PARAXIAL COUPLING OF PROPAGATING MODES INTHREE-DIMENSIONAL WAVEGUIDES WITH RANDOM
BOUNDARIES∗
LILIANA BORCEA† AND JOSSELIN GARNIER‡
Abstract. We analyze long range wave propagation in three-dimensional random waveguides.The waves are trapped by top and bottom boundaries, but the medium is unbounded in the tworemaining directions. We consider scalar waves, and motivated by applications in underwater acous-tics, we take a pressure release boundary condition at the top surface and a rigid bottom boundary.The wave speed in the waveguide is known, but the top boundary has small random fluctuations thatcause significant cumulative scattering of the waves over long distances of propagation. To quantifythe scattering effects, we study the evolution of the random amplitudes of the waveguide modes. Weobtain that in the long range limit they satisfy a system of paraxial equations driven by a Brownianfield. We use this system to estimate three important mode-dependent scales: the scattering meanfree path, the cross-range decoherence length, and the decoherence frequency. Understanding thesescales is important in imaging and communication problems, because they encode the cumulativescattering effects in the wave field measured by remote sensors. As an application of the theory, weanalyze time reversal and coherent interferometric imaging in strong cumulative scattering regimes.
Key words. waveguides, random media, asymptotic analysis
AMS subject classifications. 76B15, 35Q99, 60F05
DOI. 10.1137/12089747X
1. Introduction. We study long range scalar (acoustic) wave propagation in athree-dimensional waveguide. The setup is illustrated in Figure 1, and it is motivatedby problems in underwater acoustics. We denote by z ∈ R the range, the maindirection of propagation of the waves. The medium is unbounded in the cross-rangedirection x ∈ R, but it is confined in depth y by two boundaries which trap the waves,thus creating the waveguide effect.
The acoustic pressure field is denoted by p(t, x, y, z), and it satisfies the waveequation(1.1)[∂2x+∂
2y+∂
2z−
1
c2(y)∂2t
]p(t, x, y, z) = f(t, x, y)δ(z), y ∈ [0, T (x, z)], x, z ∈ R, t > 0,
in a medium with wave speed c(y). The excitation is due to a source located in theplane z = 0, emitting the pulse f(t, x, y). The medium is quiescent before the sourceexcitation,
(1.2) p(t, x, y, z) = 0, t� 0.
∗Received by the editors November 2, 2012; accepted for publication (in revised form) April 9,2014; published electronically June 24, 2014.
http://www.siam.org/journals/mms/12-2/89747.html†Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043 (borcea@
umich.edu). This author’s work was partially supported by AFSOR grant FA9550-12-1-0117, ONRgrants N00014-12-1-0256, N00014-09-1-0290, and N00014-05-1-0699, and NSF grants DMS-0907746and DMS-0934594.
‡Laboratoire de Probabilites et Modeles Aleatoires & Laboratoire Jacques-Louis Lions, UniversiteParis Diderot, 75205 Paris Cedex 13, France ([email protected]). This author’swork was supported in part by ERC Advanced Grant Project MULTIMOD-267184.
Fig. 1. Schematic of the problem setup. The system of coordinates has range origin z = 0 atthe source. The rigid bottom boundary y = 0 is assumed flat, and the pressure release top boundaryhas fluctuations around the value y = D. The cross-range x and the range z are unbounded, thatis, (x, z) ∈ R2.
The bottom of the waveguide is assumed rigid,
(1.3) ∂yp(t, x, y = 0, z) = 0,
and we take a pressure release boundary condition at the perturbed top boundary,
(1.4) p(t, x, y = T (x, z), z) = 0.
Perturbed means that the boundary y = T (x, z) has small fluctuations around themean depth D,
(1.5) |T (x, z)−D| � D.
We choose this setup for simplicity. The results extend readily to other boundaryconditions and to fluctuating bottoms. Such boundaries were considered recentlyin [1, 14], in two-dimensional waveguides. Extensions to media with small (x, y, z)-dependent random fluctuations of the wave speed can also be made using the tech-niques developed in [15, 8, 11, 12, 13].
The goal of our study is to quantify the effect of scattering at the surface. Becausein applications it is not feasible to know the boundary fluctuations in detail, we modelthem with a random process. The solution p(t, x, y, z) of (1.1)–(1.4) is therefore arandom field, and we describe in detail its statistics at long ranges, where cumulativescattering is significant. We use the results for two applications: time reversal andsensor array imaging.
Our method of solution uses a change of coordinates to straighten the boundary.The transformed problem has a simple geometry but a randomly perturbed differentialoperator. Its solution is given by a superposition of propagating and evanescentwaveguide modes of the unperturbed waveguide, with random amplitudes. We showthat in the long range limit these amplitudes satisfy a system of paraxial equationsthat are driven by a Brownian field. The detailed characterization of the statistics ofp(t, x, y, z) follows from this system. It involves the calculation of the mode-dependentscattering mean free path, which is the distance over which the modes lose coherence;the mode-dependent decoherence length, which is the cross-range offset over whichthe mode amplitudes decorrelate; and the mode-dependent decoherence frequency,which is the frequency offset over which the mode amplitudes decorrelate. Thesescales are important in studies of time reversal and imaging, because they dictate
the resolution of focusing and the robustness (statistical stability) of the results withrespect to realizations of the random fluctuations of the boundary.
We use the characterization of the statistics of the acoustic pressure field to studythe refocusing of the waves in the time reversal process and the resolution of imagingthe source with a remote array of sensors. In time reversal the waves received at thearray are reversed in time and reemitted in the medium where they propagate backto the original source and refocus. The interesting result is that the refocusing isimproved (has superresolution) in random waveguides, especially for long recordingtimes that capture many modes arriving at the array, and it is robust with respectto the realization of the random boundary, as long as the array has a large enoughaperture, or the emitted signals have large enough bandwidth. The superresolutionproperty of the time reversal process in random media has received much attention,beginning with the work in [10, 16], and has been analyzed mathematically in detailfor wave propagation in open space [2, 3, 20] and in two-dimensional waveguides [12].Here we analyze it for the three-dimensional random waveguides defined above anddescribe in detail the improved resolution and robustness of the refocusing.
Imaging is very different from time reversal. While in time reversal the wavespropagate physically in the real medium from the array to the source, in imagingthey are propagated analytically or computationally. Because the boundary fluctu-ations are not known, the propagation is done in a fictitious waveguide with planar(unperturbed) boundary. We show that the resulting imaging function is not usefulfor imaging sources at long distances from the array, beyond the scattering mean freepaths of all the waveguide modes. The strong cumulative scattering at the surfacecauses large random fluctuations of the wave field p (the waveguide noise) that exceedits expectation (the signal). The signal to noise ratio (SNR) is low, and the imagesare unreliable; they lack statistical stability. Note that the array data corresponds tothe pressure measured for a single realization of the random boundary. The imageis formed by processing these data, and statistical stability means that the result isessentially independent of the boundary fluctuations, which are unknown.
We show that imaging can be carried out at long ranges by propagating localcross-correlations of the array data. Local means that the cross-correlations are com-puted for the data projected on one waveguide mode at a time and for nearby cross-ranges and frequencies. The superposition of the local cross-correlations propagatedanalytically or computationally to the search domain in the fictitious waveguide withunperturbed boundary forms an imaging function that is similar to the coherentinterferometric (CINT) one introduced and analyzed in [7, 5, 6, 4] for imaging inopen random environments. We analyze in detail the resolution limits of the imagingmethod and show that it is statistically stable under the same conditions as timereversal: for large enough apertures or bandwidths. However, while cumulative scat-tering improves the refocusing in time reversal, it impedes imaging. The resolutionis worse than in unperturbed waveguides, and it does not improve by increasing therecording time.
The paper is organized as follows: We begin in section 2 with the unperturbedproblem, in ideal (unperturbed) waveguides with planar boundaries, and introducethe mode decomposition of the wave field. We consider sources that emit a beam alongthe range direction and introduce in section 3 the paraxial scaling regime for beampropagation and the random model of the boundary fluctuations. The analysis ofbeam propagation in random waveguides is in section 4. We may view it as a pertur-bation of that in ideal waveguides, in the sense that the wave field can be decomposed
in the same waveguide modes. However, the modes are coupled by scattering at therandom boundary and have random amplitudes. The random boundary fluctuationsare small, but they have a significant cumulative scattering effect at long ranges, asdescribed in section 5. The statistics of the wave field at long ranges are described insection 6. The results are summarized in section 7 and are used in sections 8 and 9for analyzing time reversal and imaging with sensor arrays. We end with a summaryin section 10.
2. The unperturbed problem and mode decomposition. The pressurefield in ideal waveguides, with planar boundaries, is given by
(2.1) po(t, x, y, z) =
∫ ∞
−∞
dω
2πpo(ω, x, y, z)e
−iωt,
with Fourier coefficients satisfying a separable problem for the Helmholtz equation[∂2x + ∂2y + ∂2z +
ω2
c2(y)
]po(ω, x, y, z) = f(ω, x, y)δ(z),(2.2)
|ω − ω0| ≤B
2, (x, z) ∈ R
2, y ∈ (0,D),
with boundary conditions
(2.3) ∂y po(ω, x, y = 0, z) = po(ω, x, y = D, z) = 0
and outgoing radiation conditions at√x2 + z2 → ∞. The Fourier transform of the
source
(2.4) f(ω, x, y) =
∫ ∞
−∞dt f(t, x, y)eiωt
is assumed to be compactly supported in [ω0 −B/2, ω0 +B/2] for any x and y. Hereω0 is the central frequency and B is the bandwidth. The spatial (cross-range) support
of the source f(ω, x, y) is larger than the wavelength, as described in detail in section3, so that it emits a beam in the range direction z.
The boundary value problem (2.2)–(2.3) for the Helmholtz equation can be solvedwith separation of variables. The solution is a superposition of N(ω) propagatingmodes, and infinitely many evanescent ones,
(2.5) po(ω, x, y, z) =
N(ω)∑j=1
φj(ω, y)uj,o(ω, x, z) +
∞∑j=N(ω)+1
φj(ω, y)vj,o(ω, x, z).
The decomposition is in the L2(0,D) orthonormal basis of the eigenfunctions φj(ω, y)of the self-adjoint differential operator in y,[
∂2y +ω2
c2(y)
]φj(ω, y) = λj(ω)φj(ω, y),
φj(ω,D) = ∂yφj(ω, 0) = 0, j = 1, 2, . . . ,
with eigenvalues λj(ω) that are simple [21].To simplify the analysis, we assume in this paper that the wave speed is homo-
The results for variable c(y) are similar in all the essential aspects, as long as thewave speed profile does not trap the eigenfunctions in the interior of the waveguide,away from the surface. The simplification brought by (2.6) amounts to having explicitexpressions of the eigenfunctions, which are independent of the frequency
(2.7) φj(y) =
√2
D cos
[π
(j − 1
2
)y
D
].
The eigenvalues are
(2.8) λj(ω) =( πD
)2 [(kDπ
)2
−(j − 1
2
)2],
where k = ω/co is the wavenumber, and only the first N(ω) of them are nonnegative:
(2.9) N(ω) =
⌊kDπ
+1
2
⌋.
The notation � stands for the integer part. We suppose for simplicity that N(ω)remains constant in the bandwidth [ω0 − B/2, ω0 + B/2] and from now on writeN(ω) = N .
