Reflection of waves generated by a point source over a randomly layered medium

WAVE MGI’ION 13 (1991) 53-87

ELSEVIER

53

REFLECllON OF WAVES GENERATED BY A POINT SOURCE OVER A RANDOMLY LAYERED MEDIUM

Werner KOHLER Department of Mathematics, Virginia Polytechnic, Institute and State University, Blacksburg, VA 24061, USA

George PAPANICOLAOU Courant Institute, New York University, 251 Mercer Street, New York, NY 10012, USA

Benjamin WHITE Exxon Research and Engineering Company, Route 22 East, Annandale, NJ 08801, USA

Received 1 November 1988, Revised 2 August 1989

1. Introduction

We considdr a randomly layered half space adjoined to a homogeneous half space at the plane interface z = 0. An acoustic source in the homogeneous medium generates a time-limited pulse which is then multiply reflected and backscattered from the random medium. We compute here the time dependent statistics of the signals recorded at receivers located on the interface z = 0.

We do nor assume that the random fluctuations in material parameters are of small magnitude, and we do not assume that the medium is statistically stationary in space. Our fundamental assumption is that of a separation of spatial scales. The random fluctuations occur on a small microscale, while their statistics, e.g., mean properties, are allowed to vary on a much larger macroscale. The pulse wavelength is assumed intermediate between the two scales, so that it can be an effective probe of the macroscopic variation only.

Using asymptotic methods for stochastic differential equations, we compute power spectra and cross power spectra for the receivers. Since the backscattered signals are not stationary random functions of time, these power spectra can only be defined for segments, or time windows, during which the signals are approximately statistically stationary. We show how these statistics change with time, i.e. the location of the time window.

In this paper we present an extension of previous, one-dimensional results [l-4] to the more realistic three dimensional, but stratified, case.

2. Formulation and scaling

2.1. The equations and their scaling

We consider acoustic wave propagation in three dimensions. The linearized equations for the velocity u( t, x) and pressure p( t, x) are

P(Z)% +vP = F(t, x1 (2.1)

0165-8641/91/%03.50 @ 1991- Elsevier Science Publishers B.V. (North-Holland)

54 W. Kohler et al. / Reflection by randomly layered medium

1 -p,+v. u=o. K(r)

(2.2)

Here p is the density and K the bulk modulus of the medium which is assumed to be layered so that p

and K are function of z only. We assume that the force term on the right of (2.1) is due to a point source

and has the form

UC, x) = S(xP(yP(z -zslf(t)e. (2.3)

The position vector x has coordinates x, y, z. The directivity vector e of the source has components e,,

e,, e3 and is a unit vector, e. e = 1. The source location is at (0, 0, zS) and f( t) is the pulse-shape function,

which is smooth and of compact support in (0, CO). We will assume that the random half space occupies

the region z < 0 so that p and K are constant for z > 0. We will also assume that the source is located in

the homogeneous medium above the random half space so that z, 20. Problem (2.1)-(2.3) is completed

by setting the velocity II and the pressure p equal to zero at t = 0.

The acoustic equations (2.1) describe propagation in a medium without dissipation. The simplest way

to introduce dissipation effects is by adding the term VU in the momentum equation yielding

p(z)u,+au+Vp=F(t,x) (2.la)

1 -p,+v * u=o. K(z)

(2.2a)

The coefficient u may depend on x and it may also depend on time in which case the term uu is interpreted

as the convolution in time of u and u.

The next step is to write the equations in dimensionless form and to scale them in an appropriate

manner. We will scale densities relative to p,,, the constant density in z> 0, and speeds relative to

co = (Ko/Po)“~ where K. is the constant bulk modulus for z > 0. We scale length in units of 1 which are

macroscopic units. If we let 0 < E CC 1 be the ratio of the pulse width (pulse duration times co) over the

macroscopic length scale I then the scaled and dimensionless form of our equation is as follows:

,0(z)(l+rl(r,s)) u,+Vp=~‘/iT(f) S(xP(y)S(z-z,b (2.4)

(2.5)

with u = 0 and p = 0 at t = 0. We now let pa(z), K,(z), and co(z) denote the (scaled, dimensionless and

macroscopically varying) mean density, bulk modulus and sound speed, respectively. In (2.4)-(2.5) n(z, 4’)

and V(Z, t) are stationary for each z, zero mean stochastic processes bounded by a constant less than one.

We will introduce more hypotheses about them later. Note that n and v are identically zero in the

homogeneous half space z > 0 where in addition, p. = 1 and K,= 1, so co= 1. Note also that e2 is the

scale of variation of the random inhomogeneities. This is the scaling in which suitable averaging occurs

over the detailed structure of the inhomogeneities since the pulse width, or order E, is broad compared

to their scale of variation. On the other hand the pulse is narrow compared to macroscopic variations so

they can be resolved. Note also the factor &3’2 on the right side of (2.4). This factor makes the total energy

released by the source independent of E. When dissipation is present as in (2.la), (2.2b) then eq. (2.4) is

modified with the addition of the term cr(z, Z/E’)U on the left side.

W. Kohler et al. / Rejection by randomly layered medium 55

The asymptotic analysis of (2.4)-(2.5) goes in three general steps: First we Fourier analyze the fields with respect to time and the transverse space variables x and y. Then we do the asymptotic analysis of the resulting stochastic differential equations. Finally we reconstruct the signal by Fourier synthesis. We write the pressure and velocity in the form

Then p*’ and u^” = iv P”, the z-component of the transformed velocity field, satisfy the ordinary differential equations

p*~_!E$= &3’26(z-zS)~(w)e3 (2.8) E

(2.9)

Here K = (Kf + K;)“2 and p and K are the normalized random density and bulk modulus before the factorization into the mean and fluctuating part shown in (2.4)-(2.5). When dissipation is present eqs. (2.8), (2.9) remain the same but p becomes complex and we have to substitute it by

From the form of (2.9) we see that we must also factor p-l into its mean and fluctuation. We write

(2.10)

with x(z, 6) a stationary for each z, zero mean bounded process like 7) and V. In the homogeneous half space z > 0 we have r) = Y = x = 0 and p. = p, = 1. We also use the notation

1 1 1 K2 -=-_--

K,(z) &(z) P*(Z) CO2

and

w=~ ( :)-y$x(z,;). y 57 0

Then we can write (2.8)-(2.9) in the form

;:-;+(*+p)@ e3’2S(z-z,)1(1+x)~(w)~~e. 1 UPI

(2.11)

(2.12)

(2.13)

(2.14)

In the asymptotic analysis of (2.13)-(2.14) we will also use the notation

5(z) = (Po(ZvG(Z)Y (2.15)

56

and

W. Kohler et al. / Rejection by randomly layered medium

(2.16)

Note that 5 is the dimensionless impedance, identically equal to one for z > 0, and T(Z) is the dimensionless travel time into the inhomogeneous half space, and is positive for z G 0. We will not use T(Z) for z > 0. Note also that since K, depends on K/W, so do /3, 5 and r but this dependence is not shown explicitly. In this paper we assume that we stay above any turning point zT where K,(z) vanishes so that K,(z) > 0 for z> zT in (2.16). Note from (2.11) that the turning points depend on K and o.

2.2. Solution in the homogeneous half space z > 0

Recall that we have assumed that the source is located in the homogeneous half space so that z, > 0. Now we shall solve (2.13)-(2.14) when z is positive. The system simplifies then to the form

p^;_iw;=o (2.17) &

2 ;$.! l_K p^“=O ‘( > & co2

(2.18)

for 0 < z < 00, z # z, along with the jump conditions

p^“(z:)-p^“(z;) = E3’2_?(m)e3 (2.19)

C’(zf) - ;“(z;) = E3/2f(w) y . (2.20)

Throughout we will assume that 1 -(K/w)~> 0 so that only propagating waves are considered. The geometrical optics scaling that we have adopted makes the evanescent waves exponentially small as E tends to zero, so they need not be considered.

In the region between the interface and the source 0 < z < z, we have

p^” = p, exp -;(1_K2w -2)~~2(z-zs))+p2exp(~~1-K2u2)1”(Z-Zs)) (2.21)

where p, and p2 are constants and from (2.17)

i.5 u^‘= -_p^; w

or

u^” = (1 - K2W-‘)“’ [ (

-p, exp -51-K20 -2)“i(Z-Z~))+p2eXp(~~1-K2~-2~1’2~Z-Z~~)].

(2.22)

The first term on the right of (2.21) is the pressure waveincident on the half space and the second is the reflected pressure wave.

W. Kohler er al. / Reflection by randomly layered medium 57

In the region above the source z > z, we have

p^’ = p. exp (

+& _2)1/2(2 - .))

(2.24)

Applying the jump conditions (2.19) and (2.20) we find that

pO=p2+SE3’2j.(0)[e3+(1-K20-2)-“20-1K’ e] (2.25)

p1 =t~“‘$(m)[-e~+(l -K~w-~)-“~w-‘K~ e]. (2.26)

Note that the amplitude of the incident pressure wave p, depends only on the source strength while the

amplitude of the pressure wave above the source depends on both the source and the reflected pressure wave.

2.3. Interface conditions at z = 0

Both p^” and u^” must be continuous at z = 0. In the region zT < z ~0, inside the random half space but above the turning point, we use (2.15) and (2.16) and let

pan = ~1/2[e-iuw/eAe _eiwr/~~~-j (2.27)

2s = ~-l/2[e-iun/EAE +ei~7/e~~]. (2.28)

By analogy with the homogeneous case we think of A” as the amplitude of an upgoing wave (T increases

with decreasing z) and B” as the amplitude of a downgoing wave. The differential equations (2.13)-(2.14)

give the following ones for A” and B”.

