Transport Theory for Acoustic Waves with Re ection and

Transmission at Interfaces

Guillaume Bal Joseph B. Keller y George Papanicolaou z Leonid Ryzhik x

March 8, 1999

Abstract

Transport theoretic boundary conditions are derived for acoustic wave re ection and trans-mission at a rough interface with small random uctuations. The Wigner distribution is used togo from waves to energy transport in the high frequency limit, and the Born expansion is used tocalculate the eect of the random rough surface. The smoothing method is also used to removethe grazing angle singularity due to the Born approximation. The results are presented in a formthat is convenient both for theoretical analysis and for numerical computations.

Contents

1 Introduction 2

1.1 The Radiative Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Half Space Problems and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 3

2 Acoustic Wave Transport 5

3 Flat Boundary 7

3.1 Re ection and Transmission of Waves in Homogeneous Media . . . . . . . . . . . . . 73.2 High Frequency Limit: Re ection and Transmission of Energy . . . . . . . . . . . . . 9

4 Rough Boundaries 10

4.1 Surface Re ection Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Re ection and Transmission at a Rough Interface . . . . . . . . . . . . . . . . . . . . 134.3 Re ection and Transmission from Small Amplitude Rough Interfaces . . . . . . . . . 144.4 Derivation of the Dirichlet and Neumann Boundary Conditions . . . . . . . . . . . . 16

5 The Initial Boundary Value Problem for the Radiative Transport Equation 17

A Derivation of the Interface Conditions 18

B Grazing Angles and the Smoothing Method 21

B.1 Smoothing Method for Dirichlet and Neumann Boundary Conditions . . . . . . . . . 22B.2 Extension to Interface Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Department of Mathematics, Stanford CA, 94305; bal@math.stanford.eduyDepartment of Mathematics, Stanford CA, 94305; keller@math.stanford.eduzDepartment of Mathematics, Stanford CA, 94305; papanico@math.stanford.eduxDepartment of Mathematics, University of Chicago, Chicago IL, 60637; ryzhik@math.uchicago.edu

1

1 Introduction

Wave propagation in weakly uctuating random media over distances long compared to the wavelength can be described by the energy transport equation. This is the radiative transport regimedescribed in the next section. Near boundaries and interfaces the waves undergo coherent or partiallycoherent re ection and transmission, according as the interfaces are smooth or randomly rough.Starting from the acoustic equations, we derive systematically, boundary conditions for the radiativetransport equation. This equation is asymptotically valid away from the boundaries and interfaces.The asymptotic limit that leads to radiative transport corresponds to weak uctuations and highfrequency waves, with the correlation length of the inhomogeneities comparable to the wave length.We use the Wigner distribution to analyze this limit, as we did in [23]. The main result of this work isthe derivation of the transport boundary conditions (1.3) and (1.4) for re ection and transmission ata rough interface. The boundary conditions are intuitively clear. They combine coherent or specularre ection and transmission, and incoherent or diuse re ection and transmission, in the form of alinear input-output relation. If there is no dissipation, as we assume, energy ux is conserved as(1.5) shows. We calculate the explicit form of the coherent and incoherent energy ux re ection andtransmission coecients and cross-sections when the uctuations of the interface are weak. Specialcases of these coecients have been obtained before in the Born approximation [5, 10, 29]. Ourresults are summarized at the end of section 1.2.

The re ection and transmission coecients, computed by a Born expansion, are singular atgrazing angles. To eliminate the singularity, we recalculate them in Appendix B by using thesmoothing method, as was done in [31].

1.1 The Radiative Transport Equation

Radiative transport is a theory that was introduced phenomenologically to describe the propagationof light intensity through the Earth's atmosphere. It has been applied successfully to many otherproblems of wave propagation in complex media. In its simplest form, a(t;x;k) denotes the angularlyresolved energy density dened for wave vector k, position x and time t. Because of interaction withthe inhomogeneous medium through which it propagates, a wave with wave vector kmay be scattered

into any other direction k0, where k =k

jkj . Energy balance leads to the transport equation

@a(t;x;k)

@t+ rk!(x;k) rxa(t;x;k) rx!(x;k) rka(t;x;k) (1.1)

=

ZRn

(x;k;k0)a(t;x;k0)dk0 (x;k)a(t;x;k):

Here n is the dimension of space (n = 2 or 3), !(x;k) is the local frequency at position x of a wavewith wave vector k, the dierential scattering cross-section (x;k;k0) is the rate at which energywith wave vector k0 is converted to wave energy with wave vector k at position x, and

(x;k) =

Z(x;k0;k)dk0 (1.2)

is the total scattering cross-section. The function (x;k;k0) is nonnegative and usually symmetricin k and k0. The left side of (1.1) is the total time derivative of a(t;x;k) at a point moving along atrajectory in phase space (x;k) and the right side describes the eects of scattering.

The transport equation (1.1) is conservative when (1.2) holds because then the total energy isindependent of time: Z Z

a(t;x;k)dxdk = const:

2

Absorption may be accounted for easily by letting the total scattering cross-section be the sum oftwo terms

(x;k) = sc(x;k) + ab(x;k)

where sc(x;k) is the total scattering cross-section given by the right side of (1.2) and ab(x;k) isthe absorption cross section.

Derivations of (1.1) from equations for wave propagation by many authors are nicely presentedin a recent review [22]. (See [23] for references). In our work [23], radiative transport theory forscalar and vector waves, including mode conversion and polarization, was derived in the followingregime:

Distances of propagation L are much larger than the wavelength ,

The medium parameters vary on a scale comparable to the wavelength,

The mismatch between the inhomogeneities and the background medium is small,

Absorption is small.

This regime arises in many physically important situations such as seismic wave propagation, whereteleseismic events can be modeled by radiative transport equations [1, 13].

1.2 Half Space Problems and Boundary Conditions

The radiative transport equation (1.1) has been derived in an unbounded domain. Boundary condi-tions at a at boundary or interface were derived for time harmonic acoustic waves for the monochro-matic transport equation without scattering. They can be used with (1.1) in a domain with curvedboundaries and interfaces varying on a scale large compared to the wave length because of theseparation of scales and weak scattering.

The problem of wave scattering from rough boundaries has been studied extensively [5, 10, 29].We show that scattering from a rough interface can be incorporated in radiative transport theoryunder the following conditions:

Wavelength small compared to the curvature of the mean interface

Roughness height smaller than or comparable to the wavelength

Correlation length of the surface uctuations comparable to the wavelength.

Then the boundary conditions for (1.1) are as follows:Let a1in(x

0;k0) and a2in(x0;k0) be the energy densities of the waves coming to the interface from

above and below, respectively, with x0 being a horizontal coordinate on the rough surface, and k0

the horizontal wave vector. Then the outgoing energy densities are given by

v1k1na1out(x

0;k0) = v1k1njR11(k0)j2a1in(x0;k0) + v2k2njT12(k0)j2a2in(x0;k0) (1.3)

+

Zdp0

v1k1n(p

0)11(x0;k0;p0)a1in(x

0;p0) + v2k2n(p0)12(x

0;k0;p0)a2in(x0;p0)

and

v2k2na2out(x

0;k0) = v2k2njR22(k0)j2a2in(x0;k0) + v1k1njT21(k0)j2a1in(x0;k0) (1.4)

+

Zdp0

v2k2n(p

0)22(x0;k0;p0)a2in(x

0;p0) + v1k1n(p0)21(x

0;k0;p0)a1in(x0;p0)

:

3

1a out

a out 2

1a in

Figure 1.1: Incident, re ected, transmitted, and diusely scattered waves at a at interface.

Flat surfaceRough surface, re ection

coecient correction

Rough surface,

scattering cross-section

Dirichlet 1 (4.27) entry 2; 2 (4.28) entry 2; 2

Neumann 1 (4.27) entry 1; 1 (4.28) entry 1; 1

Interface (3.11) (4.23) (4.22)

Table 1.1: References to the explicit formulas for the re ection and transmission coecients andscattering cross-sections

The rst and second terms on the right in (1.3) and (1.4) represent coherent re ection and trans-mission respectively, and the integrals represent diuse or incoherent scattering. The coecients inthese equations satisfy the energy ux conservation law

1 = jR11(k0)j2 + jT21(k0)j2 +

Zdp0

11(p

0;k0) + 21(p0;k0)

; (1.5)

with a similar relation holding for R22, T12, 12 and 22. This is the analog of (1.2) for surfacescattering. Here vj are the acoustic wave speeds in medium j, and kjn is the normal component ofthe normalized wave vector kj = kj=jkj j where kj has tangential component k0; j = 1; 2.

