Bulletin of the Seismological Society of America. Vol. 64, No. 6, pp. 1685-1696. December 1974

REFLECTIONS, RAYS, A N D REVERBERATIONS

BY B. L. N. KENNETT*

ABSTRACT

The connection is established between conventional matrix methods for layered media and the reflection and transmission properties of a single layer. This inter- relation is then used to set up an iterative approach to the calculation of reflection and transmission coefficients in multilayered media. This approach lends itself to a ray interpretation and allows estimates of errors involved in taking truncated partial ray expansions to be made. The special effects due to a free surface are also considered.

INTRODUCTION

As techniques for calculating theoretical seismograms based on ray methods have been developed, it has become apparent that no more than a limited set of the total ray expansion for a layered medium can possibly be computed. A number of authors have therefore looked at the errors involved in taking certain prescribed sets of partial rays. Mtiller (1970) and Chapman (1974) have looked at errors involved in the neglect of multiple reflections in the context of Exact ray theory. Hron, Kanasewich and Alpa- slan (1974); Kanasewich, Alpaslan and Hron (1973) have looked at errors involved in using asymptotic ray theory.

In this paper, we consider an iterative approach to the calculation of reflection and transmission coefficients in multilayered media. This technique is derived from the conventional propagator matrix methods for layered media by establishing the connec- tion between the layer matrices and the reflection and transmission properties of the layers. This approach lends itself to a ray interpretation and enables simple criteria to be set up to allow computation of a partial ray expansion to within a specified error toler- ance. The special effects introduced by a free surface are also considered in detail.

WAVE PROPAGATION IN A MULTILAYERED MEDIUM

We consider a medium comprized of horizontal isotropic layers, and for simplicity will confine our attention to two-dimensional problems in which stresses and displace- ments are independent of the coordinate y. We take a plane-wave decomposition of the elastic-wave field

u(x, z) = 2~ ~(k, z) exp (ikx) dk (1)

and assume a time-dependence of the form exp (-i~ot); pulses may then be generated by Fourier synthesis.

The stress-displacement vector B, whose elements are the horizontal and vertical components of displacement u, w and the stresses zxz, r~z; i.e.,

B = [u, w, ~xz, ~z] r, (2)

is continuous across any horizontal interfaces. I f we consider a single plane-wave component of the elastic field, the stress-displacement fields, for this component, at

*On leave at: Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, California 92037.

1685

1686 B . L . N . KENNETT

the top and bottom of a sequence of layers are related by

B(k, z l) = P(k, z l, z,) B(k, z,), (3)

where P is the resultant propagator for the sequence (Gilbert and Backus 1966, Kennett 1972). This propagator may be decomposed into layer contributions

H(k, zl) = Pl(k , z l, z2) Pz(k, z2, z3). • • P , - l ( k , z,_ 1, z,) B(k, z,). (4)

For uniform layers these layer propagators are identical to the Haskell layer matrices (Haskell, 1953). These propagators include the effect of all the reflections and reverbera- tions within the sequence of layers.

For locally homogeneous media at the boundaries of the sequence of layers, we are able to express the stress-displacement vectors B(k, z~) and B(k, z,) in terms of the local upgoing and downgoing P and S waves. Thus for example

B(k, zl) = To(k ) V(k, z , ) , (5)

where V is the vector whose components are the amplitudes of upgoing and downgoing P and S waves

V = [4) U, ~h U, 4) °, ~o]r = [V U, VOlt, (6)

where ~b U'° refer to P and ~u,o to S waves. The columns of the matrix To(k) are the propagation eigenvectors for the bounding homogeneous medium, which it is convenient to take in a form normalized with respect to energy flux in the z direction. Similarly we ma2~ express B(k, z,) as

B(k, z,) = T,(k) V(k, z,), (7) so that equation (3) becomes

iro(k) V(k, z,) = e(k, z~, z.) r .(k) V(k, z.). (8)

We now have a connection between the upgoing and downgoing waves on the two sides of the sequence of layers, i.e.,

vo( ,z, i

set Q = T o- l(k) P(k, Zl, z,) T,(k) where we have submatrices.