(2.29)

dB’ -= dz

a’+(P’p.?+5BK;‘)B’]+d ~e-‘2”‘l’A’. (2.30)

We apply now the continuity conditions for p^’ and u^” at z = 0. This implies that

&(O-)“‘(AE(O-) - B”(O-)) =p, exp (’

T (1 - K’W -2)1’2zr) +p2 exp (-$ (1 - K’w-2)“2z.) (2.31)

e(O-)-“‘(A(0-) + B(O-)) = (1 - K2W e2)‘i2[ -p, exp(: (1 - KZO~)~/~Z,)

+p2exp -@(~-K~c~-‘)~/~z~ ( . E )I Let r be the pressure reflection coefficient defined by

r=zexp (

F (1 _ K20-2)lV2z~) .

Similarly, let the reflection coefficient for z = O- be defined by

A”(O-) _= e-iG B”(O-) ’

(2.32)

(2.33)

(2.34)


The modulus of this reflection coefficient is one because the random half space is totally reflecting. This

is true in general but especially obvious if there is a turning point at some zT > --CO. Now from (2.31)-(2.34)

we see that

r-1 ~=1(0-)(1-K*0 -*)I/* (.CY) .

Let

5(0_)(1- K*w-*)“* - 1

5=5(0-)(1-K*O-*)1/*+1.

Then

r=-“. -i*+!J e-‘+J- 1

Note that for (1 -K~w-*)>O,I~~< 1 and r is well defined. Note also that from (2.15) and (2.11)

[

do-) I I/*

[(o-) = K,l(O-)_p;l(O-)K2W-2 f

(2.35)

(2.36)

(2.37)

(2.38)

In the special case where p,,(O-) = K,,(O-) = p,(O-) = 1 we see that 5 = 0 and so r = e-“. We refer to this

special case as the matched case.

2.4. Integral representation for the rejected pressure

The reflected pressure is given by the inverse Fourier transform (2.6).

x p2 exp -f (I- K20J-2)1’2Zs) dw dK, dK2.

But by (2.33)

(1 - K*c~-~)"~Z,

(2.39)

(2.40)

and p1 is given by (2.26). This is then the integral representation of the reflected pressure field. The main

purpose of this paper is to compute the local pressure correlation function

A(x,y, t; X,7, r) =lim E{p,(x+eX,y+ejj,O, t+ef)p,(x,y,O, t)}. E-r0

(2.41)

Note that the spatial and temporal offsets in (2.41) are O(E) on the macroscopic scale and thus are

O(1) quantities on the intermediate scale. We will use the representation (2.39) to obtain an integral

representation for A. This is a straightforward calculation that proceeds as follows. We multiply two

integrals like (2.39) with arguments as in (2.41). It is convenient to conjugate one of the integrals, which

W. Kohler et al. / Reflection by randomly layered medium 59

is permissible since pr is real. The variables of integration are denoted by w(l), K(1*), K$” and w(*), K(1*),

K$*). We now make a change of variables in this six dimensional integral by letting

eh @(‘) = w -- “(2) = eh

2’ W+-

2

(2) _ 4 El El2

2, Kl - K, +- (2) _

- 2, (‘lEK -1

K2 2 2, K2 K2+-. 2

(2.42)

Let rCi), i = 1, 2 be th e reflection coefficient (2.37) when w, K, and K* are replaced by the expressions in

(2.42). In Section 3 we shall show that the limit

E{T”‘r’*‘*} exp(i(ht - I,x - 12y)) dh dl, dl, (2.43)

exists and we will show how to compute A. In terms of A we can express A, after a few simple calculations,

in the form

dw dK, dK2exp(i(K,%+K2y-~f3)Ij(W)12

x[e3-(1-K2W~2)~“20-1K’ e]’

where K = (K;+ K:)“*.

(2.44)

Note that through (2.26) and the change of variables (2.42), the C3 factor that appears in (2.39) has

been eliminated. This shows why the factor .z3’* was appropriate for scaling the source strength in (2.4).

3. Two frequency limit theorem and derivation of the transport equations

3.1. Equations for the reflection coeficient

If A” and B” are any solutions of (2.29)-(2.30) with B” # 0 let

r” = A”/ B”.

Then r’ satisfies the Riccati equation

(3.1)

+--[e ’ d5 i*un/c _e-i2mr/E 25 dz

(n*l. (3.2)

We are interested in the reflection coefficient at z = 0 and in (2.34) we noted that r” has modulus one

because random half spaces are totally reflecting. The reflected signal, (2.39) with 0 G t 6 T < co, depends

on the properties of the half space only up to a certain depth, -L 4 z c 0, with L proportional to T. This

is a consequence of the hyperbolicity of (2.4)-(2.5). In the frequency domain however, r” depends for

each o on the properties of the whole half space. It is convenient therefore to assume that outside a slab


-L s z s 0 the reflection coefficient is known and then to determine it at z = 0 using the equation for r”. The choice of r” in z G -L does not affect signals in the time interval OS t s T, if L is large enough. It is particularly convenient then to let

r”=e-‘*‘, -J_.<z<O

with $,‘(--L, w) =O. Then tj’(z, W) satisfies

(3.3)

~=+[rn+~cos(*E+~)]-~!$sin($.+~), -LSz<O, +‘(-L,o)=O. (3.4)

where

m = W’Pol) + UG’P), n = &-‘po7J - g-K;‘p). (3.5)

From (2.43) and (2.37) we see that we must compute the limit as E tends to zero of expressions like

.(r(o-,,-~)P*(o-,,+~)} = E{T(exp(-iJI’(O-, w - &/2)))r(exp(+i+“(O-, 0 +&h/2)))} (3.6)

where r is defined by (2.37) and w and h are held fixed.

3.2. Two frequency asymptotics

To calculate the asymptotic limit of (3.6) as E tends to zero we use the asymptotic analysis of stochastic equations ([l, 21 and references therein) of the form (3.4) and o and K having the two sets of values defined in (2.42). Let I,@‘(Z) be the process defined by (3.4) and w and K equal to o(j) and K(‘), j = 1,2.

From the asymptotic theory we know that as E tends to zero ( I,#“( z), tiC2’( z)) tends to a diffusion Markov process. The backward Kolmogorov equation for this diffusion can be obtained easily, as was done in [2], so we will omit the details here. Let T and [ be defined by (2.15) and (2.16), respectively and let

A = K2W-‘(K-2K’ I-o-‘h) z” 5~;’ da (3.7)

with K = (K, , K~), I = (I,, IJ. Define, as in [2]

I

03

~lnln = E{m(z, z’)m(z, z’+r)} dr 0

m

%I = I

E{n(z, z’)n(z, z’+r)} dr. 0

Then the generator of the diffusion process (ICI”‘, $I”‘) has the form,

L, = 4w2( Q,, + f a,,) a2 a2

+ a$“’ &p’ &p &p

>

+ 8w2[ CY,, +&,,, cos(1@“-#~‘+2A -2h7)] a2

w”’ WC*).

(3.8)

(3.9)

(3.10)

W Kohler et al. / Reflection by randomly layered medium 61

We have indicated dependence of the generator on the independent variable z by the subscript z. Travel time T in (3.10) is given by (2.16) and (2.11), where o and K now correspond to the center frequency and wavenumber defined by (2.42). From the asymptotic theory we know that

lim E{g(e-ilL”‘(O-))g(e+i~(2)(O-) 1 ‘p’(-L) = ‘p, ‘p(-L) = ‘p} = V(-L, p, ‘p) (3.11) E-+0

where V(z, $(I), $‘*‘) satisfies the backward Kolmogorov equation

av d,+L’v=o, -Lsz<O, V(O-, tic’), +(*)) = g(e-ig”‘)g(e+i,‘2’). (3.12)

For the processes defined by (3.4), where +,“‘(-L) = +,‘*‘(-L) = 0, the quantity of interest (3.6) is simply V(-L, 0,O). More generally, if the starting values at z = -L in (3.4) are themselves random variables, (3.6) is obtained by evaluating E{ V( - L, JI”‘( - L), $‘*‘( -L))}.

3.3. Transformations of the generator

By changing variables in (3.12) and applying the Fourier integral transform as on the right side of (2.43) we can get a convenient expression for calculating A. In fact we want to show that

A(r, t, K, co)= lim F z wpq(z, t, r; o, K) z-r-cc p=o q=o

(3.13)

where r = (x, y), K = (K,, K*) and Ww satisfies the hyperbolic system,

-2w*~,“[-(p-l)(q-l)w~-‘~-‘+(p*+q~)w~~-(p+l)(q+1)w~+‘~+~]=O

wpqlz,o = T,T,S( t)G(x)S(y)

(3.14)

(3.15)

r, = { Idi’-ip+‘,

p=o

p= 1,2,. . . . (3.16)

In (3.16) I is the impedance defined by (2.36). The limit in (3.13) is tantamount to letting -L in (3.4) recede to --oo. In this limit, the choice of r’ in z s -L becomes irrelevant for all time, i.e. 0 s t < 00. If a turning point exists at zT, --CO < zT < 0, the limit in (3.13) need only be taken down to zr.

