MULTISCALE MODEL.SIMUL c › ... › MOVING_MEDIA.pdf · the source s(t,x~), we obtain two wave propagation regimes. The ﬁrst is called the wide beam regime because the cone of

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MULTISCALE MODEL. SIMUL. c� 2019 Society for Industrial and Applied MathematicsVol. 17, No. 1, pp. 31–67

WAVE PROPAGATION AND IMAGING IN MOVING RANDOM

MEDIA⇤

LILIANA BORCEA† , JOSSELIN GARNIER‡ , AND KNUT SOLNA§

Abstract. We present a study of sound wave propagation in a time dependent random mediumand an application to imaging. The medium is modeled by small temporal and spatial randomfluctuations in the wave speed and density, and it moves due to an ambient flow. We develop atransport theory for the energy density of the waves, in a forward scattering regime, within a cone(beam) of propagation with small opening angle. We apply the transport theory to the inverseproblem of estimating a stationary wave source from measurements at a remote array of receivers.The estimation requires knowledge of the mean velocity of the ambient flow and the second-orderstatistics of the random medium. If these are not known, we show how they may be estimated fromadditional measurements gathered at the array, using a few known sources. We also show how thetransport theory can be used to estimate the mean velocity of the medium. If the array has largeaperture and the scattering in the random medium is strong, this estimate does not depend on theknowledge of the statistics of the random medium.

Key words. time dependent random medium, Wigner transform, transport, imaging

AMS subject classifications. 76B15, 35Q99, 60F05

DOI. 10.1137/18M119505X

1. Introduction. We study sound wave propagation in a time dependentmedium modeled by the wave speed c(t, ~x) and density ⇢(t, ~x) that are random per-turbations of the constant values co and ⇢o. The medium is moving due to an ambientflow, with velocity ~v(t, ~x) that has a constant mean ~vo and small random fluctuations.The source is at a stationary location and emits a signal in the range direction denotedhenceforth by the coordinate z, as illustrated in Figure 1.1. The signal is typically apulse defined by an envelope function of compact support, modulated at frequency !o.It generates a wave that undergoes scattering as it propagates through the randommedium. The goal of the paper is to analyze from first principles the net scatteringat long range and to apply the results to the inverse problem of estimating the sourcelocation and medium velocity from measurements of the wave at a remote, stationaryarray of receivers.

Various models of sound waves in moving media are described in [16, Chapter 2]using the linearization of the fluid dynamics equations about an ambient flow, followedby simplifications motivated by scaling assumptions. Here we consider Pierce’s equa-tions [16, section 2.4.6] derived in [19] for media that vary at longer scales than the

⇤Received by the editors June 18, 2018; accepted for publication (in revised form) November 7,2018; published electronically January 3, 2019.

http://www.siam.org/journals/mms/17-1/M119505.htmlFunding: The first author’s research is supported in part by the Air Force O�ce of Scientific

Research under award FA9550-18-1-0131 and in part by the U.S. O�ce of Naval Research underaward N00014-17-1-2057. The third author’s research is supported in part by the Air Force O�ce ofScientific Research under award FA9550-18-1-0217 and NSF grant 1616954.

†Department of Mathematics, University of Michigan, Ann Arbor, MI 48109 ([email protected]).‡Centre de Mathematiques Appliquees, Ecole Polytechnique, 91128 Palaiseau Cedex, France

([email protected]).§Department of Mathematics, University of California at Irvine, Irvine, CA 92697 (ksolna@math.

uci.edu).

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32 LILIANA BORCEA, JOSSELIN GARNIER, AND KNUT SOLNA

receiver array

z~vo

source

Fig. 1.1. Illustration of the setup. A stationary source emits a wave in the range direction z, ina moving medium with velocity ~v(t, ~x) that has small random fluctuations about the constant mean

~vo. The orientation of ~vo with respect to the range direction is arbitrary. The wave is recorded by

a stationary, remote array of receivers.

central wavelength �o = 2⇡co/!o of the wave generated by the source. Pierce’s modelgives the acoustic pressure

(1.1) p(t, ~x) = �⇢(t, ~x)Dt�(t, ~x)

in terms of the velocity quasi-potential �(t, ~x), which satisfies the equation

(1.2) Dt

h 1

c2(t, ~x)Dt�(t, ~x)

i�

1

⇢(t, ~x)r~x ·

h⇢(t, ~x)r~x�(t, ~x)

i= s(t, ~x)

for spatial variable ~x = (x, z) 2 Rd+1 and time t 2 R, with natural number d � 1.Here x 2 Rd lies in the cross-range plane, orthogonal to the range axis z. Moreover,r~x and r~x· are the gradient and divergence operators in the variable ~x and

Dt = @t + ~v(t, ~x) ·r~x

is the material (Lagrangian) derivative, with @t denoting the partial derivative withrespect to time. The source is modeled by the function s(t, ~x) localized at the originof range and with compact support. Prior to the source excitation there is no wave

(1.3) �(t, ~x) ⌘ 0, t ⌧ 0,

but the medium is in motion due to the ambient flow.Sound wave propagation in ambient flows due to wind in the atmosphere or ocean

currents arises in applications like the quantification of the e↵ects of temperaturefluctuations and wind on the rise time and shape of sonic booms [4] or on radio-acoustic sounding [12], monitoring noise near airports [21], acoustic tomography [14],and so on.

Moving media also arise in optics, for example, in Doppler velocimetry or anemom-etry [6, 7], which uses lasers to determine the flow velocity ~vo. This has applicationsin wind tunnel experiments for testing aircraft [10], in velocity analysis of waterflow for ship hull design [13], in navigation and landing [1], and in medicine andbioengineering [15]. A description of light propagation models used in this contextcan be found in [8, Chapter 8].

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MOVING RANDOM MEDIA 33

Much of the applied literature on waves in moving random media considers eitherdiscrete models with Rayleigh or Mie scattering by moving particles [8] or continuummodels described by the classic wave equation with wave speed c(0, ~x�~vt). These useTaylor’s hypothesis [11, Chapter 19], where the medium is “frozen” over the durationof the experiment and simply shifted by the uniform ambient flow. A transport theoryin such frozen-in media is obtained, for example, in [11, Chapter 20] and [16, Chapter8], in the paraxial regime where the waves propagate in a narrow angle cone aroundthe range direction. The formal derivation of this theory assumes that the randomfluctuations of the wave speed are Gaussian and uses the Markov approximation,where the fluctuations are �-correlated in range, i.e., at any two distinct ranges, nomatter how close, the fluctuations are assumed uncorrelated.

In this paper we study the wave equation (1.2) with coe�cients c(t, ~x), ⇢(t, ~x),and ~v(t, ~x) that have random correlated fluctuations at spatial scale ` and temporalscale T . These fluctuations are not necessarily Gaussian. We analyze the solution�(t, ~x) and therefore the acoustic pressure p(t, ~x) in a forward scattering regime,where the propagation is within a cone (beam) with axis along the range direction z.The analysis uses asymptotics in the small parameter " = �o/L ⌧ 1, where L is therange scale that quantifies the distance between the source and the array of receivers.Pierce’s equations (1.1)–(1.2) are justified for small �o/`⌧ 1. By fixing �o/` or lettingit tend to zero, independent of ", and by appropriate scaling of the spatial support ofthe source s(t, ~x), we obtain two wave propagation regimes. The first is called the widebeam regime because the cone of propagation has finite opening angle. The second isthe paraxial regime, where the cone has very small opening angle. We use the di↵usionapproximation theory given in [9, Chapter 6] and [17, 18] to study both regimes andobtain transport equations that describe the propagation of energy. These equationsare simpler in the paraxial case and we use them to study the inverse problem oflocating the source. Because the inversion requires knowledge of the mean velocity~vo of the ambient flow and the second-order statistics of the random medium, we alsodiscuss their estimation from additional measurements of waves generated by knownsources.

The paper is organized as follows. We begin in section 2 with the mathematicalformulation of the problem. Then we state in section 3 the transport equations. Theseequations are derived in section 5 and we use them for the inverse problem in section 4.We end with a summary in section 6.

2. Formulation of the problem. We study the sound wave modeled by theacoustic pressure p(t, ~x) defined in (1.1) in terms of the velocity quasi potential �(t, ~x),the solution of the initial value problem (1.2)–(1.3). The problem is to characterizethe acoustic pressure p(t, ~x) in the scaling regime described in section 2.2 and thenuse the results for localizing the source and estimating the mean medium velocity ~vo.

2.1. Medium and source. The coe�cients in (1.2) are random fields, definedby

~v(t, ~x) = ~vo + V �v~⌫⇣ t

T,~x� ~vot

`

⌘,(2.1)

⇢(t, ~x) = ⇢o exph�⇢⌫⇢

⇣ t

T,~x� ~vot

`

⌘i,(2.2)

c(t, ~x) = coh1 + �c⌫c

⇣ t

T,~x� ~vot

`

⌘i�1/2,(2.3)

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where co, ⇢o are the constant background wave speed and density, ~vo is the constantmean velocity of the ambient flow, and V is a velocity scale (of the order of |~vo|) thatwill be specified later. Equations (2.1)–(2.3) describe a randomly perturbed medium,with typical spatial structures of the size of the order of ` and lifetimes of the orderof T , advected with the ambient flow with mean velocity ~vo. It is possible to considerthat the perturbations are steady, i.e., ⌫⇢ and ⌫c do not depend explicitly on timeand ~⌫ = 0, as discussed briefly in section 3.3. However, we assume a more generalmodel where the perturbations are unsteady and correlated, to be consistent with thebasic equations satisfied by the perturbations [16, Chapter 2], moreover, to capturetypical physical regimes when the medium decorrelates in time. The fluctuations in(2.1)–(2.3) are given by the random stationary processes ~⌫, ⌫⇢, and ⌫c of dimensionlessarguments and mean zero

(2.4) E⇥~⌫(⌧, ~r)

⇤= 0, E[⌫⇢(⌧, ~r)] = 0, E[⌫c(⌧, ~r)] = 0.

We assume that ~⌫ = (⌫j)d+1j=1 , ⌫⇢, and ⌫c are twice di↵erentiable, with bounded deriva-

tives almost surely, have ergodic properties in the z direction, and are correlated, withcovariance entries

(2.5) E⇥⌫↵(⌧, ~r)⌫�(⌧

0, ~r 0)⇤= R↵�(⌧ � ⌧ 0, ~r � ~r 0).

Here the indices ↵ and � are either 1, . . . , d+ 1, or ⇢, or c. The covariance is an evenand integrable symmetric matrix valued function, which is four times di↵erentiableand satisfies the normalization conditions

(2.6) R↵↵(0,0) = 1 or O(1),

Z

Rd⌧

Z

Rd+1

d~rR↵↵(⌧, ~r) = 1 or O(1).

The scale T in definitions (2.1)–(2.3) is the correlation time, the typical lifespan ofa spatial realization of the fluctuations, and ` is the correlation length, the typicallength scale of the fluctuations. The dimensionless positive numbers �v, �⇢, and �cquantify the standard deviation of the fluctuations. They are of the same order andsmall, so definitions (2.2) and (2.3) can be approximated by

⇢(t, ~x) ⇡ ⇢oh1 + �⇢⌫⇢

⇣ t

T,~x� ~vot

`

⌘i, c(t, ~x) ⇡ co

h1�

�c2⌫c⇣ t

T,~x� ~vot

`

⌘i,

with co and ⇢o close to the mean wave speed and density. The exponential in (2.2) andthe inverse of the square root in (2.3) are used for convenience because some importante↵ective properties of the medium are defined in terms of E[log ⇢] and E[c�2], whichare equal to log ⇢o and c�2

o .The origin of the coordinates is at the center of the source location, modeled by

(2.7) s(t, ~x) = �se�i!otS

⇣ t

Ts,x

`s

⌘�(z),

for ~x = (x, z), using the continuous function S of dimensionless arguments and com-pact support. The length scale `s is the radius of the support of s(t, ~x) in cross-rangeand the time scale Ts is the duration of the emitted signal. Note that s(t, ~x) ismodulated by the oscillatory exponential at the frequency !o. We call it the centralfrequency because the Fourier transform of s(t, ~x) with respect to time is supportedin the frequency interval |! � !o| O(1/Ts). The solution �(t, ~x) of (2.2) dependslinearly on the source, so we use �s to control its amplitude.