The propagating components in (2.5) satisfy the two-dimensional Helmholtz equa-tion
with outgoing, radiation conditions at√x2 + z2 → ∞. The evanescent components
solve
(2.11)[∂2x + ∂2z − β2
j (ω)]vj,o(ω, x, z) = Fj(ω, x)δ(z), j > N,
with decay condition vj,o(ω, x, z) → 0 at√x2 + z2 → ∞. Here we introduce the
coefficients of the source profile in the basis of the eigenfunctions
(2.12) Fj(ω, x) =
∫ D
0
dy φj(y)f(ω, x, y), j ≥ 1,
and the mode wavenumbers
(2.13) βj(ω) =√|λj(ω)| =
π
D
√∣∣∣∣(kDπ)2
−(j − 1
2
)2∣∣∣∣, j ≥ 1.
We assume that none of the βj(ω) vanishes in the bandwidth, so that there are nostanding waves. That is to say,
(2.14)kDπ
= N + α(ω)− 1
2, α(ω) ∈ (0, 1) for all ω ∈ [ω0 −B/2, ω0 +B/2].
3. The paraxial scaling regime and the random model. We define insection 3.1 the paraxial scaling regime and the random model of the boundary fluc-tuations. The source has a cross-range profile that extends over many wavelengthsand emits a beam along the range axis z. The beam propagation in ideal waveguidesis described in section 3.2. We use it as a reference to compare with the result inrandom waveguides derived in section 4.
3.1. Scaling and the random boundary fluctuations. The source is of theform
(3.1) fε(t, x, y) = f(t, εx, y),
where
(3.2) ε =λ0r0
� 1
is a small dimensionless parameter defined as the ratio of the central wavelength λ0and the transverse width r0 of the source. Standard diffraction theory gives that theRayleigh length for a beam with initial width r0 is of the order of
r20/λ0 = λ0/ε2.
The Rayleigh length is defined as the distance along the z axis from the beam waistto the place where the beam area is doubled by diffraction. To capture order onediffraction effects we analyze the wave field at O(ε−1) cross-range scales, similar tor0, and at range scale Lε, similar to the Rayleigh length
(3.3) Lε/λ0 = O(ε−2).
The boundary fluctuations are modeled with a random process μ:
(3.4) T ε(x, z) = D[1 + ε3/2μ (εx, εz)
], z ∈ (0, L/ε2).
The process μ is bounded, zero-mean, stationary, and mixing, meaning in particularthat its covariance is integrable.1 Because our method of solution flattens the bound-ary by changing coordinates, we require that μ be twice differentiable, with almostsurely bounded derivatives. Its covariance function is given by
(3.5) R (ξ, ζ) = E [μ(ξ′ + ξ, ζ′ + ζ)μ(ξ′, ζ′)] ,
and we denote by Ro(ξ) its integral over ζ,
(3.6) Ro(ξ) =
∫ ∞
−∞dζ R(ξ, ζ).
Our assumption on the differentiability of μ implies thatRo is four times differentiable.Note that ξ = 0 is the maximum of the integrated covariance Ro(ξ), so we have
(3.7) R′o(0) = 0.
As seen in (3.4) we assume that the correlation length ε of the fluctuations ofthe boundary is of the same order as the beam width
(3.8) ε = ε−1 ∼ r0,
so that there is nontrivial interaction between the random boundary and the wavebeam. The amplitude of the fluctuations is scaled to order of ε3/2λ0, in order toobtain a cumulative scattering effect of order one after the propagation distance Lε.
1More precisely, μ is a f-mixing process, with f ∈ L1/2(R+), as stated in [17, sect. 4.6.2].
Weaker fluctuations have a negligible effect, and stronger fluctuations cause so muchscattering that coherent imaging at range scale Lε is no longer possible. In (3.8) wedenote by the scaled (order one) correlation length defined by
(3.9) Ro(0) = σ2 ,R′′
o (0)
Ro(0)= − 1
2,
and σ is the typical amplitude of μ.We use the hyperbolicity of the problem to truncate mathematically the boundary
fluctuations to the range interval (0, L/ε2). The upper bound L/ε2 is the maximumrange of the fluctuations that can affect the waves up to an observation time scaledas O
(L/(ε2co)
). The lower bound 0 in the range interval coincides with the location
of the source. It is motivated by two facts: First, we observe the waves at positiveranges. Second, the backscattered field is negligible in the scaling regime definedabove, as we show later in section 5.3.
3.2. Beam propagation in ideal waveguides. We rename the field in theparaxial scaling defined above as
(3.10) pεo(t,X, y, Z) = po
(t,X
ε, y,
Z
ε2
).
Its Fourier coefficients are given by the scaled version of (2.5),
(3.11) pεo(ω,X, y, Z) =
N(ω)∑j=1
φj(y)uεj,o(ω,X,Z) +
∞∑j=N(ω)+1
φj(y)vεj,o(ω,X,Z),
with propagating mode amplitudes uεj,o satisfying the scaled equation (2.10), with the
source replaced by Fj(ω, εx = X). They can be written as
uεj,o(ω,X,Z) = −1
ε
∫ ∞
−∞dX ′ Fj(ω,X
′)Go
(βj(ω),
X −X ′
ε,Z
ε2
)in terms of the outgoing Green’s function
Go
(βj(ω), x, z
)=i
4H
(1)0
[βj(ω)
√x2 + z2
].
Here H(1)0 is the Hankel function of the first kind, and because ε� 1, we can use its
asymptotic form for a scaled range Z > 0:
i
4H
(1)0
[βj(ω)
√(X −X ′)2
ε2+Z2
ε4
]≈ 1
4
⎡⎣ 2i
πβj(ω)√
(X−X′)2ε2 + Z2
ε4
⎤⎦1/2
× exp
[iβj(ω)
√(X −X ′)2
ε2+Z2
ε4
]
≈ ε
2
√i
2πβj(ω)Zexp
{iβj(ω)
[Z
ε2+
(X −X ′)2
2Z
]}.
The propagating components of the wave field become
for j = 1, . . . , N . The evanescent components are obtained similarly from (2.11):
vεj,o(ω,X,Z) ≈ ej,o(ω,X,Z) exp
[−βj(ω)
Z
ε2
],
and
(3.13) ej,o(ω,X,Z) = −1
2
√1
2πβj(ω)Z
∫ ∞
−∞dX ′ exp
[−βj(ω)(X −X ′)2
2Z
]Fj(ω,X
′)
for j ≥ N + 1. These modes are exponentially damped and can be neglected.In summary, the paraxial approximation of the wave field is given by
(3.14) pεo(ω,X, y, Z) ≈N∑j=1
φj(y)aj,o(ω,X,Z)eiβj(ω) Z
ε2 .
It is a superposition of forward-going modes, which are quasi-plane waves propagatingin the range direction z, with slowly varying amplitudes aj,o given by (3.12). Theysolve the paraxial equations
(3.15)[2iβj(ω)∂Z + ∂2X
]aj,o(ω,X,Z) = 0, j = 1, . . . , N,
with initial conditions
(3.16) aj,o(ω,X,Z = 0) = aj,ini(ω,X) :=1
2iβj(ω)Fj(ω,X), j = 1, . . . , N.
4. Wave propagation in random waveguides. To analyze beam propagationin random waveguides we introduce in section 4.1 a change of coordinates that flattensthe random boundary. The mapped wave field satisfies a wave equation perturbed bya differential operator with random coefficients defined by the process μ. We showin section 4.2 that the solution can be written as a superposition of the unperturbedwaveguide modes with random amplitudes that solve paraxial equations driven bythe random process μ. They model the cumulative scattering effects of the randomboundary and are analyzed in section 5 in the paraxial scaling regime defined insection 3.
4.1. Change of coordinates. Consider the change of coordinates from (x, y, z)to (x, η, z), with
(4.1) η =yD
T ε(x, z),
which straightens the boundary y = T ε(x, z) to η = D for any x ∈ R and z ∈ (0, L/ε2).The pressure field in the new coordinates is denoted by
derived from (1.1) and (4.2) using the chain rule. Here ∇ and Δ are the gradient andLaplacian operators in (x, z) and f ε is the source of the form (3.1).
When substituting the model (3.4) into (4.4), we obtain that P satisfies a ran-domly perturbed problem(4.5)[
∂2x + ∂2z +(1− 2ε3/2μ(εx, εz)
)∂2η + k2 + h.o.t.
]P (ω, x, η, z) = f(ω, εx, η)δ(z).
The higher-order terms h.o.t. are
h.o.t. = rε1(εx, η, εz)∂2η + rε2(εx, η, εz)∂
2ηx + rε3(εx, η, εz)∂
2ηz + rε4(εx, η, εz)∂η,
with functions
rε1 =ε3μ2(3 + 2ε3/2μ)
(1 + ε3/2μ)2+ ε5η2
(∂ξμ)2 + (∂ζμ)
2
(1 + ε3/2μ)2,
rε2 = −2ε5/2η∂ξμ
1 + ε3/2μ,
rε3 = −2ε5/2η∂ζμ
1 + ε3/2μ,
rε4 = 2ε5η(∂ξμ)
2 + (∂ζμ)2
(1 + ε3/2μ)2− ε7/2η
∂2ξμ+ ∂2ζμ
1 + ε3/2μ,
evaluated at arguments (εx, η, εz). These terms are called higher-order because theyturn out to be negligible in the limit ε→ 0 considered in section 5.
4.2. Wave decomposition. Equation (4.5) is not separable, but we can stillwrite its solution in the L2(0,D) basis of the eigenfunctions (2.7). The expansion issimilar to (2.5):
(4.6) P (ω, x, η, z) =N∑j=1
φj(η)uj(ω, x, z) +∑j>N
φj(η)vj(ω, x, z).
We define the forward- and backward-going wave mode amplitudes aj and bj by
Definition (4.7) implies that the complex mode amplitudes also satisfy
(4.9) ∂zaj(ω, x, z)eiβj(ω)z + ∂zbj(ω, x, z)e
−iβj(ω)z = 0.
Equations (4.8) and (4.9) uniquely specify the propagating mode amplitudes. Theyeach satisfy a single boundary condition in the range (0, L/ε2) of the fluctuations. Toderive these boundary conditions, let us observe that because the boundary is flat out-side (0, L/ε2), the radiation (outgoing conditions) implies that the mode amplitudessatisfy
aj(ω, x, z = 0−) = 0,(4.10)
bj(ω, x, z = L/ε2) = 0.(4.11)
The last equation is the boundary condition for bj. The boundary value aj(ω, x, z =0+) follows from the jump conditions across the plane z = 0 of the source in (4.5).We have
[uj ]0+
0− = 0, [∂z uj]0+
0− = Fj(ω, εx),
with Fj defined by (2.12). This gives
[aj + bj ]0+
0− = 0, iβj [aj − bj ]0+
0− = Fj(ω, εx),
and therefore
(4.12) aj(ω, x, 0+) =
1
2iβj(ω)Fj(ω, εx).
Substituting (4.6) into (4.5) and using the orthonormality of the eigenfunctionsφj , we find that the wave mode amplitudes solve paraxial equations coupled by therandom fluctuations in z ∈ (0, L/ε2),(
Note that rq,εjl = O(ε5/2) for q = 1, 2, 3.The leading coupling matrix in (4.13)–(4.14) is given by
(4.15) qjl = 2
∫ D
0
dη φj(η)φ′′l (η) = −2
( πD
)2(j − 1
2
)2
δjl.