First we change variables in (3.12) by letting

6 = +(*) - $(‘I - 24 + 2hr , l# J(‘p+‘p’) (3.17)

Then (3.12) becomes

a*V A a*V ~+z(h~~-Af)~+4o2a,.~+~2~~~(~+cosj)~~+4o2a..(1-cos~)~=o (3.18)

(3.19)

where the primes denote differentiation with respect to z. Next we expand V in a Fourier series

V= ? f Pexp(-ip$“‘+iq#‘*))= f f VPqexp(-i(p_q)G)exp(i(p+q)$/2). (3.20) p=o q=o p=o q=o

62 W. Kohler et al. / ReJection by randomly layered medium

Substituting this into (3.18) and expanding (3.19) gives the following equations for VP4

a VP4 -+i(p+q)(hr’-A’)VPq-4~2~,,(p-~)2VPq

a2

+20*a,,{(p~+q*)V”-(p-l)(q-1)V~-‘~-’-(p+l)(q+1)V~+‘~+‘}=O (3.21)

vpqlZ=O = rprq (3.22)

where r,, is defined by (3.16). To go from (3.21), (3.22) to (3.14), (3.15) we have to invert the Fourier transform as in (2.43)

1 asmm

wpq =(2?r)3 _m -_m -I I I VP9 exp(i( ht - l,x - 12y)) dh dl’ dl,. (3.23)

-a

From (2.15), (2.16) and (3.7) we see that

h+A’=-htK;‘+o-k I&.

Equations (3.14), (3.15) follow from (3.21)-(3.24).

(3.24)

3.4. The matched case

When the interface impedance I defined by (2.36) is zero, (e.g. if p,,(O-‘) = K,(O-) = p,(O-) = l), the determination of /i in (2.43) via (3.13)-(3.15) simplifies considerably. In fact we need only consider the diagonal terms WN = WNN and then

13WN -+~p;'W-'K . V,WN

at 1 +202cu,,[(N-1)2WN-‘-2N2WN+(N+1)2WN+’]=0 (3.25)

wfz=,= fjNl~(t)S(x)S(y) (3.26)

with

li(r,t,K,W)= lim W”(Z,t,r;w,K). (3.27) z---m

Except for the r gradient term in (3.25), which arises because we are considering a point source and not a plane wave incident on the half space, eqs. (3.25)-(3.27) are exactly the same as the ones in [2]. When K = 0, which is the normal incidence case, they have been analyzed extensively [5,6].

4. Statistically homogeneous medium

4.1. The two-point, two frequency correlation functions

In this section we assume that the medium is statistically homogeneous, i.e., that p(z) and K(z) are stationary random functions of z. Then po, K,,, p, etc. are constants independent of z. In this case the two-point, two frequency correlation function can be written in quadrature. Following [2] we reformulate the limiting Markov process equations as K.olmogorov forward equations (KFE) instead of Kolmogorov backward equations (KBE) as was done in Section 3. The two-point, two frequency correlation function

W Kohler et al. / Reflection by randomly layered medium 63

is then written as an integral with respect to the invariant (steady-state) probabilities, which are obtained from the z-independent KFE. To see this, we first examine the KBE formulation of Section 3.

Recall (3.18), (3.19). We first solve the following KBE for V(z, 4, @)

where primes denote differentiation with respect to z. The final condition for (4.1) is

V(O_, lj, 9) = [

-exp(-i(9-$))+5 -exp(i(G++t+C))+J

exp(-i(!P-_t$))l-1 I[ exp(i(q++$))l-1 1 (4.2)

E{r(‘)rC2)*} is then

E{T”‘P2’*} = lim V(z, 6, +). ,?---co

(4.3)

In the homogeneous case, the coefficients in (4.1) are independent of z. From (3.24) we have that

hi’-A’=T[-hKo’+p;‘w-‘K. I]. (4.4)

The probabilistic interpretation of these equations is the following: Let 4(z), @(z) be Markovian with KBE (4.1). Then E{T’1’r’2’*} is the limiting conditional expectation

E{r”‘r’2”}= lim E z---o0

(4*5)

As z + --CO the dependence of the conditional expectation on the initial conditions ?8, I$ vanishes. From a physical point of view, the reflection coefficient at the interface becomes independent of the termination at z as the termination point recedes to infinity. The relevant probability density P for (@(O-), $(O-)) is obtained from the KFE, the adjoint of (4.1), by setting dP/dz = 0. The computations are simplified if we first make the change of variable

u = cot( G/2) (4.6)

so that

is_ a+i e--

( > a-i ’

In terms of u, $, (4.1) becomes

; V-(~+-P’)(~+U~); V+~W~U,,-$ V

a2 . + 2w2cz,,a~( 1+ &--I - v+ 2w2a,,

a?P2 (4.8)

64 W. Kohler et al. / Rejlection by randomly layered medium

Therefore the probability density P(z, a, @) satisfies the KFE

~=(h71-A’)$[(l+u2)P]+40z ( CT2 a2P ~lnnl +t%” -

> -+2Ja,,-

1+a2 aY2 a [l+o2,;P).

aa ( (4.9)

The invariant density is found by setting dP/az = 0 in (4.9). A @ independent solution is then easily

found [2]. It is

where

(h+-A’)

a = - 2W2ff,* .

(4.10)

(4.11)

A normalized joint probability of (a, 9) is then P(a)/21r. That is, in steady-state, a; ?@ are statistically

independent; 9 has a uniform [0,27~) distribution and u has density p

Using (4.10) in (4.5) we may now write

2n

~{p’p’*} = I m

duF(ujk I ( d+a -exp(-i(@-f4b)+ --oo 0 Y

-exp(i( + ++$)) + 5

exp(-i(!P-f+))l-1 exp(i(?P++$))j-1 >

where ei’ 2 = ei’.

is given by (4.7). The inner integral in (4.12) can be written as a contour integral by

It becomes

(4.12)

letting

(4.13)

This last equality comes from summing the residues at the simple poles 2 = 0, Z = l e”“, which are inside

the unit circle. Note that [&‘I < 1 from (2.36). Substituting (4.13) for the inner integral in (4.12) then yields

after substituting from (4.7)

E{T”‘T’2)*} = 1 + 2i c [u:i(&, .

(4.14)

Putting (4.10) into (4.14) and interchanging the orders of integration yields

[u+s sgn(a)+i][a+s sgn(a)-i]

(4.15)

The inner integral in (4.15) may now be evaluated by residues. It has simple poles at u = --s sgn(a)*i,

i( 1 + l’)/( 1 - C2). Closing the contour in the lower half plane picks up a contribution from u = --s sgn a -i.

Note again that [[I< 1. We then obtain the expression

l2 E{J+r’2’*} = Ia] lrn ela/s

s sgn(a)+2i - ( > [ 1 l-l2 ds

0 1 . s sgn(a)+2i -

( > 1 -c2

(4.16)

W. Kohler et al. / Reflection by randomly layered medium

4.2. Fourier transform of the two-point, two frequency correlarion function

We next compute (cf. (2.43))

65

/i(x,y,t;~,,~~,~)=lim(2*)-~ S-r0

E{T”‘T’2’*} exp(i(hr - l,x - 12y)) dh dl, dl,.

(4.17)

From (4.7) we note that the only dependence of the asymptotic behavior of E{T”‘T’2’*} on (h, I,, l,) is through the parameter a which, upon combining (4.11), (4.4) becomes

a z-f& (K,’ _P;1K2W-2)-1/2(K;1h -p;‘g-‘K. I)_ (4.18) nn

where K,, po, p, are all evaluated at O-. Consequently we make the change of variables in (4.17)

41 = 1,) 92= 12, q3 = a. (4.19)

Then (4.17) becomes

$X, y, t; K,, K2, W) = (2T)-‘2W2~,,,co(1 - &jl;‘K2W-2)“2

x 8(&jl;‘W-‘fK, -X)8(&&0-‘fK2--y) ,5{r(1)r(2)*} eiY4 dq,

(4.20)

where

‘)’ = 2W2a,,Co( 1 - &~;‘K2&J-2)“2f. (4.21)

Substituting (4.16) with a = q3 gives

where the last equality follows from closing the contour in the upper half plane, obtaining a residue from the double pole at s = iy. Putting (4.22) into (4.20) then yields

/i(X, y, t; K1, K2, 0) = (I- ~O~;‘K2~-2)“2~(~O~;‘o-LtK1 -X)

x 8(&p;‘O-‘tKz--y) 4w2a,,co

[y+2(1-52)-‘]2 (4.23)

where Ko, pIco are evaluated at z = O-, i.e. in the random medium, as they are in (4.21), the formula for y.

66 W. Kohler et al. / Rejection by randomly layered medium

4.3. The two-point, two-time correlation function

For simplicity we consider here only the case z, = 0, the configuration where the source is at the interface. From (2.41) and (2.44) the two-point, two-time correlation function is

E{p,(x+&~,y+&~,O,t+&T)}=n(x,y,t;~,7,,t)

exp(-ioT) exp(i( Krf + KJ)

x&o)12 * [e3-(1-K2~-2)-“2w-1K. e]’

x&,Y, t; KI, K2, o) do dK, dK2. (4.24)

We will substitute (4.23) into (4.24). First note that the point masses of the b-functions in (4.23) are at

PlWX Pl”Y

K’=KOfs K*=- _* * (4.25) Kot

Therefore, in (4.24) we may replace

K2 r*d

>=a where r* = x2 + y*.