We derive explicit formulas for the surface dierential cross sections ij and the re ection andtransmission coecients. For simplicity, we consider a homogeneous background medium. A gener-alization to an inhomogeneous medium is presented in [2]. The results remain essentially unchangedbut their derivation is substantially more involved. The equation numbers for the explicit formulasare given in Table 1.1.

Interface conditions (1.3) and (1.4) are derived when the adjacent media are homogeneous. Weexpect that they remain valid when the media are random. This is because waves are not trappednear the boundary when the uctuations of the interface are small. A mathematical justication of

4

this assumption is not available at present.

2 Acoustic Wave Transport

We recall how transport equations for the phase space energy densities are constructed [23, 14, 24].For simplicity we assume here that the space domain is IR3 (n = 3). The acoustic equations for thevelocity v and pressure p are

@v

@t+rp = 0 (2.1)

@p

@t+r v = 0:

This system may be written as a symmetric hyperbolic system (with summation over repeatedindices):

A(x)@u

@t+Dj @u

@xj= 0; (2.2)

where u = (v; p), and x 2 IRn. The matrix A(x) = diag(; ; ; ) is symmetric and positivedenite and the matrices Dj are symmetric and independent of x and t. We consider high frequencysolutions of (2.2). Physically this means that the typical wave length of the initial data is muchsmaller than the overall propagation distance L, so that " = =L 1. The spatial energy densityfor the solutions of (2.2) is given by

E(t;x) = v2

2+p2

2=

1

2(A(x)u(t;x) u(t;x)) = 1

2Aij(x)ui(t;x)uj(t;x) (2.3)

and the ux F(x) by

F i(t;x) = pv =1

2(Diu(t;x) u(t;x)): (2.4)

We have the energy conservation law

@E@t

+r F = 0: (2.5)

and thus the total energy is conserved:

d

dt

ZE(t;x)dx = 0: (2.6)

The high frequency limit "! 0 of the energy density E(t;x) is described in terms of the Wignertransform, which is dened by

W"(t;x;k) =

1

2

n Zeikyu"(t;x "y=2)u"(x+ "y=2)dy; (2.7)

where u"(t;x) is the solution of (2.2). The Wigner transformW" is a 44 Hermitian matrix. Its limitas " ! 0 is called the Wigner distribution and is denoted by W (t;x;k). The limit Wigner matrixis not only Hermitian but also positive denite. The limit energy density and ux are expressed interms of W (t;x;k) by

E(t;x) = 1

2

ZTr(A(x)W (t;x;k))dk

5

and

Fi(t;x;k) =1

2

ZTr(DiW (t;x;k))dk:

The limit Wigner distribution may be decomposed into dierent wave modes in a way thatgeneralizes the plane wave decomposition in a homogeneous medium. The dispersion matrix of thesystem (2.2) is dened by

L(x;k) = A1(x)kiDi =

0BB@

0 0 0 k1=0 0 0 k2=0 0 0 k3=

k1= k2= k3= 0

1CCA : (2.8)

It has one double eigenvalue !1 = !2 = 0 and two simple eigenvalues

!f = vjkj ; !b = vjkj ; (2.9)

where jkj =qk21 + k22 + k23 and v is the sound speed

v =1p: (2.10)

The corresponding basis of eigenvectors is

b1 =1p(z(1)(k); 0)t; b2 =

1p(z(2)(k); 0)t;

bf = (kp2;

1p2

)t; bb = (kp2; 1p

2)t; (2.11)

where the vectors k, z(1)(k) and z(2)(k) form an orthonormal triplet. The eigenvectors b1(k) andb2(k) correspond to transverse advection modes, orthogonal to the direction of propagation. Thesemodes do not propagate because !1 = !2 = 0. The eigenvectors bf (k) and bb(k) represent forwardand backward acoustic waves, which are longitudinal , and which propagate with the sound speed vgiven by (2.10).

The limit Wigner distribution matrix W (t;x;k) has the form [23]:

W (t;x;k) =2X

=1

W ij(t;x;k)b

i(k)bj(k) + af (t;x;k)bf (k)bf (k) + ab(t;x;k)bb(k)b

b (k): (2.12)

The rst term corresponds to non-propagating modes and may be set to zero without loss of gener-ality. The last two terms correspond to forward and backward propagating sound waves. The scalarfunctions af and ab are related by af (t;x;k) = ab(t;x;k), and af satises the Liouville equation

@a

@t+rk! rxarx! rka = 0: (2.13)

The functions af and ab can be interpreted as phase space energy densities since they are non-negative (because the matrix W (t;x;k) is non-negative) and

E(x) = 1

2

Zdk[af (t;x;k) + ab(t;x;k)] =

Zdkaf (t;x;k):

6

The ux is given by

F =v

2

Zdk[kaf (t;x;k) kab(t;x;k)] = v

Zdkkaf (t;x;k): (2.14)

The radiative transport equation (1.1) arises when the density and compressibility are randomand varying on the scale of the wave length, so we assume they have the forms

! (1 +p"1(

x

")); ! (1 +

p"1(

x

")):

The random processes 1 and 1 are mean zero space homogeneous with power spectral densitiesR, R, and cross spectral density R. The radiative transport equation for a(t;x;k) = af (t;x;k)has the form

@a

@t+vk rxa jkjrxv rka =

v2jkj22

Z(vjkj vjk0j)[a(k0) a(k)]

n(k k0)2R(k k0) + 2(k k0)R(k k0) + R(k k0)

odk0: (2.15)

This equation is of the form (1.1). A systematic derivation using the Wigner distribution is givenin [23].

Equation (1.1) has been derived from equations governing particular wave motions by variousauthors, such as Stott [26], Watson et.al. [32, 33, 19], Barabanenkov et.al. [3], Besieris and Tappert[6], Howe [16], Ishimaru [17] and Besieris et. al. [7], with a recent survey presented in [4]. See also[12] for similar results. These derivations also determine the functions !(x;k) and (x;k;k0) andshow how a is related to the wave eld. In [23], (1.1) and these functions are derived as a specialcase of a more general theory.

3 Flat Boundary

3.1 Re ection and Transmission of Waves in Homogeneous Media

We will derive transport theoretic boundary conditions assuming that the media adjacent to theinterface are homogeneous, with the interface at in this section, and random, in later sections.

In this section we recall brie y the transmission and re ection of time harmonic, acoustic planewaves at a plane interface. The time harmonic solutions of (2.1) with frequency ! satisfy

rp i!v = 0 (3.1)

r v ip = 0:

We consider the high frequency regime, and replace accordingly ! ! !=" with " 1. Then (3.1)becomes

"rp i!v = 0 (3.2)

"r v ip = 0:

Consider a medium composed of two homogeneous half spaces in IRn, characterized by constantdensities and compressibilities i and i, i = 1; 2. Here i = 1 in the upper half space xn > 0 andi = 2 in the lower one xn < 0. The velocity of propagation is vi = 1=

pii in each half space.