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To be able to set radiation conditions for the wave field resolved over frequencies,we make the mathematical assumption that the random fluctuations of ~v(t, ~x), ⇢(t, ~x),and c(t, ~x) are supported in a domain of finite range that is much larger than L. Inpractice this assumption does not hold, but the wave equation is causal and withfinite speed of propagation, so the truncation of the support of the fluctuations doesnot a↵ect the wave measured at the array up to time O(L/co).

2.2. Scaling regime. Because the fluctuations of the coe�cients (2.1)–(2.2) aresmall, they have negligible e↵ect on the wave at short range, meaning that �(t, ~x) ⇡�o(t, ~x), the solution of (1.2)–(1.3) with constant wave speed co, density ⇢o, andvelocity ~vo. We are interested in a long range L, where the wave undergoes manyscattering events in the random medium and �(t, ~x) is quite di↵erent from �o(t, ~x).We model this long range regime with the small and positive, dimensionless parameter

(2.8) " =�oL

⌧ 1

and use asymptotics in the limit "! 0 to study the random field �(t, ~x).The relation between the wavelength, the correlation length, and the cross-range

support of the source is described by the positive, dimensionlessparameters

(2.9) � =�o`, �s =

�o`s

,

which are small but independent of ". The positive, dimensionless parameter

(2.10) ⌘ =T

TL

determines how fast the medium changes on the scale of the travel time TL = L/co.The duration of the source signal is modeled by the positive, dimensionless pa-

rameter

(2.11) ⌘s =Ts

TL,

which is independent of ". The Fourier transform of this signal is supported in thefrequency interval centered at !o and of length (bandwidth) O(1/Ts), where

(2.12)1

Ts=

1

⌘sTL⌧

1

"TL=

co"L

=co�o

= O(!o).

Thus, the source has a small bandwidth in the "! 0 limit.Our asymptotic analysis assumes the order relation

(2.13) "⌧ min{�, �s, ⌘, ⌘s},

meaning that we take the limit " ! 0 for fixed �, �s, ⌘, ⌘s. The standard deviationsof the fluctuations are scaled as

(2.14) �c =p"��c, �⇢ =

p"��⇢, �v =

p"��v,

with �c, �⇢, �v = O(1) to obtain a O(1) net scattering e↵ect.

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The ambient flow, due, for example, to wind, has much smaller velocity than thereference sound speed co. We model this assumption with the scaling relation

(2.15) |~vo|/V = O(1), where V = "co.

Although V ⌧ co, the medium moves on the scale of the wavelength over the durationof the propagation

(2.16) V TL = VL

co= "L = �o < `,

so the motion has a O(1) net scattering e↵ect. Slower motion is negligible, whereasfaster motion gives di↵erent phenomena than those analyzed in this paper.

We scale the amplitude of the source as

(2.17) �s =1

"⌘sL

⇣�s"

⌘d

to obtain �(t, ~x) = O(1) in the limit " ! 0. Since (2.2) is linear, any other sourceamplitude can be taken into account by multiplication of our wave field with thatgiven amplitude.

Note that in section 3.2.2 we consider the secondary scaling relation

(2.18) � ⇠ �s ⌧ 1,

corresponding to the paraxial regime, where the symbol “⇠” means of the same order.Moreover, in section 4.3 we assume ⌘/⌘s ⌧ 1 corresponding to a regime of statisticalstability. In this secondary scaling regime we let

|~vo| = O

✓"co⌘�

◆(2.19)

to obtain the distinguished limit in which the medium velocity impacts the quantitiesof interest.

3. Results of the analysis of the wave field. We show in section 5 that inthe scaling regime described in (2.8)–(2.17), the pressure is given by

p(t, ~x) ⇡ i!o⇢o

Z

O

d!dk

(2⇡)d+1

a(!,k, z)p�(k)

e�i(!o+!)t+i~k·~x(3.1)

for ~x = (x, z) and O = {! 2 R} ⇥ {k 2 Rd, |k| < ko}, where the approximationerror vanishes in the limit " ! 0. This expression is a Fourier synthesis of forwardpropagating time-harmonic plane waves (modes) at frequency !o + !, with wave

vectors ~k defined by

(3.2) ~k =�k,�(k)

�, �(k) =

pk2o � |k|2, ko = 2⇡/�o.

The scattering e↵ects in the random medium are captured by the mode amplitudes,which form a Markov process

�a(!,k, z)

�(!,k)2O

that evolves in z, starting from

(3.3) a(!,k, 0) = ao(!,k) =i�sTs`ds2p�(k)

bS(!Ts, `sk).

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This process satisfies the conservation relation

(3.4)

Z

O

d!dk��a(!,k, z)

��2 =

Z

O

d!dk��ao(!,k)

��2 8z > 0.

The statistical moments of�a(!,k, z)

�(!,k)2O

are characterized explicitly in the

limit "! 0, as explained in section 5.7 and Appendix A. Here we describe the expec-tation of the amplitudes, which defines the coherent wave, and the second momentsthat define the mean Wigner transform of the wave, i.e., the energy resolved overfrequencies and direction of propagation.

3.1. The coherent wave. The expectation of the acoustic pressure (the coher-ent wave) is obtained from (3.1) using the mean amplitudes

(3.5) E[a(!,k, z)] = ao(!,k) exp [i✓(!,k)z +D(k)z] .

These are derived in section 5.7.1, with ao(!,k) given in (3.3). The exponentialdescribes the e↵ect of the random medium, as follows.

The first term in the exponent is the phase

(3.6) ✓(!,k) =ko�(k)

⇣ !co

�vo

co· k⌘+

�2⇢

8�(k)`2�~rR⇢⇢(0, ~r)|~r=0

and consists of two parts. The first part models the Doppler frequency shift anddepends on the cross-range component vo of the mean velocity ~vo = (vo, voz). Itcomes from the expansion of the mode wavenumber

r⇣ko +

! � vo · k

co

⌘2� |k|2 ⇡ �(k) +

ko(! � vo · k)

co�(k),

in the limit "! 0, using the scaling relation (2.15) and ! ⌧ !o obtained from (2.12).The second part is due to the random medium and it is small when � ⌧ 1, i.e.,�o ⌧ `.

The second term in the exponent in (3.5) is

D(k) = �k4o`

d+1

4

Z

|k0|<ko

dk0

(2⇡)d1

�(k)�(k0)

Z

Rd

dr

Z1

0drz e

�i`(~k�~k0)·~r

⇥

h�2cRcc(0, ~r) +

�2⇢

4(ko`)4�2

~rR⇢⇢(0, ~r)��⇢�c(ko`)2

�~rRc⇢(0, ~r)i,(3.7)

where we used the notation ~r = (r, rz) and definition (3.2). This complex exponentaccounts for the significant e↵ect of the random medium, seen especially in the termproportional to Rcc which dominates the other ones in the � ⌧ 1 regime. Becausethe covariance is even, the real part of D(k) derives from

(3.8)

Z

Rd+1

d~rRcc(0, ~r)e�i`(~k�~k0)·~r =

Z

R

d⌦

2⇡eRcc

�⌦, `(~k � ~k0)

�,

where

(3.9) eRcc(⌦, ~q) =

Z

Rd⌧

Z

Rd+1

d~rRcc(⌧, ~r)ei⌦⌧�i~q·~r

� 0

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is the power spectral density of ⌫c. This is nonnegative by Bochner’s theorem, soRe

⇥D(k)

⇤< 0 and the mean amplitudes decay exponentially in z, on the length scale

(3.10) S (k) = �1

Re⇥D(k)

⇤ ,

called the scattering mean free path. Note that |k|, |k0| = O(1/`) in the support of

eRcc in (3.8) and that by choosing the standard deviation �c as in (2.14), we obtainfrom (3.7)–(3.10) that S (k) = O(L) in the " ! 0 followed by the � ! 0 limit. Thisshows that the decay of the mean amplitudes in z is significant in our regime. It isthe manifestation of the randomization of the wave due to scattering in the medium.

3.2. The Wigner transform. The strength of the random fluctuations of themode amplitudes is described by the Wigner transform (energy density)

W (!,k,x, z) =

Zdq

(2⇡)deiq·(r�(k)z+x)E

ha⇣!,k +

q

2, z⌘a⇣!,k �

q

2, z⌘i

,(3.11)

where the bar denotes complex conjugate and the integral is over all q 2 Rd such that|k ± q/2| < ko. The Wigner transform satisfies the equation

⇥@z �r�(k) ·rx

⇤W (!,k,x, z) =

Z

O

d!0dk0

(2⇡)d+1Q(!,!0,k,k0)

⇥W (!0,k0,x, z)

� W (!,k,x, z)⇤,(3.12)

for z > 0, with initial condition

(3.13) W (!,k,x, 0) = |ao(!,k)|2�(x).

The integral kernel in (3.12) is called the di↵erential scattering cross section. It isdefined by

Q(!,!0,k,k0) =k4o`

d+1T

4�(k)�(k0)

h�2ceRcc +

�2⇢

4(ko`)4�2

�!r R⇢⇢ ��c�⇢(ko`)2

��!r Rc⇢

i,(3.14)

where the power spectral densities in the square bracket are evaluated as

eRcc = eRcc

�T (! � !0

� (~k � ~k0) · ~vo), `(~k � ~k0)�,(3.15)

and similarly for the other two terms, which are proportional to the Fourier transformof �2

~rR⇢⇢ and �~rRc⇢. The total scattering cross section is defined by the integral of(3.14) and satisfies

⌃(k) =

Z

O

d!0dk0

(2⇡)d+1Q(!,!0,k,k0) =

2

S (k).(3.16)

Note that the last two terms in the square brackets in (3.14) are small in the� ⌧ 1 regime, because 1/(ko`) = �/(2⇡) ⌧ 1 and �⇢/�c = O(1). If �⇢/�c were large,of the order ��2, then these terms would contribute. However, this would change onlythe interpretation of the di↵erential scattering cross section and not its qualitativeform.

3.2.1. The radiative transfer equation. The evolution equation (3.12) forthe Wigner transform is related to the radiative transfer equation [5, 20]. Indeed, weshow in Appendix D that W (!,k,x, z) is the solution of (3.12)–(3.13) if and only if

(3.17) V (!, ~k, ~x) =1

�(k)W (!,k,x, z)�

�kz � �(k)

�, ~k = (k, kz),

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solves the radiative transfer equation

r~k⌦(~k) ·r~xV (!, ~k, ~x) =

Z

Rd+1

d~k0

(2⇡)d+1

Zd!0

2⇡S�!,!0, ~k, ~k0

�⇥V (!0, ~k0, ~x)

� V (!, ~k, ~x)⇤

(3.18)

with ⌦(~k) = co|~k| and the scattering kernel

(3.19) S�!,!0, ~k, ~k0

�=

2⇡c2ok2o

�(k)�(k0)Q(!,!0,k,k0)��⌦(~k)� ⌦(~k0)

�.

The initial condition is specified at ~x = (x, 0) by

(3.20) V (!, ~k, (x, 0)) = |ao(!,k)|2�(x)�(kz � �(k))

with ao(!,k) defined in (3.3).This result shows that the generalized (singular) phase space energy (3.17) evolves

as in the standard three-dimensional radiative transfer equation, but it is supportedon the phase vectors with range component

(3.21) kz = �(k), |k| < ko.

Indeed, if ~k0 = (k0,�(k0)) and ~k = (k, kz), then

��⌦(~k)� ⌦(~k0)

�=

1

co��|~k|� ko

�=

koco�(k)

��kz � �(k)

�,

so the evolution of V (!, ~k, ~x) is confined to the hypersurface in (3.21). Physically, thismeans that the wave energy is traveling with constant speed in a cone of directionscentered at the range axis z.

3.2.2. Paraxial approximation. The paraxial approximation of the Wignertransform is obtained from (3.12)–(3.13) in the limit

� = �o/`! 0 so that �/�s = finite,

as explained in section 5.8. In this case the phase space decomposition of the initialwave energy given by (3.3) and (3.13) is supported in a narrow cone around the rangeaxis z, with opening angle scaling as

�o`s

= �s ⌧ 1.

Moreover, from the expression (3.14) of the di↵erential scattering cross section and(3.15) we see that the energy coupling takes place in a small cone of di↵erentialdirections whose opening angle is

�o`

= � ⌧ 1.