I f we consider a downgoing wave incident at z = then

VV(k, zl) = ~?D(k) VU(k, zl) , VU(k, z,)

v U (k, z,)] (9)

v ~ (k, z.) i

and partitioned this into 2 by 2

z~, and no upgoing wave in z > z,

= ~-D(k) VU(k, z 0 , (10)

where ~?o and 77- D are the matrices .of the reflection and transmission coefficients for both P and S waves for downward propagation

RD = [r"° rp~ 77-o = (,po tp~ (11) \r=~ rsUJ ' \ t~ . t=uJ "

Prom equ~tion (9) we see that these reflection and transmission matrices are related to the subpartitions of Q by

T o = Qzz 1

~u = Q l z Q 2 2 . (12)

REFLECTIONS, RAYS, AND REVERBERATIONS 1687

If we now consider an upward-going wave incident on z = z,, we find that the upward reflection and transmission matrices for the layer sequence can be expressed as

2r u = QII -Q12Q21Q21

R u = - QEEIQ21 (13)

From the relations (12) and (13) we are able to recast equation (9) into a form which shows clearly the relation of the propagation characteristics to the reflection and trans- mission properties of the sequence of layers

z, 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 4 )

\ V D (k, z 1)] - ~-Dlff~U ~-D 1 \ V D (k, Zn):

On combining equations (9) and (14), we are able to derive a generally valid expression for the propagator matrix in terms of the reflection and transmission matrices for upward and downward wave propagation

If we have an infinitesimal source in the sequence of layers, the effect of the source may be included by allowing a discontinuity in the stress-displacement vector H across the horizontal plane containing the source (see, e.g., Hudson 1969)

B(k, z s - ) - B(k, z s + ) = S(z~) (16)

where S is the source contribution. Then the counterpart of equation (3) is

B(k, z~) = P(k, z~, z.) B(k, z.)+ P(k, zt , Zs) S(zs) (17)

and by splitting the propagator from ZI to z. at the source plane into two factors we have

B(k, zl) = P(k, zl , zs) P(k, zs, z.) B(k, z.)+ P(k, Zl, z~) S(zs). (18)

The propagator representation (15) for the regions (z l, z~), (z~, z.) allows the effect of the source to be expressed in terms of the reflection and transmission properties in the two parts of the layered sequence above and below the source plane. More general sources may be obtained by superposition.

AN ITERAT1VE SCHEME FOR REFLECTION AND TRANSMISSION COEFFICIENTS

(a) Direct iteration. We consider, initially, a single homogeneous layer bounded by two homogeneous layers for which

B(k, Zl) -= Pl(k, zl , z2) B(k, z2). (19)

The propagator matrix for the uniform layer may be written as (Hudson 1969, Dunkin 1965)

Pl(k, zl , z2) = T~(k) exp (A(z~ - z 2 ) ) T ; l(k) (20)

where exp (A(z I - z 2 ) ) is a diagonal matrix containing the phase effects appropriate to the propagation of the upgoing and downgoing P and S waves through the layer. We may write this propagator

Pl(k, Za, Z2) = Ul(k, zl) U~ l(k, Z2) (21)

1688 B . L . N . KENNETT

absorbing into the U the phases appropriate to each interface, i.e.,

U(k, z) = T(k) exp (Az). (22) Similarly we may set

l](k, Zl) = Uo(k, zl) W(k, zl) = To(k) Y(k, Zl) (23)

and then, in terms of the phase-related amplitude vector W, equation (19) becomes

W(k, zl) = Uo l(k, zl) Ul(k, zl) U'f l(k, z2) U2(k, z2) W(k, z2). (24)

If we consider just the interface z = z~ between the media "0" and "1" we have

(~01 _01/_01"~- 1--01 _01/~Oh- 1 x I v --K D ~I D ) KU K D [I D )

Uo l(k, zl) Ul(k, Zl) ........................................................... (25) \ -- (TO1) - 1RO1 (TO1) -1 ]

using phase-related reflection and transmission coefficient matrices (Scholte 1956, Cisternas, Betancourt and Leiva 1973). These are coefficients phased to a reference level z = 0, which is not necessarily coincident with the interface; the relation of these co- efficients to the usual definition is given in the Appendix. Similarly, if we consider the interface at z = z2,