It takes this simple diagonal form, because we assumed a homogeneous backgroundspeed co. If we had a variable speed c(y), the matrix {qjl} would not be diagonal,and the modes with j �= l would be coupled. However, the results of the asymptoticanalysis below would still hold, because the coupling would become negligible in thelimit ε→ 0 considered in section 5, due to rapid phase terms exp(i(±βj±βl)z) arisingin the right-hand sides of (4.13), (4.14).
The equations for the evanescent components are obtained similarly,
with higher-order terms as in (4.13)–(4.14), and they are augmented with the decayconditions vj(ω, x, z) → 0 as
√x2 + z2 → ∞ for all j ≥ N + 1. The equations for the
evanescent mode amplitudes are similar to those encountered in [1]. These amplitudeswere shown to vanish as ε → 0 in [1, sect. 3.3] in a regime that was similar to thataddressed in this paper. We therefore neglect them in the following.
5. The limit process. We characterize next the wave field in the asymptoticlimit ε → 0. We begin with the paraxial long range scaling that gives significant netscattering and then take the limit. The scaling has already been described in section3.1.
5.1. Asymptotic scaling. We obtain from (4.13)–(4.15) that the propagatingmode amplitudes satisfy the system of partial differential equations
(2iβj∂z + ∂2x e−2iβjz∂2xe2iβjz∂2x −2iβj∂z + ∂2x
)(ajbj
)= ε3/2qjjμ(εx, εz)
(1 e−2iβjz
e2iβjz 1
)(ajbj
)−
N∑l=1
(e−i(βj−βl)z e−i(βj+βl)z
ei(βj+βl)z ei(βj−βl)z
)[r1,εjl (εx, εz) + r2,εjl (εx, εz)∂x
+ r3,εjl (εx, εz)∂z](al
bl
)−
N∑l=1
βlr3,εjl (εx, εz)
(ie−i(βj−βl)z −ie−i(βj+βl)z
iei(βj+βl)z −iei(βj−βl)z
)(albl
)(5.1)
for j = 1, . . . , N .Because the leading-order term of the right-hand side in (5.1) is small, of order
ε3/2, and has zero statistical expectation, it follows from [11, Chap. 6] that there isno net scattering effect until we reach ranges of order ε−2. Thus, we let
(5.2) z = Z/ε2,
with scaled range Z independent of ε. The source directivity in the range directionsuggests observing the wave field on a cross-range scale that is smaller than that inrange. We choose it as
(5.3) x = X/ε,
with scaled cross-range X independent of ε, to balance the two terms in the paraxialoperators in (5.1).
Our goal is to characterize the ε→ 0 limit of the mode amplitudes in the paraxiallong range scaling regime (5.2)–(5.3). We denote them by
and obtain from (5.1)–(5.3) that they satisfy the scaled system(2iβj∂Z + ∂2X e−2iβjZ/ε2∂2Xe2iβjZ/ε2∂2X −2iβj∂Z + ∂2X
)(aεjbεj
)
=1
ε1/2μ
(X,
Z
ε
)qjj
(1 e−2iβjZ/ε2
e2iβjZ/ε2 1
)(aεjbεj
)
−N∑l=1
(e−i(βj−βl)Z/ε2 e−i(βj+βl)Z/ε2
ei(βj+βl)Z/ε2 ei(βj−βl)Z/ε2
)
×[1
ε2r1,εjl
(X,
Z
ε
)+
1
εr2,εjl
(X,
Z
ε
)∂X + r3,εjl
(X,
Z
ε
)∂Z
](aεlbεl
)−
N∑l=1
1
ε2βlr
3,εjl
(X,
Z
ε
)(ie−i(βj−βl)Z/ε2 −ie−i(βj+βl)Z/ε2
iei(βj+βl)Z/ε2 −iei(βj−βl)Z/ε2
)(aεlbεl
)(5.5)
for j = 1, . . . , N , with initial conditions
(5.6) aεj(ω,X, 0) = aj,ini (ω,X) :=1
2iβj(ω)Fj(ω,X)
and end conditions
(5.7) bεj(ω,X,L) = 0.
Note that the higher-order terms in (5.5) (i.e., the terms with rq,εjl , q = 1, 2, 3) are at
most of order O(ε1/2).
5.2. The random propagator. Let us rewrite (5.5) in terms of the randompropagator matrixPε(ω,X,X ′, Z) ∈ C2N×2N , the solution of the initial value problem
∂ZPε(ω,X,X ′, Z) =
[I− Lε
(ω,X,Z
)]−1
×[
1
ε1/2μ
(X,
Z
ε
)H
(ω,X,
Z
ε2
)+G
(ω,X,
Z
ε2
)+Kε
(ω,X,Z
)]Pε(ω,X,X ′, Z),
Pε(ω,X,X ′, 0) = δ(X −X ′)I.(5.8)
Here I is the 2N × 2N identity matrix, δ(X) is the Dirac delta distribution in X , andG and H are 2N × 2N matrices with entries given by partial differential operatorsin X with deterministic coefficients. Kε is a 2N × 2N matrix with entries given bypartial differential operators in X , with random coefficients. Lε is a 2N × 2N matrixwhose entries are random bounded coefficients. The norm of the matrix is smallerthan one for ε small enough, so that the matrix I − Lε is indeed invertible. We candefine these matrices from (5.5) once we note that the solution
Here bε(ω,X ′, 0) is the vector of backward going amplitudes at the beginning of therandomly perturbed section of the waveguide, and it can be eliminated using theboundary identity
(5.11)
(aε(ω,X,L)
0
)=
∫dX ′ Pε(ω,X,X ′, L)
(aε(ω,X ′, 0)bε(ω,X ′, 0)
).
The initial conditions aε(ω,X ′, 0) are given in (5.6).We obtain from (5.5) that H, G, Kε, and Lε have the block form
H =
(Ha Hb
Hb Ha
), G =
(Ga Gb
Gb Ga
),
Kε =
(Kε,a Kε,b
Kε,b Kε,a
), Lε =
(Lε,a Lε,b
Lε,b Lε,a
),(5.12)
where the bar denotes complex conjugation. The blocks Ha, Hb, Ga, and Gb arediagonal, with entries
(5.13) Hajl = − i δjl qjj
2βj, Hb
jl = − i δjl qjj2βj
e−2iβjZ/ε2
and
(5.14) Gajl =
i δjl2βj
∂2X , Gbjl =
i δjl2βj
e−2iβjZ/ε2∂2X
for j, l = 1, . . . , N . The entries of the diagonal blocks Ha and Ga depend only onthe mode indices and the frequency, via βj(ω). The entries of the off-diagonal blocksHb and Gb are rapidly oscillating, due to the large phases proportional to Z/ε2. Theelements of the matrices Kε,a and Kε,b given by
(5.15)
Kε,ajl =
i
2βje−i(βj−βl)Z/ε2
[1
ε2r1,εjl
(X,
Z
ε
)+
1
εr2,εjl
(X,
Z
ε
)∂X +
i
ε2βlr
3,εjl
(X,
Z
ε
)],
(5.16)
Kε,bjl =
i
2βje−i(βj+βl)Z/ε2
[1
ε2r1,εjl
(X,
Z
ε
)+
1
εr2,εjl
(X,
Z
ε
)∂X − i
ε2βlr
3,εjl
(X,
Z
ε
)]are of order ε1/2. The elements of the matrices Lε,a and Lε,b given by
(5.17)
Lε,ajl =
i
2βje−i(βj−βl)Z/ε2r3,εjl
(X,
Z
ε
), Lε,b
jl =i
2βje−i(βj+βl)Z/ε2r3,εjl
(X,
Z
ε
)are of order ε5/2.
The symmetry relations satisfied by the blocks in H, G, Kε, and Lε imply thatthe propagator has the form
5.3. The diffusion limit. The limit of Pε as ε → 0 is a multidimensionalMarkov diffusion process, with entries satisfying a system of Ito–Schrodinger equa-tions. This follows from the diffusion approximation theorem [18, 19] (see also [11,sect. 6.3.4]) applied to system (5.8) that we rewrite as
∂ZPε(ω,X,X ′, Z) =
[1
ε1/2μ
(X,
Z
ε
)H
(ω,X,
Z
ε2
)+G
(ω,X,
Z
ε2
)+ Kε
(ω,X,Z
)]Pε(ω,X,X ′, Z),
Pε(ω,X,X ′, 0) = δ(X −X ′)I,(5.19)
with
Kε(ω,X,Z
)=[I− Lε
(ω,X,Z
)]−1Lε(ω,X,Z
)×[
1
ε1/2μ
(X,
Z
ε
)H
(ω,X,
Z
ε2
)+G
(ω,X,
Z
ε2
)]+[I− Lε
(ω,X,Z
)]−1Kε(ω,X,Z
).
The matrix Kε has the structure
Kε =
(Kε,a Kε,b
Kε,b Kε,a
),
and its entries are at most of order ε1/2.Computing the generator of the limit process with the formula given in the diffu-
sion approximation theorem, we obtain that due to the fast phases in the off-diagonalblocks Hb and Gb and due to the convergence to zero of Kε,b, the complex blockRε converges to zero as ε → 0. This means that the forward- and backward-goingamplitudes decouple as ε → 0, and this implies that there is no backscattered fieldin the limit, because the backward-going amplitudes bε are set to zero at Z = L.Equation (5.10) simplifies as
(5.20) aε(ω,X,Z) =
∫dX ′Tε(ω,X,X ′, Z)aε(ω,X ′, 0),
where the initial conditions aε(ω,X ′, 0) are given in (5.6). The complex matrixTε(ω,X,X ′, Z) is called the transfer process, because it gives the amplitudes of theforward-going modes at positive ranges Z in terms of the initial conditions at Z = 0.The limit transfer matrix is described in the next proposition.
Proposition 5.1. As ε → 0, Tε(ω,X,X ′, Z) converges weakly and in distribu-tion to the diffusion Markov process T(ω,X,X ′, Z). This process is complex and diag-onal matrix valued, with diagonal entries Tj(ω,X,X ′, Z) solving the Ito–Schrodingerequations
Equations (5.21) are uncoupled, but they are driven by the same Brownian field B(X,Z),satisfying
(5.23) E [B(X,Z)] = 0, E [B(X,Z)B(X ′, Z ′)] = min{Z,Z ′}Ro(X −X ′),
with Ro defined in (3.6). Thus, the transfer coefficients Tj are statistically correlated.The weak convergence in distribution means that we can calculate the limit ε→ 0
of statistical moments of Tε, smoothed by integration over X against the initialconditions, using the Markov diffusion defined by (5.21)–(5.22). In applications wehave a fixed ε� 1, and we use Proposition 5.1 to approximate the statistical momentsof the amplitudes of the forward-going waveguide modes.
The proof of Proposition 5.1 is obtained by the application of the diffusion-approximation theorem [11, sect. 6.3.4]. This theorem states that Tε converges toa diffusion Markov process and it gives the explicit form of the infinitesimal gener-ator. One can then check that this generator is that of the diagonal matrix whosediagonal entries satisfy the Ito–Schrodinger equations (5.21).