The wavenumber jkij of a plane wave is related to the frequency ! by the dispersion relation

! = vijkij for i = 1; 2: (3.3)

7

Any solution of (3.2) in xn > 0 can be decomposed into a sum of incoming and outgoing waves,

ui"(x) = uiI;"(x) + uiR;"(x): (3.4)

When the medium is homogeneous

uiI;"(x) =

Zdk0

(2")(n1)=2exp [i

k0 x0"

] exp [iikinxn"

] i"(k0) bif (k

i) (3.5)

uiR;"(x) =

Zdk0

(2")(n1)=2exp [i

k0 x0"

] exp [iikinxn"

] i"(k0) bif (k

i+): (3.6)

The wave vector k0 is the horizontal component of k and kin, i = 1; 2 is its normal component:

ki = (k0;ikin); kin(k0) =

qjkij2 jk0j2; 1 = 1; 2 = 1 (3.7)

We consider only propagating waves because the energy in evanescent waves is exponentially small inthe high frequency limit, and vanishes as "! 0. Therefore we assume that the normal componentsof the wave vectors k1n and k2n are real. Then the support of the amplitudes i" and

i" is uniformly

inside the ball fjk0j < !=vig. Then uiI;", the Fourier transform of uiI;", and the corresponding

amplitude i"(k0) are related by

uiI;"(k

") = (2")(n+1)=2 i"(k

0) bi(ki) (kn + ikin): (3.8)

An analogous relation holds for the re ected waves.The amplitudes i" and i" are not independent of one another. They are related through the

interface conditions at xn = 0, expressed by the continuity of the normal velocity and pressure

u1n(x0; 0) = u2n(x

0; 0)

p1(x0; 0) = p2(x0; 0):(3.9)

Equations (2.11) and (3.5-3.6) give the following relations for the wave amplitudes (for i 6= j)

(kin)1=2(ki)1i"(k

0) = Ri(k0) (kin)1=2(ki)1i"(k

0) + T j(k0) (kjn)1=2(kj)1j"(k

0): (3.10)

HereRi is the Fresnel re ection coecient for medium i, and T j is the Fresnel transmission coecientfrom medium j into medium i. They are given by (see e.g. [18, 8])

Ri(k0) =jkin ikjn

jkin + ikjn; T i(k0) =

2(ijkinkjn)

1=2

jkin + ikjn; i 6= j: (3.11)

Thus R2(k0) = R1(k0) and T 1(k0) = T 2(k0). Moreover, the conservation of energy ux impliesthat

(Ri(k0))2 + (T i(k0))2 = 1 ; i = 1; 2: (3.12)

The angles of incidence and refraction i and j are related by Snell's law

sin i

vi=

sin j

vj: (3.13)

8

3.2 High Frequency Limit: Re ection and Transmission of Energy

To derive interface conditions on a at a at boundary we consider the Wigner transform of theincoming and outgoing waves. We denote byW [u] the limit as "! 0 ofW"[u"], the Wigner transformof u" dened by (2.7). The directions of the incident and re ected waves are not overlapping. Thenby the general theory [14, 24], they decouple in the high frequency limit "! 0 so that

W [ui](x;k) = W [uiI ](x;k) + W [uiR](x;k): (3.14)

This also may be checked directly using (3.5) and (3.6). We will now show that

W [uiI ](x;k) = W n1[i] (k0;[x0 xnknk0]) (kn + ikin) b

i(ki)bi(ki)

W [uiR](x;k) = W n1[i] (k0;[x0 xnknk0]) (kn ikin) b

i(ki+)bi(ki+);

(3.15)

where W n1[i"] is the limit as "! 0 of the (n 1)-dimensional Wigner transform of i".Let us compute W [uiI ](x;k). By virtue of the relation [14, 24]

W"[f"(:)](x;k) = W"[1

(2")n=2f"(

:

")](k;x);

and (3.8), we have

W"[uiI;"](x;k) =

2"

(2)n

Zdpi"(k

0 +"p0

2)i" (k

0 "p0

2)(kn + "

pn2+ ikin(k

0 + "p0

2)) (3.16)

(kn "pn2+ ikin(k

0 "p0

2)) exp (ip x)bi((k+ "

p

2)i)b

i((k "p

2)i):

From the dispersion relation (3.3) we deduce

(kn + "pn2)2 =

!2

(vi)2 (k0 + "

p0

2)2

(kn "pn2)2 =

!2

(vi)2 (k0 "

p0

2)2:

Then

jkj2 + j"p2j2 =

!2

(vi)2; knpn = k0 p0:

On the other hand one readily checks that for constants c1 and c2 we have

(kn +"pn2

+ c1) (kn "pn2

+ c2) = (kn +c1 + c2

2) (pn c2 c1

"):

By using this relation and the change of variables pn ! "pn in (3.16), we get for " small,

W [uiI ](x;k) 1

(2)n1

Zdp0i"(k

0 +"p0

2)i" (k

0 "p0

2) exp i[p0 (x0 xn

knk0)]

bi(ki)bi(ki) (kn + ikin):

(3.17)

When "! 0 in (3.17) it yields the rst equation in (3.15), and the second equation in (3.15) can bederived similarly.

9

The explicit representation (3.15) implies that the Wigner matrix of the incoming waves has theform

W iI = aiI(x;k)b

i(k)bi(k)

with aiI supported on the wave vectors k with kn = ikin. In the interior, the function aiI satisesthe transport equation

k rxaiI = 0: (3.18)

SimilarlyW i

R = aiR(x;k)bi(k)bi(k):

The function aiR is supported on the hemisphere of outgoing wave vectors with kn = ikin and itsatises the same transport equation (3.18).

Since the coecients Ri and T i do not depend on ", we deduce from (3.10) that

kin(ki)2

W n1[i] =kin(ki)2

jRi(k0)j2 W n1[i] +kjn(kj)2

jT j(k0)j2 W n1[j ]

+

qkin

ki

qkjn

kj[Ri(k0) T j(k0) +Ri(k0) T j(k0)] W n1[i; j ]:

(3.19)

By virtue of (3.15), the rst two terms on the right side of (3.19) are related to the incident energyW [uiI ]. The third term, involving W n1[i; j ], corresponds to the correlation between amplitudesof waves coming to the surface from opposite sides. We assume that they are uncorrelated, so that

W n1[i; j ] = 0: (H1)

Condition (H1) is not satised in general; in particular not for two incoming plane waves. How-ever it is satised when the wave packets arriving at the interface from dierent sides have uncorre-lated random phases. This is the case in the radiative transport regime when waves arriving at theinterface have undergone independent multiple scattering.

When (H1) holds, (3.19) shows that the ai satisfy the interface conditions

vikinai(x0; 0;k0; ikin) = jRi(k0)j2 vikinai(x0; 0;k0;ikin) + jT j(k0)j2 vj kjnaj(x0; 0;k0;jkjn): (3.20)

Note that evanescent waves disappear in the high frequency regime. Therefore aj(x0; 0;k0; kjn) = 0when kjn is imaginary. For incident angles below the critical angle (obtained through Snell's law(3.13)), the re ection coecient jRi(k0)j2 = 1 and no energy is transmitted. These relations alsohold if the two adjacent media are deterministic but inhomogeneous, as is shown in [24]. We expectthat they hold when the adjacent media are random with weak uctuations.

4 Rough Boundaries

4.1 Surface Re ection Operator

Let us assume that the boundary of the domain or the interface between two subdomains is a surfacegiven by (x0; h(x0)). First we describe the re ected and transmitted waves generated by an incidentwave, and then derive boundary conditions for the phase space energy density in the high frequencylimit.

The rst part of this problem has been studied at length in the past. (See [5, 11, 29].) Forsimplicity we deal rst with re ected waves without transmission, so the superscript corresponding

10

to the medium can be dropped. Consider an incident plane wave, with k00 the horizontal componentof the wave vector and with amplitude " = 1:

in(x) = exp [ik00 x0 ikn(k00)xn] b(k):

In the region xn > hmax, the scattered wave can be decomposed into a sum of outgoing plane waves:

sc(x) =

Zdk0 S(k0;k00) exp [ik

0 x0 + ikn(k0)xn] b(k+):

The kernel S(k0;k00) is the scattering amplitude for waves of incident horizontal wave number k00scattered to waves of horizontal wave number k0. It is determined by the height h(x0) of the surface.S can be obtained by solving an integral equation that follows from the boundary conditions.

Let the boundary vary on the scale of the wavelength, let " be proportional to the wave length,and let the scaled boundary h" be given by

h"(x0) = "h(

x0

"): (4.1)

Here h is a bounded function. This scaling implies that the roughness height, the scale length ofthe boundary, and the wavelength are all of the same order. Then S is independent of ". In thehigh frequency limit an incident wave has scaled horizontal wave vector k0=". Then the change ofvariables x! x=" yields the same equation for S as when " = 1. However, the incoming wave stilldepends on " through the amplitude ".