We need also the explicit expression for a,,,,. Substituting (4.26) into (3.5) we have that

(4.26)

(4.27)

Therefore

I

00

aPI” = E{n(O-, O)n(O-, a)} da 0

is the following function of r*/ t2

a,, (~)=~f+-GG-l$ ~om~{rl(O-,(L)r)o-,~)}d~ lP0 m

-4K, 0 I E{dO-, 0)(40-, u) -PJG’ $xW, a))) da

1Po m -4K, 0 I E{dO-, ~)(dO-, 0) -P,&’ $x(0-, O))l du

++poK,' I-p,K,+ ‘j-’

I 00

X E{( v(O-, 0) -p, K,’ ; x(0-, O))( v(O-, a) - p, K,' 5 x(0-, a))} do. (4.28) 0

Similarly, from (4.21) we have that

t. (4.29)


From (2.36), (4.26) 6 is the following function of r*/ t*

( ( r2 l/2

> (

2 l/2

POCO

l(r’/t*) = 1 - d&i2 7 - l-p&+

> >

( (

2 I/2

> (

r2 112

POCO 1 - P:G* + + l-p,zQ ) )

67

(4.30)

Now, substitution of (4.29), (4.30) into (4.23) yields the following expression for (4.24)

E{P,(X + e% Y + e?, 0, t + EOPAX, y, 0, t)I

-~~[b,-(*-p:K,*~)-1’2p*~~~(~~,+Se,)li t On)

o,, ($) I_exp( -iw ( I-p2K,’ [fi+fJ])) f(w)[*

(4.31)

4.4. An illustrative special case

As an illustrative special case, we consider a vertical source alignment

e, = e2 = 0, e3= 1, so that L?=Z().

We also assume that p is nonrandom and c,(O-) = 1 so that

po(O_) = K,‘(O_) = p;‘(o-).

As in section (4.3), we consider z, = 0. Substituting (4.33) into. (2.15), we have

t(o-) = (1- K*w-*)-“*.

Therefore, from (2.36) and (4.34)

~=(~(o-)(1-K2W-2)1’2-1)/(~(o-)(1-K20-2)1’2+1)=o.

Now substitution of (4.32), (4.33) and (4.35) into (4.31) yields

E{Pr(x+E%Y++xO, t+4Pr(X,Y,O, t)I

(4.32)

(4.33)

(4.34)

(4.35)

(4.36) w*

Since p is nonrandom, 77 = ,y = 0. This fact, together with (4.33) reduces (4.28) to

cz~,,($)=;(l-$)-‘o, o=JOmE[.(O-,O)v(O-,o)]do (4.37)

68 W. Kohler et al. / Rejection by randomly layered medium

and this has been used in (4.36). Note from (4.37) that (Y,, = 0 if Y = 0, i.e., (Y,,, vanishes in the absence of noise. If r = r, (Y,, diverges. However the analysis is not valid if r = t. Other properties are also easily seen from (4.36). For instance, the kernel does not vanish as [w( + co, but approaches [4(2rr)3 f4a,,( r*/ t*)( l- r2/ t*)“*]-I. The kernel does vanish, however, as t + 00 with r*/ t* fixed. As t + O+ with r*/ t* constant and f- (xl t)Z - (y/ t)J constant, the correlation function diverges as t-*.

The Fourier inversion (4.36) may also be interpreted as a convolution. For brevity, let

(4.38)

Then (4.36) may be written as

e-iwr~(co)12 O4 (l+ .2w*)*

dw. (4.39)

The kernel has the inverse Fourier transform

4 -iwr w

e dw= (2a26J2+ 1)

(1+aZw2)* (1 + u*W*)* 1 e -ior do

( )I ,_M U

(4.40)

The second term in brackets in (4.40) is obtained for r> 0 by closing the contour in the lower half plane, thereby obtaining residues from the double pole at o = -ia-‘. For r< 0 note that this form is an even

function of r. Let F(t) be the inverse transform of VI’

F(t) =& 1-1 v(w)[* emiw’dw. (4.41)

Then from (4.38), (4.39)

U2

- 4(2p)*t3u4 F(T) - F(T)* {$(3_;)}]

where * denotes convolution. As an explicitly calculable case, consider

f(t) = exp(-t2/2T2)

(J;;7y2 *

Then

v(w)j2=2J;;Texp(-f.02T2)

and so

F(t) = exp(-t2/4T2).

(4.42)

(4.43)

(4.44)

(4.45)

W. Kohler et al. / Rejection by randomly layered medium 69

Substitution of (4.45) into (4.42) yields

E{p,(x+ e% Y + G, 0, r + &0P,(X, Y, 0, t)l

” [exp( --$)-J(r,a)-J(-T,fz)] - 4(21r)‘t~a~

(4.46)

where

J(T, a) = expy~)) a [(,_;+L$) &erfc(f-_L) -yexp(_(_&_gi), . (4.47)

Now as t + CO with r/t and T fixed, a2 - t + 00. Therefore from (4.47)

Hence

E{p,(x+e%,y+ej,O, t+d)p,(x,y,O, t)}-const. * F(T) - C4 as t+o~, r/t fixed.

(4.48)

(4.49)

4.5. The coherent field

Up to this point we have focused attention solely upon the correlation function of the reflected pressure at the interface z = 0; the mean or coherent reflected pressure E{p,(x, y, 0, t)} has been ignored. In the general (mismatched) case, however, this coherent component will be nonzero and must be taken into account.

In Section 6, we study a more general configuration, that of a random half space containing a planar, deterministic discontinuity at z = -L<O. By letting L+O, we can recover the present configuration of interest, i.e. the mismatched random half space, as a special case. As in the preceding sections, we shall assume for simplicity that the source is at the interface z = 0. Using the results of Section 6 (c.f. (6.15)), we obtain the following asymptotic expression for the coherent reflected pressure

E{p,(x, y, 0, t)} - --;(2TT)-3&-3’2

(K,X+KlJ’-Of) &IJ) dw dK, dK2. >

(4.50)

This expression is the response of a deterministic half space characterized by reflection coefficient l(O). Therefore, the reflection of coherent energy at the interface is governed solely by the local, interfacial values of the functions pO, p1 and K,, (c.f. (2.36), (2.38)). Note that the integral in (4.50) is singular and should be interpreted as a distribution.

Consider now the behavior of the first and second moments of the reflected pressure, i.e. (4.50), (4.23) and (4.24), as functions of the reflection coefficient l. At one extreme, in the matched medium case where 5 = 0, the coherent reflected pressure vanishes and (4.24), with offsets X, y and i set equal to 0, represents the covariance of the reflected pressure. At the other extreme, suppose that the bulk modulus of the random half space could somehow be made to vanish; noting (2.36) and (2.38), this would cause d to approach -1. From a physical point of view, increasing the interfacial mismatch “chokes off” the

70 W Kohler et al. / Reflection by randomly layered medium

penetration of acoustic energy. In particular, the fluctuations and coda of the reflected pressure should decrease as the mismatch increases. In the limit, when 6 = -1, the reflection process should be totally deterministic.

From (4.50) we note that the coherent reflected pressure behaves as expected; it approaches the negative of the incident pressure as 3+ -1. From (4.23), (4.24), on the other hand, we see that the second moment reduces to 0 as l-, -1. We do not obtain the square of the incident field in this limit. The explanation for this apparent inconsistency lies in the fact that (4.24) is valid only for values of t that are O(1) and positive. In other words, (4.24) is not valid within the O(E) initial layer, during which the incident pulse illuminates the interface. In this context, then, (4.24) makes good physical sense. This expression represents coda, which should tend to zero as the mismatch increases.

To capture the behavior of the correlation function within the O(E) initial layer, one must rescale the problem. In this new scaling (a Strong Law of Large Numbers scaling), one obtains an asymptotic limit that is basically deterministic.

5. Reflection from a finite thickness slab

5.1. Problem form@ation

We shall now consider the reflection of acoustic energy by a randomly layered medium of finite thickness. We shall assume that the random slab occupies the region -L s z s 0. For simplicity, both ends of the

slab will be assumed to be matched and the point source will be assumed to lie at the interface. Because of the hyperbolic nature of the problem, the matched slab problem should have a solution

that is closely related to its counterpart for the matched semi-infinite medium. The coherent reflected pressure at z = 0 should be zero while the correlation function of the reflected pressure should, at least for some initial macroscopic time interval, be identical to its semi-infinite medium counterpart.

This basic similarity will be established. However, our use of Fourier transforms (e.g. (2.6)) introduces some complications into the analysis. The most fundamental difference is the fact that for each spectral component, the reflection coefficient r” = A&/B” will no longer be unimodular; we must account for both the amplitude and the phase of this process.

The formulation of the matched slab problem proceeds as in Section 2. Equations (2.29) and (2.30) again describe the behavior of A”, B’ within the slab region, -L s z s 0. The radiation condition at z = -cc

and the matched condition at z = -L lead to the boundary condition

A&(-L) =0 (5.1)

while the matched condition at z = 0 allows us to express the ratio of the reflected to incident pressure

as (c.f. (2.21), (2.31), (2.32))

PZ/P, = -A”(O)IB”(O). (5.2)

We shall follow the approach of [ 1,7] and introduce the fundamental matrix for the underlying linear system (2.29), (2.30). This propagator matrix, call it Y, has the structure

y= “_ b

[ 1 b ci’ lal*-lblZ= 1

and can be parameterized by three real variables 0, II: C#J in the following way

(5.3)

a = exp(i( 4 - + + 7r)/2) cash i, b=exp(-i(4+$+?r)/2)sinhi. (5.4)


with 0 < 8 <co, 0 d JI < 21r, 0 c 4 c 4~. The evolution of the complex matrix-valued process Y = Y” can thus be described by the real-valued (V, $‘, 4”) process. Equations (5.3) and (5.4), when substituted into the linear system defined by (2.29), (2.30), leads to the following differential equations for the parametric variables

~8’=-2~nsin(,‘+207/E)+1dfcos(I’+207/E) 5 dz

-+2++ n coth 0’ cos(t,V+207/e)]-1 d5coth 8” sin(+‘+2Wr/e) 5 dz

;@=-20 n csch 6’ COS(I~‘+~COT/E)-i ficsch 8” sin(t,V+2mr/e). &

(5.5)

We impose the initial conditions

eE(--L) = @(-L) = +“(-L) =o (5.6)

so that, in particular, Y(-L) reduces to the identity matrix. In terms of these variables, the quantity of interest (5.2) becomes

A’(O)/B”(O) = 6(0)/d(O) = eeirlr’ tanh( P/2). (5.7)

Compare (5.7) with (2.34). Note that if tanh(0’/2)+ 1, we recover the unimodular reflection coefficient for the random half space.