When comparing the Ito–Schrodinger equations (5.21) to the deterministic Schro-dinger equations (3.15) satisfied by the amplitudes in the ideal waveguides, we see thatthe random boundary scattering effect amounts to a net diffusion, as described by thelast two terms in (5.21). We show next how this leads to loss of coherence of thewaves, that is, to exponential decay in range of the mean field. We also study thepropagation of energy of the modes and quantify the decorrelation properties of therandom fluctuations of their amplitudes.
6. Statistics of the wave field. We begin in section 6.1 with the analysis ofthe coherent field. Explicitly, we estimate the mean forward-going mode amplitudesin the paraxial long range regime. Traditional imaging methods rely on these beinglarge with respect to their random fluctuations. However, this is not the case, becauseE[aεj(ω,X,Z)
]decay exponentially with Z, at rates that increase monotonically with
mode indices j. The second moments of the amplitudes do not decay, but there isdecorrelation over the modes and the frequency and cross-range offsets, as shown insections 6.3 and 6.4. Understanding these decorrelations is key to designing timereversal and imaging methods that are robust at low SNR. Robust means that wavefocusing in time reversal or imaging is essentially independent of the realization ofthe random boundary fluctuations; it is statistically stable. Low SNR means that thecoherent (mean) field, the “signal,” is faint with respect to its random fluctuations,the “noise.”
6.1. The coherent field. The mean modal amplitudes are
E[aεj(ω,X,Z)
]≈∫dX ′
E [Tj(ω,X,X ′, Z)]aj,ini (ω,X ′) ,(6.1)
with mean transfer matrix satisfying the partial differential equation
with units of length. Here we used definitions (2.13), (2.14), and (3.9) and obtained(6.3) by taking expectations in (5.21). Its solution is given by
(6.4) E [Tj(ω,X,X ′, Z)] =
√βj(ω)
2πiZexp
[− Z
Sj(ω)+iβj(ω)(X −X ′)2
2Z
],
and the mean modal amplitudes are obtained from (6.1) and (5.6),
E[aεj(ω,X,Z)
]≈ −1
2
√i
2πβj(ω)Z
∫dX ′ Fj(ω,X
′)
× exp
[− Z
Sj(ω)+iβj(ω)(X −X ′)2
2Z
]= aj,o (ω,X,Z) exp
[− Z
Sj(ω)
],(6.5)
with aj,o the solution of the paraxial wave equation (3.15)–(3.16) in the ideal wave-guide.
The mean wave field follows from (4.6), after neglecting the evanescent part,
(6.6) E
[P
(ω,X
ε, η,
Z
ε2
)]≈
N∑j=1
φj(η)aj,o (ω,X,Z) exp
[− Z
Sj(ω)+ iβj(ω)
Z
ε2
].
It is different from the field in the ideal waveguides,
(6.7) po
(ω,X
ε, η,
Z
ε2
)≈
N∑j=1
φj(η)aj,o (ω,X,Z) exp
[iβj(ω)
Z
ε2
],
because of the exponential decay of the mean mode amplitudes, on range scales Sj(ω).
6.2. High-frequency and low-SNR regime. We call the length scales Sj(ω)themode-dependent scattering mean free paths, because they give the range over whichthe modes become essentially incoherent, with low SNR,
SNRj,ω =
∣∣E [aεj(ω,X,Z)]∣∣√E[|aεj(ω,X,Z)|2
]−∣∣E [aεj(ω,X,Z)]∣∣2 ∼ exp
[− Z
Sj(ω)
]� 1
if Z � Sj(ω).(6.8)
The second moments E[|aεj(ω,X,Z)|2
]are calculated in the next section, and they
do not decay with range. This is why (6.8) holds.The scattering mean free paths decrease monotonically with mode indices j, as
shown in (6.3). The first mode encounters less often the random boundary and hasthe longest scattering mean free path
(6.9) S1(ω) =32D2
σ2π2
[(N + α(ω)− 1/2)2 − 1/4
]≈ 32D2N2
σ2π2 .
The highest indexed mode scatters most frequently at the boundary, and its scatteringmean free path
is much smaller than S1(ω) when N is large. To be complete, we also have
Sj(ω) ≈ S1(ω)1− s4
s41
N4if j = �sN, s ∈ (0, 1),
and
Sj(ω) ≈ S1(ω)1
(2j − 1)4if j = o(N).
Our analysis of time reversal and imaging is carried in a high-frequency regime,with waveguide depth D larger than the central wavelength λo or, equivalently, withN � 1. We also assume a low-SNR regime, with scaled range Z exceeding thescattering mean free path of all the modes, so that none of the amplitudes aj iscoherent. This is the most challenging case for sensor array imaging, because thewave field measured at the sensors is dominated by noise. We model the low-SNRregime using the dimensionless large parameter
(6.11) γ =Z
S1(ω0)� 1
and observe from (6.3) that
(6.12)Z
Sj(ω0)≥ γ � 1 for all j = 1, . . . , N.
6.3. The second moments. The quantification of SNR and the analysis of timereversal and imaging involves the second moments of the mode amplitudes. Recallthat
(6.13) aεj(ω,X,Z) ≈∫dX ′T ε
j (ω,X,X ′, Z)aj,ini(ω,X ′),
with T εj the diagonal entries of the transfer matrix Tε. To calculate the second mo-
ments, we estimate E[T εj T ε
l
]. The equations for T ε
j (ω1, X1, X′1, Z)T ε
l (ω2, X2, X ′2, Z)
follow2 from the forward scattering approximation of (5.8),
∂ZT εj T ε
l ≈[
i
2βj(ω1)∂2X1
− i
2βl(ω2)∂2X2
]T εj T ε
l
− i
2ε1/2
[qjj μ(X1, Z/ε)
βj(ω1)− qll μ(X2, Z/ε)
βl(ω2)
]T εj T ε
l(6.14)
for Z > 0, with initial condition
(6.15) T εj (ω1, X1, X
′1, 0)T ε
l (ω2, X2, X ′2, 0) = δ(X1 −X ′
1)δ(X2 −X ′2).
Their statistical distribution is characterized in the limit ε → 0 by the diffusionapproximation theorem [18, 19]; see also [11, Chap. 6]. It is the distribution ofTj(ω1, X1, Z)Tl(ω2, X2, Z), with Tj the limit transfer coefficients in Proposition 5.1.This gives the approximate relation
E
[aεj(ω1, X1, Z)aεl (ω2, X2, Z)
]≈∫dX ′
1
∫dX ′
2 aj,ini (ω1, X′1) al,ini (ω2, X ′
2)
× E
[Tj(ω1, X1, X
′1, Z)Tl(ω2, X2, X ′
2, Z)].(6.16)
The calculation of E[TjTl
]is given in Appendix A. We summarize the results in
Propositions 6.1–6.3.
2Here we neglected the higher-order terms that play no role in the limit ε → 0.
6.3.1. The single mode and frequency moments. It is easier to calculatethe diagonal moments, with j = l, and the same frequency ω1 = ω2 = ω. We havethe following result proved in Appendix A.
Proposition 6.1. For all j = 1, . . . , N and all the frequencies ω ∈ [ω0−B/2, ω0+B/2],
E
[Tj(ω,X1, X
′1, Z)Tj(ω,X2, X ′
2, Z)]
=βj(ω)
2πZexp
{iβj(ω)[(X1 −X ′
1)2 − (X2 −X ′
2)2]
2Z
− 2Z
Sj(ω)
∫ 1
0
dsCo
[(X1 −X2)s+ (X ′
1 −X ′2)(1 − s)
]},(6.17)
with kernel Co defined by
(6.18) Co(X) = 1− Ro(X)
Ro(0).
The general second moment formula does not have an explicit form in arbitraryregimes. But it can be approximated in the low-SNR regime (6.11). The expression(6.17) also simplifies in that regime, as stated in the following proposition, which weprove below.
Proposition 6.2. In the low-SNR regime (6.11) and under the assumption X ′1 =
X ′2 = X ′, the right-hand side in (6.17) is essentially zero, unless
(6.19)|X1 −X2|
�√
3Sj(ω)
γ S1(ω)� 1,
and the moment formula simplifies to
E
[Tj(ω,X1, X
′, Z)Tj(ω,X2, X ′, Z)]
≈ βj2πZ
exp
[iβj [(X1 −X ′)2 − (X2 −X ′)2]
2Z− (X1 −X2)
2
2X2d,j(ω)
],(6.20)
with
(6.21) Xd,j(ω) =
√3Sj(ω)
2Z=
√3Sj(ω)
2γ S1(ω)� .
If the initial points X ′1 and X ′
2 are different, but still close enough to satisfy
Proof. We see from definitions (3.6) and (6.18) that Co(X) ≈ 1 for |X | � .Therefore, ∫ 1
0
dsCo
[(X1 −X2)s
]≈ 1 if |X1 −X2| � ,
and the right-hand side in (6.17) becomes negligible, of order O(e−2Z/Sj
)� 1. In the
case |X1 −X2| ∼ we obtain similarly that the damping term is of order Z, and theright-hand side in (6.17) is exponentially small. It is only when |X1 −X2| � thatthe moment does not vanish. Then, we can approximate the kernel Co in the integralwith its first nonzero term in the Taylor expansion around zero, using the relations
(6.24) Co(0) = 0, C′o(0) = 0, and C′′
o (0) = −R′′o (0)
Ro(0)=
1
2,
that follow from (3.9)–(3.7). We have∫ 1
0
dsCo
[(X1 −X2)s
]≈ |X1 −X2|2
6 2if |X1 −X2| � ,
and the right-hand side in (6.17) is of the order exp[− |X1−X2|2Z3�2Sj
]. This gives the
condition (6.19), and the simpler moment formula (6.20) follows.Essentially the same proof applies in the case X ′
1 �= X ′2, because we can still
expand the integrand in (6.17) by assumption (6.22).
6.3.2. The two mode and frequency moments. The general second momentformula is derived in Appendix A, in the low-SNR regime (6.11). It has a complicatedexpression that we do not repeat here, but it simplifies for nearby frequencies, as statedbelow.
Proposition 6.3. The modes decorrelate under the low-SNR assumption (6.11):
(6.25) E
[Tj (ω1, X1, X
′1, Z) Tl (ω2, X2, X ′
2, Z)]≈ 0 if j �= l
for any two frequencies ω1, ω2 and cross-ranges X1, X2. The modes also decorrelatefor frequency offsets that exceed
(6.26) Ωd,j(ω) =Sj(ω)β
2j (ω)
2
Z2|β′j(ω)|
=βj(ω)
|β′j(ω)|
Sj(ω)βj(ω) 2
γ2S21 (ω)
,
where β′j(ω) is the derivative of βj(ω) with respect to ω. For much smaller frequency
6.4. Decorrelation properties. We already stated the decorrelation of themodes in Proposition 6.3. But even for a single mode, we have decorrelation overcross-range and frequency offsets.
The decoherence length of mode j is denoted by Xd,j(ω), and it is defined in(6.21). It is the length scale over which the second moment at frequency ω decayswith cross-range. It follows from (6.21) that Xd,j is much smaller than the correlationlength, for all the modes, and that it decreases monotonically with j. The first modehas the largest decorrelation length
(6.29) Xd,1(ω) =
√3
2γ,
because it scatters less often at the boundary. The decoherence length of the highestmode is much smaller in high-frequency regimes with N � 1,
(6.30) Xd,N(ω) = Xd,1(ω)
√SN (ω)
S1(ω)≈
8
√3α(ω)
γN−5/2.