12 _12r_12x-1~12, _12t~12x-I / TU ~KD LID ) KU i KD ~.ID J

Ui-l(k, z2) U2(k, z2) ........................................ i ................... (26) 12 -- 1 12 ! 12 - 1

\ --(TD ) RU i (TD) /

where the reflection coefficients are again phased relative to the same reference level (z = 0). From equation (14), the overall transmission and reflection coefficients for this layer may be obtained by multiplying together the two matrices (25) and (26)

( T U - - ~ D T D I ~ u ~DT/~I / ............................................. Uo- l(k, zl) Ul(k, Zl) U1 - a(k, z2) U2(k, z2). (27)

\ / This gives results which depend on the coefficients at both the interfaces z = Zl and

Z ~ Z 2 TD _12rtr _01_121-1 TO 1

t o [ u - - K v t t o J

TU _01r. --12-011-1 T12 -~ ZU L U - - t t D KU .I

~D RO1 _O°l_12rTr __01__121- 1 TOD1 = .91-1U K D [ l l - - . K U K D J

~U = R12 "~- T12 [ ff --KU--0I-- 12a- 1 K D J KU-01--12Iu (28)

where ~ is the 2 by 2 unit matrix. We see that the upward and downward propagation coefficients have the appropriate reciprocal forms. We see that the matrix overall re- flection coefficient An is nonlinearly related to Ro 12 and this is reflected in the fact that for an infinitesimal thickness of layer the reflection coefficient equation reduces to a matrix Ricatti equation.

Although the above results were derived for a single layer between two homogeneous layers, they may be extended without any further analysis to a stack of layers. Consider the layers between the planes z = z~ and z = z, then

W(k, zl) = Uo l(k, zl) Ul(k, Zl) Ql(k, z2, z.) W(k, z.) (29)

and the overall reflection and transmission coefficients for the stack of layers may be found from equation (28) by replacing gl 2, w 12 for the interface z = z 2 by the resultant coefficients for z 2 < z < z.. We thus see that we may extend knowledge of the reflection

REFLECTIONS, RAYS, AND REVERBERATIONS 1689

and transmission coefficients upward (or downward) a layer at a time. Consider the interfaces z = z j and z -- z j _ 1; let the overall phase-related coefficient matrices at a level be denoted by ~, ~- and the phase-related coefficient matrices for a single interface be R, T, then we have the iterative schemes:

(a) for downward propagation

TD(Zj-1) = ~-D(Zj)[ ~ --Ru(Zj-1) ~£~D(Zj)]-1 TD(Zj_ 1)

ffS~D(Zj- I) = RD(Zj- 1)q-Tu(Zj - 1)ffS~D(Zj)[~--Ru(Zj - 1)~S~D(Zj)] - 1 TD(Zj_ 1) (30)

(b) for upward propagation

~l-u(Zj- 1) = Tu(Zj- 1)[ ~ - ~D(Zj)Ru(Zj- 1)] - 1 ~l-u(Zj )

g:~u(Zj - 1) : ~u(Zj)+ rD(Zj)[g--RU(Zj-1)g2~D(Zj)]- 1Ru(Zj- 1) ru(Zj) (31)

relating the resultant reflection and transmission coefficients at the j - l t h interface to those at thejth.

These equations for an iterative development of reflection and transmission coefficients represent an extension of well-known results in scalar-wave propagation which have been applied to geophysical situations by Wuenschel (1960), Treitel and Robinson (1966) among others. For an acoustic wave, the matrix of reflection coefficients reduces to a single component and so we have, for example, for downward wave propagation

TD(Zj- 1) = T o ( z i ) t v (z j - 1) 1 -- rv(z j_ 1) RD(Zj)

R D ( Z j - 1 ) = rD(Z j - 1 ) + to(Z J - I ) R D ( z j ) t v ( z J - 1) (32) 1 - - r v ( z j _ I )RD(Zj)

for coefficients phased relative to the reference level z = O. From such expressions, one may make a ray interpretation of the resultant reflection coefficients and we shall consider this further in the next section.