In the paraxial regime (3.12) simplifies to@z +

k

ko·rx

�W (!,k,x, z) =

Z

Rd

dk0

(2⇡)d

Z

R

d!0

2⇡Qpar(!

0,k0)

⇥W (! � !0� k0

· vo,k � k0,x, z)� ⌃parW (!,k,x, z),(3.22)

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where we obtained from definition (3.2) and the scaling relations (2.10), (2.15) thatin the limit � ! 0,

�(k) ! ko, `��(k)� �(k0)

�� ! 0, T��voz(�(k)� �(k0))

�� ! 0.

The di↵erential scattering cross section becomes

Qpar(!,k) =k2o�

2c `

d+1T

4eRcc

�T!, `k, 0

�,(3.23)

and the total scattering cross section is

⌃par =

Z

Rd

dk0

(2⇡)d

Z

R

d!0

2⇡Qpar(!

0,k0) =�2c `k

2o

4R(0,0) =

2

Spar,(3.24)

where Spar is the scattering mean free path in the paraxial regime and

(3.25) R(⌧, r) =

Z

Rdrz Rcc(⌧, ~r), ~r = (r, rz).

The initial condition is as in (3.13), with ao defined in (3.3),

(3.26) W (!,k,x, 0) =�2sT

2s `

2ds

4ko

�� bS(Ts!, `sk)��2�(x).

Note that the right-hand side of (3.22) is a convolution, so we can write theWigner transform explicitly using Fourier transforms, as explained in Appendix C.The result is

W (!,k,x, z) =�2sTs`ds4ko

Z

R

d⌦

2⇡

Z

Rd

dK

(2⇡)d|bS(⌦,K)|2

Z

Rdt

Z

Rd

dy

⇥

Z

Rd

dq

(2⇡)dexp

⇢i⇣! �

⌦

Ts

⌘t� iy ·

⇣k �

K

`s

⌘+ iq ·

⇣x�

K

ko

z

`s

⌘

+�2c `k

2o

4

Z z

0dz0

hR

⇣ t

T,y �

qko(z � z0)� vot

`

⌘� R(0,0)

i),(3.27)

and we use it next in the inverse problem of estimating the source location and themean flow velocity ~vo.

3.3. Steady perturbations. Here we discuss briefly the case of steady mediumperturbations (2.1)–(2.3), where ~⌫ = 0 and ⌫⇢ and ⌫c are advected by ~vo but do notdepend explicitly on time. This case can be studied directly or it can be obtained asthe limit T ! +1 of the previous results:

• The mean amplitudes decay as (3.5) and the expressions of the phase ✓(!,k)and complex damping term D(k) are given by (3.6) and (3.7), respectively.

• The Wigner transform (3.11) satisfies (3.12) with the singular di↵erentialscattering cross section

Q(!,!0,k,k0) =2⇡k4o`

d+1

4�(k)�(k0)��! � !0

� (~k � ~k0) · ~vo

�

⇥

h�2cbRcc +

�2⇢

4(ko`)4\�2�!r R⇢⇢ �

�c�⇢(ko`)2

\��!r Rc⇢

i�`(~k � ~k0)

�,(3.28)

where ~k = (k,�(k)) and

bRcc(~q) =

Z

Rd+1

d~rRcc(0, ~r)e�i~q·~r.D

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• In particular, in the paraxial regime, if we introduce

cW (!,k,x, z) = W (! + k · vo,k,x, z),

then we find that it satisfies the following equation in which ! is frozen:

⇥@z +

k

ko·rx

⇤cW (!,k,x, z) =

Zdk0

(2⇡)dbQpar(k

0)⇥cW (!,k � k0,x, z)

�cW (!,k,x, z)⇤,

for z > 0, with the initial condition cW (!,k,x, 0) = |ao(! + k · vo,k)|2�(x)and with the reduced di↵erential scattering cross section

bQpar(k) =k2o�

2c `

d+1

4bRcc(`k, 0).

This is the transport equation derived in [2, section III.C] in the absence ofambient flow.

4. Application to imaging. In this section we use the transport theory in theparaxial regime, stated in section 3.2.2, to localize a stationary in space time-harmonicsource in a moving random medium with smooth and isotropic random fluctuations,from measurements at a stationary array of receivers. The case of a time-harmonicsource is interesting because it shows the beneficial e↵ect of the motion of the randommedium for imaging. In the absence of this motion, the wave received at the arrayis time-harmonic, it oscillates at the frequency !o, and it is impossible to determinefrom it the range of the source. The random motion of the medium causes broadeningof the frequency support of the wave field, which makes the range estimation possible.

We consider a strongly scattering regime, where the wave received at the array isincoherent. This means explicitly that the range L is much larger than the scatteringmean free path Spar or, equivalently, from (3.24),

(4.1)�2c `k

2oL

4R(0,0) � 1.

We also suppose that

(4.2)⌘

⌘s=

T

Ts⌧ 1

to ensure that the imaging functions are statistically stable with respect to the real-izations of the random medium. We begin in section 4.1 with the approximation ofthe Wigner transform (3.27) for a time-harmonic source, in the strongly scatteringregime. This Wigner transform quantifies the time-space coherence properties of thewave, as described in section 4.2. Then, we explain in section 4.3 how we can estimatethe Wigner transform from the measurements at the array. The source localizationproblem is discussed in section 4.4 and the estimation of the mean medium velocityis discussed in section 4.5.

4.1. Wigner transform for time-harmonic source and strong scattering.

To derive the Wigner transform for a time-harmonic source, we take the limit Ts ! 1

in (3.27), after rescaling the source amplitude as

(4.3) �s = �/pTs, � = O(1).

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We assume for convenience1 that the source has a Gaussian profile,

(4.4)

Z

Rd⌦

�� bS(⌦,K)��2 = (2⇡)de�|K|

2

,

so we can calculate explicitly the integral overK in (3.27). We obtain after the changeof variables y = ⇠ + (q/ko)z that

W (!,k,x, z) =�2`ds⇡

d/2

4ko(2⇡)d

Z

Rdt

Z

Rd

d⇠

Z

Rd

dq exp

(i!t�

|⇠|2

4`2s� i⇠ · k + iq ·

⇣x�

k

koz⌘

+�2c `k

2o

4

Z z

0dz0

hR

⇣ t

T,⇠ + q

koz0 � vot

`

⌘� R(0,0)

i).(4.5)

Note that the last term in the exponent in (4.5) is negative, because R is maximalat the origin. Moreover, the relation (4.1) that defines the strongly scattering regimeimplies that the integrand in (4.5) is negligible for t/T � 1 and |⇠+q/koz0�vot|/` � 1.Thus, we can restrict the integral in (4.5) to the set

n(t, ⇠, q) 2 R2d+1 : |t| ⌧ T,

��⇠ +q

koz0 � vot

�� ⌧ `o

and approximate

(4.6) R(⌧, r) ⇡ R(0,0)�↵o

2⌧2 �

#o2|r|2

with ↵o,#o > 0. Here we used that the Hessian of R evaluated at the origin is negativedefinite and because the medium is statistically isotropic, it is also diagonal, with theentries �↵o and �#o. We obtain that

�2c `k

2o

4

"R

⇣ t

T,⇠ + q

koz0 � vot

`

⌘� R(0,0)

#⇡ �

↵

2

⇣ t

T

⌘2�#

2

✓|⇠ + q

koz0 � vot|

`

◆2

with the positive parameters

(4.7) ↵ = ↵o�2c `k

2o

4, # = #o

�2c `k

2o

4.

Substituting in (4.5) and integrating in z0 we obtain

W (!,k,x, z) ⇡�2`ds⇡

d/2

4ko(2⇡)d

Z

Rdt

Z

Rd

d⇠

Z

Rd

dq exp

⇢i!t�

↵z

2

⇣ t

T

⌘2�#z

2`2��⇠ � vot

��2

�|⇠|2

4`2s� i⇠ · k �

#z2

2`2(⇠ � vot) ·

q

ko�#z3

6`2

��q

ko

��2+ iq ·

⇣x�

k

koz⌘�

.(4.8)

The imaging results are based on this expression. Before we present them, we studythe coherence properties of the transmitted wave and define the coherence parameterswhich a↵ect the performance of the imaging techniques.

1The results extend qualitatively to other profiles but the formulas are no longer explicit.

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4.2. Time-space coherence. Let us define the time-space coherence function(4.9)

C(�t,�x,x, z) =�o

2⇡(co⇢o)2

Z

Rdt p(t+�t,x+�x/2, z)p(t,x��x/2, z)ei!o�t

and obtain from (3.1) that in the paraxial regime

C(�t,�x,x, z) ⇡

Z

R

d!

2⇡

Z

Rd

dk

(2⇡)d

Z

Rd

dq

(2⇡)da(!,k + q/2, z)a(!,k � q/2, z)

⇥ expniq · [zr�(k) + x] + i�x · k � i!�t

o.(4.10)

Moreover, in view of (3.11) and the fact that we average in time so that the statisticalfluctuations of C are small (see Remark 4.1) we have

C(�t,�x,x, z) ⇡ EhC(�t,�x,x, z)

i

⇡

Z

R

d!

2⇡

Z

Rd

dk

(2⇡)dW (!,k,x, z)e�i!�t+i�x·k.(4.11)

This shows formally that we can characterize the Wigner transform as the Fouriertransform of the coherence function

(4.12) W (!,k,x, z) ⇡

Z

Rd�t

Z

Rd

d�xC(�t,�x,x, z)ei!�t�i�x·k.

Using the expression (4.5) of the Wigner transform in (4.11) we find after evalu-ating the integrals that

C(�t,�x,x, z) ⇡�2`ds

22+d/2koRdz

exp⇥i'(�t,�x,x, z)]

⇥ exp

�

�t2

2T 2z

�|x|2

2R2z

�|�x|2

2D21z

�|Hz�x� vo�t|2

2D22z

�(4.13)

with phase

(4.14) '(�t,�x,x, z) =kox ·

h⇣1 + #z

⇣`s`

⌘2⌘�x� #z

⇣`s`

⌘2vo�t

i

zh1 + 2

3#z⇣

`s`

⌘2i

and coe�cients

Tz =T

p↵z

, Rz =z

p2`sko

✓1 +

2`2s3D2

z

◆1/2

, Dz =`

p#z

,

(4.15)

D1z =2Dz

3⇣1 +

`2s6D2

z

⌘�1/2, D2z = Dz

0

@1 + 2`2s

3D2z

1 + `2s6D2

z

1

A1/2

, Hz = 1�1

2⇣1 + `2s

6D2z

⌘ .

(4.16)

The decay of the coherence function in �x models the spatial decorrelation ofthe wave on the length scale corresponding to the characteristic speckle size. This is

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quantified by the length scales D1z and D2z, which are of the order of Dz. We callDz the decoherence length and obtain from (2.14) and (4.7) that it is of the order ofthe typical size ` of the random fluctuations of the medium,

(4.17) Dz =`

p#z

=`

⇡p#o

rL

z= O(`).

The decay of the coherence function in �t models the temporal decorrelation ofthe wave, on the time scale

(4.18) Tz =T

p↵z

=T

⇡p↵o

rL

z= O(T ),

where we used definitions (2.14) and (4.7). We call Tz the decoherence time and notethat it is of the order of the life span T of the random fluctuations of the medium.

The decay of the coherence function in |x| means that the waves propagate in abeam with radius Rz, which evolves in z as described in (4.15) and satisfies

(4.19) Rz ⇡

8><

>:

zp2`sko

for `s ⌧ Dz,r#

3

z3/2

ko`for `s � Dz.

This shows that the transition from di↵raction based beam spreading to scatteringbased beam spreading happens around the critical propagation distance

(4.20) z⇤ =1

#

⇣ ``s

⌘2= L

⇣�s�

⌘2 1

⇡2#o.

This expression is derived from equation `s = Dz? and definitions (2.14) and (4.7),and it shows that z?/L is finite in our regime.2

Note that when z � z⇤, i.e., Dz ⌧ `s, the coe�cients (4.16) become

D1z ⇡p2`s, D2z ⇡ 2Dz, Hz ⇡ 1,(4.21)

and the coherence function satisfies

(4.22)|C(�t,�x,x, z)|

|C(0,0,0, z)|⇡ exp

✓�

�t2

2T 2z

�|x|2

2R2z

�|�x|2

4`2s�

|�x� vo�t|2

8D2z

◆.