The decorrelation frequency is derived in Appendix A.2. It is given by (6.26) or,more explicitly, by
(6.31) Ωd,j(ω) ≈ωσ2π3
64γ2
(
λ
)3
[(N + α(ω)− 1
2
)2 − (j − 12
)2]5/2N9(j − 1/2)4
,
it is much smaller than ω for all the modes, and it decreases monotonically with j,starting from
(6.32) Ωd,1(ω) ≈ωσ2π3( /λ)3
4γ2N4.
7. The forward model. Let us gather the results and summarize them in thefollowing model of the pressure field:(7.1)
P
(ω,X
ε, η,
Z
ε2
)≈
N∑j=1
φj(η)
2iβj(ω)eiβj(ω) Z
ε2
∫dX ′ Tj(ω,X,X ′, Z)
∫ D
0
dη′φj(η′) f(ω,X ′, η′),
where the symbol ≈ stands for approximate, in distribution. That is to say, thestatistical moments of the random pressure field P are equal to those of the right-hand side in the limit ε→ 0. The first and second moments follow from Propositions5.1–6.3. In our analysis of time reversal and imaging we take small frequency offsets,satisfying |ω| � Ωd,j(ω), so that we can use the simpler moment formula (6.28).
The computation of the fourth moments of the transfer coefficients is quite in-volved. We estimate some of them in Appendix B for a particular combination of themode indices and arguments. These moments are used in the next sections to showthe statistical stability of the time reversal and coherent interferometric imaging func-tions.
We analyze next time reversal and imaging in the low-SNR regime defined insection 6.2. We assume for convenience that the source (3.1) has the separable form
meaning that the same pulse f(t) is emitted from all the points in the support of thenonnegative source density ρ. We scale this support with the dimensionless parametersθX and θη and normalize the source by
(7.3)
∫dX ′
θX
∫dη′
θηρ
(X −X
θX,η − η
θη
)= 1.
To analyze the resolution of time reversal and imaging, we study in detail the case ofa source density localized around the point (X, η, 0). In an abuse of terminologywe may say that we study the “point spread” time reversal and imaging functions,because f(t,X, η) has small support. Note, however, that it is not a point source.Recalling the definition (3.1), we see that f(t,X, η) is the source after the cross-rangescaling by ε. The actual source has cross-range support θX/ε and θη, with positiveparameters θX and θη that are small but independent of ε.
The coefficients
(7.4) Fj(ω,X) =f(ω)
θXθη
∫ D
0
dη φj(η)ρ
(X −X
θX,η − η
θη
)are proportional to the Fourier coefficients f(ω) of the pulse and are thus supportedin the frequency interval [ωo −B/2, ωo +B/2]. We write this explicitly by letting
(7.5) f(t) = e−iωotϕ(Bt), f(ω) = B−1ϕ
(ω − ωo
B
),
with function ϕ of dimensionless arguments and Fourier transform ϕ supported inthe interval [−1/2, 1/2]. The pulse f(t) is large in a time interval proportional toB, and we distinguish two regimes: the broadband regime with B � ε2ωo, and thenarrowband regime with B ≤ ε2ωo. The comparison with ε2 is because the source isat range ZA/ε2 from the array, and the modes arrive at time intervals of order 1/ε2.Broadband pulses have smaller support than these travel times, meaning that we canobserve the different arrivals of the modes, at least in the ideal waveguides. In anycase, we assume that B � ωo so that we can fix the number of propagating modes tothat at the central frequency, as explained in section 2.
8. Time reversal. Let us denote by D(t,X, η) the pressure field measured in atime window ψ(t/T ε) at an array A, with aperture modeled by the indicator function
(8.1) 1A(X, η) = 1AX (X)1Aη (η),
at range zA = ZA/ε2. Here X is the scaled cross-range in the array, related to thecross-range x by x = X/ε, and AX ⊂ R and Aη ⊂ [0,D] are intervals in X and η. Thewindow ψ is a function of dimensionless arguments, of support of order one, and T ε
denotes the length of time of the measurements. Because the waves travel distancesof order ε−2, we scale T ε as T ε = T/ε2, with T of order one.
In time reversal the array takes the recorded field D(t,X, η), time reverses it, andemits D(T ε − t,X, η) back in the medium. We study in this section the resolution ofthe refocusing of the waves at the source, in the high-frequency and low-SNR regimedescribed in section 6.2. We compare the results with those in ideal waveguides, toshow the improved refocusing in random waveguides. The resolution analysis includesthat of statistical stability, which quantifies the robustness of the refocusing withrespect to different realizations of the random boundary fluctuations.
The small frequency shifts ε2u/T are due to the time scaling, and we can neglect
them in the source terms Fj and in the argument of the amplitude factor 1/βj.We are interested in the wave at the range of the source, observed at coordinates
(xs = Xs/ε, ηs, zs = 0). We can model it by
O(t,Xs, ηs) =
N∑j=1
φj(ηs)
∫dω
2π
eiβj(ω)ZAε2
−iωt
2iβj(ω)
×∫dX
∫dη φj(η) D
TR(ω,X, η) Tj(ω,X,Xs, ZA),
using reciprocity. Note the similarity with (7.1), except that the source is now atthe array, which we approximate as a continuum, instead of a discrete collection ofsensors. This approximation is convenient for the analysis, because sums over thesensors are replaced by integrals over the X and η apertures, of lengths |AX | and|Aη|. We assume henceforth that
(8.6) AX =
[−|AX |
2,|AX |2
], Aη = [η1, η2] ⊂ [0,D]
and define the matrix
(8.7) Γjl =
∫ D
0
dη 1Aη (η)φj(η)φl(η)
that models the coupling of the modes in the expression of refocused field, due to afinite aperture. It becomes the identity when the array has full aperture Aη = [0,D].
Moreover, assuming a tightly supported source density around (X, η), modeled by
small θX and θη, and recalling that the bandwidth B of f(ω) satisfies B � ωo, we cansimplify the result as
O(t,Xs, ηs)≈N∑
j,l=1
Γjlφj(η
s)φl(η)
4βj(ωo)βl(ωo)ψ
(β′l(ωo)ZAT
)×∫dω
2πf(ω)eiω(T ε−t)+i[βj(ω)−βl(ω)]
ZAε2
×∫ |AX |/2
−|AX |/2dX Tj
(ω,X,Xs, ZA
)Tl(ω,X,X, ZA
).(8.8)
Here we used the differentiability of Tl with respect to the frequency argument, whichis not difficult to deduce using the type of analysis described in section 5.3, andevaluated the integral over u, the inverse Fourier transform of the recording windowψ. Note that β′
l(ωo)ZA are the scaled travel times of the waveguide modes, so onlythe first NT modes that arrive within the time support of the window ψ contributein (8.8). They satisfy
(8.9) β′l(ωo)ZA ≤ T, l = 1, . . . , NT .
8.2. Time reversal in ideal waveguides. The observed wavefield in idealwaveguides follows from (8.8) and the expression
(8.10) Tj,o(ω,X,X ′, Z) =
√βj(ω)
2πiZexp
[iβj(ω)(X −X ′)2
2Z
]of the ideal transfer coefficients, defined by the Green’s functions of the paraxialoperator in (3.15). We obtain that
Oo(t,Xs, ηs)≈ |AX |
8πZA
N∑j=1
NT∑l=1
Γjlφj(η
s)φl(η)
4√βj(ωo)βl(ωo)
×∫ |AX |/2
−|AX |/2
dX
|AX |eiβj(ωo)(X−Xs)2
2ZA − iβl(ωo)(X−X�)2
2ZA
×∫dh
2πϕ(h) ei
[βj(ωo+Bh)−βl(ωo+Bh)
]ZAε2
+i(ωo+Bh)(T ε−t),(8.11)
where we used the definition (7.5) of the emitted pulse and the assumption B � ωo.
The refocusing of Oo depends on the bandwidth. If the emitted signal is broad-band, with B � ε2ωo, then we see a refocused pulse at the discrete set of observationtimes
tεjl = tj,l/ε2, tj,l = T + [β′
j(ωo)− β′l(ωo)]ZA.
Only one term of the sum in (8.11) contributes at time tεj,l for j �= l, so there is norefocusing in the depth coordinate η. The refocusing in cross-range is dictated by theFresnel-type integral in X , which peaks at search locations
Xs =βl(ωo)
βj(ωo)X �= X.
The wave field refocuses at the source only at time T ε, corresponding to j = l above,where all the diagonal terms contribute to the sum in (8.11). Thus, we define thetime reversal function as
(8.12) J TR,bb
o (Xs, ηs) = Oo(Tε, Xs, ηs),
where the index “bb” stands for broadband, and obtain that
J TR,bb
o (Xs, ηs)≈ |AX |ϕ(0)8πZA
NT∑j=1
Γjjφj(η
s)φj(η)
4βj(ωo)
× eiβj(ωo)
[(Xs)2−(X�)2]2ZA sinc
[βj(ωo)(X
−Xs)|AX |2ZA
],(8.13)
where sinc(s) = sin(s)/s. This function is focused at Xs = X and ηs = η, as wediscuss in more detail in section 8.4, where we compare the resolution with that inrandom waveguides.
If the emitted signal is narrowband, say with B = ε2B and scaled bandwidth Bof order ωo, we can rewrite (8.11) as
Oo(t,Xs, ηs) ≈ |AX |
8πZA
N∑j=1
NT∑l=1
Γjlφj(η
s)φl(η)
4√βj(ωo)βl(ωo)
×∫ |AX |/2
−|AX |/2
dX
|AX |eiβj(ωo)(X−Xs)2
2ZA − iβl(ωo)(X−X�)2
2ZA
× ei[βj(ωo)−βl(ωo)
]ZAε2
+iωo(Tε−t) ϕ
×[B(T − ε2t) +B(β′
j(ωo)− β′l(ωo))ZA
].(8.14)
Here all the terms contribute to the sum, but we have oscillations over the index offsetsj − l, due to the O(ε−2) phase of the exponential. Thus, we expect that the leadingcontribution comes from the diagonal terms in (8.14), which refocus around the timet = T ε. The narrowband time reversal function, defined as above by Oo(T
The diagonal part is the same as (8.13), but the off-diagonal terms deteriorate therefocusing at the source. These terms are large when the array has small apertureAη. The best refocusing occurs for full aperture arrays, where the matrix Γjl is theidentity.
8.3. Time reversal in random waveguides. If the time reversal process isstatistically stable, then we can estimate the refocusing of the wave by studying themean of (8.8). We refer to the next section for the analysis of the statistical stability.