In order to express conservation of energy ux at the boundary in a natural way we introducethe amplitudes for energy uxes:

~" =qvkn "; ~" =

qvkn ":

These are energy ux densities, as can be seen from the expression (2.14) for the total ux.Consider the general incoming and outgoing plane waves given by (3.5), (3.6) with the incident

and re ected amplitudes related by

~"(k0) =

Zdk00 S(k

0;k00)~"(k00): (4.2)

The incoming and outgoing Wigner transforms are given in (3.15) with W n1[~"] being the knownincoming energy ux. The unknown W n1[ ~] is given by

W n1[ ~](k0;x0) = lim"!0

Zdp0 exp [ip0 x0]

Zdk00S(k

0 "p0

2;k00)~"(k

00)

Zd~k00S(k

0 + "p

2; ~k00)~"(

~k00)

=

Zdp0 exp [ip0 x0]

Zdk00S(k

0;k00)~"(k00)

2 +O(");

since S does not depend on ". It is clear from this expression that we cannot write W n1[ ~] only interms of W n1[~] for a general scattering kernel S.

We consider random boundaries that are stationary, i.e. statistically invariant under translation.The physical problem is not changed by translation and the incident and re ected waves change phaseonly. Then the scattering operator for a surface shifted by a vector d0 must satisfy the relation

Sh(x0d0)(k0;k00) = exp [i(k0 k00) d0]Sh(x0)(k0;k00): (4.3)

11

Since hh(x0)i, the mean over all realizations, is a constant, we conclude from (4.3) that

hSi(k0;k00) = exp [i(k0 k00) d0]hSi(k0;k00):

Because this relation is true for all shifts d0 we conclude that hSi has the form

hSi(k0;k00) = R(k0)(k0 k00): (4.4)

Here R(k0) is the coherent re ection coecient.Consider now the uctuation S of the scattering amplitude:

S(k0;k0) = R(k0)(k0 k00) + S(k0;k00): (4.5)

The product of two uctuations are given by

(k0;k00;a0;a00) = S

k0 a0

2;k00

a002

S

k0 +

a0

2;k00 +

a002

:

Again, because of translational invariance, the average of is proportional to (a0a00). Thus thereexists a kernel E such that:

hi(k0;k00;a0;a00) = E(k0;k00;a0)(a0 a00): (4.6)

Conservation of energy means that for every incident eld ~(k0),Zj~(k0)j2dk0 =

Zj~(k0)j2dk0:

Squaring and averaging (4.2), using (4.5), yield then

jR(k0)j2 +Zjk0

0j<!=v

(k00;k0) dk00 = 1; jk0j < !=v: (4.7)

Here(k00;k

0) = E(k00;k0; 0) (4.8)

is the dierential cross section for the incoherent or diuse surface scattering.We are now in a position to calculate the average energy associated with the re ected waves in

terms of the incident energy. We have

hW n1[ ~]i(k0;x0)= lim

"!0

Zdp0 exp [ip0 x0]h

Zdk00S(k

0 "p0

2;k00)~"(k

00)

Zd ~k00S(k

0 + "p0

2; ~k00)~"( ~k00)i

= lim"!0

Zdp0 exp [ip0 x0]

Zdk00

Zd ~k00hS(k0 "

p0

2;k00)S(k

0 + "p0

2; ~k00)i~"(k00)~"( ~k00):

Because of (4.4)(4.6) we have

< S(k0 "p0

2 ;k00)S(k

0 + "p0

2 ;~k00)i = jR(k0)j2(k0 "p

0

2 k00)(k0 + "p

0

2 ~k00)

+ E(k0; k00+ ~k002 ; "p0) (k00 ~k00 "p0) + O(")

12

since R is independent of ". Since E is also independent of ", we get

hW n1[ ~]i(k0;x0) = jR(k0)j2hW n1[~]i(k0;x0)+

Zdp0 exp [ip0 x0]

Zdk00E(k0;k00 + "p0; 0)~"(k

00)~"(k

00 "p0) + O(")

= jR(k0)j2hW n1[~]i(k0;x0) +Zdk00 (k

0;k00)hW n1[~]i(k00;x0):

This implies the following boundary condition for the average scalar energy a(x;k) at xn = 0:

vkn(k0) a(x0; 0;k0; kn) = jR(k0)j2vkn(k0) a(x0; 0;k0;kn)

+

Zdk00 (k

0;k00) vkn(k00) a(x

0; 0;k00;kn(k00)); kn > 0:(4.9)

Here R(k) dened by (4.4) is the re ection coecient and (k;k0) dened by (4.8) is the dierentialscattering cross-section of the boundary.

4.2 Re ection and Transmission at a Rough Interface

Let us now consider two homogeneous half spaces separated by a rough interface h. The scalarscattering operator S must be replaced by a two by two matrix-valued operator (Sij)1i;j2. Whena wave of unit amplitude is incident in medium j, Sij determines the amplitude of the wave scatteredinto medium i. As before, we introduce the ux amplitudes ~i" and

~i":

~i" =qvikin

i";

~i" =qvikin

i" =

p!

ki

qkin

i": (4.10)

We consider general incoming and outgoing plane waves given by (3.5) and (3.6), with the incidentand re ected amplitudes related by

~i"(k0) =

Zdk00 S

ii(k0;k00)~i"(k

00) +

Zdk00 S

ij(k0;k00)~j"(k

00):

Again we deduce from translational invariance that

Sij(k0;k00) = Rij(k0)(k0 k00) + Sij(k0;k00): (4.11)

Moreover,hi;kli(k0;k00;a0;a00) = E i;kl(k0;k00;a0)(a0 a00); (4.12)

where

i;kl(k0;k00;a;a0) = Sikk0 a0

2;k00

a002

Sil

k0 +

a0

2;k00 +

a002

:

Therefore we have

hSik(k0 "p0

2 ;k00)S

il(k0 + "p0

2 ;~k00)i = Rik(k0)R

il(k0)(k0 "p

0

2 k00)(k0 + "p

0

2 ~k00)

+ E i;kl(k0; k00+ ~k002 ; "p0) (k00 ~k00 "p0) + O(")

Here Rik(k0) are the re ection coecients.We denote the dierential scattering cross-sections by

i;kl(k0;k00) = E i;kl(k0;k00; 0): (4.13)

13

Then as above, we obtain

hW n1[ ~i]i(k0;x0) =2X

k;l=1

Rik(k0)Ril(k0)W n1[~k; ~l](k0;x0)

+2X

k;l=1

Zdk00

i;kl(k0;k00)Wn1[~k; ~l](k00;x0):

When the decorrelation condition (H1) stated in the previous section holds, we let ij = i;jj andwe get the incoming-outgoing energy ux relations

vikin(k0) ai(x0; 0;k0; ikin) =

2Xj=1

jRij(k0)j2 vj kjn(k0) aj(x0; 0;k0;jkjn)

+2X

j=1

Zdk00

ij(k0;k00) vj kjn(k

00) a

j(x0; 0;k00;jkjn(k00)):(4.14)

Here Rij is dened by (4.11) and ij by (4.13).

4.3 Re ection and Transmission from Small Amplitude Rough Interfaces

We now specialize to re ection and transmission at a rough interface with small random uctuations.The interface is given by xn = "h(x

0

" ), where is another small parameter. The random processh(y0) has mean zero and is stationary, with covariance function Q(y0) dened by

hh(x0 + y0)h(x0)i = Q(y0):

The power spectral density Q(p) of the surface uctuations is the Fourier transform of Q:

Q(y0) =

Zdp0

(2)n1eip

0y0

Q(p0): (4.15)

To simplify the calculations, we assume that the scattering operator S, a 2 2 matrix, has anexpansion of the form

S = S0 + S1 + 2S2 + o(2): (4.16)

Here S0(k0;k00) = R(k0)(k0 k00) is the scattering operator of the unperturbed at surface. Clearly

S1 has mean zero because it is linear in h. Then

R(k0) = R0(k0)(k0 k00) + 2R2(k

0)(k0 k00) + o(2)

S = S1 +O(2):(4.17)

Then (4.13) implies that depends only on S1 up to o(2). Moreover R depends only on S2 through

its statistical average R2.The details of the small computation of the re ection and transmission coecients is given in

Appendix A. The results are as follows. The leading order term is

R0 =

0B@ R TT R

1CA : (4.18)

14

The Fresnel coecients R and T are given in (3.11). They can also be written in the form

R =21 2221 + 22

; T =21221 + 22

; (4.19)

where

i =

skini:

Now we dene the auxiliary matrices P , M and N by

P =

0B@ 1 2

11 1

2

1CA ; M(p0;k0) =

0BB@ (k1)2 p0 k0

k1n(k0)

1(k0)

(k2)2 p0 k0k2n(k

0)2(k

0)

k1n(k0)1

1 (k0) k2n(k0)1

2 (k0)

1CCA (4.20)

and N(p0;k0) = P1(p0)V0M(p0;k0); where

V0 =

0B@ 1 0

0 1

1CA :

Then Bij is given by B(p0;k0) = 12 [R0(k

0)N(p0;k0) +N(p0;k0)R0(k0)].