5.2. The coherent reflected pressure

As already mentioned, we anticipate that the coherent reflected pressure at the interface z = 0 should be zero. We have already seen in Section 4.5 that this quantity vanishes for the matched semi-infinite medium. Since the slab that we now consider is matched at both boundaries, there is no physical mechanism present that can generate a nonzero coherent return. Using (5.2) and (5.7), we can represent the coherent reflected pressure at z = 0 as

E{ p,(x, y, 0, t)} = ;(21T)-3&-3’2 IP,~_~~_~exp(f(I(IX+K2~-O1))

x E 1

em’“‘(‘) tanh$ (0) f(u) do dK, dK2. 1

(5-g)

In order to determine the behavior of E{e-‘““O’ tanh P/2(0)} as E + 0, we apply the asymptotic theory

outlined in [2] to stochastic initial value problem (5.5), (5.6). We obtain the result that

lim E e+O {

emis’ tanh $ (0) 1 @(-IL) = 0, I,V(-L) = $, 4&(-L) = c$} = V(-L, 0, $, 4)

where V is a solution of the final value problem

(5.9)

$ v+L,v=o, lim V = e-‘” tanh 4 (5.10) 2-O

In (5.10), the generator L, of this backward Kolmogorov equation has the form

coth ed+csch ed 2

)I 2

a* a4 +4&,$

ae2. (5.11)


Once V is determined, we can compute the quantity of interest as

lim E E-0 {

e-‘“‘(O) tanh T (0) I

= V( -L, 0, 0,O).

Problem (5.10) has the explicit solution

(5.12)

V(z,0,$,4)=e-‘#tanhzexp -4~’ { I

0 (a,,+&,,)ds .

1 (5.13)

Z

Therefore, V( -L, 0, 0,O) and, consequently, E{ pl(x, y, 0, t)} both vanish (as anticipated). In Section 6.2, when coherent return from a (completely reflecting) random half space is discussed, the

same exponential factor, i.e. exp{-40* JY ( LY,, +&,,) ds}, will appear (c.f. (6.10)). The appearance of common structure will be present in the correlation function discussion as well. This seems ultimately due to the underlying hyperbolicity of the problem, the fact that different physical configurations can

have solutions that, on some finite initial time interval at least, are the same.

5.3. Correlation function of the reflected pressure at the interface z = 0

To determine the correlation function at z = 0 for the finite random slab, we must again consider (2.43). The relevant expression for the current problem, however, is

&y,t;K,,K*,W)=lim exp(-i(+“‘(O)-t,!k*‘(O))) tanhy(0)

xtanhF(0) expi(ht-I,x-I,y)dhdl,dl,. I

(5.14)

This expression, when inserted into (2.44), will give us the asymptotic behavior of E{ p,(x + ~3, y + ~7, 0,

t +e?)p,(x, y, 0, t)}. It thus becomes necessary for us to consider the two-frequency asymptotics of

eqs. (5.5), (5.6). Consideration of the two-frequency (@, +‘) process actually suffices. For this problem

the analog of (3.11) becomes

lim E E-O

exp(-i(+,“‘(O) - +,‘*‘(O))) tanh F (0) tanh $! (0) 1 oCi’(z) = @, +,“‘(z) = t,kCi),

b(‘)(z) = +ci), i = 1, 2 I

= V(Z, fj(‘), f$*), $(I), $(*), 4(l), +(*))

where V is the solution of the final value problem

i v+L,v=o, e(l) e(2)

lim V = e-i(@“-J”2’) tanh T tanh T . Z-DO

(5.15)

(5.16)

Note that the ( )-superscript again corresponds to the notation introduced in (2.42). The backward

equation generator L, in (5.16) is given by

L, = 2W2Lx,” a*

ae(l)afj(l) +coth e(t)--&+(coth P--$+csch @‘)+)’

a* -I- a~(*) ae(*) +coth e(*)--&+(coth e(*)--&+csh P$j)*

+ 2 cos( @‘) -@*‘+2A -2h7)


x coth eC2) --&+ csch oC2) --&)I

-2sin(t+G”- I//‘~‘+ 24 - 2h7)

X coth 0(r) d+ csch e(r) d a*(') a4("

(5.17)

In order to solve (5.16), (5.17), we first introduce change of variables (3.17). We next observe that in the transformed variables V will depend upon z, 8 (‘) Oc2) and $ only. Therefore, the transformed problem , becomes

; V+2(hr’-A’) 2 V+~W’CZ,,, ae,l$;Ooj+coth e(l’d 2

de(l) +coth’ e(l) - d+

a2

a*’ ae(2) a@(2)

+coth 8’2’--&+coth2 8’2’-$

+2cosj [

a2 se(l) de(z) - coth B(l) coth eC2) a2

&P 1 -2sinJ

K coth e’“-&+coth eC2’-&

> 11 5 w v=o lim V=ei4tanhytanh$. Z--O (5.18)

A start for the solution of (5.18) can be obtained by considering (3.20) in the matched case, i.e. when p = q. In this case, the expansion V = Cp”o Vpp exp(ip$) can be interpreted as an expansion in powers of the final data exp i$. This data represents the product of the input reflection coefficient at (o(l), K$‘), KY)) and its COmpleX Conjugate at (wC2), ~~ , K$~)). Th’ (2) is o b servation (together with final condition (5.18)) suggests that we look for a solution of (5.18) having the form

V= : VP p=o (5.19)

with VP a function of .z, h, f, w, K. When (5.19) is substituted into (5.18), the VP functions are found to

satisfy the final value problem

vplp=o = sp,, (5.20)


which is precisely the same problem as that obtained for the matched semi-infinite random medium (c.f. (3.21), (3.22) with p = q and 5 = 0). Consequently, when inverse Fourier transform (3.23) is computed, the resulting functions Wp will satisfy (3.25), (3.26) (with p = N).

Because of initial conditions (5.6), the expectation in the integrand of (5.14) becomes (c.f. (5.15))

lim E e(‘)(o)

exp( -i( +“‘(O) - $1’2’(O))) tanh 2 tanh @2)(O) - = V(_L, @(I) = $0) = e(2) = 41’2’ =O) E’O 2 I

= q-L). (5.21)

Therefore

$x, J’, 1; K1, K2, 0) = k@(-L, t, r; 0, K). (5.22)

Note that as the slab thickness L approaches infinity, (5.22) becomes (3.27), the corresponding result for the matched semi-infinite medium.

In retrospect, the success of approach (5.19) and the fact that we again obtain problem (3.25), (3.26) is not surprising. For both problems, i.e. the semi-infinite medium and the finite slab, the same Riccati equation can be used to describe the evolution of the reflection coefficient r”; in the former case the reflection coefficient is unimodular while in the latter case it is not. This Riccati equation, moreover, can be used to transform the problem into one involving an infinite-dimensional, first order, linear system, where the dependent variable is the infinite-dimensional vector

[P) (P)2, (P)3,. . .) (P)“). . .I’.

In this discussion, we have restricted consideration to the matched slab for reasons of simplicity. Analyzing the case where mismatches exist at z = 0 and/or z = -L is conceptually straightforward but computationally more involved. In Section 6 we discuss the case of a single interior discontinuity.

6. Random half space with an interior discontinuity

6.1. Problem formulation

We now consider a randomly layered half space whose background variation undergoes an abrupt change, or discontinuity, at some fixed depth, say z = -L. For such a configuration, the reflection coefficient will again be unimodular, since the random half space neither dissipates energy nor permits any radiation loss at z = -cc. However, acoustic energy that penetrates into this half space will undergo reflection from the discontinuity as well as multiple scattering from the random layering. The discontinuity will reflect both coherent and incoherent energy; therefore, we shall evaluate both the coherent reflected pressure and the correlation function of this reflected pressure at the interface z = 0. For simplicity, we shall assume that the medium is matched at z = 0 (i.e. l(O-) = 0), so that the only discontinuity present exists at z = -L. As before, we assume that no turning points occur in the region of interest.

The formulation and scaling of the problem proceeds as in Section 2. The solution in the homogeneous half space z> 0 is identical to that of Section 2.2, while the interface conditions at z =0 are actually simplified by our assumption that l(O-) = 0. In each of the two regions -L < z < 0 and -co < z < -L, A’

and B” satisfy (2.29), (2.30). Since the reflection factor r’ is unimodular, we again define (c.f. (3.1)-(3.3))


Then @(z, w) satisfies differential equation (3.4) in each of the two regions -L< z < 0 and --OO < z < -L.

What we require is an interface condition at z = -L, connecting the values of this phase function on both sides of the discontinuity.

This relation is obtained by requiring that pressure and the normal component of velocity be continuous at z = -L. Let f subscripts denote corresponding one-sided limits at -L; for example [+ = lim e(z).

Z--L+

Then, from (2.27), (2.28) we obtain:

[Y2[exp(-ifr(-L))A:-exp(ifT(-L))B:]

=[N’[exp(-ifr(-L))AI-exp(ifi(-L))BF]

&“‘[exp(-ifT(-L))A:+exp(iir(-L))B:]

=&11/2[exp(-i~r(-L))Al+exp(i~~(-L))B?].