Using the moment formula (6.28) we obtain
E [O(t,Xs, ηs)] ≈ |AX |8πZA
NT∑j=1
Γjjφj(η
s)φj(η)
4βj(ωo)
× eiωo(Tε−t) ϕ(B(T ε − t))e
iβj(ωo)[(Xs)2−(X�)2]
2ZA
× e− (Xs−X�)2
2X2d,j
(ωo) sinc
[βj(ωo)(X
−Xs)|AX |2ZA
],(8.16)
where we evaluated the integrals over the bandwidth and X . The sum has onlydiagonal terms, so unlike in the ideal waveguide, refocusing occurs only at time t = T ε,independent of the bandwidth. The time reversal function is defined by J TR(Xs, ηs) =O(t = T ε, Xs, ηs), and its mean is given by
E [J TR(Xs, ηs)]≈ |AX |ϕ(0)8πZA
NT∑j=1
Γjjφj(η
s)φj(η)
βj(ωo)eiβj(ωo)
[(Xs)2−(X�)2]2ZA
× e− (Xs−X�)2
2X2d,j
(ωo) sinc
[βj(ωo)(X
−Xs)|AX |2ZA
].(8.17)
The coefficients Γjj are nonnegative,
Γjj =
∫ D
0
dη 1Aη(η)φ2j (η)
=|Aη|D +
η2D sinc
[2π(j − 1/2)
η2D
]− η1
D sinc[2π(j − 1/2)
η1D
]≥ 0,
where we used the expression (2.7) of the eigenfunctions φj(η) and the aperture Aη =[η1, η2] ⊂ [0,D]. If the array has full aperture Aη = [0,D], then Γjj = 1.
8.4. Comparison of refocusing resolution in random and ideal wave-guides. When we compare the expression of (8.17) with that in ideal waveguides,we see that the array aperture Aη does not play a big role in the refocusing in therandom waveguide. More importantly, we show next that the refocusing is improved,which is known as superresolution.
Cross-range resolution. We observe in (8.17) that modes contribute differentlyto the focusing in cross-range X , with resolution
(8.18) |Xs −X| � ΔX,j := min
{Xd,j(ωo),
2πZAβj(ωo)|AX |
}.
The first argument of the minimum quantifies the support of the Gaussian in (8.17)and the second argument the support of the sinc function. The latter gives the cross-range resolution in the ideal waveguide for broadband signals and arbitrary apertureAη, or for narrow-band signals and full aperture Aη = [0,D].
Recall from (6.21) and (6.29) that Xd,j decreases monotonically with j,
Xd,j(ωo) ≈Xd,1(ωo)
4(j − 1/2)2
[(N + α(ωo)− 1/2)2 − (j − 1/2)2
]1/2N
,
Xd,1(ωo) =
√3
2γ,(8.19)
whereas
(8.20)2πZA
βj(ωo)|AX | ≈2ZAD|AX |
[(N + α(ωo)− 1/2)2 − (j − 1/2)2
]−1/2
increases with j. Thus, in the high-frequency regime with N � 1, the cross-rangeresolution for the high-order modes is determined by the decorrelation length, evenfor large apertures. The cross-range resolution of the first modes may be determinedby the aperture, but only if it is large enough,
(8.21) |AX | � 2ZAD
√2γ
3N.
In conclusion, the cross-range resolution is better in random waveguides thanin ideal ones. The modes with higher indices give the best cross-range resolution,but they travel at lower speed. Thus, the focusing improves when we increase therecording time, because the array can capture the late arrivals of the high-ordermodes (see Figure 2). We may expect that the wave field can be sharply refocusedeven for small apertures |AX |, but we shall see in section 8.5 that large aperturesincrease the statistical stability (i.e., robustness) of the refocusing with respect todifferent realizations of the random boundary.
Depth resolution. To study the focusing in η, we evaluate (8.17) at cross-rangeXs = X. We have
E [J TR(X, ηs)] ≈ |AX |ϕ(0)8πZA
NT∑j=1
Γjjφj(η
s)φj(η)
βj(ωo)
≈ J TR,bb
o (X, ηs).(8.22)
The result is the same as in ideal waveguides if the emitted signals are broadband.The time reversal function for narrowband signals in ideal waveguides is
J TR,nb
o (X, ηs)≈ |AX |8πZA
N∑j=1
NT∑l=1
Γjlφj(η
s)φl(η)
4√βj(ωo)βl(ωo)
× ei[βj(ωo)−βl(ωo)]ZAε2 ϕ
[B(β′
j(ωo)− β′l(ωo))ZA
].(8.23)
Its diagonal (j = l) part is the same as (8.22), but the off-diagonal terms may have avisible effect for small apertures Aη.
The sum in (8.22) is maximum at ηs = η, because all the terms are positive.The point spread function is smaller at other depths, because of cancellations in thesum of the oscillatory terms. We can make this more explicit in the high-frequencyregime, with N � 1, if we write
Fig. 2. Depth profile (left) and cross-range profile (right) of the mean point spread function forthe time reversal functional. Here ZA = 100, � = 1, σ = 0.25, ko = 60, D = 1 (so that N = 19).The array diameter |AX | is supposed to be smaller than the critical value (8.21), which is about 220.NT is the cut-off number (modes smaller than NT are recorded and reemitted). Note that the highmodes play an important role. The larger NT is, the better the resolution.
and interpret (8.22) as a Riemann sum, which we then approximate with an integral.Consider for simplicity the full aperture case, where
E [J TR(X, ηs)] ≈ |AX |ϕ(0)8πZA
NT∑j=1
2
Dβjcos
[(j − 1
2
)koη
s
N
]cos
[(j − 1
2
)koη
N
]
≈ |AX |8π2ZAN
NT∑j=1
cos[(j−1/2)
N ko(η − η)
][1− (j−1/2)2
N2
]1/2≈ |AX |
8π2ZAΛNT /N
[ko(η
s − η)], Λα(x) =
∫ α
0
dscos(sx)√1− s2
.(8.25)
The function Λα becomes proportional to the Bessel function of first kind J0 as α→ 1;more explicitly, we have Λ1(x) = (π/2)J0(x) so that
E [J TR(X, ηs)] ≈ |AX |ϕ(0)16πZA
J0 [ko(ηs − η)] if NT ≈ N.(8.26)
We can then estimate the depth resolution as the distance between the peak of J0,which occurs when ηs = η, and its first zero, which occurs when ko|ηs − η| ≈ 2.4(first zero of J0). Therefore, the depth resolution of time reversal with full apertureis equal to the diffraction limit
(8.27) |ηs − η| � Δη :=2.4
ko
if the array records the waves long enough to capture almost all the propagatingmodes. The resolution deteriorates if NT is much smaller than N . Indeed forsmall α we have Λα(x) ≈ (α/π) sinc(αx), and therefore the depth resolution isΔη ≈ πN/(koNT ) (see Figure 2).
8.5. Statistical stability. We now show that the time reversal function is sta-tistically stable, meaning that the refocusing of the wave at the original source locationdoes not depend on the realization of the random medium but only on its statistical
distribution, and J TR is approximately equal to its expectation in the vicinity of thesource, and in particular at Xs = X and ηs = η.
We restrict the analysis of statistical stability to the case of full aperture, wherethe calculations are simpler because the coupling matrix Γjl becomes the identity. Weobtain from (8.8) and the definition of J TR as O(t = T ε, Xs, ηs) that
J TR(X, η) ≈NT∑j=1
φ2j (η)
β2j (ω)
∫dω
2πBϕ
(ω − ωo
B
)∫ |AX |/2
−|AX |/2dX |Tj(ω,X,X, ZA)|2 .
Let us rewrite this equation as
J TR(Xs, ηs) =
∫dω
2πBϕ
(ω − ωo
B
)MTR(ω,X, η)
and compute first the variance of the frequency dependent kernel MTR,
Var [MTR(ω,X, η)] =
NT∑j,J=1
φ2j (η)
β2j (ω)
φ2J (η)
β2J(ω)
∫∫ |AX |/2
−|AX |/2
× dXdY{E[|Tj(ω,X,X, ZA)|2 |TJ(ω, Y,X, ZA)|2
]− E
[|Tj(ω,X,X, ZA)|2
]E[|TJ(ω, Y,X, ZA)|2
]}.
From Appendix B (first case) we find that the variance is much smaller than thesquare expectation when |AX | � , and therefore the kernel MTR is equal to itsmean approximately. The results contained in Appendix B (second case) also showthat if |AX | < , the variance of MTR is large, and therefore that time reversalrefocusing may be unstable in the narrow-band regime.
There is, however, another mechanism that can ensure statistical stability of J TR
if the array is small. Indeed, if the bandwidth of the emitted signal is larger than thedecorrelation frequency, then the variance
Var [J TR(X, η)] =
∫dω
2πB
∫dω′
2πBϕ
(ω − ωo
B
)ϕ
(ω′ − ωo
B
)× Cov
[MTR(ω,X, η),MTR(ω′, X, η)
]is small because the covariance of the point spread function at two frequencies becomesapproximately zero if the frequency gap is large enough. Therefore, if the pulse haslarge bandwidth, then the time reversal focal spot is statistically stable, even for smallarrays.
9. Imaging. The improved and stable focusing of the time reversal process inthe random waveguide is due the backpropagation of the time-reversed field DTR
in exactly the same waveguide. Time reversal is a physical experiment, where thewaves can be observed in the vicinity of the source as they refocus. In imaging wehave access only to the data measured at the array, and the backpropagation to thesearch points is synthetic. Because we cannot know the fluctuations of the boundary,we simply ignore them in the synthetic backpropagation and obtain the so-calledreversed-time migration imaging function. We analyze it in section 9.1 and showthat it does not give useful results in the low-SNR regime defined in section 6.2.In particular, we show that the images are not statistically stable with respect to
realizations of the fluctuations. Stability can be achieved by imaging with local cross-correlations of the array measurements. Local means that we recall the decorrelationproperties of the random mode amplitudes described in section 6.4, and cross-correlatethe measurements over receivers located at nearby cross-ranges X , and projected onthe same eigenfunctions. The resulting coherent interferometric imaging method isanalyzed in section 9.2.
9.1. Reverse-time migration. The reverse-time migration function is givenby the time-reversed data DTR propagated (migrated) in the ideal waveguide to thesearch points (xs, ηs, zs) =
(Xs
ε , ηs, Z
s
ε2
). Its mathematical expression follows from
(3.14), with amplitudes (3.12) replaced by
aj,o(ω,X,Z)�∫dX ′Tj,o(ω,X,X ′, Z)
1
2iβj(ω)
∫ D
0
dη φj(η)DTR(ω,X ′, η),(9.1)
with ideal transfer coefficients given by (8.10). We obtain
J M(Xs, ηs, Zs)=
N∑j=1
φj(ηs)
∫dω
2π
eiβj(ω)ZA−Zs
ε2−iωt
2iβj(ω)
∫dX
∫dη φj(η)
× DTR(ω,X, η) Tj,o(ω,X,Xs, ZA − Zs)∣∣∣t=T ε
,(9.2)
with the right-hand side evaluated at the same time t = Tε as in time reversal.We assume again a tightly supported source density normalized by (7.3) and
substitute the model (8.4) of DTR into (9.2) to obtain
JM(Xs, ηs, Zs) =
∫dω
2πBϕ
(ω − ωo
B
)MM (ω,Xs, ηs, Zs) ,(9.3)
with frequency-dependent kernel
MM (ω,Xs, ηs, Zs)≈N∑j=1
NT∑l=1
Γjlφj(η
s)φl(η)
4βj(ωo)βl(ωo)eiβj(ω)
ZA−Zs
ε2−iβl(ω)
ZAε2
×∫ |AX |/2
−|AX |/2Tj,o(ω,X,Xs, ZA − Zs
)Tl(ω,X,X, ZA
).(9.4)
The derivation of this kernel is similar to that of (8.8) in section 8.1, and we recallfrom there that NT is the number of modes that arrive within the recording time ofthe window ψ.