The treatment above can be extended to layered inhomogeneous media, since, in general, in an inhomogel~eous media the propagator can be expressed as

P ( k , z t, Zm) = U ( k , z t ) U - l(k, zm) (33)

where the columns of U are independent solutions of the elastodynamic equations (Gilbert and Backus 1966). For weak inhomogeneity, these independent solutions can be identified as largely upgoing and downgoing P and S waves (Richards 1971) and thus reflection and transmission coefficients may be defined at the bounding interfaces. Then the results above will hold with the new forms of U and reflection coefficients found from, e.g., equations (26) and (12). Such a technique would be very suitable for an inhomogeneous region bounded by first-order discontinuities. However, the method will be inapplicable if the plane wave has a turning point in the inhomogeneous region.

In order to apply the iterative schemes (30), (31) to a set of layers one will have to calculate the reflection and transmission coefficients for both upward and downward propagation at each successive interface. However, for first-order discontinuities, we may make use of the symmetry relations between reflection and transmission coefficients

o D t ° t ° 0 a r D v u u for each and only need to find the ten quantities tpp, tp,, sp, ~,, rpp, r~p, ,,, rpp, r~p, r~s,

interface. After the above analysis was completed, the equations (30), (31) were found to be

1 6 9 0 B . L . N . K E N N E T T

equivalent to those of Altman and Cory (1969) for coupled ionospheric wave propagation, although the method of derivation is rather different.

(b ) R e v e r s e i t e r a t i o n . The iterative relations (30) and (31) may also be used to recover information about the reflection coefficients at different levels from an overall reflection coefficient. Suppose for example that the downward reflection coefficient matrix at the j - l th interface is known the from (30.ii)

~D(Zj - 1) = RD(Zj- 1) +Tu(Zj - 1)~:~D(Zj)[ U --Ru(Zj- 1)~:~D(Zj)]- 1TD(Zj- 1)" ( 3 4 )

Then introducing the effective reflection coefficient matrix at the j - lth interface

~E(Zj- 1) = ~D(Zj- 1 ) - RD(Zj- 1) (35) we have

g~E(zj - 1)To l ( z j - 1) = T v ( Z j - X ) ~ D ( Z j ) [ g - - R v ( Z j - X)~o(Zj)]- 1 (36)

This matrix equation may be solved for the reflection coefficient matrix at the j th inter- face to yield

RD(Zj) = RE(Zj- 1)[TD(Zj - 1)Tu(Zj - 1) +TD(Zj- 1)RE(Zj - 1 ) T D I(Zj - I)Ru(Zj - 1)] - 1 (37)

where all the quantities on the right-hand side are to be evaluated at the level z = z j _ 1.

The expression on the right-hand side of (37) may be regarded as a full dereverberation operator for the effective reflection coefficient matrix to remove the effect of all internal multiples and interconversions. However, such a scheme requires knowledge of all the reflection and transmission coefficients (Ro, RD, TO, TD) at the j - lth interface and will be difficult to apply in practice.

For acoustic wave propagation the expression (37) simplifies to

R D ( Z j ) = R E ( Z j - 1)/( 1 --rD2(Zj - 1 ) - r D ( Z j - 1 ) R E ( z j - 1)) (38)

since t u t D = 1 - - r D 2 and r U = - r D. This expression may be rewritten as

R D ( Z j ) = [ R D ( z j - 1) -- r D ( Z j - X)]/[ 1 -- r D ( Z j - 1 ) R D ( z j - 1)1 (39)

giving a relatively simple frequency-domain dereverberation operator.