Thus, the spatial spreading and decorrelation of the wave field for z � z⇤ are governedby the parameters Rz, `s, and Dz, with Rz given by the second case in (4.19) andDz given in (4.17). These parameters scale with the propagation distance z < L asRz ⇠ z3/2 and Dz ⇠ z�1/2. The temporal decorrelation is on the scale Tz ⇠ z�1/2.

4.3. Estimation of the Wigner transform. Suppose that we have a receiverarray centered at (xo, z), with aperture in the cross-range plane modeled by theappodization function

(4.23) A (x) = exp

✓�|x� xo|

2

2({/ko)2

◆.

2Recall from section 3.2.2 that the paraxial regime is obtained in the limit � ! 0 so that�/�s = `s/` remains finite. Here we allow the ratio `s/` to be large or small but independent of �,which tends to zero.

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The linear size of the array is modeled by the standard deviation {/ko, with dimen-sionless { > 0 defining the diameter of the array expressed in units of �o.

Recalling the wave decomposition (3.1) and that �(k) ⇠ ko in the paraxial regime,we define the estimated mode amplitudes by

aest(!,k, z) =koe�i�(k)z

i!o⇢o

Z

Rdt

Z

Rd

dxA (x)p(t,x, z)ei(!+!o)te�ik·x

=

✓{2

2⇡k2o

◆d/2 Z

Rd

dek a(!,k + ek, z)ei[�(k+ek)��(k)]z+iek·xo�

{2|ek|2

2k2o .(4.24)

With these amplitudes we calculate the estimated Wigner transform

West(!,k,x, z) =

Z

Rd

dq

(2⇡)deiq·(r�(k)z+x)aest

⇣!,k +

q

2, z⌘aest

⇣!,k �

q

2, z⌘

(4.25)

and obtain after carrying out the integrals and using the approximationh�⇣k +

q

2

⌘� �

⇣k �

q

2

⌘iz ⇡ q ·r�(k)z

that

(4.26) West(!,k,x, z) ⇡⇣ {2

⇡k2o

⌘d/2e�

k2o|x�xo|2

{2

Z

Rd

dK e�

{2|K|2

k2o W (!,k +K,x, z).

We can now use the expression (4.8) in this equation to obtain an explicit approx-imation for West. Equivalently, we can substitute (4.12) in (4.26) and obtain afterintegrating in K that

West(!,k,x, z) ⇡ e�k2o|x�xo|2

{2

Z

Rd�t

Z

Rd

d�xC(�t,�x,x, z)ei!�t�i�x·k�k2o|�x|2

4{2

(4.27)

with C given in (4.13).

Remark 4.1. Note from (4.9) that the time integration that defines the coherencefunction is over a time interval determined by the pulse duration Ts, which is largerthan the coherence time T of the medium by assumption (4.2). If we interpret the waveas a train of Ts/T pulses of total duration T , each individual pulse travels throughuncorrelated layers of medium because the correlation radius of the medium ` is muchsmaller than coT . This follows from the fact that `/(coT ) = "/(⌘�) and "⌧ �⌘. Thus,C(�t,�x,x, z) is the superposition of approximately Ts/T uncorrelated componentsand its statistical fluctuations are small by the law of large numbers. Moreover, weconclude from (4.27) that the estimated Wigner transform is approximately equal toits expectation, up to fluctuations of relative standard deviation that is smaller thanpT/Ts.

4.4. Source localization. We now show how we can use the estimated Wignertransform to localize the source. Recall that we use the system of coordinates withorigin at the center of the source. Thus, the location (xo, z) of the center of thearray relative to the source is unknown and the goal of imaging is to estimate it.We begin in section 4.4.1 with the estimation of the direction of arrival of the wavesat the array, and then describe the localization in range in section 4.4.2. These twoestimates determine the source location in the cross-range plane, as well.

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4.4.1. Direction of arrival estimation. We can estimate the direction ofarrival of the waves from the peak (maximum) in k of the imaging function

ODoA(k, z) =

Z

R

d!

2⇡West(!,k,xo, z),(4.28)

determined by the estimated Wigner transform at the center of the array of receivers.If the medium were homogeneous, the maximum of k 7! ODoA(k, z) would be at thecross-range wave vector k⇤ = ko

xoz , and the width of the peak (the resolution) would

be 1/(p2{). However, cumulative scattering in the random medium gives a di↵erent

result, as we now explain.Substituting (4.27) in (4.28) and using the expression (4.13), we obtain after

evaluating the integrals that

ODoA(k, z)

maxk0 ODoA(k0, z)= exp

(�

1

2#2DoA(z)

��k � k(z)

ko

��2)

(4.29)

with(4.30)

#DoA(z) =

8<

:1

3D2zk

2o

0

@1 + `2s

2D2z

1 + 2`2s3D2

z

1

A+1

2{2

9=

;

1/2

, k(z) = koxo

z

0

@1 + `2s

D2z

1 + 2`2s3D2

z

1

A .

Therefore, the maximum of k 7! ODoA(k, z) is at the cross-range wave vector k(z)and the width of the peak (the resolution) is determined by #DoA(z). This resolutionimproves for larger array aperture (i.e., {) and deteriorates as z increases. Dependingon the magnitude of z relative to the critical range z⇤ defined in (4.20), we distinguishthree cases:

1. In the case z ⌧ z⇤, i.e., `s ⌧ Dz, the intensity travels along the deterministiccharacteristic, meaning that ODoA(k, z) peaks at

(4.31) k(z) ⇡ koxo

z.

However, the resolution is worse than in the homogeneous medium,

(4.32) #DoA(z) ⇡

⇢1

3D2zk

2o

+1

2{2

�1/2

,

with Dz defined in (4.17).2. In the case z � z⇤, i.e., `s � Dz, the peak of ODoA(k, z) is at the cross-range

wave vector

(4.33) k(z) ⇡3

2ko

xo

z,

and the resolution is

(4.34) #DoA(z) ⇡

⇢1

4D2zk

2o

+1

2{2

�1/2

.

Here the peak corresponds to a straight line characteristic, but with a di↵erent slopethan in the homogeneous medium. The resolution is also worse than in the homoge-neous medium.

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3. In the case z = O(z⇤), the characteristic can no longer be approximated bya straight line, as seen from (4.30). Nevertheless, we can still estimate the sourceposition from the observed peak k(z), provided that we have an estimate of therange z. The resolution of the estimate of k(z) is #DoA(z) given by (4.30) that isbounded from below by (4.34) and from above by (4.32).

Remark 4.2. Note that (4.30) is a decreasing function of the array diameter {/ko,as long as this satisfies {/ko

p2Dz. Thus, increasing the aperture size beyond the

critical valuep2Dz does not bring any resolution improvement.

4.4.2. Range estimation. The results of the previous section show that thedirection of arrival estimation is coupled with the estimation of the range z in general,with the exception of the two extreme cases 1 and 2 outlined above.

To estimate the range z, we use the imaging function

Orange(t, z) =

Z

R

d!

2⇡e�i!t

Z

Rd

dk

(2⇡)dWest(!,k,xo, z) ⇡ C(t,0,xo, z),(4.35)

derived from (4.27). Substituting the expression (4.13) of the coherence function inthis equation we obtain

|Orange(t, z)|

maxt0 |Orange(t0, z)|= exp

⇢�

t2

2#2range(z)

�(4.36)

with

#range(z) = Tz

8<

:1 +|vo|

2T

2z

D2z

0

@1 + `2s

6D2z

1 + 2`2s3D2

z

1

A

9=

;

�1/2

.(4.37)

As a function of t, this peaks at t = 0 and its absolute value decays as a Gaussian, withstandard deviation #range(z). If we know the statistics of the medium (the decoherencetime Tz and length Dz) and the magnitude of the cross-range velocity |vo|, then wecan determine the range z by estimating the rate of decay of Orange(t, z). Note thatthe array dimameter {/ko plays no role for the range estimation.

Remark 4.3. We can also estimate the mean velocity ~vo = (vo, voz) from (4.36)by considering di↵erent beam orientations in the case that the source locations andalso the medium statistics (the decoherence time Tz and length Dz) are known. Thatis to say, with three known beams we can get the vector ~vo, and then we can useit to localize the unknown source using the direction of arrival and range estimationdescribed above. See also section 4.5 for a more detailed analysis of the velocityestimation.

Remark 4.4. If the decoherence time Tz and length Dz are not known, they canalso be estimated using additional known sources. Definitions (4.17)–(4.18) show thatDzz1/2 and Tzz1/2 are constant with respect to z. Once estimated, these constantscan be used in the imaging of the unknown source.

4.5. Single beam lateral velocity estimation. We observe from (4.29) and(4.36)–(4.37) that the source localization depends only on the Euclidian norm |vo| ofthe cross-range component of the mean velocity of the medium. We show here that vo

can be obtained with only one beam and, when the receiver array is large and z � z⇤

i.e., `s � Dz, the velocity estimate is independent of the medium statistics and thesource location.

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The estimation of vo is based on the imaging function

Ov(y, t, z) =

Z

R

d!

2⇡e�i!t

Z

Rd

dk

(2⇡)deik·y

Z

Rd

dxWest(!,k,x, z)

⇡ exp⇣�

k2o |y|2

4{2

⌘Z

Rd

dx exp⇣�

k2o |x� xo|2

{2

⌘C(t,y,x, z).(4.38)

Substituting the expression (4.13) of the coherence function and carrying out theintegrals we obtain that

|Ov(y, t, z)| ⇡�2⇡d/2({`s)d

22+dkd+1o Ad

z

exp

⇢�

t2

2T 2z

�|y � sztvo|

2

2m2zA

2z

�|tvo|

2

n2zA2z

�|xo|

2

4A2z

�(4.39)

with the e↵ective apperture

A2z =

1

4

h⇣ {ko

⌘2+⇣ z

ko`s

⌘2⇣1 +

2`2s3D2

z

⌘i(4.40)

and dimensionless parameters

m2z =

8

1 + 23D2

z

�z

ko`s

�2⇣1 + `2s

2D2z

⌘+� {ko`s

�2⇣1 + 2`2s

D2z

⌘+�ko{�2� z

ko`s

�2⇣1 + 2`2s

3D2z

⌘ ,

n2z =m2

z

sz(qz � sz/2),

sz =m2

z

4D2z

h⇣ {ko

⌘2+

1

2

⇣ z

ko`s

⌘2⇣1 +

`2s3D2

z

⌘i, qz =

⇣{ko

⌘2+⇣

zko`s

⌘2⇣1 + `2s

6D2z

⌘

2⇣

{ko

⌘2+⇣

zko`s

⌘2⇣1 + `2s

3D2z

⌘ .

These depend on the radii {/ko of the array and `s of the source, the decoherencelength Dz, and the ratio z/(ko`s) that quantifies the cross-range resolution of focusingof a wave using time delay beamforming at a source of radius `s.

To estimate vo we can proceed as follows. First, we estimate for each time tthe position ymax(t) that maximizes y 7! Ov(y; t, z). Second, we note from (4.39)that ymax(t) should be a linear function in t of the form ymax(t) = szvot. Therefore,we can estimate szvo with a weighted linear least squares regression of ymax(t) withrespect to t. In practice sz is likely unknown. However, in the case of a large receiverarray with radius satisfying

(4.41){ko

� max

⇢z

ko`s,

z

koDz

�,

and for z � z?, so that `s � Dz, we obtain that sz ⇡ 1. Thus, the least squaresregression gives an unbiased estimate of vo.

In view of (4.39), the least squares regression can be carried out over a timeinterval with length of the order of min(Tz, nzAz/|vo|). Beyond this critical time thefunction Ov vanishes. Therefore, as long as |vo| < nzAz/Tz, the velocity resolution is

resv =mzAz

szTz⇡

Dz

Tz,(4.42)

where the approximation is for a large array and `s � Dz. If |vo| is larger thannzAz/Tz, then the resolution is reduced to

resv =mzAz

sznzAz/|vo|⇡

Dz

Tz

|vo|Tz

nzAz.(4.43)

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5. Analysis of the wave field. To derive the results stated in section 3, webegin in section 5.1 with a slight reformulation, which transforms (1.2) into a formthat is more convenient for the analysis. We scale the resulting equation in section 5.2,in the regime defined in section 2.2, and then we change coordinates to a moving framein section 5.3. In this frame we write the wave as a superposition of time-harmonicplane waves with random amplitudes that model the net scattering in the randommedium, as described in section 5.4. We explain in section 5.5 that the backwardgoing waves are negligible and use the di↵usion approximation theory in section 5.6to analyze the amplitudes of the forward going waves, in the limit "! 0. We end insection 5.8 with the paraxial limit.