Bij =(p0 k0)i(p0)j(k0) + ij(1 2)12(p

0)12(k0)i0(k

0)j0(p0)

(21 + 22)(p0)(21 + 22)(k

0); (4.21)

where i0 = 2 if i = 1 and vice-versa, and

(p0 k0) =(k2)2 k0 p0

2 (k1)2 k0 p0

1:

The interface conditions (3.9) are continuity of the normal velocity and pressure. Then from(4.13) and (4.17) we nd, after a long computation given in Appendix A, that the scattering cross-section tensor is

ij(p0;k0) = 4B2ij(p

0;k0)Q(p0 k0): (4.22)

The part R2 of the coherent re ection coecient may be split into a sum of two terms

R2 = R2;0 +R2;1;

which are given by

R2;0 =T2((k1)2 (k2)2) Q(0)

0B@ 0 1

1 0

1CA

R2;1(p0) = 2

Zdk0N(p0;k0)B(k0;p0)Q(p0 k0):

(4.23)

Conservation of energy ux (4.7) has to be satised at the interface. This means that energyincident upon the interface is either re ected, transmitted, or scattered. From (4.17)

jRij j2 = jRij0 + 2Rij

2 j2 = (Rij0 )

2 + 22Rij0 R

ij2 + o(2): (4.24)

15

Therefore up to o(2) conservation of energy ux for each p0 is given by

1 = jR11(p0)j2 + jR21(p0)j2 + 2Zdk0[11(k0;p0) + 21(k0;p0)] (4.25)

1 = jR22(p0)j2 + jR12(p0)j2 + 2Zdk0[22(k0;p0) + 12(k0;p0)]:

In the limit = 0 this simplies to (3.12). The 2 terms in (4.25) give the identity

0 = 2(T R212;0 +RR11

2;1 + T R212;1) +

Zdk0[11(k0;p0) + 21(k0;p0)]

0 = 2(T R122;0 RR22

2;1 + T R122;1) +

Zdk0[22(k0;p0) + 12(k0;p0)]:

A straightforward though tedious computation shows that this is indeed true when we use theformulas given above. Note that the scattering cross section (4.22) is positive for every incident andre ected or transmitted angle. Therefore, when the incident angle is smaller than the angle of totalre ection for the at interface, only diuse energy is transmitted.

To summarize, we have derived boundary conditions for the transport equation (3.18) in twoadjacent homogeneous half spaces from the plane wave decomposition (3.5) and (3.6) for both atand rough interfaces, with small uctuations. For slowly varying heterogeneous media with at orsmoothly varying interfaces such conditions are derived in detail in [24]. For weakly uctuatingrandom media or heterogeneous media with random rough interfaces no such derivation has beengiven. In the high frequency limit the boundary conditions for the transport equation must remainvalid, but this is dicult to justify. When the uctuations of the interface are small, the interaction ofthe volume scattering with the surface uctuations must be of higher order and therefore negligible.

4.4 Derivation of the Dirichlet and Neumann Boundary Conditions

When 1=2 ! 0, the interface corresponds to a hard boundary with vanishing normal velocityand Neumann boundary condition for waves incident from medium 1, and to a soft boundary withvanishing pressure and Dirichlet boundary condition for waves incident frommedium 2. ThenR ! 1,T ! 0, and

R0 =

0B@ 1 0

0 1

1CA :

Moreover

S1(p0;k0) =

0BB@ 2i (k1)2 p0 k0

(k1n(p0)k1n(k

0))1=2h(p0 k0) 0

0 2i(k2n(p0) k2n(k0))1=2 h(p0 k0)

1CCA : (4.26)

As is to be expected in this limit, R2;0 = 0 and R2 = R2;1 is given by

R2(p0) =

0BB@ 2

Zdk0

((k1)2 k0 p0)2k1n(k

0)k1n(p0)

Q(p0 k0) 0

0 2

Zdk0k2n(k

0)k2n(p0)Q(p0 k0)

1CCA : (4.27)

The dierential scattering tensor , deduced from the expression for S1, is

(p0;k0) =

0BB@ 4

((k1)2 k0 p0)2k1n(k

0)k1n(p0)

Q(p0 k0) 0

0 4k2n(k0)k2n(p

0)Q(p0 k0)

1CCA : (4.28)

16

1

γ

γ

γ

2Ω

Ω1

2

0

Figure 5.1: Geometry of the media

Here 11 is the dierential scattering cross-section for the Neumann problem in the rst medium,and 22 is the dierential scattering cross-section for the Dirichlet problem in the second medium.Conservation of energy ux also holds:

0 = 2Rii0R

ii2;1(p

0) +

Zdk0 ii(p0;k0); i = 1; 2:

5 The Initial Boundary Value Problem for the Radiative Transport

Equation

Now we summarize the initial, boundary, and interface conditions for the radiative transport equationin two adjacent plane layers 1 and 2 with rough boundaries, separated by a rough interface, as isshown in Figure 5.1. Radiative transport theory is valid in the interior of each layer. The boundaryand interface conditions are those for high frequency time harmonic waves derived in the previoussections. We now use them in the time domain, assuming that time variations in the transportregime are slow compared to the oscillation frequency "1. This is so if the frequency spread of thewaves about a carrier frequency is narrow. At the outer boundaries we impose a Neumann conditionat 0 and a Dirichlet condition at 2.

The transport equation in i is

@ai(t;x;k)

@t+ viki rxa

i(t;x;k) =

ZR3

i(x;k;k0)hai(t;x;k0) ai(t;x;k)

idk0; i = 1; 2: (5.1)

The scattering cross-sections are given by

i(x;k;k0) =M(vi)2jkj2

2(vijkj vijk0j) (5.2)

n(k k0)2Ri

(k k0) + 2(k k0)Ri(k k0) + Ri

(k k0)o:

The mean boundaries 0, 1 and 2 are the at surfaces xn = L0, xn = 0 and xn = L2

respectively. At 0 we require p = 0 and at 2, vn = 0. Then the conditions at 0 and 2 are givenrespectively by

a1(t;x0; L0;k0; kn) =

h1 2R11

2;1(k0)2

ia1(t;x0; L0;k

0;kn)+ 2

Zdp011(k0;p0)a1(t;x0; L0;p

0; p1n)(5.3)

17

a2(t;x0;L2;k0; kn) =

h1 2R22

2;1(k0)2

ia2(t;x0;L2;k

0;kn)+ 2

Zdp022(k0;p0)a2(t;x0;L2;p

0; p2n):(5.4)

Here kn < 0, pn > 0 at 1, and kn > 0, pn < 0 at 2, so that the part on the left representsthe outgoing or scattered wave, and the part on the right corresponds to the incoming waves. Thematrices R2;1 and appearing in (5.3) and (5.4) are given by (4.27) and (4.28).

The interface condition at 1 for the outgoing eld a1(t;x0; 0;k0; k1n) with kn > 0 is

a1(t;x0; 0;k0; k1n)=jRj2 + 2RR11

2;12a1(t;x0; 0;k0;k1

n) +

jT j2 + 2T (R12

2;0+R12

2;1)2a2(t;x0; 0;k0; k2

n)

+2Z

dp011(k0;p0)a1(t;x0; 0;p0;p1n) + 2

Zdp012(k0;p0)a2(t;x0; 0;p0; p2

n): (5.5)

Here p1n < 0 and p2n > 0 correspond to incoming waves. The condition on a2 is similar to (5.5).The matrices R2;0, R2;1 and are given by (4.23) and (4.22). These interface conditions are validunder the assumption that waves incident on the interface from the two sides are uncorrelated.This assumption is valid provided that the widths of the layers L1, L2 are much larger than thecorresponding scattering mean free paths

lj =vj

j; j = 1; 2:

The total scattering cross-section j is dened by (1.2).