RAYS AND REFLECTIONS

If we return to the simple model of a layer sandwiched between two uniform media, we have from equation (28.iii) that the overall reflection coefficient matrix for downward propagation is given by

~)D ~ 0 1 - - ~ 0 1 ~ 1 2 r ~ _ 0 1 _ 1 2 1 - 1TO1 " : K D " i - I U K D [ / / - - K U K D J (40)

If the inverse matrix is expanded as a power series we have

~D ~O1--_01_ 12r~ ~ (-01_ 12xql_O1 (41) = K D --[-1 U K o [ / / -~ kKU K n ) J l o ,

q = l

and this latter expansion corresponds to the set of physical rays in this layer. The first three terms of this expansion are

b~ D ,.~ RO1 ._L_O 1~ 1 2 _ 0 1 - - _ 0 1 _ 1 2 _ 0 1 _ 12--01 . 1 U K D I D - ' f - I U R D K U R D I D

..~ ~ 0 1 _ 1 2 ~ 01~ 1 2 _ 0 1 _ 1 2 ~ 0 1 (42) l U K D K U K D K U K D 1 D .

The first term of this sequence is simply the set of reflection coefficients which would be obtained from the "01" interface. The other terms represent the effect of waves which

REFLECTIONS, RAYS, AND REVERBERATIONS 1691

have passed through the "01" interface and been reflected successively at the two bounding interfaces of the layer before re-emerging through the "01" interface. Thus this expansion includes all multiple reflections and interconversions in the layer; this is perhaps best illustrated by considering the expression for a single reflection coefficient. In order to simplify the notation, we set, e.g., (Rl2)pp = Rpp; (T°l)pp = tpp and denote coefficients for upward propagation by an overbar. With up to three reflections in the layer we obtain

Rpp = rpp + (i,,pRpp + tpsRsp)(tpp + (~ppRpp + ~sRsp)tpp + (?pl,Rp, + ~psR,s)t,p )

+ (tppRps + 7p,R,,) (t,p + (e, pRpp + ?ssRsp)tpp + (espRps + fssRss)t,p}, (43)

and the phase effects are included in the phase-related coefficients. For an N-layered medium Cisternas et al., (1973) have described a procedure whereby

all the physical rays in the medium may be generated by taking the inverse of a 4N by 4N matrix of the form

[ I - X ] - I = I+ ~ X q. (44) q=l

However, it is somewhat easier to generate the system of rays a layer at a time. This may be achieved by using an expansion of the matrix inverse in the iterative method of equations (30) and (31) to give, for example,

~D(Zj - 1) = RD(Zj- 1 ) + TU(Zj- 1)ffZ~O(Zj)[ ~ + ~ (Ru(Zj-1)ff2~D(Zj)}q]Vo(2j - 1), q= l

To(z j - 1) = To(zj)[q + ~ {Rv(z 2- x)Ro(Zj)}q]To(Z 2- 1). (45) q=l

In practice, for many layers it becomes impracticable to keep track of all reflections and interconversions, and it is necessary to impose some upper limit on the number of reverberations to be considered in any one layer. Such a limitation may be achieved by truncating the sequence (44) after a specified number of terms or by similar truncation of the iterative expressions in (45). The iterative technique has the advantage that the truncation level may be specified independently for each layer and a larger number of terms taken if reverberation is likely in any particular layer. A convenient working criterion for truncation of these series is that for N layers and overall accuracy level e, then at least Q terms should be taken, where Q is such that

[rR] e <__ e/N. (46)

Here r, R are the moduli of the largest reflection coefficients, for the plane-wave com- ponent under consideration, at the roof and floor of the layer; thus

Q > In (e/N)/ln(rR). (47)

The importance of the higher terms, in particular the effects of interconversions, has recently been forcibly demonstrated by Kanasewich et al. (1973) and Hron et al. (1974).

If the ray phases are now reinstated using the relations given in the Appendix, the truncated partial ray expansion may be used as the basis of the Exact ray method and for a pulse source the contribution for each partial ray determined by the Cagniard-de Hoop technique (Helmberger 1968, Miiller 1970).