9.1.1. The mean imaging function. Let us take for simplicity the case of fullaperture in depth, where the coupling matrix Γjl given by (8.7) becomes the identity.We obtain from (9.4), the moment formula (6.4), and the evaluation of the integralover X that
The result is almost the same as for time reversal in the ideal waveguide, except forthe damping coefficients exp [−ZA/Sj ] .
The sinc function in the mean kernel gives the focusing in cross-range, with mode-dependent resolution
(9.6) |XS −X| � ΔX,j :=2πZA
βj(ωo)|AX | .
The best resolution is for the first mode, which has the largest wavenumber β1(ωo) ≈πN/D ≈ ko, and gives the Rayleigh cross-range resolution
(9.7) ΔX,1 ≈ 2πZAko|AX | .
The focusing of the kernel MM in range can be due only to the summation ofthe rapidly oscillating terms exp
[−iβjZs/ε2
], at least for large enough N . But these
terms are weighted by exp[−ZA/Sj], which decay fast in j. The first term dominatesin (9.5), evaluated at Xs = X and ηs = η, so the mode diversity does not lead tofocusing in range, as is the case in ideal waveguides. The mean reverse-time migrationfunction peaks at Zs = 0 because of the integral over the bandwidth in (9.2), and therange resolution is of the order ε2/[β′
1(ωo)B].When we evaluate (9.5) at Zs = 0 and Xs = X, we obtain
E [MM(ω,Xs = X, ηs, Zs = 0)] ≈ |AX |8πZA
NT∑j=1
φj(ηs)φj(η
)
βj(ωo)e−ZA/Sj(ωo).(9.8)
This is a sum of the oscillatory functions
φj(ηs)φj(η
) =1
D
{cos
[π
(j − 1
2
)(ηs − η)
D
]+ cos
[π
(j − 1
2
)(ηs + η)
D
]}multiplied by positive weights, which are small and decay fast in j. The first termdominates in (9.8), and there is no depth resolution at all. We show next that thesesmall weights also indicate the lack of statistical stability of the reverse-time migrationfunction.
9.1.2. Stability analysis. To assess the stability of the reverse-time migration,we calculate its variance at the source location
and we recall from Proposition 6.3 that only the diagonal terms j = l contribute tothe expectation. We also assume a small bandwidth B � Ωd,j for all the modes j, sothat we can use the simpler moment formula (6.28). We obtain
E
[|J M(X ′, η′, 0)|2
]≈ |ϕ(0)|2
(8πZA)2
NT∑j=1
φ4j(η)
β2j (ωo)
∫∫ |AX |/2
−|AX |/2dX1dX2e
− (X1−X2)2
2X2d,j
(ωo)(9.10)
and, integrating in X1 and X2, under the assumption that the decoherence lengthsXd,j are much smaller than the array aperture,
E
[|J M(X ′, η′, 0)|2
]≈ |ϕ(0)|2|AX |
√2π
(8πZA)2
NT∑j=1
Xd,j(ωo)φ4j (η
)
β2j (ωo)
.(9.11)
The second moment (9.11) is clearly much larger than the square of the mean(9.9), which is exponentially small in range. Therefore, the variance of JM is largenear the peak of its expectation. Although the mean of the imaging function isfocused at the source, it cannot be observed because it is dominated by its randomfluctuations. The reverse-time migration lacks statistical stability with respect to therealizations of the random fluctuations of the boundary of the waveguide.
The calculations above are for a small bandwidth, satisfying B � Ωd,j for j =1, . . . , NT . The calculations are more complicated for a larger bandwidth, but theconclusion remains that reverse-time migration is not stable with respect to differentrealizations of the random boundary fluctuations.
9.2. Coherent interferometric imaging. The main idea of the coherent inter-ferometric (CINT) imaging approach is to backpropagate synthetically to the imagingpoints the local cross-correlations of the array measurements, instead of the measure-ments themselves. By local we mean that because of the statistical decorrelationproperties of the random mode amplitudes described in section 6.4, we cross-correlatethe data D(ω,X, η) at nearby frequencies and cross-ranges X , after projecting it onthe subspace of one eigenfunction φj at a time. The projection gives the coefficients
(9.12) Dj(ω,X) =
∫ D
0
dη φj(η)D(ω,X, η),
which are directly proportional to the coefficients Fj of the source only in the caseof an array spanning the entire depth of the waveguide. We assume this case here,because it simplifies the analysis of the focusing and stability of the CINT function.We also assume as in the previous sections a source supported tightly around thelocation (X, η).
The model of the coefficients (9.12) is
Dj(ω,X)≈ 1AX (X)φj(η
)
2iβj(ωo)ψ
(β′j(ωo)ZAT
)× 1
Bϕ
(ω − ωo
B
)eiβj(ω)ZA/ε2Tj
(ω,X,X, ZA
),(9.13)
and we cross-correlate them at cross-ranges satisfying |X1 − X2| ≤ Xd,j(ωo) and atfrequency offsets
Each term in the sum focuses at the source, with resolution
(9.21) |Xs −X| � ΔX,j :=ZA
βj(ωo)Xd,j(ωo)
defined by the standard deviation of the Gaussian in (9.20). The number of modesparticipating in the sum is determined by the length of the recording time window, asbefore, but each mode is weighted by the correlation length Xd,j(ωo), which decreasesmonotonically with j. The first mode has the largest contribution in (9.20) and givesthe best cross-range resolution. Since its wavenumber is approximately β1(ωo) ≈πN/D ≈ ko,
(9.22) ΔX,1 ≈ λoZAXd,1(ωo)
is comparable to the classic Rayleigh resolution for an array of aperture equal toXd,1(ωo) (see Figure 3).
The cross-range resolution (9.22) is worse than that of time reversal. Scatteringat the random boundary is beneficial to the time reversal process, and the more modesrecorded, the better the result. However, scattering impedes imaging, and the bestcross-range resolution is achieved with the first mode. Even with this mode, theresolution is worse than that in ideal waveguides 2πZA/(ko|AX |), because Xd,1 �|AX |.
Range focusing. When we evaluate the mean CINT kernel (9.18) at the cross-range Xs = X, we obtain
E [MCINT(ω,X, Zs)] ≈NT∑j=1
φ2j (η)
64π4ZA(ZA − Zs)
∫dω 1Ω(ω)e
−i[βj(ω+ ω2 )−βj(ω− ω
2 )]Zs
ε2
×∫∫ |AX |/2
−|AX |/2dX1dX2 e
−iβj(ωo)(X1−X2)Zs
ZA(ZA−Zs)(X1+X2
2 −X�)− (X1−X2)2
2X2d,j
(ωo) .(9.23)
Because we integrate over ω the rapidly oscillating integrand, at scale ε2, we havefrom the method of stationary phase that (9.23) is large for
Zs = ε2ζs,
with ζs independent of ε. Recall the assumption (9.14) of the frequency offsets. Themean kernel becomes
(9.24)
E[MCINT(ω,X, ε2ζs)
]≈ Ω|AX |(2π)1/2
64π4Z2A
NT∑j=1
Xd,j(ωo)φ2j (η
)sinc[β′j(ωo)Ωζ
s],
and we define the mode-dependent scaled range resolution by
(9.25) |ζs| � Δζ,j :=1
Ωβ′j(ωo)
.
This is similar to the classic range resolution defined as the distance traveled by thewaves during the duration of the pulse. Here the propagation speed 1/β′
dependent and, instead of the pulse, we have the time window of duration ∼ 1/Ω,with Fourier transform 1Ω(ω). The first mode has the largest weight Xd,1 in (9.24),and it determines its range resolution, as illustrated in Figure 3. This is similar towhat we saw above in the analysis of cross-range resolution. However, while the firstmode is best for focusing in cross-range, it gives the worst range resolution, becauseit has the largest speed 1/β′
1 ≈ co. The CINT imaging function (9.15) is not optimalfor range focusing. We need weights that emphasize the contribution of the higher-order modes, which travel at speed 1/β′
j � co, and give better range resolution.
Depth estimation. One natural way to estimate the depth η would be toconsider the full CINT imaging function
J CINT(Xs, ηs, Zs) =
N∑j=1
J CINT
j (Xs, Zs)φ2j (ηs),
with J CINT
j (Xs, Zs) defined by (9.16). However, when we evaluate the mean of thisexpression at Xs = X and Zs = 0, we obtain
E
[J CINT(X, ηs, 0)
]≈ Ω|AX |(2π)1/2‖f‖2
64π4Z2A
NT∑j=1
Xd,j(ωo)φ2j (η
)φ2j (ηs),(9.26)
where we used (7.5) and let ‖f‖2 be the mean square norm of the emitted signal. Theexpression (9.26) is a sum of positive terms, and it does not have a peak at the depthof the source (see Figure 4).
Because of scattering at the random boundary the modes are decoupled, and wecannot speak of coherent imaging in depth. We work instead with the squares of themode amplitudes, i.e., intensities. Incoherent imaging means estimating the depth ofthe source based on the mathematical model (9.26). More explicitly, we can estimateη by solving the least squares minimization problem
(9.27) minηs
NT∑j=1
∣∣∣∣J CINT
j (X, Z)− Ω|AX |(2π)1/2‖f‖264π4Z2
AXd,j(ωo)φ
2j (η
s)
∣∣∣∣2 ,where the estimators X and Z of the cross-range X and range offset Z = 0 of thesource have been determined as the location of the maximum of (9.15) (see Figure 4).
9.2.2. Statistical stability. The analysis of statistical stability of the CINTfunction is basically the same as that of time reversal. The function is stable whenevaluated in the vicinity of the source location if the array has large aperture |AX | � .We have seen in the previous section that a large aperture does not improve thefocusing of E [J CINT]. The cross-range resolution is limited by the decoherence length.But a large aperture is needed for the CINT function to be statistically stable.
Another way of achieving statistical stability of CINT is to have a pulse withlarge bandwidth. This was already noted in the discussion of statistical stability oftime reversal in section 8.5.
Note that the statistical stability of CINT relies on computing correctly the localcross-correlations of the measurements at the array. By this we mean that the cross-range and frequency offsets in the correlations should not exceed the decoherencelength and frequency. Moreover, the cross-correlations should be with one mode ata time. This can be done with arrays that span the whole depth of the waveguide,
Fig. 3. Range profile and cross-range profile of the mean point spread function for the CINTfunctional. Here ZA = 100, � = 1, σ = 0.25, ko = 60, D = 1 (so that N = 19), and the cut-offfrequency is Ω/c = 1. NT is the cut-off number (modes smaller than NT are recorded and reemitted).Note that the high modes do not play any role.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
η
dept
h pr
ofile
(a.u
.)
NT=5NT=10NT=19
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
η
dept
h pr
ofile
(a.u
.)NT=5NT=10NT=19
Fig. 4. Depth profile with (9.26) (left) and with (9.27) (right) for the CINT functional. In theright picture we plot the reciprocal of the square root of the function in (9.27). Here ZA = 100,� = 1, σ = 0.25, ko = 60, D = 1 (so that N = 19). NT is the cut-off number (modes smaller thanNT are recorded and reemitted). Note that the high modes do not play any role.
because the coupling matrix Γjl becomes the identity when |Aη| = D. If the aperture|Aη| is small, there are large mode index offsets |j−l| for which Γjl �= 0. Consequently,there are many terms of the form TjTl, with j �= l, that participate in the expressionof the imaging function. Since only the diagonal terms are correlated, we obtain thatJ CINT has large variance when |Aη| � D.