5.1. Transformation of the wave equation. Let us define the new potential

(5.1) (t, ~x) =

p⇢(t, ~x)p⇢o

�(t, ~x),

and substitute it in (1.2) to obtain the wave equation

Dt

1

c2(t, ~x)Dt (t, ~x)

��

Dt (t, ~x)Dt ln ⇢(t, ~x)

c2(t, ~x)��~x (t, ~x) + q(t, ~x) (t, ~x)

= �s

p⇢(t, ~x)p⇢o

e�i!otS⇣ t

Ts,x

`s

⌘�(z),(5.2)

for t 2 R and ~x = (x, z) 2 Rd+1, where �~x is the Laplacian operator and

q(t, ~x) =�~x

p⇢(t, ~x)p

⇢(t, ~x)�

1

c2(t, ~x)

(D2

t

p⇢(t, ~x)p

⇢(t, ~x)�

1

2[Dt ln ⇢(t, ~x)]

2

)

�1

2Dtc

�2(t, ~x)Dt ln ⇢(t, ~x).(5.3)

The initial condition (1.3) becomes

(5.4) (t, ~x) ⌘ 0, t ⌧ �Ts.

5.2. Scaled wave equation. We use the scaling regime defined in section 2.2and denote with primes the dimensionless, order one variables

(5.5) ~x = L~x0, t = TLt0.

We also let

(5.6) ~vo = V ~v0

o, co = coc0

o, !o = !o!0o

2⇡,

where the constants c0o = 1 and !0o = 2⇡ are introduced so that the scaled equation is

easier to interpret.In the scaled variables, and using the source amplitude (2.17), the right-hand side

in (5.2) becomes

�s

p⇢(t, ~x)p⇢o

e�i!otS⇣ t

Ts,x

`s

⌘�(z) =

⇥1 +O(

p")⇤

"⌘sL2

⇣�s"

⌘de�i

!0o" t0S

⇣ t0

⌘s,�sx0

"

⌘�(z0).

(5.7)

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We also have from definitions (2.1)–(2.3) that the random coe�cients take the form

~v(t, ~x)

V= ~v0(t0, ~x0) = ~v0

o +p"� �v ~⌫

⇣ t0

⌘,~x0

� "~v0ot

0

"/�

⌘,(5.8)

⇢(t, ~x)

⇢o= exp

p"� �⇢ ⌫⇢

⇣ t0

⌘,~x0

� "~v0ot

0

"/�

⌘�,(5.9)

c2oc2(t, ~x)

=1

(c0o)2

1 +

p"� �c ⌫c

⇣ t0

⌘,~x0

� "~v0ot

0

"/�

⌘�(5.10)

with scaled standard deviations �c, �v, �⇢ defined in (2.14).The solution of (5.2) must have variations in t0 and ~x0 on the same scale as the

source term and the coe�cients (5.8)–(5.10), meaning that @t0 ⇠ 1/", |r~x0 | ⇠ 1/".From (2.8)–(2.15) we obtain that in the scaled variables we have

(5.11) Dt =1

TLc0oD"

t0 with D"t0 = @t0 + "~v0(t0, ~x0) ·r~x0 .

Equation (5.9) gives

Dt ln ⇢(t, ~x) =

p"��⇢TL

D"t0⌫⇢

⇣ t0

⌘,~x0

� "~v0ot

0

"/�

⌘=

O(p")

TL

and

D2t

p⇢(t, ~x)p

⇢(t, ~x)=

p"��⇢2T 2

L

(D"t0)

2⌫⇢⇣ t0

⌘,~x0

� "~v0ot

0

"/�

⌘+"��2

⇢

4T 2L

D"

t0⌫⇢⇣ t0

⌘,~x0

� "~v0ot

0

"/�

⌘�2

=O(

p")

T 2L

.

From (5.10) we get

Dt

1

c2(t, ~x)

�=

p"��c

(coc0o)2TL

D"t0⌫c

⇣ t0

⌘,~x0

� "~v0ot

0

"/�

⌘=

O(p")

coL,

and q defined in (5.3) takes the form

(5.12) q(t, ~x) =�5/2�⇢2"3/2L2

Q"

⇣ t0

⌘,~x0

� "~v0ot

0

"/�

⌘+O("2)

�

with

(5.13) Q"(⌧, ~r) = �~r⌫⇢(⌧, ~r) +

p"��⇢2

|r~r⌫⇢(⌧, ~r)|2 .

Substituting in (5.2) and multiplying both sides by "L2, we obtain that the po-tential denoted by 0(t0, ~x0) in the scaled variables satisfies

"

8<

:

h1 +

p"� �c⌫c

�t0

⌘ ,~x0

�"~v0ot

0

"/�

�i

(c0o)2

@2t0 +2"

(c0o)2~v0

o ·r~x0@t0 ��~x0

9=

; 0(t0, ~x0)

+�⇢�5/2

2p"

Q"⇣ t0

⌘,~x0

� "~v0ot

0

"/�

⌘ 0(t0, ~x0) ⇡

1

⌘s

⇣�s"

⌘de�i

!0o" t0S

⇣ t0

⌘s,

x0

"/�s

⌘�(z0)(5.14)

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with initial condition obtained from (1.3) and (5.1),

(5.15) 0(t0, ~x0) ⌘ 0, t0 ⌧ �⌘s.

The approximation in (5.14) is because we neglect O(p") terms that tend to zero in

the limit " ! 0. Note in particular that the random perturbations ~⌫ of the velocityof the flow appear in these terms and are negligible in our regime.

All variables are assumed scaled in the remainder of the section and we simplifynotation by dropping the primes.

5.3. Moving frame. Let us introduce the notation ~vo = (vo, voz) for the scaledmean velocity of the ambient flow and change the range coordinate z to

(5.16) ⇣ = z � "vozt.

We denote the potential in this moving frame by

(5.17) u"(t,x, ⇣) = (t,x, ⇣ + "vozt)

and obtain from (5.14) that it satisfies the wave equation

"

8<

:

h1 +

p"� �c⌫c

⇣t⌘ ,

x�"vot"/� , �⇣

"

⌘i

c2o@2t +

2"

c2ovo ·rx@t ��x � @2⇣

9=

;u"(t,x, ⇣)

+�⇢�5/2

2p"

Q"⇣ t

⌘,x� "vot

"/�,�⇣

"

⌘u"(t,x, ⇣) ⇡

⇣�s"

⌘d e�i!o" t

⌘sS⇣ t

⌘s,�sx

"

⌘�(⇣ + "vozt),

(5.18)

where again we neglect the terms that become negligible in the limit " ! 0. Thegradient rx and Laplacian �x are in the cross-range variable x 2 Rd.

5.4. Wave decomposition. The interaction of the waves with the randommedium depends on the frequency and direction of propagation, so we decomposeu"(t,x, ⇣) using the Fourier transform

(5.19) bu"(!,k, ⇣) =

Z

Rdt

Z

Rd

dxu"(t,x, ⇣)ei�

!o" +!

�t�ik

" ·x

with inverse

(5.20) u"(t,x, ⇣) =

Z

R

d!

2⇡

Z

Rd

dk

(2⇡")dbu"(!,k, ⇣)e�i

�!o" +!

�t+ik

" ·x.

The Fourier transform of (5.18) is

��2(k)

"� 2ko

⇣ !co

�vo · k

co

⌘� "@2⇣

�bu"(!,k, ⇣)�

⌘�1/2�d

p"

Z

R

d!0

2⇡

Z

Rd

dk0

(2⇡)d

bu"(!0,k0, ⇣)

k2o �cb⌫c

⇣⌘�! � !0

� (k � k0) · vo

�,k � k0

�,�⇣

"

⌘

��2�⇢2

bQ"⇣⌘�! � !0

� (k � k0) · vo

�,k � k0

�,�⇣

"

⌘�

⇡e�i !⇣

"voz

"⌘svozqS⇣�

⇣

"⌘svoz,k

�s

⌘,(5.21)

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where

(5.22) �(k) =p

k2o � |k|2, ko =!o

co,

andqS(⌧,) =

Z

Rd

dr S(⌧, r)e�i·r.

Note that the right-hand side in (5.21) is supported at |k| = O(�s), so by keeping�s small, we ensure that �(k) remains real valued in our regime. Physically, thismeans that bu"(!,k, ⇣) is a propagating wave, not evanescent.

Note also that if the mean velocity ~vo is orthogonal to the range direction, thesource term satisfies

limvoz!0

e�i !⇣"voz

"⌘svozqS⇣�

⇣

"⌘svoz,k

�s

⌘= bS

⇣⌘s!,

k

�s

⌘�(⇣)

in the sense of distributions, where

bS(!,) =Z

Rd⌧

Z

Rd

dr S(⌧, r)ei!⌧�i·r.

We introduce

a"(!,k, ⇣) =hp�(k)

2bu"(!,k, ⇣) +

"

2ip�(k)

@⇣bu"(!,k, ⇣)ie�i�(k) ⇣

" ,(5.23)

a"�(!,k, ⇣) =

hp�(k)

2bu"(!,k, ⇣)�

"

2ip�(k)

@⇣bu"(!,k, ⇣)iei�(k)

⇣" ,(5.24)

so that we have the decomposition

bu"(!,k, ⇣) =1p�(k)

ha"(!,k, ⇣)ei�(k)

⇣" + a"

�(!,k, ⇣)e�i�(k) ⇣

"

i,(5.25)

and the complex amplitudes a" and a"�

satisfy the relation

(5.26) @⇣a"(!,k, ⇣)ei�(k)

⇣" + @⇣a

"�(!,k, ⇣)e�i�(k) ⇣

" = 0.

This gives that

@⇣bu"(!,k, ⇣) =ip�(k)

"

ha"(!,k, ⇣)ei�(k)

⇣" � a"

�(!,k, ⇣)e�i�(k) ⇣

"

i(5.27)

and, moreover, that

@2⇣ bu"(!,k, ⇣) = ��2(k)

"2bu"(!,k, ⇣) +

2ip�(k)

"@⇣a

"(!,k, ⇣)ei�(k)⇣" .(5.28)

The decomposition in (5.25) and (5.20) is a decomposition of u"(t,x, ⇣) into asuperposition of plane waves with wave vectors

(5.29) ~k± = (k,±�(k)),

where the plus sign denotes the waves propagating in the positive range directionand the negative sign denotes the waves propagating in the negative range direction.

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The amplitudes a" and a"�

of these waves are random fields, which evolve in rangeaccording to (5.26) and the equation

@⇣a"(!,k, ⇣) ⇡

iko(! � vo · k)

co�(k)

ha"(!,k, ⇣) + a"

�(!,k, ⇣)e�2i�(k) ⇣

"

i

+i⌘�1/2�d

2p"

Z

R

d!0

2⇡

Zdk0

(2⇡)d

k2o �cb⌫c

⇣⌘�! � !0

� (k � k0) · vo

�,k � k0

�,�⇣

"

⌘

��2�⇢2

bQ"⇣⌘�! � !0

� (k � k0) · vo

�,k � k0

�,�⇣

"

⌘�

⇥1p

�(k)�(k0)

a"(!0,k0, ⇣)ei

⇥�(k0)��(k)

⇤⇣" + a"

�(!0,k0, ⇣)e�i

⇥�(k0)+�(k)

⇤⇣"

�

+i

2p�(k)"⌘svoz

qS⇣�

⇣

"⌘svoz,k

�s

⌘e�i !⇣

"voz�i�(k) ⇣

" ,

(5.30)

derived by substituting (5.25)–(5.28) into (5.21).