Acknowledgement

We thank John G. Watson for the use of his calculations in Appendix B.2. The work of Bal,Keller and Papanicolaou was supported by AFOSR grant F49620-98-1-0211 and by NSF grantDMS-9622854. The work of Ryzhik was supported in part by the MRSEC Program of the NationalScience Foundation under Award Number DMR-9400379.

A Derivation of the Interface Conditions

We shall now calculate the re ection coecients (4.18) and (4.23), and the scattering cross sectiontensor (4.22). We note rst that the normal vector to the surface [x

0

" ; h(x0

" )] is

x0

"; h(

x0

")

= C

ez rh"(x

0

")

= C

ez (rh)(x

0

")

; (A.1)

where C is a normalization constant. Using the plane wave decomposition (3.5) and (3.6), thedenition (2.11) of the eigenvector bf , and the ux amplitudes (4.10), we obtain

Zdk0ei

k0x0

" ei1k1

nh(x

0

")k

0 0 1k1nnk1n

1 ~1" +

Zdk0ei

k0x0

" ei1k1

nh(x

0

")k

0 0 + 1k1nnk1n

1 ~1"

=

Zdk0ei

k0x0

" ei2k2

nh(x

0

")k

0 0 2k2nnk2n

2 ~2" +

Zdk0ei

k0x0

" ei2k2

nh(x

0

")k

0 0 + 2k2nnk2n

2 ~2"

Zdk0ei

k0x0

" ei1k1

nh(x

0

") 1

1 ~1" +

Zdk0ei

k0x0

" ei1k1

nh(x

0

") 1

1~1"

=

Zdk0ei

k0x0

" ei2k2

nh(x

0

") 1

2 ~2" +

Zdk0ei

k0x0

" ei2k2

nh(x

0

") 1

2~2" :

(A.2)

18

Here i is dened by

i =

skini:

The incoming energy ux amplitudes ~i" are not random.To calculate the re ected wave amplitudes ~i" using a Born expansion in the height of the rough

boundary, we need the following intermediate expansions:

exp [iknh(x0

")] = 1 + iknh(

x0

") 1

2(kn)

2h2(x0

") 2 + o(2)

exp [iknh(x0

")](k n)C1 = kn +

i(kn)

2h(x0

") k0 rh(x

0

")

+

1

2(kn)

3h2(x0

") iknh(

x0

")k0 rh(x

0

")

2 + o(2)

~i" = ~";0 + ~";1 + 2 ~";2 + o(2):

(A.3)

Using (A.3) in (A.2), we nd at zeroth order in

~1";01 ~1"1 = ~2";02 + ~2"2

~1";011 + ~1"

11 = ~2";0

12 + ~2"

12 :

(A.4)

We can now invert (A.4) to obtain

~";0 = R0 ~" =

0B@ R TT R

1CA ~": (A.5)

Here ~" = (~i") and~";k = (~i";k), with k = 0; 1; 2, are two-component vectors, and R and T are

given by

R =21 2221 + 22

; T =21221 + 22

: (A.6)

At the rst order in in equation (A.2) we obtain

Zdk0 exp [i

k0 x0"

]

( ik1nh(

x0

") k0 (rh)(x0" )

k1n

!1(~

1" +

~1";0) + 1 ~1";1

)

=

Zdk0 exp [i

k0 x0"

]

( ik2nh(

x0

") k0 (rh)(x0" )

k2n

!2(~

2" +

~2";0) 2 ~2";1

)

Zdk0 exp [i

k0 x0"

]

ik1nh(

x0

")1

1 (~1" +~1";0) + 1

1~1";1

=

Zdk0 exp [i

k0 x0"

]

ik2nh(

x0

")1

2 (~2" ~2";0) + 12

~2";1

:

(A.7)

Denoting by h the Fourier transform of h, we have

h(x0

") =

Zexp (i

0 x0"

)h(0)d0; (rh)(x0

") =

Zi0 exp (i

0 x0"

)h(0)d0:

19

Now we replace ~";0 in (A.7) by its expression (A.5), and compute the Fourier transform of (A.7)with respect to x0, denoting the conjugate variable p0:

1 ~1";1(p

0) +

Zdk0h(p0 k0) 1 (ik

1n i

k0 (p0 k0)

k1n) ((1 +R)~1" + T ~2")

= 2 ~2";1(p0) +Zdk0h(p0 k0) 2 (ik

2n i

k0 (p0 k0)

k2n) (T ~1" + (1R)~2")

11

~1";1(p0) +

Zdk0h(p0 k0) 1

1 ik1n ((1 +R)~1" + T ~2")

= 12

~2";1(p0) +

Zdk0h(p0 k0) 1

2 ik2n (T ~1" + (1 +R)~2"):

(A.8)

For m = 1; 2 we have

ikmn ik0 (p0 k0)

kmn= i

(km)2 k0 p0kmn

:

Using the denition of M given in (4.20) we can rewrite (A.8) in vector form:

P (p0) ~";1(p0) =

Zdk0 ih(p0 k0)(M(p0;k0)~"(k

0) + V0M(p0;k0) ~";0)(k0): (A.9)

From (A.6) we have

1 (1 +R) =231

21 + 22; 2 (1R) =

23221 + 22

along with similar expressions for T . Therefore, from (A.8) we get

(1 ~1";1 + 2 ~

2";1)(p

0) = 2i

Zdk0h(p0 k0)(p0 k0) (1 ~

1" + 2 ~

2")

(21 + 22)(k0)

(11

~1";1 12

~2";1)(p0) = 2i

Zdk0h(p0 k0)(1 2)

(122 ~

1" 2

21 ~

2")

(21 + 22)(k0)

(A.10)

where

(p0 k0) =(k2)2 k0 p0)

2 ((k1)2 k0 p0)

1:

Using the solution (A.6) of (A.5) we obtain in vector form

(~";1)(p0) =

Zdk0S1(p

0;k0)(~")(k0) = 2i

Zdk0h(p0 k0)B(p0;k0)(~")(k

0) (A.11)

where B is given by (4.21).The terms of order 2 in (A.2) yield

Zexp (

k0 x0"

) 1dk0 =

Zexp (

k0 x0"

) 2dk0 (A.12)

where the vectors m are

m1

= m[(1

2(km

n)2h2(

x0

") + ih(

x0

")k0 rh(

x0

"))m(~" + ~m

";0) + (ikm

nh(x0

")

k0 rh(x0

")

kmn

) ~m";1

+ m ~m";2

]

m2

= 1

m[

1

2(km

n)2h2(

x0

")(~m

+ ~m

";0) + (imkm

nh(x0

")) ~m

";1+ ~m

";2]:

(A.13)

20

We average the interface conditions (A.12) before inverting them.We note rst that

hh2(x0

")i =

ZQ(0)d0 = Q(0)

hh(x0

") k0 rh(x

0

")i =

Zik0 0 Q(0)d0 = k0 rQ(0):

We also note the following Fourier transforms for m = 1; 2

F1x0!p0h

Zdk0m(k

0) exp [ik0 x0="](ikmn h(x0

")

k0 rh(x0

")

kmn) ~m";1i

= 2Zdk0m(k

0)(km)2 p0 k0

kmn (k0)

Q(k0 p0)(B(k0;p0)~"(p0))m

F1x0!p0h

Zdk0m(k

0) exp [ik0 x0="]ikmn h(x0

") ~m";1i

= 2Zdk0m(k

0)kmn (k0)Q(k0 p0)(B(k0;p0)~"(p

0))m

Here F1x0!p0 is the inverse Fourier transform with conjugate variables x0 and p0.