Alternatively, the results may be used directly in the frequency domain as an adjunct to the reflectivity method (Fuchs and M/filler 1971); for a particular frequency if a ray interpretation is not required we may work directly with the iterative equations (30) and (31). For plane waves traveling at an angle of only a few degrees (< 10 °) to the

1692 B . L . N . KENNETT

vertical, interaction of wave types is not important and P and S waves may be treated as being essentially uncoupled. Then we may use the simple forms

o ..~ _ %p)Rpp(zj)/( 1 _ rppROp(zj)), Rpp(Zj_ 1) - rpp-t- (1 2 D

R,,(z j _ o 1) ~- r,s+ (1 - r2)RO(zj)/(1 - r ,sR°(z j ) ) ,

Rps = Rsp '~ O. (48)

where rpp, r,, are both evaluated at z = z j_ 1 ; these expressions all for reverberation but not interconversions, and, in general, near normal incidence internal multiples will be small. For larger angles of incidence, corresponding to important propagation paths for large source-receiver separations, reverberation and interconversion phenomenon become more important and then the equations (30) and (31) should be used. In the present reflectivity methods, the layers above the reflection zone only contribute inter- facial transmission losses to the computed seismograms. However, it is possible to make improved allowances for reverbative structures, such as low-velocity zones in these layers, by using the iterative scheme to calculate more accurate transmission effects for such structures.

THE EFFECT OF A FREE SURFACE

Most measurements in seismology are made at the Earth's surface, which to an extremely good approximation may be represented as a free surface at which the stress components zxz and Zz~ vanish. This free surface requires slightly different treatment from the interfaces considered earlier. We consider a multilayered sequence bounded above by the free surface z = 0 and underlain by a uniform half-space in z > zt~-

In the absence of sources, the stress-displacement vector at the free surface B(k, o) = [f4o, ~'o, o, o] T and that at the top of the half-space are related by

~(k , o) = P(k , o, zN) ~(k ,zN). (49)

In terms of the phase-related amplitude vectors W we have

Wl(k, o) = U? l(k, o) P(k , o, ZN) UN(k, z~) WN(k, ZN)

= O(k , o, zN) WN(k, ZN) (50)

and the matrix Q is related to the phase-related transmission and reflection coefficients for upward and downward propagation in 0 < z < z N, by equation (14). The free- surface stress-displacement vector is thus given by

~(k , o) = Ul (k , o) Q(k , o, zN) WN(k, ZN)

= g ( k , o, ZN) WN(k, ZN). (51)

If we have sources present and assume that there are only outgoing waves in the lower half-space, WN(k, ZN) = [O, O, %, Cs] T and then

• ! lR21 R22

0 0

° ° .

¢p

Cs

sl] $2

+ .. .}

$4 J

(52)

in terms of the 2 by 2 subpartitions of R. The vector S is best described as the stress and displacements produced at the surface by direct propagation from the source, the total

REFLECTIONS, RAYS, AND REVERBERATIONS 1693

wave field then adjusts itself so that the induced stresses are neutralized; thus

At the free surface B(k, o) = [~o, o] r, and Wl(k, o) = [~u, rVo] r in terms of upward and downward propagating waves. We write U~j for the 2 by 2 partitioned submatrices of U, and so obtain

(fto°)=(Ux-)--!U--k2-)(wv ~ . (54)

Thus

I~ O = -- U - 1U21~ U = ffy~?u (55)

where ~ is the matrix of free-surface reflection coefficients. We introduce the matrices

My = Ul1, Mo = U12, No = U22, (56)

where N o is the stress transformation for downgoing waves and Mu, Mo are the displace- ment transformations of upgoing and downgoing waves. Then the partitioned submatrices of~? have the form, on using (14), (51), and (56);

R12 = U I 1 Q 1 2 + U 1 2 Q 2 2 ,

U D D

R22 = U 2 1 Q 1 2 + U 2 2 Q 2 2 ,

= ND(ff_. ff~ff~D) ~D 1 (57)

From equation (53) the free surface displacements for this source problem are given by

( ) : --1N-1($3)"{- ( s l ) , (58' I~oU° -- (Muffed + Mo)[ff -- ff~0q~O] O $4 $2

and the secular function for the half-space has the form

det (No -- ND~ff~D) = 0, (59)

while it would appear that det No = 0 is always a root, there is a compensating singu- larity in det (~- ~ o ) . This expression for the surface displacements will be valid for an arbitrarily inhomogeneous medium provided that the elastic properties asymptote to uniformity at great depths.