5.5. Forward scattering approximation. Equation (5.30) shows that the am-plitudes a" are coupled to each other and to a"

�. In our scaling regime, where the cone

of directions of propagation has small opening angle controlled by the parameter �s,and where the covariance (2.5) of the fluctuations is smooth, the coupling betweena" and a"

�becomes negligible in the limit " ! 0. We refer to [2, section C.2] and [3,

section 5.2] for a more detailed explanation of this fact.Using the assumption that the random fluctuations are supported at finite range

(see section 2), we require that the wave be outgoing at |⇣| ! 1. This radiationcondition and the negligible coupling between a" and a"

�in the limit " ! 0 imply

that we can neglect the backward going waves, and we can write

(5.31) bu"(!,k, ⇣) ⇡a"(!,k, ⇣)p

�(k)ei�(k)

⇣" , ⇣ > O(").

The starting value of a"(!,k, ⇣) is determined by the source term in (5.30), whichcontributes only for ⇣ = vozO("). For such small ⇣, we can change variables ⇣ = "⇠in (5.30) and obtain that

@⇠a"(!,k, "⇠) =

i

2p�(k)⌘svoz

qS⇣�

⇠

⌘svoz,k

�s

⌘e�i !⇠

voz�i�(k)⇠ +O(

p").

Integrating in ⇠ and using that a"(!,k, ⇣) vanishes for ⇣ ⌧ �O("), we obtain that

a"(!,k, "⇠) ⇡i

2p�(k)⌘svoz

Z

Rd⇠ qS

⇣�

⇠

⌘svoz,k

�s

⌘e�i !⇠

voz�i�(k)⇠

=i

2p�(k)

bS⇣⌘s(! + �(k)voz),

k

�s

⌘.

We use this expression as the initial condition for the forward going amplitudes

(5.32) a"(!,k, 0+) ⇡i

2p�(k)

bS⇣⌘s(! + �(k)voz),

k

�s

⌘

and drop the source term and the backward going amplitudes a"�

in (5.30) for range⇣ > 0.

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5.6. The acoustic pressure field in the Markovian limit. By definitions(1.1), (5.1), (5.17), and the scaling relations (5.5), the acoustic pressure is

p(TLt, Lx, Lz) ⇡ �⇢oTL

@tu"(t,x, z � "vozt).

Furthermore, (5.31) and the Fourier decomposition (5.20) give that

p(TLt, Lx, Lz)

2⇡co⇢o/�o⇡

Z

R

d!

2⇡

Z

Rd

dk

(2⇡")dia"(!,k, z)p

�(k)e�i

�!o" +!+�(k)voz

�t+i (k,�(k))

" ·(x,z),

where we have used (5.30) to write a"(!,k, z � "vozt) = a"(!,k, z) +O(p").

The shifted scaled frequency ! + �(k)voz appears in the initial condition (5.32),and the random processes b⌫c and bQ" in (5.30) depend on k/�. Thus, it is convenientto introduce the variables

(5.33) ⌦ = ⌘⇥! + �(k)voz

⇤, K = k/�,

and rewrite the expression of the pressure as

p(TLt, Lx, Lz)

!o⇢o⇡

Z

R

d⌦

2⇡⌘

Z

Rd

dK

(2⇡"/�)dA"(⌦,K, z)p

�(�K)e�i

�!o" +⌦

⌘

�t+i (�K,�(�K))

" ·(x,z)

(5.34)

with redefined amplitude

(5.35) A"(⌦,K, z) = ia"⇣⌦⌘� �(�K)voz, �K, z

⌘+ o(1).

The o(1) term, which tends to zero as "! 0, is used in this definition so that we havean equal sign in the evolution equation for A", derived from (5.30), after neglectingthe backward going amplitudes,

@zA"(⌦,K, z) =

iko�(�K)

h ⌦

⌘co�~vo

co·��K,�(�K)

�iA"(⌦,K, z)

+i

2

r�

"

Z

R

d⌦0

2⇡

Z

Rd

dK 0

(2⇡)dA"(⌦0,K 0, z)p�(�K)�(�K 0)

ei⇥�(�K0)��(�K)

⇤z"

⇥

hk2o �cb⌫c

⇣⌦� ⌦0

� ⌘��K � �K 0,�(�K)� �(�K 0)) · ~vo,K �K 0,

�z

"

⌘

��2�⇢2

bQ"⇣⌦� ⌦0

� ⌘��K � �K 0,�(�K)� �(�K 0)) · ~vo,K �K 0,

�z

"

⌘i(5.36)

for z > 0. The initial condition (5.32) becomes

(5.37) A"(⌦,K, 0+) = Ao(⌦,K) = �1

2p�(�K)

bS⇣⌘s⌘⌦,

�

�sK⌘.

5.7. The Markovian limit. Let L2(O,C) be the space of complex-valued,square-integrable functions defined on the set

(5.38) O = {⌦ 2 R}⇥ {K 2 Rd, �|K| < ko}

and denote A"(z) = (A"(⌦,K, z))(⌦,K)2O for z � 0. From (5.36) we obtain theconservation of energy relation

(5.39) @z

Z

O

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so the Markov process A"(z) 2 L2(O,C) lives on the surface of the ball with centerat the origin and " independent radius RA defined by

(5.40) R2A =

Z

O

d⌦dK |A"(⌦,K, z)|2 =

Z

O

d⌦dK |Ao(⌦,K)|2.

We describe the Markovian limit " ! 0 in Appendix A. The result is that theprocess of A"(z) converges weakly in C([0,1), L2) to a Markov process whose in-finitesimal generator can be identified. The first and second moments of the limitprocess are described below.

5.7.1. The mean amplitude. The expectation of A"(⌦,K, z) in the limit"! 0 is given by

(5.41) lim"!0

E⇥A"(⌦,K, z)

⇤= Ao(⌦,K)ei✓(⌦,K)z+D(K)z,

where

✓(⌦,K) =ko

�(�K)

h ⌦

⌘co�~vo

co· (�K,�(�K))

i+

�3�2⇢

8�(�K)�~rR⇢⇢(0, ~r)

��~r=0

(5.42)

is a real phase and

D(K) =�

Z

|K0|<ko/�

dK 0

(2⇡)d1

4�(�K)�(�K 0)

Z

Rd

dr

Z1

0drz e

�i�K�K0, �(�K)��(�K0

)

�

�·~r

⇥

(k4o �

2cRcc(0, ~r) +

�4�2⇢

4�2

~rR⇢⇢(0, ~r)� k2o�2�c�⇢�~rRc⇢(0, ~r)

)(5.43)

with ~r = (r, rz). Moreover, Re⇥D(K)

⇤< 0, because

Z

Rd+1

dr e�i ~K·~r

(k4o �

2cRcc(0, ~r) +

�4�2⇢

4�2

~rR⇢⇢(0, ~r)� k2o�2�c�⇢�~rRc⇢(0, ~r)

)� 0

is the power spectral density of the process

X(t, ~r) = k20�c⌫c(t, ~r)��⇢�2

2�~r⌫⇢(t, ~r)(5.44)

in the variable ~r (for fixed t), which is nonnegative by Bochner’s theorem. Thus, themean amplitude decays on the range scale

(5.45) S (K) = �1

Re⇥D(K)

⇤ ,

called the scattering mean free path. In the relatively high frequency regime thedamping is mainly due to the fluctuations of the wave speed, while in the relativelylow frequency regime the damping is mainly due to the fluctuations of the density.This damping is the mathematical manifestation of the randomization of the wavedue to cumulative scattering.

Recall that we have assumed �⇢ = O(1). Therefore, in the regime � ⌧ 1, the Rcc

term dominates in (5.43).

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5.7.2. The mean intensity. The expectation of the intensity

(5.46) I(⌦,K, z) = lim"!0

E⇥|A"(⌦,K, z)|2

⇤

satisfies

@zI(⌦,K, z) =

Z

O

d⌦0

2⇡

dK 0

(2⇡)dQ(⌦,⌦0,K,K 0) [I(⌦0,K 0, z)� I(⌦,K, z)] ,(5.47)

for z > 0, with initial condition obtained from (5.37):

(5.48) I(⌦,K, 0) = |Ao(⌦,K)|2.

Denoting the power spectrum of X(t, ~r) in (5.44) by P (⌦, ~K), and letting

⇣e⌦, ~K

⌘=

✓⌦� ⌘

��K,�(�K)) · ~vo,K,

�(�K)

�

◆,

the kernel in (5.47) is given by

Q(⌦,⌦0,K,K 0) =P (e⌦� e⌦0, ~K � ~K 0)

4�(�K)�(�K 0),

that is explicitly

Q(⌦,⌦0,K,K 0) =

(k4o �

2ceRcc +

�4�2⇢

4

|K �K 0

|2 +

⇣�(�K)� �(�K 0)

�

⌘2�2

eR⇢⇢

+ k2o�2�c�⇢

|K �K 0

|2 +

⇣�(�K)� �(�K 0)

�

⌘2�eR⇢c

�1

4�(�K)�(�K 0),

(5.49)

where eRcc is the power spectral density (3.9), evaluated as

eRcc = eRcc

⇣⌦�⌦0

�⌘��K��K 0,�(�K)��(�K 0)) ·~vo,K�K 0,

�(�K)� �(�K 0)

�

⌘,

and similarly for eR⇢c and eR⇢⇢.Note that the kernel satisfies

(5.50)

Z

O

d⌦0

2⇡

dK 0

(2⇡)dQ(⌦,⌦0,K,K 0) = �2Re[Q(K)] =

2

S (K),

where S (K) is the scattering mean free path defined in (5.45).

5.7.3. The Wigner transform. The wave amplitudes decorrelate at distinctfrequencies ⌦ 6= ⌦0 and wave vectors K 6= K 0, meaning that

(5.51) lim"!0

E⇥A"(⌦,K, z)A"(⌦0,K 0, z)

⇤= lim

"!0E⇥A"(⌦,K, z)] lim

"!0E[A"(⌦0,K 0, z)

⇤.

The right-hand side is the product of the means of the mode amplitudes, which decayon the range scale defined by the scattering mean free path (5.45).

However, the amplitudes are correlated for |⌦�⌦0| = O(") and |K�K 0

| = O(").We are interested in the second moment

EhA"

⇣⌦,K +

"q

2, z⌘A"

⇣⌦,K �

"q

2, z⌘i

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whose Fourier transform in q gives the energy density resolved over frequencies anddirections of propagation. This is the Wigner transform defined by

W "(⌦,K,x, z) =

Z

Rd

dq

(2⇡)deiq·(r�(�K)z+x)E

A"

⇣⌦,K +

"q

2, z⌘A"

⇣⌦,K �

"q

2, z⌘�

.

(5.52)

We show in Appendix A.3 that the Wigner transform converges in the limit "! 0 toW (⌦,K,x, z), the solution of the transport equation

[@z �r�(�K) ·rx]W (⌦,K,x, z) =

Z

O

d⌦0

2⇡

dK 0

(2⇡)dQ(⌦,⌦0,K,K 0)

⇥ [W (⌦0,K 0,x, z)�W (⌦,K,x, z)] ,(5.53)


(5.54) W (⌦,K,x, 0) = |Ao(⌦,K)|2�(x).

The transport equation (3.12) in the physical scales is obtained from (5.54) asexplained in Appendix B. In the next section we will show how this equation sim-plifies in the paraxial regime, when � ⌧ 1. This is the result used for the imagingapplications discussed in section 4.

5.8. The paraxial limit. Equation (5.53) shows that the energy is transportedon the characteristic

(5.55) x = ��K

�(�K)z,

parametrized by z, and depending on the wave vectorK. Here |x|/z = O(�) quantifiesthe opening angle of the cone (beam) of propagation with axis z. We write thisexplicitly as

(5.56) X = x/�, where |X| = O(1).

The paraxial regime corresponds to a narrow beam, modeled by � ! 0 and

(5.57) � = �/�s = O(1).

At the range z = 0 of the source we have from (5.54) and (5.37) that

(5.58) W (⌦,K,x, 0) =

�� bS�⌘s

⌘ ⌦,�K��2

4�dko�(X),

and to obtain a finite limit as � ! 0 we rescale the Wigner transform as

(5.59) W(⌦,K,X, z) = �dW (⌦,K, �X, z).

