A S T I C M E D I A 651

Real (a~m) ik if j + k even integer

Imaginary (aRm)j~ i f j + k odd integer

The same is true also for the product matrix A . . For a phase velocity, c, less than or equal to the half-space shear velocity #~, we see that the quantities defined in equation (68) are also real or imaginary for all real k. We now express the imaginary quantities as a real quantity (designated by an asterisk superscript) multiplied by i or

L =iL*, G=iG*, M=iM*, and S = i S *

and thus

FR(% k) = NK + L'M*

which is real for all real k and c < fl~ • Taking the ratio of {40}R~ to {~0}Rj, we obtain from equation (79)

{~0}Rj _ i K HI(2>(kRj r)

o r

K . K

Thus at horizontal ranges large compared to the wavelength, the surface displacements are either prograde elliptical or retrograde elliptical dependent on whether the real ratio (K/L*) is positive or negative respectively. This large distance result is the same as obtained by Haskell (1953) for the homogeneous case of plane two- dimensional Rayleigh waves:

u0] _ K ~o~ L

Rewriting equation (79) in Haskell's notation

Evaluating the residue contributions of the integral representations for ~1 (r,O,D),_P=,(r,O,D),P~zo,(r,O,D),q~2(r,O,D), ffJ~(r,O,D),P=~(r,O,D) and P,.,~(r, O, D) by using equations (53) and (50) we find that

652 B U L L E T I N OF T H E S E I S M O L O G I C A L S O C I E T Y OF A M E R I C A

{ 71 ~A.,~I,} {q.ffr, O,D)}.~ = - - i (A~.I),2+ ~o ~,

• { ff)o } .1 HI(2) ( k . j r)/Ho(2) (kR i r)

-- --i V~'l(D---~) 1 {¢o}.ig~(2)(k.ir)/Ho(2)(k.~r) L @o J.~

,o.,{. o, o),~,= (~A.,)~ + [ : ] . (A..)~i} ,oo,.

L wo J~

{P.,.I(r,O,D)}R, = ilcRj (AR.,)32 + -~o ~i

= p ~ l ( D ) ] /~o/ @o ni iG~L c J . ~

{ [~f~O] (ARal)41 } {P.ol(r,O,D)}.~ = Gj (A~.I)~ + ~ -~-

X {@o} .~//1 (:) (/~., r ) / H o (2) (l~.j r)

= kR, F~~'(D~)I {ff~o}.~H,(2)(k.,r)/Ho(2)(k.,r) (vo

and

(8o)

{~2(r, 0, D)}Rj = {~( r , 0, D)}Ri

{~@-, o, D)}~, = {~dr, o, D)}~ (81)

{/5=~(r, O, D)}R~ = { P ~ ( r , O, D)}Rj

From the set of equations (81), we see that there is no discontinuity in displacement or stress across the source plane z = D, for the residue contributions. Thus from equation (80) we obtain

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o:roq~t

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(88) ao

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(06)

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(86)

(~6)

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(96)

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pu~

a

spiot.£ (rL) q~noaqc~ (6S) suo!c~nbo :~m.s_Tl

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'aaojaq s~ 's.~ (,)N oaoq~a

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