For a surface layer with P- and S-wave speeds ~', fl' and density p' we have from the results in the Appendix that

det No = - e,'e#'#' 2 (I" 2 + 4k 2 v,'v#') = - e ,e#R(k) (60)

where

}'a t = [((D2/g'2)--k2]l/2, Vfl' ~- [((D2/flt2)--k2]l/2; In v~', v#' > 0

r = ( o ~ 2 / B ' 2 ) - 2 k 2, ~' = p'B '2

e~' = (2/p'o93v~') 1/2, e#' = (2/p'~o3vp') 1/2

and this determinant is simply a multiple of the Rayleigh denominator for a uniform

1694 B.L.N. KENNETT

half-space with the same properties. The secular function for the multilayered half-space has the form

det (No-- N~Do)

= ]~/, 2 ([2kv,'e,'(1 + rpp) + Fe#'r~p] [2kv#'e#'(1 + r~) - Fe ,'rp~]

+ [rea'(1 - r,~) - 2kv,% ~'rps] [Fe,(1 + rpp) - 2kv#'ea'r~p ] ) (61)

in terms of the downward reflection coefficients for the half-space. We see that the secular function departs further from the Rayleigh denominator as the

inhomogeneity of the half-space, and thus the reflection coefficients increase. The re- flection coefficients appearing in (61) can be conveniently calculated using the iterative scheme in equation (30).

If we consider only that part of the displacement corresponding to the first term in the overall ray expansion, i.e., we neglect waves previously reflected at the free surface, we obtain

ff~o k S 4 / \ S 2 J

\sU (62)

and in this case the free surface effects appear principally through the matrix N~ 1. This equation represents the extension of the reflectivity method to an arbitrary source distribution. The elements of the matrix F in (62) have the form

# t

• 2 , i ~2 ~ '~ F11 = l[2k v# (I + r p p ) + V # F(1-rs ,)+2kv# r , p - k F - - rps]/R(k),

t

FI 2 i[kF(l+rpp)_2kv ,v#,(l_rss)+Fv # e#' 2 , = ' - - rsp-}- 2k v~ - - rps]/R(k),

# #

F21 = i [2kv , ' v i (1 - rpp) -kr (1 +rs,)+ 2k2va' ~ % + Fv~' ~ , rp,]/R(k) ,

# #

F22 = i[Fv~:(1 -rpp)+ 2k2v~'(1 + r ~ ) + k F ~ r~p-2kv~ '2 e.~ rp~]/R(k) , (63)

and we see that it is easy to separate out the direct source contribution and that reflected from the underlying structure.

In some cases, for example when there is a thin low-velocity layer at the surface, near- surface reverberations will be important. It is then necessary to take more terms in the series expansion of (58) so that

--(Mu~?D+MD)(0+ ~ ( ~ U q o ) q ) N ~ l S a + sl_ (64) mo q=l ks4~ \ s 2 /

and the number of terms to be considered can be found by using criteria such as (47)•

APPENDIX

Phase-Related Reflection and Transmission Coefficients

For a medium with P- and S-wave speeds :~, fl and density p, for a plane wave with

REFLECTIONS, RAYS, AND REVERBERATIONS 1695

horizontal wave number k we introduce the quantities

v, = [(~2/~2)-k211/2, va = [(~2/f12)-k211/2, In v,, v a __> 0,

F = (~2 / f l z ) -2k2 , # = pfl2,

e, = [2/p~o3v~] 1/2, ea = [2/po~avtj] 1/2, (65)

the last two are related to the normalizat ion employed. Then the matr ix of p ropaga t ion eigenvectors normalized to unit energy flux in the z direction

ike~ ivaea T(k) = -iv~e~ ikea

21~kv,e, #Fea -~ tFe , 21~kvaea

and the diagonal matrix A(k) is

ike ~ -- ivaea i iv,e, ikea ] -2l~kv~e~ l~Fet~ I -I~Fe~ -21~kv~ep j

(66)

A(k) = diag [ - i v y , - iva, ivy, ivp]. (67)

The phase-related eigenvector matrix at the level z, U(k, z) is then given by

U(k, z) = T(k) exp [A(k)z]. (68)