We also change variables in (5.53),

⌦� ⌦0� ⌘

��(K �K 0),�(�K)� �(�K 0)

�· ~vo ⌦0, K �K 0 K 0,

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and obtain the transport equation

@z +

K

�(�K)·rX

�W(⌦,K,X, z) =

k4o4

Z

O

d⌦0

2⇡

dK 0

(2⇡)d1

�(�K)�(�K 0)

⇥

h�2ceRcc

⇣⌦0,K 0,

�(�K)� �(�K � �K 0)

�

⌘+O(�2)

i

⇥

hW (⌦� ⌦0

� ⌘�vo ·K0� ⌘voz(�(�K)� �(�K � �K 0)),K �K 0,X, z)

� W(⌦,K,X, z)i

(5.60)

for z > 0 and a finite � ⌧ 1, where O(�2) denotes the eR⇢c and eR⇢⇢ terms in thekernel (5.49).

Recall that � ⌧ 1 so, in order to observe a significant e↵ect of the ambient motion,we rescale the transversal speed as

(5.61) vo =Vo

⌘�with |Vo| = O(1).

With a similar scaling of the range velocity

(5.62) voz =Voz

⌘�with |Voz| = O(1),

we obtain that the range motion plays no role in (5.60) as � ! 0, because

�(�K) = ko +O(�2), �(�K)� �(�K � �K 0) = O(�2).

The transport equation satisfied by the Wigner transform W(⌦,K,X, z) in theparaxial limit � ! 0 is

h@z +

K

ko·rX

iW(⌦,K,X, z) =

k2o4

Z

Rd

dK 0

(2⇡)d

Z

R

d⌦0

2⇡�2ceRcc(⌦

0,K 0, 0)

⇥⇥W (⌦� ⌦0

�K 0· Vo,K �K 0,X, z)�W(⌦,K,X, z)

⇤,(5.63)


(5.64) W(⌦,K,X, 0) =

�� bS�⌘s

⌘ ⌦,�K��2

4ko�(X).

The transport equation (3.22) in the physical scales is obtained from (5.63) using thescaling relations explained in Appendix B.

6. Summary. We introduced an analysis of sound wave propagation in a timedependent random medium which moves due to an ambient flow at speed ~v(t, ~x) andis modeled by the wave speed c(t, ~x) and mass density ⇢(t, ~x). The random fields~v(t, ~x), c(t, ~x), and ⇢(t, ~x) have small, statistically correlated fluctuations about theconstant values ~vo, co, and ⇢o, on the length scale ` and time scale T and withthe fluctuations modeled in terms of stationary random fields. The analysis startsfrom Pierce’s equation, which is obtained from the linearization of the fluid dynamics

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equations about an ambient flow, and applies to waves with central wavelength �o ⌧ `.The excitation is from a stationary source with radius `s, which emits a narrowbandsignal of duration Ts.

The analysis is in a forward wave propagation regime to a large distance (range)L � `, within a cone with small opening angle. Using the di↵usion approxima-tion theory, we showed that the coherent part (the expectation) of the wave decaysexponentially in L/S and quantified the frequency and wavevector dependent scat-tering mean free path S . We also derived transport equations for the energy density(Wigner transform) of the wave, which show explicitly the e↵ect of the ambient flowand net scattering in the time dependent random medium.

We used the wave propagation theory to study the inverse problem of localizing(imaging) the source from measurements at a stationary array of receivers located atrange L. This study is in the regime of paraxial wave propagation, where the Wignertransform can be computed explicitly, and assumes a large range L � S , so that thewave is incoherent due to strong scattering in the random medium. The temporalvariation of the medium is at time scale T ⌧ Ts, and it has two beneficial e↵ects forimaging. First, it causes broadening of the bandwidth of the recorded waves, whichleads to improved travel time estimation and consequently, better range resolution.Second, it allows a robust (statistically stable) estimation of the Wigner transformfrom the array measurements. We presented an explicit analysis of imaging basedon this Wigner transform and showed how one can estimate the source location, themean velocity ~vo, and the statistics of the random medium.

Appendix A. The Markovian limit theorem. In this appendix we obtainthe " ! 0 limit of the Markov process A"(z) = (A"(⌦,K, z))(⌦,K)2O , which lies onthe surface of the sphere with radius RA given in (5.40). The set O is defined by(5.38). The process A"(z) starts from

(A.1) A"(0) =

�

1

2p�(�K)

bS⇣⌘s⌘⌦,

�

�sK⌘!

(⌦,K)2O

,

which is independent of ", and evolves at z > 0 according to the stochastic equation

(A.2)dA"

dz= G

⇣z",z

"

⌘A" +

1p"F

⇣z",z

"

⌘A".

Here G and F are integral operators

[G(z, ⇣)A](⌦,K) =

Z

O

d⌦0dK 0 G(z, ⇣,⌦,⌦0,K,K 0)A(⌦0,K 0),

[F(z, ⇣)A](⌦,K) =

Z

O

d⌦0dK 0 F (z, ⇣,⌦,⌦0,K,K 0)A(⌦0,K 0),

with kernels depending on the random processes ⌫c(⌧, ~r) and Q"(⌧, ~r). Recall thedefinition (5.13) of Q"(⌧, ~r). We rewrite it here as

(A.3) Q"(⌧, ~r) = Q(0)(⌧, ~r) +

p"��⇢2

Q(1)(⌧, ~r)

with

(A.4) Q(0)(⌧, ~r) = �~r⌫⇢(⌧, ~r) and Q(1)(⌧, ~r) = |r~r⌫⇢(⌧, ~r)|2 .

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The kernel G has a deterministic part supported at ⌦0 = ⌦ and K 0 = K and arandom part determined by Q(1),

G(z, ⇣,⌦,⌦0,K,K 0) =iko

�(�K)

h ⌦

⌘co�~vo

co·��K,�(�K)

�i�(⌦0

� ⌦)�(K 0�K)

�i�3�2

⇢

8(2⇡)d+1p�(�K)�(�K 0)

exp�i⇥�(�K 0)� �(�K)

⇤⇣

⇥ bQ(1)⇣⌦� ⌦0

� ⌘��K � �K 0,�(�K)� �(�K 0)) · ~vo,K �K 0, �z

⌘.(A.5)

The kernel F is determined by ⌫c and Q(0),

F (z, ⇣,⌦,⌦0,K,K 0) =ip�

2(2⇡)d+1p�(�K)�(�K 0)

exp�i[�(�K 0)� �(�K)]⇣

⇥

k2o �cb⌫c

⇣⌦� ⌦0

� ⌘��K � �K 0,�(�K)� �(�K 0)) · ~vo,K �K 0, �z

⌘

��2�⇢2

bQ(0)⇣⌦� ⌦0

� ⌘��K � �K 0,�(�K)� �(�K 0)) · ~vo,K �K 0, �z

⌘�.

(A.6)

The random process A"(z) is Markov with generator

L"f(A,A) =

Z

O2

1p"F⇣z",z

",⌦,⌦0,K,K 0

⌘ �f

�A(⌦,K)A(⌦0,K 0)d⌦0dK 0d⌦dK

+

Z

O2

1p"F⇣z",z

",⌦,⌦0,K,K 0

⌘ �f

�A(⌦,K)A(⌦0,K 0)d⌦0dK 0d⌦dK

+

Z

O2

G⇣z",z

",⌦,⌦0,K,K 0

⌘ �f

�A(⌦,K)A(⌦0,K 0)d⌦0dK 0d⌦dK

+

Z

O2

G⇣z",z

",⌦,⌦0,K,K 0

⌘ �f

�A(⌦,K)A(⌦0,K 0)d⌦0dK 0d⌦dK,

where �f/�A(⌦,K) denotes the variational derivative, defined as follows. If ' is asmooth function and

f(A,A) =

Z· · ·

Z'(⌦1, . . . ,⌦n+m,K1, . . . ,Kn+m)

nY

j=1

A(⌦j ,Kj)

⇥

n+mY

j=n+1

A(⌦j ,Kj)n+mY

j=1

d⌦jdKj ,

then we have

�f

�A(⌦,K)=

nX

l=1

Z· · ·

Z'(⌦1, . . . ,⌦n+m,K1, . . . ,Kn+m) |

⌦l=⌦,Kl=K

nY

j=1,j 6=l

A(⌦j ,Kj)

⇥

n+mY

j=n+1

A(⌦j ,Kj)n+mY

j=1,j 6=l

d⌦jdKj

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and

�f

�A(⌦,K)=

n+mX

l=n+1

Z· · ·

Z'(⌦1, . . . ,⌦n+m,K1, . . . ,Kn+m) |

⌦l=⌦,Kl=K

nY

j=1

A(⌦j ,Kj)

⇥

n+mY

j=n+1,j 6=l

A(⌦j ,Kj)n+mY

j=1,j 6=l

d⌦jdKj .

The linear combinations of such functions f form an algebra that is dense in C(L2)and is convergence determining. We can also extend the class of functions to includegeneralized functions ' of the form

'(⌦1, . . . ,⌦2n,K1, . . . ,K2n) =�(⌦1, . . . ,⌦n,K1, . . . ,Kn)

⇥

nY

j=1

�(⌦n+j � ⌦j)�(Kn+j �Kj),

where � is a smooth function.Applying the di↵usion approximation theory described in [9, Chapter 6] and [18,

17], we obtain the limit generator

Lf(A,A) =

Z1

0d⇣ lim

Z!1

1

Z

Z Z

0dh

Z

O4

d⌦0

1dK0

1d⌦0

2dK0

2d⌦1dK1d⌦2dK2

⇥

nE⇥F (0, h,⌦1,⌦

0

1,K1,K0

1)F (⇣, ⇣ + h,⌦2,⌦0

2,K2,K0

2)⇤

⇥�2f

�A(⌦1,K1)�A(⌦2,K2)A(⌦0

1,K0

1)A(⌦0

2,K0

2)

+ E⇥F (0, h,⌦1,⌦

0

1,K1,K0

1)F (⇣, ⇣ + h,⌦2,⌦0

2,K2,K0

2)⇤

⇥�2f

�A(⌦1,K1)�A(⌦2,K2)A(⌦0

1,K0

1)A(⌦0

2,K0

2)

+ E⇥F (0, h,⌦1,⌦

0

1,K1,K0

1)F (⇣, ⇣ + h,⌦2,⌦0

2,K2,K0

2)⇤

⇥�2f

�A(⌦1,K1)�A(⌦2,K2)A(⌦0

1,K0

1)A(⌦0

2,K0

2)

+ E⇥F (0, h,⌦1,⌦

0

1,K1,K0

1)F (⇣, ⇣ + h,⌦2,⌦0

2,K2,K0

2)⇤

⇥�2f

�A(⌦1,K1)�A(⌦2,K2)A(⌦0

1,K0

1)A(⌦0

2,K0

2)o

+

Z1

0d⇣ lim

Z!1

1

Z

Z Z

0dh

Z

O3

d⌦0

1dK0

1d⌦1dK1d⌦0dK 0

⇥

nE⇥F (0, h,⌦0,⌦0

1,K0,K 0

1)F (⇣, ⇣ + h,⌦1,⌦0,K1,K

0)⇤ �f

�A(⌦1,K1)A(⌦0

1,K0

1)

+ E⇥F (0, h,⌦0,⌦0

1,K0,K 0

1)F (⇣, ⇣ + h,⌦1,⌦0,K1,K

0)⇤ �f

�A(⌦1,K1)A(⌦0

1,K0

1)o

+ limZ!1

1

Z

Z Z

0dh

Z

O2

d⌦dKd⌦0dK 0

nE⇥G(0, h,⌦,⌦0,K,K 0)

⇤

⇥�f

�A(⌦,K)A(⌦0,K 0) + E

⇥G(0, h,⌦,⌦0,K,K 0)

⇤ �f

�A(⌦,K)A(⌦0,K 0)

o.