Let l( - L) denote the following interface reflection coefficient

5(-L) 25 5-+5+’

(6.2)

(6.3)

Note that if L approaches 0, l( -L) reduces to (2.36), since t+ in this case becomes (1 - K~o-~)-“*. Using (6.1)-(6.3), we obtain the following interface condition for the reflection coefficient

>> = (6.4)

6.2. The coherent rejected pressure

The coherent reflected pressure at the interface z = 0 is given by (c.f. (2.6)) 00 cc

I I I

cc

E{P,(% Y, 0, t)) = 1/(27d3 exp(i(K,X+K2y-Ot)/E) -m --oo --m

x E{$:(K,, K2, 0, w)} dw dK, dK2 (6.5)

where p^F denotes the transformed reflected pressure. We shall assume, for simplicity, that the source is vertical (e = ZJ and located at the interface (z, = O+). Then, noting (2.26) and recalling that the interface at z = 0 is matched, we obtain

E{$(KI, K2,0,0)} =$~3’2~(:(W)E{e-i~c’o~}. (6.6)

The angle process $‘(z) is a solution of (3.4) in each of the two regions -L < z < 0 and -cc < z < -L.

Applying the asymptotic limit of Section 3 to this process leads to the following generator 2

L,=402(umm+fann)-$. (6.7)


In terms of this generator, we obtain the following asymptotic characterization

lim E{e-i*“‘o’I $“(-L+) = +} = V(-L, +) E-+0

where V is a solution of the final value problem

; v+ L,V=O, lim V( 2, $) = e-‘“. 2-O

(6.8)

(6.9)

Noting (6.7), the solution of (6.9) is

{ I

0

V(z, $) = exp -4w2 (a,,,,,, +;a,,,) ds e-‘“. (6.10) Z 1

Solution (6.10) is consistent with the underlying physics. The functions (Y,, and (Y,,, are both positive,

since they are both power spectra at zero wavenumber. Therefore, V(z, 1,4,) decreases as the conditioning

point z decreases. In other words, as the conditioning point z recedes more deeply into the random half

space, the intervening random slab (from z to 0) plays a correspondingly greater role in “randomizing”

the conditioned +’ process, i.e. in driving +‘(O) toward a uniform distribution.

The asymptotic behavior of E{exp(-iti’(O is given by

E{exp(-i+,“(O))} = E{ V(-L, JI”(-L+))} (6.11)

where the latter expectation is computed with respect to the limiting distribution of the reflection coefficient

phase at z = -L+. Interface condition (6.4), however, can be used to transform the reflection coefficient

to z = -L-. Moreover, since exp(-i+‘(-LJ) represents the reflection coefficient of the infinitely thick

random slab (extending from -cc to -L), the asymptotic distribution of I,F(-L-) is uniform. Therefore,

since

1 - I

2n -exp(-i+)+exp(i2ur(-L)ls)5dti = exp(i2w~(_L),E)5

27F 0 l-exp(-i(2~r(-L)/e++))5 (6.12)

it follows from (6.6) and (6.10)-(6.12) that

E{i?(K,, K21 0, w)}--$E~‘~~(w) exp a,,+fa.,)ds}exp(i2:7(-L))& (6.13)

When (6.13) is, in turn, inserted into (6.5), we obtain the following expression for the asymptotic behavior

of the coherent reflected pressure at z = 0

E{p,(x, y, 0, t)}- -;(21T-3r-3’2 I_",I_",I_",exp(~(K~x+n,y)exp(-i~(r-2?(-L)))

(ff,, &u) dw dK1 dK2. (6.14)

Much of the relevant physics is apparent in (6.14). If the discontinuity location recedes to -00, the mean

coherent reflected pressure at z = 0 will tend to zero (since +,“(O) tends to a uniform distribution). In the

general case, the incident coherent wave suffers attenuation and smearing (due to “randomization” or

loss of coherence) during its two-way transit to the discontinuity and back. The strength of the return

signal is proportional to the degree of mismatch at the discontinuity (i.e. 5). Lastly, the reflected coherent

wave involves a time lag, the delay being equal to the two-way transit time to the discontinuity and back.


Note that if we let L approach zero in (6.14), we obtain the mean reflected pressure from a random (mismatched) half space characterized by interface reflection coefficient 5 at z = 0. In this special case

E{Pr(X, Y, 0, t)l- -%27r)- E 3 -3’2 irn irn rrn exp(~(K,x+Kly--Ot))&~)dW dK, dKZ. (6.15)

Equation (6.15) represents the reflected coefficient l.

J-m J-cc J-CC \E /

pressure from an effective medium characterized by reflection

6.3. Correlation function for the reflected pressure

For the configuration that we are considering, the two-point, two-time correlation function of the reflected pressure is again given by (2.41), (2.43) and (2.44), with e3 = 1 (that is K * e=O) and z, = 0. Therefore

(6.16)

E{T”‘T’2’*} exp(i(ht - 1,x- 1,~)) dh dl, dl,

(6.17)

x&Y, f; KI, K2, w) dw dK, dK2. (6.18)

Moreover, since the interface at z = 0 is matched, it follows from (2.37) that

E{T”‘T’2’*} = E{exp(-i(#“‘(O) - $“‘(O)))}. (6.19)

The discontinuity at z = -L will make its presence felt through its effect upon expectation (6.19). Noting (3.11), (3.12), we obtain that

‘,‘r-:: E{exp(-i($,“‘(O)- @2’(0)))(+“‘(-L+) = t,b(‘), t,b’2’(-L’) = +bc2’} = V(-L, +(‘), +I(‘)) (6.20)

where V is the solution of the final value problem

; v+ L,V=O, -L<ZSO

V(0, I+P, $,‘2’) = exp( -i( +,“‘(O) - $“‘(O)))

with the generator L, given by (3.10). We again introduce change of variables (3.17) and expand V in the Fourier series

(6.21)

v= f VN eiN&_ N=O

(6.22)


Substitution of (6.22) into (6.21) leads to the following final value problem for VN (c.f. (3.21), (3.22))

$ VN+i2N(h~‘-A’)VN-2~2~,,{(N-l)2VN~’-2~2VN+(N+1)2VN~‘}=O

VNlz+=8N,,. (6.23)

From (6.20) and (6.22), it follows that

lim E{exp(-i(+“‘(O) - t+G*‘(O)))} = F V”(-L)E{exp(iA$(-L’))} (6.24) E-+0

where, from (3.17)

N=O

tj&L+)=$,‘*‘(-L+)--I//“‘(-L+)-2A(-L)+2hr(-L). (6.25)

The latter expectation in (6.24) is taken with respect

($“‘(-L+), t&(*)(-L’)). We again use interface condition (6.4) to

across the discontinuity. Let

$(1)=+CI(‘)-h7+A, $c2) = $(*’ + hr - A.

Then, from (6.4) we obtain

to the limiting joint distribution of

transform the reflection coefficient phase

(6.26)

exp(i&(-L+)) - [

exp(iP)-lexp(-i20r(-L)/e) N exp(i@‘)-5 exp(i20r(-L)/.e) N

1 -exp(i(2Wr(-_)/e + @‘))l I[ - 1 1 -exp(-i(2Wr(-_)/c + $?‘))l

=; f a: exp i2 f (p - q)r(--L) ( >

exp(i(p$?- q$‘)) (6.27) p=o q=o

where terms that go to zero as E + 0 have been deleted on the right side of (6.27). Note that

P~(*‘-q~“‘=(p-q)~++(p+q)~. (6.28)

Using (6.27), (6.28), we can represent (6.24) in terms of a conditional expectation, where the conditioning

point z now lies in the interval --o;) < z < -L. We then obtain the desired expectation (6.24) by letting the

conditioning point recede to --CO. To implement this, let VN pq be the solution of the final value problem

(c.f. (3.21))

V~Jr_L=aPN4exp iZz(p-q)r(-L) >

. (6.29)

One can show that

lim VpN4=0 unless p=q=O. z--CC

(6.30)

Therefore, in terms of these functions

Fz E{exp(-i(+(‘)(0)-(Cr’2’(0)))}= NEo V”(-L)vo,O(-00). (6.31)


For brevity, let Vg = VP,. Note from (6.29) that this diagonal set evolves as a decoupled or separate subsystem. Therefore, one need only solve the diagonal subsystem of (6.29) and evaluate lim,,_, v”,. As in (3.23), let

VN exp(i(ht - I,x - lz~)) dh dl, dl,

V”, exp(i(ht - f,x - &y)) dh dl, df,. (6.32)

The functions WN are solutions of (3.25), (3.26) while the functions W$, are solutions of

(6.33) +2w*a,,[(p-1)2W~‘-2~*W~+(~+1)*WPN+’]=0

wpNlz,-L = a$$s(t)s(x)s( y).

In terms of these functions, it follows from (6.17), (6.19) that

&.Y,r;WW’J)= : I a3

II I

co

WN(-L, t-s, r-r’; 0, K)

N=O 0 -co -co

x ~N(-m, s, r’; o, K) dr’ ds (6.34)

where we have used the fact that WN and W$, vanish when t<O. Equation (6.34), when inserted into the integrand of (6.18), gives the asymptotic behavior of the reflected pressure correlation function.