The phase-related reflection coefficients derived f rom (25) are related to the normal coefficients (denoted by a tilde) for the interface z = H between media "1" and "2" by

and

and

(a) for downward propaga t ion

o ~o exp (2iv~lt¢) rpp = rpp

r ° = ~° o exp [i(vp 1 + v,l)tc]

o = (?o) , exp [i(v/~ 1 + v~l)/¢] rps

o ~o exp (2iva't Q rss

D D too = ?~o exp [i(v i _ v,2)x]

t ° = Top exp [i(v , ' -va2)x]

to~ = ?Op, exp [i(v p' - v ,2)x]

t ° = T ° exp [i(vp I-vp2)~c]

(b) for upward p ropaga t ion

v ~v exp (2iv,2x) rpo = rpp

r ~ = ?v o exp [i(vp 2 + v~2)K]

v = (~vo), exp [ i (va2+v,2)x] rps

v ~v exp (2iva2tc) rss = rss

v = ( t o o ) , tpo

U = (tOps), lsp

to% = ( t O ) *

= (t,°,) *

(69)

(70)

(71)

(72)

1696 B.L.N. KENNETT

In these expressions we have made use of the symmetry relations between the reflection and t ransmission coefficients; an asterisk indicates complex conjugat ion.

ACKNOWLEDGMENT

I would like to thank Dr V. ~erven~, for many useful discussions. This work was performed during the tenure of a Research Fellowship from Emmanuel College, Cambridge.

REFERENCES

Altman, C. and H. Cory (1969) The generalized thin film optical method in electromagnetic wave propa- gation, Radio Sci. 4, 459-469.

Chapman, C. H. (1974). A generalized ray theory for inhomogeneous media, Geophys. J. 36, 673-704. Cisternas, A., O. Betancourt, and A. Leiva (1973). Body waves in a "real Earth", Part I, Bull. Seism. Soc.

Am. 63, 145-156. Dunkin, J. W. (1965). Computation of modal solutions in layered elastic media at high frequencies, Bull.

Seism. Soc. Am. 55, 335-358. Fuchs, K. and G. Miiller (1971). Computation of synthetic seismograms by the reflectivity method and

comparison with observations, Geophys. J. 23, 417-433. Gilbert, F. and G. E. Backus (1966). Propagator matrices in elastic wave and vibration theory, Geophysics

31,326-332. Haskell, N. A. (1953). The dispersion of surface waves on multilayered media, Bull. Seism. Soc. Am. 43,

17-34. Helmberger, D. V. (1968). The crust-mantle transition in the Bering Sea, Bull. Seism. Soc. Am. 58, 179-214. Hron, F., E. R. Kanasewich and T. Alpaslan (1974). Partial ray expansion required to suitably approxi-

mate the exact wave solution, Geophys. J. 36, 607-626. Hudson, J. A. (1969). A quantitative evaluation of seismic signals at teleseismic distances: I. Radiation

from point sources, Geophys. J. 18,233-249. Kanasewich, E. R., T. Alpaslan and F. Hron (1973). The importance of S-wave precursers in shear-wave

studies, Bull. Seism. Soc. A m. 63, 2167-2176. Kennett, B. L. N. (1972). Seismic waves in laterally inhomogeneous media, Geophys.J. 27, 301-325. Miiller, G. (1970). Exact ray theory and its application to the refection of elastic waves from vertically

inhomogeneous media, Geophys. J. 21,261-273. Richards, P. G. (1971). Elastic-wave solutions in stratified media, Geophysics 36, 798-809. Scholte, J. G. J. (1956). On seismic waves in a spherical Earth, Konink. Ned. Metereol. Inst. Med. 65. Treitel, S. and E. A. Robinson (1966). Seismic-wave propagation in layered media in terms of com-

munication theory, Geophysics 31, 17-32. Wuenschel, P. C. (1960). Seismogram synthesis including multiples and transmission coefficients,

Geophysics 25, 106-129.

DEPARTMENT OF APPLIED MATHEMATICS AND THEORETICAL PHYSICS UNIVERSITY OF CAMBRIDGE SILVER STREET, CAMBRIDGE CB3 9EW, ENGLAND

Manuscript received May 15, 1974.