Taking the Fourier transform of the interface conditions (A.12), we get the following 22 system

1(p0)[(1

2(k1n)

2Q(0) + ip0 rQ(0))(~1" +~1";0)(p

0)]

2Zdk01(k

0)(k1)2 p0 k0

k1n(k0)

Q(p0 k0)(B(k0;p0)~"(p0))1 + 1(p

0)h~1";2i

= 2(p0)[(1

2(k2n)

2Q(0) + ip0 rQ(0))(~2" ~2";0)(p0)]

2Zdk02(k

0)(k2)2 p0 k0

k2n(k0)

Q(p0 k0)(B(k0;p0)~"(p0))2 2(p

0)h~2";2i

11 (p0)[1

2(k1n)

2Q(0)(~1" +~1";0)]

2Zdk01

1 (k0)k1n(k0) Q(p0 k0)(B(k0;p0)~"(p

0))1 + 11 (p0)h~1";2i

= 12 (p0)[1

2(k2n)

2Q(0)(~2" +~2";0)]

+2

Zdk01

2 (k0)k2n(k0) Q(p0 k0)(B(k0;p0)~"(p

0))2 + 12 (p0)h~2";2i:

(A.14)

We solve (A.14) for h~";2i and write the result in terms of the tensor R2. Then we split the tensorR2 into two parts, R2 = R2;0 +R2;1, and we obtain the formulas (4.23).

B Grazing Angles and the Smoothing Method

As has been pointed out by several authors [28, 10, 30, 31] the re ection coecient blows up asthe incidence direction becomes parallel to the mean surface in the case of the Neumann boundarycondition. As can be seen from (4.27), the term R11

2;1 is innite for grazing angles, since kn(p0) = 0.

This shortcoming can be avoided by using the smoothing method instead of the Born expansion.For the Neumann boundary condition, the smoothing method gives the re ection coecient [31]

R(k) =1 + 2Q1 2Q (B.1)

21

where

Q = Zdk0

((k1)2 k0 p0)2k1n(k

0)k1n(p0)

Q(p0 k0): (B.2)

The Born expansion for the re ection coecient gives, when (4.27) is used,

R(k) = 1 + 22Q+O(3):

Thus, (B.1) agrees with the Born expansion to O(3), but it is not singular at grazing angles. Notethat Q is negative since it involves only propagating waves for which kin is real. Therefore there ection coecient R given by (B.1) ranges between 0 and 1.

The smoothing method gives for the scattering cross-section [31]

11ren(p0;k0) =

4

(1 2Q(k0))2((k1)2 k0 p0)2k1n(k

0)k1n(p0)

Q(p0 k0): (B.3)

This coecient is nonsingular at grazing angles.The reason why the second Born approximation (4.27) is singular at grazing incidence is that

it includes only single and double scattering. At grazing incidence, all the singly scattered wavescombine in phase in the direction of the re ected wave, and their sum is innite. Double scatteringalone does not reduce these waves enough to make the sum converge. The smoothing method takesinto account some multiple scattering of all orders|that corresponding to the ladder diagrams ofdiagrammatic resummation methods [25, 27]|and this reduces the scattered waves enough to maketheir sum converge.

B.1 Smoothing Method for Dirichlet and Neumann Boundary Conditions

The smoothing method is an alternative to the Born expansion which includes ladder diagrams. Wewill present it in a form suitable for re ection from a random surface with a Dirichlet or Neumannboundary condition, as in [31].

Vanishing pressure (soft boundary condition) and vanishing normal velocity at the rough bound-ary (hard boundary condition) give equations of the form

K() ~ = K()~: (B.4)

Here ~ and ~ are the incident and re ected wave amplitudes and the sign is for the Dirichlet and+ for the Neumann boundary condition. The operator K for the Dirichlet and Neumann conditionsis

[KD()~g] (p0) = (2M)n

Zdx0eip

0x0Zdk0eik

0x0eiknh(x0)1(k0)~g(k0)

[KN ()~g] (p0) = (2M)n

Zdx0eip

0x0Zdk0eik

0x0eiknh(x0)(1 +

k0 rh(x0)kn

)(k0)~g(k0):

Expansion in powers of gives

K() = 1 + K1 + 2K2 +O(3): (B.5)

Plugging (B.5) into (B.4) yields

(I K1 + 2K2) ~ = (I + K1 + 2K2)~+O(3): (B.6)

22

We denote by < f > the average over realizations of a random quantity f . We denote the uctuation by f 0 = f < f >. We now use this notation to invert (B.6) approximately. Note thatthe incoming energy ux amplitude ~ is independent of h. We also note from (4.26) that < K1 >= 0.

Averaging (B.6) over the realizations yields

(I + 2 < K2 >) < ~ > < K 01~0 > +2 < K 0

2~0 >= (I + 2 < K2 >)~+O(3): (B.7)

Subtracting (B.7) from (B.6) and keeping terms through O() we get an expression for the uctuatingpart of the re ected wave amplitude

~0 = (K 01 <

~ > K 01 ~) +O(2): (B.8)

Finally we use (B.8) for ~0 in (B.7) and obtain

([I + 2 < K2 >] 2 < K 01K

01 >) <

~ > = ([I + 2 < K2 >] + 2 < K 01K

01 >)~+O(3): (B.9)

Dropping the error and noting that < K2 > and < K 01K

01 > are scalars, we get for the mean re ected

amplitude

< ~ >=(1 + 2 < K2 > +2 < K 0

1K01 >)

1 + 2 < K2 > 2 < K 01K

01 >

~:

The factor < K2 > is not singular at grazing angles, while < K 01K

01 > is singular in the Neumann

case. We can therefore divide both the numerator and denominator in (B.9) by 1 + 2 < K2 >, usethe approximation [1 + 2 < K2 >]

1 1 2 < K2 > and obtain to order 3,

< ~ >=(1 + 2 < K 0

1K01 >)

1 2 < K 01K

01 >

~: (B.10)

Upon dening Q =< K 01K

01 >, (B.10) becomes (B.2) for the Neumann boundary condition. For the

Dirichlet condition the smoothing method is not necessary since the Born approximation is boundedindependently of the angle of incidence.

Using (B.8) for ~0 and (B.10) for < ~ > allows us to recompute the correlation function< W n1[ ~] >. By doing so we check that the recomputed scattering cross-section for the Neumannboundary condition is (B.3). We can interpret the smoothing method as a way of renormalizing theresults of the Born approximation.

B.2 Extension to Interface Problems

We will now use the smoothing method to obtain nonsingular interface re ection and transmissioncoecients, following unpublished calculations of John G. Watson. Continuity of pressure andnormal velocity give interfacial conditions of the form0

B@ K11()1 K12()2K21()1

1 K22()12

1CA ~ =

0B@ K11()1 K12()2

K21()11 K22()1

2

1CA ~: (B.11)

Here ~ = (~i), ~ = (~i), i = 1; 2, and we have used the following denitions:

Kij() = 1 + Kij1 + 2Kij

2 +O(3)hKij

1 () i(p0) =

Zh(p0 k0)mj

i (p0;k0) (p0)dp0h

< Kij2 () > ( )

i(k0) =

1

2(mj

1)2Q(0) (k0):

(B.12)

23

Only the real part of K1j has been written since the imaginary part does not contribute, and

mj1 = ijkjn; mj

2(p0;k0) = i

(kj)2 p0 k0jkjn(k0)

:

These expressions are obtained from (A.2).We note that R0 = P1V0P where P is dened in (4.20). Therefore, if we multiply both sides of

(B.11) by P1, we can write it in the form

(I + R1 + 2R2) ~ = R0 (I R1 + 2R2) ~; (B.13)

where R1 and R2 are dened by using (B.12) in (B.11) and adopting matrix notation.We can now use the smoothing method, as in the previous section. With the notation of the

previous section for average and uctuation, we have

~0 = [R01 <

~ > +R0R01~] +O(2); (B.14)

and then up to terms of order 3

< I + 2R2 >< ~ > 2 < R01R

01 ><

~ > = R0 < I + 2R2 > ~+ 2 < R01R0R

01 > ~: (B.15)

Again < R2 > is a local multiplication operator and we can multiply both sides of (B.15) by(I + 2R2)

1. This yields, to order 3

< ~ > = [I 2 < R01R

01 >]

1[R0 + 2 < R01R0R

01 > +2(R0 < R2 > < R2 > R0)]~ (B.16)

uniformly in angles, including grazing angles.The matrices appearing in (B.16) can be related to the quantities already computed in section