6.4. Piecewise constant backgrounds

The calculations are simplifiied somewhat when the macroscale background variation in both regions is constant. In particular, systems (6.29) or (6.33) need not be solved since the joint invariant density of (@, 4) at z = -L- is explicitly known. Noting (6.27), (6.28) and (4.7)

E{exp(iN&-L+))}= f $ ayexp i2f(p-q)T(-L) p=o q=o ( >

-&

exp(i(p-q)$ d@ 1-1 (~)‘p+q”2P(~) da

Using (4.10) and contour integration

where, as in (4.18)

a 2$&y _p;lK*g-*)--l/*(Kgl~ _-p;lw-lK. I)*

nn

(6.35)

(6.36)

(6.37)


The parameters po, p, , Ko, K, , etc. in (6.37) correspond to the region z < -L. We again introduce change

of variables (4.19) into the integral expression for A, i.e. (4.17). Note, from (6.27), that the coefficients

a#’ are independent of II, l2 and h. Let y be defined as in (4.21). Then, we obtain

&Y, t; KI, K2r W)

dq, dq, dqJN(-L)

xexp{i(K,p;‘w-‘t~,-x)q,+i(K,p;‘w-’t~,-y)q,+iyq,}. (6.38)

Several observations can be made that will help to simplify the calculations. The first of these follows

from (6.23); when viewed as a function of q, , q2 and q3, VN is a function of q3 only since (c.f. (4.11), (4.19))

/+-A’= -202a,,q3. (6.39)

Therefore, the variables WN defined by (6.32) have the form

cc oc m WN = (2T)_3

I I I VN exp(i(hr - 1,x - I,y)) dh dl, dl,

-CC -CC --oD

VN exp{i(Kop;‘w-‘k, - x)q,

+i(Kop;‘w-‘fK2-y)+iYq3} da dq2 dq3 00

= (2?r-1yfS(Kop;‘W-l fK1-X)8(KoP;‘W-‘fK2-y) VN eiyq3 dq,. -cc

(6.40)

Define

I

03

tiN(Z, 7; W, K) = (27F)-’ VN eiyq3 dq,. -m

Then, from (6.23) it follows that r?/” is solution of

5 +N-4&nnN; ~N-2~2cz,,{(N-l)2~N-‘-2N2~N+(N+1)2~N+1}=0

@Nl,=o=6N,,6(y), with G”=O if y<O.

In terms of these new CI” functions

(6.42)

WN = yt-‘~?(K~p;‘~-‘t~, -X)6(K,p;‘w-‘t~,-y)+~.

The second observation relevant to (6.38) is the fact that

(6.43)

(6.44)


This result is readily established by means of contour integration. Using (6.43) and (6.44), we can recast (6.38) as

i(X, J’, ?; K,, Kz, W) = ~t-‘8(&jl;‘CO-‘tK, -X)6(&&t-‘tK~-~)

m co

x c c a: r?l”(-L, y-s;w,lc) 2pP

N=Op=l (s+2)P+‘ds* (6.45)

Substitution of (6.45) into (6.18) leads to the following expression for the asymptotic behavior of the correlation function

ds dw (6.46)

where the argument K is evaluated at K, = K;‘p,wt-‘x, ~~ = K;‘p,wr-‘y.

7. Illumination by a bounded beam

In this section we consider the illumination of the randomly layered half space by a spatially confined (or bounded) beam. We use the methods of Brekhovskikh [8] to recast the problem into an equivalent one involving an effective source distribution upon some reference plane z = z,. Since the underlying problem is linear, this equivalent extended source problem is a straightforward extension of the point-source analysis of the previous sections.

Assume that we are interested in using a temporally-pulsed, spatially bounded acoustic signal to probe the random medium. In general, the signal will impinge obliquely upon the interface z = 0. In order to effectively probe the macroscale structure of the random half space, the temporal and spatial variations of this incident signal should be O(1) on the intermediate scale. We shall form this incident excitation in the following way (cf. Fig. 1).

Temporolly pulsed

plone wove

\ Z’Zs

\ \

\, Incident beom

Screen with aperture

Randomly -l_oyered Holf -Space

Fig. 1. Beam formed by plane wave incident upon an aperature in a screen.


Arbitrarily select a reference plane z = z, > 0. If we properly model the incident signal in the region

0 < z < z,~, the solution of the complete problem in this region will provide everything we need. Assume

that a screen, containing a small (nominally circular) aperture, is placed at z = z,. The zero-thickness

screen will be thought of as a perfect acoustic absorber. Assume that a temporally pulsed, acoustic plane

wave is obliquely incident upon the screen from above. The Kirchhoff approximation, used to model

propagation through the aperture, creates the desired incident excitation. The region z < zS (and hence

the random half space) sees only that a bounded beam is incident upon it.

For definiteness, we shall assume that the pulsed plane wave that impinges upon the screen has a

Gaussian variation: the (scaled) incident pressure associated with this pulsed plane wave is:

Pf”J y, z, t) = &-2’3 exp(-(t-ysinS+zcos6)2/e2~z) (7.1)

where y, z, t represent nondimensional macroscale variables and .E is an 0( 1) constant. We have referenced

time so that the peak pressure impinges upon the origin at t = 0. To create the beam, we insert the screen

(with an assumed circular aperture) at z = z,. Using the Kirchhoff approximation, the actual scaled pressure

incident upon the region z < z, becomes:

xz+y2< &*I2 otherwise.

(7.2)

Since I is an assumed 0( 1) constant, the circular aperture has radius I when measured on the intermediate

scale. Taking temporal and transverse spatial Fourier transforms of (7.2) leads to:

&(K,, K2, z,, W) = E '/*2~'/22;n-'lexp -if

(

zScosS)exp(-$w')l,(Rl)

R =[Kf+(Kz--W sin s)2]1’2. (7.3)

Here J, denotes the Bessel function of order one. To obtain (7.3), we have (as in previous sections) let

o = w ‘/ E, K, = K {/ E, K2 = K* = K h/ E and then have dropped the primes. In the Fourier domain, p^kC represent

the “amplitude” of the incident pressure wave at z = z,~. The incident pressure in the region 0 < z < z, is

then given by:

Pl~“c(K,,K2,Z,W)=Pl~“c(K,,K2,Z,,w)eXp -if(1-K20-2)“2(Z-Zs) , ( >

o<Z<Z,. (7.4)

To relate these ideas to the work of the prior sections, consider now a new problem in which the random

half space is illuminated by a source distribution concentrated in the plane z = z,. For definiteness, let

e = z,,, i.e. the source distribution is assumed to have a constant vertical direction. The basic equations

become:

P(Z)& +vp =f(x, y, tM(z - z,)zo, (p(z)c(z))_‘p,+v * u=o. (7.5)

We now repeat the nondimensionalizing, e-scaling and Fourier transformation steps of the previous

sections; solving for the fields above the interface, we obtain an incident pressure in 0 < z < z, of the form

(cf. (2.26)):

%“c(K ,,K2,Z,W)=-$E3’2j(K,,K+)eXp -if(1-K20-2)1’2(Z-Zs) >

f( KI,K*, W)’ f(x,y, t) exp(iwt-i(K,x+ K*Y)) dt dx dy. (7.6)


Note that in the special case of the point source, f(x,~, t)=f(t)a(x)a(y) and _&K,, K~, W)=_!(U). A comparison of (7.4) and (7.6) enables us to relate the two problems. The bounded beam problem that we posed corresponds to the following choice of effective source distribution.

K19 K2, W) = -4P”2~f2-‘1 eXp -i~r,cosS)exp(-$w’)J,(LU). (7.7)

Having formulated this effective source distribution, the rest of the analysis parallels that of the previous sections. With e3 = 1 (and K * e = 0), one need only replace f(u) by !(K, , K~, o) as given by (7.7). The solution that we will obtain will depend upon z,, the location of the beam-forming screen. This is not

surprising; the radiation passing through the aperture will not (as suggested by the schematic nature of Fig. 1) simply propagate in collimated ray-optical fashion to the interface. Diffractive effects will broaden it in the region 0 < z < z,.

This suggests that we can best model the bounded incident beam of interest by letting z, + O’, i.e. by letting the screen approach the interface from above. In this case, tl-.e beam problem solution can be obtained by a simple modification of (4.24); we need only set e3 = 1, K * e = 0 and replace [T(u)[’ by

Ij(‘G, K2, 412, where .f(‘Q 9 K2, o) is given by (7.7), to obtain:

OD m 00 E{p,(x+Gy+-tJ,O, t+EGPr(X,Y,O, [)I-t

I I I exp( -iwl+ i( K$-!- ~~7))

--oo -m -cc

X?L2RZ12 eXp(-~‘w’/2).f~(~,)~(X, y, t; K,, K2, w) do dK, dK2. (7.8)

8. Conclusions

We have formulated and solved the problem of calculating the statistics of the reflected pressure at the surface of a randomly layered half space when waves from a pulsed point source impinge on it. The working hypothesis of our theory is separation of scales which are: the 0( 1) macroscopic scale, the O(E) pulse width and the 0(a2) width of the random layering, where E is a small parameter.

The main result is equations (3.14)-(3.16) for the space-time local power spectral density for the reflected pressure signal at the surface. In the special case of a randomly layered half space that is statistically homogeneous and matched to the nonrandom half space we have solved these equations explicitly to obtain (4.36). Various other results that can be derived within the present framework are given in Sections 5, 6 and 7.

The physical significance of the results is of course the same .as that of the simpler, plane wave case analyzed previously [l-6] but it is more difficult to see here. We are conducting at present extensive numerical simulations similar to the one in [5,6] and will report on the results in the near future.

Appendix. A comparison of the electromagnetic and acoustic point source problems

The discussion thus far has dealt solely with the acoustic problem. In this Appendix, we shall examine the extent to which the electromagnetic point source problem can be reduced to a superposition of appropriately defined acoustic problems.