(A.7)

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The expectations in the expression of the generator can be computed with

E⇥b⌫c(⌦,K, ⇣)b⌫c(⌦0,K 0, 0)

⇤= (2⇡)d+1�(K +K 0)�(⌦+ ⌦0) bRcc(⌦,K, ⇣),(A.8)

E⇥b⌫c(⌦,K, ⇣)b⌫c(⌦0,K 0, 0)

⇤= (2⇡)d+1�(K �K 0)�(⌦� ⌦0) bRcc(⌦,K, ⇣),(A.9)

E⇥b⌫c(⌦,K, ⇣)b⌫c(⌦0,K 0, 0)

⇤= (2⇡)d+1�(K �K 0)�(⌦� ⌦0) bRcc(⌦,K, ⇣),(A.10)

E⇥b⌫c(⌦,K, ⇣)b⌫c(⌦0,K 0, 0)

⇤= (2⇡)d+1�(K +K 0)�(⌦+ ⌦0) bRcc(⌦,K, ⇣),(A.11)

and similarly for b⌫⇢. Note here that both ⌫c and bRcc are real. We also have

bQ(0)(⌦,K, z) =�� |K|

2 + @2z�b⌫⇢(⌦,K, z)(A.12)

and

E⇥ bQ(1)(⌦,K, z)

⇤= �(2⇡)d+1�(⌦)�(K)�~rR⇢⇢(0, ~r)

��~r=0

.(A.13)

A.1. The mean amplitude. To calculate the mean of the limit process, we let

f(A,A) =

Z

O

d⌦dK '(⌦,K)A(⌦,K)

so that�f

�A(⌦1,K1)= '(⌦1,K1),

�f

�A(⌦1,K1)= 0,

and all second variational derivatives are zero.From the expression (A.7), definitions (A.5)–(A.6), and the expectations (A.8)–

(A.13) we obtain

Lf(A,A) =

Z

O

d⌦dK⇥i✓(⌦,K) +D(K)

⇤'(⌦,K)A(⌦,K)(A.14)

with ✓ and D given in (5.42) and (5.43). This gives the result (5.41).

A.2. The mean intensity. To characterize the mean intensity of the limit pro-cess, we let

f(A,A) =

Z

O

d⌦dK '(⌦,K)|A(⌦,K)|2

=

Z

O2

d⌦dKd⌦0dK 0 '(⌦,K)�(⌦� ⌦0)�(K �K 0)A(⌦,K)A(⌦0,K 0)

so that

�f

�A(⌦1,K1)= A(⌦1,K1)'(⌦1,K1),

�f

�A(⌦1,K1)= A(⌦1,K1)'(⌦1,K1),

�2f

�A(⌦1,K1)�A(⌦2,K2)= '(⌦2,K2)�(⌦2 � ⌦1)�(K2 �K1),

and all other second variational derivatives are zero.Using the expectation (A.13) and definition (A.5) in (A.7), we obtain that

the G dependent terms make no contribution. Furthermore, using the expectations

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(A.8)–(A.11) and (A.12) we get

Lf(A,A) =�

Z

O

d⌦1dK1

(2⇡)d+1'(⌦1,K1)|A(⌦1,K1)|

2

Z

O

d⌦0

1dK0

1Q(⌦1,⌦0

1,K1,K0

1)

+

Z

O

d⌦1dK1

(2⇡)d+1'(⌦1,K1)

Z

O

d⌦0

1dK0

1 |A(⌦0

1,K0

1)|2Q(⌦1,⌦

0

1,K1,K0

1)

with kernel defined in (5.49). This gives the equation satisfied by the mean intensity.

A.3. Wave decorrelation and the Wigner transform. To study the secondmoments at distinct frequencies ⌦, ⌦0 and wave vectors K and K 0, we let

f(A,A) =

Z

O2

d⌦dKd⌦0dK 0 '(⌦,⌦0,K,K 0)A(⌦,K)A(⌦0,K 0).

Then, we have

�f

�A(⌦1,K1)=

Z

O

d⌦0dK 0 A(⌦0,K 0)'(⌦1,⌦0,K1,K

0),

�f

�A(⌦1,K1)=

Z

O

d⌦0dK 0 A(⌦0,K 0)'(⌦0,⌦1,K0,K1),

�2f

�A(⌦1,K1)�A(⌦2,K2)= '(⌦1,⌦2,K1,K2),

and all other second variational derivatives are zero.Substituting in (A.7) and using the expectations (A.8)–(A.13) we obtain that

Lf(A,A) =

Z

O

d⌦dK

Z

|K0|<ko

dK 0 '(⌦,⌦0,K,K 0)

⇥⇥i✓(⌦,K)� i✓(⌦0,K 0) +D(K) +D(K 0)

⇤A(⌦,K)A(⌦0,K 0)(A.15)

with ✓(⌦,K) and D(K) defined in (5.42) and (5.43). This gives the decorrelationresult (5.51).

Finally, to study the Wigner transform, we use (5.36) to obtain an evolutionequation for

W"(⌦,K, t,x, z) =

Z

R

dw

2⇡

Z

R

dq

(2⇡)de�iwt+iq·[x+r�(K)z]

⇥A"⇣⌦+

"w

2,K +

"q

2, z⌘A

"⇣⌦�

"w

2,K �

"q

2, z⌘,

and then analyze the limit "! 0 of W" with the same approach as described in this

appendix. The Wigner transform (5.52) is

W "(⌦,K,x, z) =

Z

RdtE

⇥W

"(⌦,K, t,x, z)⇤,

and this converges in the limit to the solution W (⌦,K,x, z) of (5.53)–(5.54).

Appendix B. The transport equation in the physical scales. To distin-guish between the scaled and unscaled variables, we resurrect the notation of section5.2 with the unscaled variables denoted by primes.

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We begin with the pressure field (5.34),

p(TLt0, Lx0, Lz0) ⇡

i!o⇢o"d

Zd!0

2⇡

Zdk0

(2⇡)da"

0(!0

� �0(k0)v0z,k0, z0)p

�0(k0)

⇥ e�i�

!0o" +!0

�t0+ik0

" ·x0+i �0(k0

)

" z0.(B.1)

The scaling relations (2.8)–(2.15) and (5.5)–(5.6) give

!0ot

0

"=

2⇡

�o/L

t

TL= !ot,

!0t0 = !0t

TL= !t, i.e., !0 = !TL,

k0

"· x0 =

k0

�o/L·x

L=

k0

�o· x = k · x, i.e., k0 = �ok,

�0(k0) =p(k0o)

2 � |k0|2 = �opk2o � |k|2 = �o�(k),

!0� �0(k0)v0oz = TL! � �o�(k)

voz(�o/L)co

= TL[! � �(k)voz],

�0(k0)

"z0 =

�o�(k)

�o/L

z

L= �(k)z.

Equation (B.1) becomes (3.1), with amplitudes

(B.2) a(!,k, z) =TLLd

p�o

a"0⇣TL(! � �(k)voz),�ok,

z

L

⌘,

satisfying the initial conditions

(B.3) a(!,k, 0) =TLLd

�o

i

2p�(k)

bS(Ts!, `sk) =i�sTs`ds2p�(k)

bS(Ts!, `sk),

derived from (2.17) and (5.32).It remains to write the transport equation (3.12) for the Wigner transform. To

do so, we obtain from definitions (5.33) and the scaling relations above that

⌦0 = ⌘[!0 + �0(k0)v0oz] = T [! + �(k)voz],

K 0 =k0

�=

�ok

�o/`= `k.

We also recall the definition (5.35) of A" in terms of a" and obtain that

W (!,k,x, z) =

Zdq

(2⇡)dexp

hiq ·

�r�(k)z + x

�iEa⇣!,k +

q

2

⌘a⇣!,k �

q

2

⌘�

=

✓TLLd

p�o

◆21

�do

Zdq0

(2⇡)dexp

hi

q0

�o/L·�r�0(k0)z0 + x0

�i

⇥ EA"

⇣⌦0

� ⌘�0(�K 0)v0oz,K0 +

q0

2, z0

⌘A"

⇣⌦0 � ⌘�(�K 0)v0oz,K

0 �q0

2, z0

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with ⌦0 and K 0 defined as above in terms of ! and k, and z0 = z/L. Since " = �o/L,we can change the variable of integration as q0 q0" and obtain

W (!,k,x, z) =T 2LL

d

�oW " 0�

⌦0� ⌘�0(�K 0)v0oz,K

0,x0, z0�

⇡ �2sT

2s `

2ds�oLd

W 0�⌦0

� ⌘�0(�K 0)v0oz,K0,x0, z0

�.(B.4)

Here the approximation is for " ⌧ 1, where we have replaced W " 0by its " ! 0

limit W0.

Using the initial conditions (5.37) and (5.54) and the scaling relations between⌦0, K 0 and ! and k, we have

W (!,k,x, 0) = �2sT

2s `

2ds�oLd

�(x/L)

4�0(k0)

�� bS⇣⌘s!

0,`s`K 0

⌘��2

= �2sT

2s `

2ds�(x)

4�(k)

�� bS�Ts!, `sk

��2,

as stated in (3.13) and (3.3). The transport equation (3.12) follows from (5.53), using

@z0 �r�0(�K 0) ·rx0 = L [@z �r�(k) ·rx] .

Appendix C. Solution of the transport equation in the paraxial regime.

To deal with the convolution in (3.22), we Fourier transform in !,k and x,

W (t,y, q, z) =

Z

R

d!

2⇡e�i!t

Z

Rd

dk

(2⇡)deik·y

Z

Rd

dx e�iq·xW (!,k,x, z).(C.1)

Using definition (3.23) of the scattering kernel and the expression (3.9) of the power

spectral density bRcc, we write

Qpar(!,k) =k2o�

2c `

d+1T

4

Z

Rdet0 ei!T et0

Z

Rd

dy0 e�i`k·y0Z

Rdz0 Rcc(et0,y0, z0)

=k2o�

2c `

4

Z

Rdt ei!t

Z

Rd

dy e�ik·yR

⇣ t

T,y

`

⌘

with R defined in (3.25). Substituting into (3.22), we obtain

h@z +

q

ko·ry

iW (t,y, q, z) =

�2c `k

2o

4

hR

⇣ t

T,y � vt

`

⌘� R(0,0)

iW (t,y, q, z),(C.2)

for z > 0, with initial condition obtained from (3.26) and (C.1)

W (t,y, q, 0) = W0(t,y) :=�2sT

2s `

2ds

4ko

Z

R

d!

2⇡e�i!t

Z

Rd

dk

(2⇡)deik·y

�� bS(Ts!, `sk)��2

=�2sTs`ds4ko

Z

R

d⌦

2⇡e�i⌦ t

Ts

Z

Rd

dK

(2⇡)deiK·

y`s

�� bS(⌦,K)��2.(C.3)

Note that this condition is independent of q.Equation (C.2) can be solved by integrating along the characteristic y = y0+

qkoz,

stemming from y0 at z = 0,

W⇣t,y0 +

q

koz, q, z

⌘= W0(t,y0)

⇥ expn�2

c `k2o

4

Z z

0dz0

hR

⇣ t

T,y0 +

qkoz0 � vot

`

⌘� R(0,0)

io.

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Substituting y0 = y � q/koz in this equation, and inverting the Fourier transform,

W (!,k,x, z) =

Z

Rdt ei!t

Z

Rd

dy e�ik·y

Z

Rd

dq

(2⇡)deiq·xW0

⇣t,y �

q

koz⌘

⇥ expn�2

c `k2o

4

Z z

0dz0

hR

⇣ t

T,y �

qko(z � z0)� vot

`

⌘� R(0,0)

io.(C.4)

The result (3.27) follows after substituting the expression (C.3) into this equation.

Appendix D. Proof of radiative transfer connection. We prove here theresult involving (3.17) and (3.18). We start by computing the di↵erent terms in (3.18):The first term is

r~k⌦(~k) ·r~xV (!, ~k, ~x) =

coko��kz � �(k)

�h�@z �rk�(k) ·rx

�W (!,k,x, z)

i.

For the second term we use that if ~k0 = (k0,�(k0)) and ~k = (k, kz), then

��⌦(~k)� ⌦(~k0)

�=

1

co��|~k|� ko

�=

koco�(k)

��kz � �(k)

�

andZ

Rd+1

d~k0

(2⇡)d+1

Zd!0

2⇡S�!,!0, ~k, ~k0

�V (!0, ~k0, ~x) =


�

⇥

Z

O

d!0dk0

(2⇡)d+1Q(!,!0,k,k0)W (!0,k0,x, z).

Similarly, for the third term we have that if ~k = (k,�(k)) and ~k0 = (k0, k0z), then

��⌦(~k)� ⌦(~k0)

�=

1

co��ko � |~k|

�=

koco�(k0)

��k0z � �(k0)

�

andZ

Rd+1

d~k0

(2⇡)d+1

Zd!0

2⇡S�!,!0, ~k, ~k0

�V (!, ~k, ~x) =


�

⇥

Z

O

d!0dk0

(2⇡)d+1Q(!,!0,k,k0)W (!,k,x, z).

Gathering the results and using (3.12), we obtain (3.18).

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