4.3 as follows. We deduce from (A.9) given in Appendix A that the operator R1 = R01 has the form

(R01g)(p

0) =

Zdk0 ih(p0 k0)N(p0;k0)g(k0):

Therefore we have

(I 2 < R01R

01 >) = I + 2W

(R0 + 2 < R01 R0 R

01 >) = R0(I 2V );

where

V (p0) =

Zdk0P1(p0)M(p0;k0)P1(k0)M(k0;p0)Q(p0 k0)

W (p0) =

Zdk0N(p0;k0)N(k0;p0)Q(p0 k0):

We rewrite expression (B.16) for the coherent re ection coecient in the form

< ~ > = [ ~R0 + 2(R0 < R2 > < R2 > R0) ] ~+O(3); (B.17)

where~R0 = (I + 2W )1R0(I 2V ): (B.18)

The dierential scattering cross-sections (4.22) also need to be recomputed. From (B.14) and(B.17) we nd that

~0 = [R01~R0 +R0R

01]+O(2):

24

Born expansion Smoothing Method

Dirichlet B.C.(4.27) entry 2; 2

(4.28) entry 2; 2

(4.27) entry 2; 2

(4.28) entry 2; 2

Neumann B.C.(4.27) entry 1; 1

(4.28) entry 1; 1

(B.1)

(B.3)

Interface Condition(4.23)

(4.22)

(B.17)

(B.20)

Table B.1: Reference to the explicit formulas for the Smoothing Method

Therefore replacing B by

~B(p0;k0) =1

2(N(p0;k0) ~R0(k

0) +R0(k0)N(p0;k0)); (B.19)

we get for the cross-sections the nonsingular result

ijren(p0;k0) = = 4 ~B2

ij(p0;k0)Q(p0 k0): (B.20)

If we expand the inverse in (B.16) to order 3 we get the standard Born expansion

< ~ > = [R0 + 2((R0 < R2 > < R2 > R0)+ < R01R0R

01 > + < R0

1R01 > R0)] ~:

The term < R01R0R

01 > + < R0

1R01R0 > corresponds to R2;1 in (4.23). These coecients blow

up at grazing angles, as in the case of the Neumann boundary condition We note that the term

(R0 < R2 > < R2 > R0), which is proportional to Q(0)

0B@ 0 1

1 0

1CA and corresponds to the

term R2;0, does not vanish. Since it does not depend on the angle of incidence, this term is alwaysbounded and does not need to be renormalized.

To summarize, the re ection and transmission coecients given in Table 1.1, valid in the Bornexpansion, are now replaced by those in Table B.1 given by the smoothing method.

References

[1] K.Aki and R.S.Wu, Multiple scattering and energy transfer of seismic waves separation ofscattering eect from intrinsic attenuation II. Application of the theory to Hindu Kush region,in Seismic wave scattering and attenuation, vol. I (R.S.Wu and K.Aki, eds), 1988, 49-80

[2] G.Bal, G.Papanicolaou and L.Ryzhik, Diuse scattering from a random surface, preprint, 1998.

[3] Yu.Barabanenkov, A.Vinogradov, Yu.Kravtsov and V.Tatarskii, Application of the theory ofmultiple scattering of waves to the derivation of the radiative transfer equation for a statisticallyinhomogeneous medium, Radiozika, 15, 1972, 1852-1860. English translation 1420-1425.

25

[4] Yu.Barabanenkov, Yu.Kravtsov, V.Ozrin and A.Saichev, Enhanced backscattering in optics,Progress in Optics 29, 1991, 67-190.

[5] F.G.Bass and I.M.Fuks, Wave Scattering from Statistically Rough Surfaces, Pergamon, NY,1978.

[6] I.M.Besieris and F.D.Tappert, Propagation of frequency modulated pulses in a randomly strat-ied plasma, Jour. Math. Phys. 14, 1973, 704-707.

[7] I.M.Besieris, W.Kohler and H.Freese, A transport-theoretic analysis of pulse propagationthrough ocean sediments, J. Acoust. Soc. Am., 72, 1982, 937-946.

[8] L.Brekhovskikh, Waves in Layered Media, Academic Press, 1980.

[9] S.Chandrasekhar, Radiative Transfer, Dover, New York, 1960.

[10] J.A.DeSanto, Theoretical methods in ocean acoustics, in Topics in current physics, Vol 8,Chap.2, Springer, New York, 1979, 7-77.

[11] J.A.DeSanto and G.S. Brown, Analytical techniques for multiple scattering from rough surfaces,in Progress in Optics XXIII (E. Wolf, ed.), Elsevier, Amsterdam, 1986, 1-62.

[12] A.Fannjiang and L.Ryzhik, Phase space transport theory for sound waves in random ows,submitted to SIAM, J. App. Math, 1998.

[13] M.C.Fehler and H.Sato, Seismic Wave Propagation and Scattering in the Heterogeneous Earth,AIP Press, 1998.

[14] P.Gerard, P.Markowich, N.Mauser and F.Poupaud, Homogenization limits and Wigner trans-forms, Comm. Pure Appl. Math., 50, 1997, 323-380.

[15] J.J.Greet and M.NietoVesperinas, Field theory for the generalized bidirectional re ectivity:derivation of Helmholtz's reciprocity principle and Kirchho's law, submitted to Jour. Acoust.Soc. Am., 1998.

[16] M.S.Howe, On the kinetic theory of wave propagation in random media, Phil. Trans. Roy. Soc.Lond. 274, 1973, 523-549.

[17] A.Ishimaru, Wave propagation and scattering in random media, vol. II, Academic Press, NewYork, 1978.

[18] L.D. Landau and E.M. Lifschitz, Fluid mechanics; Pergamon Press, NY, 1987.

[19] C.W.Law and K.Watson, Radiation transport along curved ray paths, Jour. Math. Phys., 11,1970, 3125-3137.

[20] L.Margerin, M.Campillo and B. van Tiggelen, Radiative transfer and diusion of waves in alayered medium: a new insight into coda Q, to appear in Geophys. Jour. Int., 1998.

[21] M.Nieto-Vesperinas, Scattering and diraction in physical optics, Wiley, New York, 1991.

[22] M.C.W.van Rossum and Th.M.Nieuwenhuizen, Multiple scattering of classical waves: frommicroscopy to mesoscopy and diusion, to appear in Phys. Rev. E.

26

[23] L.Ryzhik, G.Papanicolaou and J.B. Keller, Transport Equations for Elastic and Other Wavesin Random Media, Wave Motion 24, 1996, 327-370.

[24] L.Ryzhik, G.Papanicolaou and J.B. Keller, Transport equations in a half space, Comm. PDE's,22, 1997, 1869-1911.

[25] P.Sheng, Introduction to wave scattering, localization, and mesoscopic phenomena, AcademicPress, 1995.

[26] P.Stott, A transport equation for the multiple scattering of electromagnetic waves by a turbulentplasma, Jour. Phys. A, 1, 1968, 675-689.

[27] B.A.van Tiggelen and R.Maynard, Reciprocity and coherent backscattering of light, in `Wavesin Complex Media', G. Papanicolaou editor, IMA Volume 96, Springer Verlag, 1997, 247-271.

[28] V.Twersky, On scattering and re ection of sound by rough surfaces, J. Acoust. Soc. Am. 29,1957, 209-225.

[29] A.Voronovich, Wave Scattering from Rough Surfaces, Springer Verlag, 1991.

[30] J.G.Watson and J.B.Keller, Re ection, scattering, and absorption of acoustic waves by roughsurfaces, J. Acoust. Soc. Am. 74 (6), 1983, 1887-1894.

[31] J.G.Watson and J.B.Keller, Rough surface scattering via the smoothing method, J. Acoust.Soc. Am. 75 (6), 1984, 1705-1708.

[32] K.Watson and J.L.Peacher, Doppler shift in frequency in the transport of electromagnetic wavesin an underdense plasma, Jour. Math. Phys11, 1970, 1496-1504.

[33] K.Watson, Multiple scattering of electromagnetic waves in an underdense plasma, Jour. Math.Phys., 10, 1969, 688-702.

27