Assume that the region z > 0 is free space (characterized by permitivity c0 and permeability CL,,) while the region z Q 0 is composed of layered media (having constitutive parameters E(Z) and p(z)). For the


present discussion, we shall asume the region z G 0 to be lossless; dissipation can subsequently be introduced in a straightforward way. We shall assume that a point current source is located z, units above the interface. Maxwell’s equations thus become

V x E = -pa,H, V.B=O, VxH=J(“)+aE t , V.D=p (A.l)

where E, D, H and B represent the electric field vector, the dielectric displacement vector, the magnetic field vector and the magnetic induction vector, respectively. The quantity p in equation (A.l) denotes volume chage density while the point source current J(‘) is assumed to be

J(‘) = J’“‘(t)S(x)S( y)S(z - z,)e (A.2)

where e = ezy,+ e3z,, is a constant unit vector. We shall first take temporal Fourier transforms of eqs. (Al); for example, let

05 &, Y, z, w) =

I E(x, y, z, t) ei”’ dt. (A.3)

-cc

The temporally-transformed version of eqs. (A.l) becomes

Vxi?=iW&, v*(~A)=o, Vx&=j’S’-iw& 1 v-(df)=~ (A.4)

where we have used the fact that k = & and 6 = &. Eqs. (A4), in turn, can be recast into a system of

equations involving only the transverse (to z) electric and magnetic fields [9]. The subscripts I and z will

be used to denote transverse and z-directed vector components, respectively. Define the vector operator

V, = %& +yc& (A.3

Then, it follows from eqs. (A.4) that

~.Vxl?=V,.(E,xzJ=iWp&, zo.aXA=V,.(~,X,x)=j~)-iiWE~=

~~x(VxIZ)=V,.&-a~&=-i~~(ljrxz~), ~x(VxA)=-~!“‘x~+iWE(~,x~) (A.6)

The first pair of eqs. (A.6) expresses the longitudinal (z) components of the electric and magnetic fields in terms of the transverse fields. Ths use of these representations in the second pair of eqs. (A.6) leads to the following equations involving only the transverse fields.

a,&=iiW~(fi,XzJ+(i~E)-‘V,[_?p)-V,. (&x&l

a,ti,=~j"~x~,-i0&x~)+(i~~)-'V,[V,~(&x~~)]. (A.7)

Note that we have used the face that E and p do not vary transversely. We shall now take transverse spatial coordinate Fourier transforms. For simplicity, we shall continue to use the superscript carat to denote the transformed field; thus, for example

00 cc

I I I?(, y, z, w) e-i(K~x+r~y) dx dy = &(K,, K~, z, co). (A.8)

-co -m

Let k = K,X,,+ Kg,,, K’S K* K and K,,= K-’ K. Since the effect of Fourier transform (A.8) is to replace V,

by iK, the spatially transformed version of eqs. (A.7) becomes

a,iZ,=iiwp(fi,x2J+(f.0E)-1K[.?~)-iK. (fi,X&)]

a,fi,=&“)x~-i~f5(&xzJ+(t.0p)-1K[iK.(~*x~)] (A.9)


where

P(K,) K*, z,w)=.P)(w)s(z-zs)e.

We shall now decompose the transverse electromagnetic vector fields in eqs. (A.9) into two pairs of

85

(A.lO)

coupled scalar fields. These two pairs of fields, corresponding to two orthogonal polarizations, will subsequently be put into correspondence with acoustic fields. The first pair of scalar fields consists of ~~ * E, and ~~ * (fi, X Zo). From eqs. (A.9) and (A.lO), we obtain

a,(&‘&)-iw (Kg’(tj,xro))=~j’“‘(o)S(z-z~)e,

~,(Ko~(Ej,~Z~))-i~&(K~~~,)=-K-‘K~~~S~(O)~(Z-Z,)~~.

(A.ll)

The second pair of fields consists of ~~ * (i, x ~0) and -K~ . I?,. From eqs. (A.9) and (A.lO) we likewise

obtain

&(-Ko’ti,)-iW

(A.12)

Before we compare eqs. (A.ll) and (A.12) with their acoustic counterpart, several points should be noted. The first is that these two pairs of scalar fields determine the total electromagnetic fields. Noting eqs. (A.6), we have

& = Ko(Ko ’ &,>+ kox Ko)(Ko ’ (2, x ko)), fi,=-KO(-K,,‘&)+(&,XK,,)(K’(fi,XZ,,))

$ = (iWe)-‘[.!y) -iK(Ko’ ($XZo))], iiz=(O+)-l[K(Ko*(&XZo))].

(A.13)

The second point to note is that separate solutions to this pair of scalar field problems (i.e. eqs. (A.ll) and (A.12)) can be used to construct relevant field correlation functions. We shall use the (1) and (2) superscripts (as in Section 2) to indicate the frequency and wavenumber offsets defined by equations (2.42); in terms of this notation, we have to leading order that

$(I) * kc*‘* - (Kg ’ f I 2 )(“( K. * i,)(‘)* + (Kg * (& x &))(‘)( K. * (i, X Zo))(*)*. I (A.14)

Therefore, one can separately compute the correlation functions associated with problems (A.ll) and (A.12) and superpose them to obtain, for example, the asymptotic behavior of

E{E(x+eX,1,+~~,0,t+~?)~E(x,y,0,t)}.

We shall now compare eqs. (A.ll) and (A.12) with the acoustic equations. Assume now that an acoustic point source is located z, units above a layered half-space. Noting eqs. (2.8) and (2.9), the unscaled transformed acoustic equations are

d,p^-iwpu^=j;S(z-z,), a.n-iw[a’-~]i=j,~s(z-z~) (A.15)

where we have assumed a more general point source excitation of the form i =$2yo+$3xo. Comparing eqs. (A.ll) and (A.15), we note the correspondence

Kg’ &ii, -K-‘K~.f’S)(W)C?++j;, Ko * (6 xz0)~6,

KK;‘.f(S)(0)Ejf*f2, p-K-', E-p. (A.16)

86 W. Kohler et al. / ReJection by randomly layered medium

On the other hand, if we compare eqs. (A.12) and (A.15), we obtain the correspondence

Ko*(hZo)t*fi, o-.i;, -Ko*fi,C,;

(A.17)

-K-‘K,P(W)e2.f+“‘jz, /.&f*p, E-K-‘. WP

It is clear from correspondences (A.16) and (A.17) that j must be taken to be a function of K as well as w. It is also very important to note that the interface boundary conditions (and radiation conditions) for the electromagnetic and acoustic problems likewise stand in correspondence. In particular, for the electromagnetic problem, the tangential vector fields $ and fi, must be continuous across z = 0. For the acoustic problem, on the other hand, pressure fi and normal velocity component u^ must be continuous across z = 0. These continuity constraints are compatible with (A.16) and (A.17).

We shall now consider the introduction of dissipation. In most electromagnetic problems (involving non magnetic materials), p = po, i.e. the permeability is that of free space. Energy dissipation occurs due to the presence of dispersive (and hence dissipative) dielectric material and/or ohmic conductivity. Both of these loss mechanisms can be introduced into Maxwell’s equations by the following modification of the dielectric constant

.s+e’+i ( >

E”+~ . 0

(A.18)

Thus, the real dielectric constant E is replaced (in the curl equation) by a complex quantity involving both the dispersive dielectric constant E’(O) +3’(w) and the electrical conductivity u,.

Consider now the impact of modification (A.18) upon correspondences (A.16) and (A.17). In (A.16), we must complexify the density in the following manner

ptp+5 (A.19)

where the (frequency-dependent) acoustic conductivity Us corresponds to a, + WE”. Note, however, that correspondence (A.16) makes the sound speed complex-valued since the bulk modulus K remains real. The (source-free) acoustic equations that arise as a consequence of (A.16) are

ad - iwpii + u,d = 0, a,8-i~[K-i-(p+~)-‘~JB=0 (A.20)

which are the same as (2.8), (2.9) with the substitution (2.9a) (and with E = 1). In correspondence (A.17) the bulk modulus and sound speed must be made complex in such a way

that the density remains real. Modification on (A.18) requires the bulk modulus to be made complex in the following manner

K’+ K-‘+iy (A.21)

where the frequency-dependent positive parameter y corresponds to &‘+a,/~. The source-free acoustic equations that arise as a result of correspondence (A.17) thus have the form

2 a$ - iwpu^ = 0, a$-io

1 K-‘+iy--& p^=O. 1 (A.22)

Note that these dissipative equations do not arise in the acoustic case when the basic equations have the form (2.la), (2.2a).


In summary, we have found that the electromagnetic point current source problem can be decomposed into two acoustic point source problems (corresponding to the two orthogonal polarizations of the transverse electromagnetic fields). Electromagnetic field correlations of interest, moreover, can be construc- ted using correlations computed in the two corresponding acoustic problems. The full analysis of the electromagnetic point source problem involving a dissipative, layered half space, however, will require the study of corresponding acoustic problems having loss mechanisms different from that considered in this paper.

Acknowledgement

The work of Werner Kohler was supported by the Air Force Office of Scientific Research under grant # AFOSR-88-0112. The work of George Papanicolaou was supported by the National Science Foundation under grant # NSF-DMS-8701895 and by the Air Force Office of Scientific Research under grant # AFOSR-86-0352.

References

[l] R. Burridge, G. Papanicolaou and B. White, “Statistics for pulse reflection from a randomly layered medium”, Siam. J. Appl.

Math. 47, 146-168 (1987).

[2] R. Burridge, G. Papanicolaou, P. Sheng and B. White, “Probing a random medium with a pulse”, Siam J. Appl. Math. 49, 582-607 (1989). An abridged version appeared in: J.R. Knopps and A.A. Lacey, eds., Non-Classical Conlinuum Mechanics,

Cambridge Univ. Press, 3-21 (1987).

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Reflection of waves generated by a point source over a randomly layered medium