In practice, the decoherence scales Xd,j and Ωd,j are likely not known explicitly.The formulas derived above are specific to our mathematical model. However, thedecoherence scales can be estimated as we form the image, using an adaptive proceduresimilar to that introduced in [5].
10. Summary. In this paper we analyze propagation of acoustic waves in three-dimensional random waveguides. The waves are trapped by top and bottom bound-aries, but the medium is unbounded in the remaining two directions. The top bound-ary has small, random fluctuations. We consider a source that emits a beam andstudy the resulting random wave field in the waveguide.
The analysis is in a long range, paraxial scaling regime modeled with a smallparameter ε. It is defined as the ratio of the central wavelength λ0 of the pulseemitted from the source and the emitted beam width r0. The range of propagation isof the order of the Rayleigh length r20/λ0 = ε−2λ0. The fluctuations of the boundaryare on a length scale that is similar to the beam width, and their small amplitude
is scaled so that they cause significant cumulative scattering effects when the wavestravel at ranges of the order of the Rayleigh length.
The wave field is given by a superposition of waveguide modes with random am-plitudes. The mode profiles are solutions of the wave equation in the ideal waveguide,with flat boundary. The scattering effects are captured by their random amplitudes.We show that in our scaling regime the mode amplitudes satisfy a system of paraxialequations driven by the same Brownian field. We use the system to calculate threeimportant mode-dependent scales that quantify the net scattering effects in the wave-guide and play a key role in applications such as imaging and time reversal. The firstmode-dependent scale is the scattering mean free path. It gives the range over whichthe mode loses its coherence, meaning that the expectation of its random amplitudeis smaller than the standard deviation of its fluctuations. The other mode-dependentscales are the decoherence length and frequency. They give the cross-range scale andfrequency offsets over which the mode amplitudes become statistically uncorrelated.
We use the results of the analysis to study time reversal and imaging of the sourcewith a remote array of sensors in a low-SNR regime. Low SNR means that the wavestravel over distances that exceed the scattering mean free paths of all the modes, sothat the random wave field measured at the array is dominated by its fluctuations.
In time reversal, the waves received at the array are time-reversed and then reemit-ted in the medium. They travel back to the source and refocus. The refocusing isexpected by the time reversibility of the wave equation, but the resolution is limited inideal waveguides by the aperture of the array. We analyze the time reversal process inthe random waveguide and show that superresolution occurs, meaning that scatteringat the random boundary improves the refocusing resolution. An essential part of theresolution analysis is the assessment of statistical stability with respect to differentrealizations of the random boundary fluctuations. We show that statistical stabilityholds if the array has large aperture and/or the emitted pulse from the source has alarge bandwidth.
Time reversal is very different from imaging. In time reversal the array measure-ments are backpropagated physically, in the real waveguide. In imaging we can onlybackpropagate the time-reversed data in software, in a surrogate waveguide. Becausewe cannot know the boundary fluctuations, we neglect them altogether, and the sur-rogate is the ideal waveguide. The resulting imaging function is called reverse-timemigration, and it does not work in low-SNR regimes. It lacks statistical stability;i.e., the images change unpredictably from one realization of the fluctuations of therandom boundary to another.
We show that robust imaging can be carried out in low-SNR regimes if we back-propagate local cross-correlations of the array measurements, instead of the measure-ments themselves. Here local means that we cross-correlate the data projected on onemode at a time, and for nearby cross-ranges and frequencies. The method is calledcoherent interferometric (CINT), because it is an extension of the CINT approachintroduced and analyzed in [7, 5, 6, 4] for imaging in open, random environments.We show that CINT images are statistically stable under two conditions: The firstcondition is the same as in time reversal, and it says that the array should have alarge aperture and/or the pulse bandwidth should be large. The second condition isthat the cross-range and frequency offsets used in the calculation of the local cross-correlations do not exceed the mode-dependent decoherence length and frequency,respectively. We derived mathematical expressions of these scales for our model. Inpractice, they can be estimated adaptively, using the image formation, with an ap-
proach similar to that in [5]. The estimation is possible because there is a trade-offbetween stability and resolution that is quantified by the decoherence scales. If weoverestimate them, we lose statistical stability. If we underestimate them, we loseresolution.
While cumulative scattering aids in time reversal, it impedes imaging. We quan-tify this explicitly in the resolution analysis of CINT. In time reversal the resolutionimproves when we record the wave field over a long time, so that we include the high-order modes that travel at a slower speed. In CINT, the best cross-range and rangeresolution is given by the first mode, which encounters the random boundary less of-ten and is thus less affected by the fluctuations. The cross-range resolution is similarto the classic Rayleigh one of range times wavelength divided by the aperture, butinstead of the real aperture we have the decoherence length of the mode. This lengthdecreases monotonically with range, because longer distances of propagation in therandom waveguide mean stronger scattering effects. Similarly, the range resolution issimilar to the classic one, of speed divided by the bandwidth, but the bandwidth isreplaced by the decoherence frequency which decreases monotonically with range.
The estimation of the depth of the source is different from that of range and cross-range. Because the modes decorrelate in the low-SNR regime, we cross-correlate thedata projected on one mode at a time, so essentially we work with intensities. Theestimation of the depth of the source from the intensities can be done by minimizingthe misfit between the processed measurements and the mathematical model. Whilethe cross-range and range estimation with CINT is done best with the first waveguidemode, the depth estimation requires many modes. Thus, we still need a long recordingtime at the array to capture the later arrival of the high-order modes.
Appendix A. Second moment calculation. The following equation is forE[Tj(ω1, X1, X
and can be integrated easily after Fourier transforming in κ and ξ. Explicitly,
(A.13) Vjj(ω, κ, ξ, Z; ξ′, ξ′) =
∫ ∞
−∞
dξ
2π
∫ ∞
−∞dκWjj(ω, ξ, κ, Z; ξ
′, ξ′)e−iκξ+iκξ
satisfies the initial value problem
[∂Z + κ∂ξ
]Vjj(ω, κ, ξ, Z; ξ
′, ξ′) = − 2
SjCo
(ξ√βj
)Vjj(ω, κ, ξ, Z; ξ
′, ξ′), Z > 0,
Vjj(ω, κ, ξ, 0; ξ′, ξ′) =
βj2πe−iκξ′δ(ξ − ξ′),(A.14)
which can be solved with the method of characteristics.We obtain that
(A.15)
Vjj(ω, κ, ξ, Z; ξ′, ξ′) =
βj2πe−iκξ′δ
(ξ − ξ′ − κZ
)exp
[− 2
Sj
∫ Z
0
dsCo
(ξ′ + κs√
βj
)],
and tracing back out transformations (A.2), (A.6), (A.8), and (A.13), we get
E
[Tj(ω,X1, X
′1, Z)Tj(ω,X2, X ′
2, Z)]=
βj2πZ
exp
[iβj [(X1 −X ′
1)2 − (X2 −X ′
2)2]
2Z
− 2
Sj
∫ Z
0
dsCo
[(X1 −X2)
s
Z+ (X ′
1 −X ′2)(1− s
Z
) ]].(A.16)
This is the result stated in Proposition 6.1.
A.1.2. Two mode moments. It is not possible to obtain a closed form solutionof (A.9), unless we make further assumptions. We consider the low-SNR regimedescribed in section 6.2 and suppose that
(A.17) |X1 −X2| � Xd,j(ω) � .
This is the condition under which the diagonal moments E[TjTj
]are not exponentially
small, by Proposition 6.2. The two mode moments cannot be larger than the diagonalones, so they are essentially zero when (A.17) does not hold.
Note that in (A.9) κ is the dual variable to ξ ∼√βj(X1 −X2), and that q is in
the support of Co, so |q| ≤ 1/ . Therefore,
|κ| ∼ 1√βj |X1 −X2|
� 1√βj � |q|√
βj,
and we can expand the Wigner transform in (A.9) around κ. The exponential canalso be expanded when
(A.18)qξ√βj
2|√βj −
√βl|
(√βj +
√βl)� X
2|√βj −
√βl|
(√βj +
√βl)
� 1,
meaning that j and l are close. We return at the end of this section to this point.
Assumptions (A.17)–(A.18) justify the approximation of the right-hand side in(A.9) by the second-order expansion in q of the product of the exponential and theWigner transform. We obtain that
(A.19)
(∂Z + κ∂ξ)Wjl ≈ − 1
βj 2√SjSl
(√βj +
√βl
2√βl
)2 [i∂κ − ξ
2(√βj −
√βl)
(√βj +
√βl)
]2Wjl
for Z > 0, with initial condition (A.10). This equation is solved in [9]. The resultfollows from the inverse Fourier transform in κ of the solution, and from (A.2), (A.6),
(A.20)
E
[Tj(ω,X1, X
′1, Z)Tl(ω,X2, X ′
2, Z)]≈√βjβl
2πZsinc−
12
⎧⎨⎩(1 + i)Z
[βj − βl
βjβl√SjSl
] 12
⎫⎬⎭× exp
⎧⎨⎩−(
1√Sj
− 1√Sl
)2
Z +i|(X1 −X ′
1)βj − (X2 −X ′2)βl|2
2Z(βj − βl)
+βjβl
(|X1 −X2|2 + |X ′
1 −X ′2|2)
(βj − βl)(1 + i)
[βj − βl
βjβl√SjSl
] 12
cot
⎡⎣ (1 + i)Z
[βj − βl
βjβl√SjSl
] 12
⎤⎦− 2βjβl(X1 −X2)(X
′1 −X ′
2)
(βj − βl)(1 + i)
[βj − βl
βjβl√SjSl
] 12
sin−1
⎡⎣ (1 + i)Z
[βj − βl
βjβl√SjSl
] 12
⎤⎦⎫⎬⎭ .
Formula (A.20) is complicated, but it can be simplified under the assumption that
(A.21)Z2|βj − βl|βjβl 2
√SjSl
� 1.
Then, we can expand the sinc, cot, and sin−1 functions in (A.20) and obtain thesimpler formula
(A.22)
E
[Tj(ω,X1, X
′1, Z)Tl(ω,X2, X ′
2, Z)]≈√βjβl
2πZ
× exp
[i(βj(X1 −X ′
1)2 − βl(X2 −X ′
2)2)
2Z
]× exp
[−(
1√Sj
− 1√Sl
)2
Z − (X1 −X2)2 + (X ′
1 −X ′2)
2 + (X1 −X2)(X′1 −X ′
2)
2√Xd,jXd,l
].
It remains to justify assumptions (A.18) and (A.21). Because of the exponentialdecay in Z, we note that the moments are essentially zero unless(
by definition (6.11), and it is satisfied only when j = l. This justifies the assumptions,and it means that the modes are essentially decorrelated.
A.2. Two frequency moments. The calculation of the two frequency momentsis exactly as in the previous section, with βj replaced by βj(ω1) and βl replaced byβj(ω2). We consider only the case j = l, because the modes decorrelate as explainedabove. The moment formula follows from (A.20), with βj replaced by βj(ω1) andβl replaced by βj(ω2), and similarly for Sj and Sl. We can simplify it under theassumption that |ω1−ω2| is sufficiently small to make first-order expansions in ω1−ω2.Let ω and ω be the center and difference frequencies
ω = ω1 − ω2, ω =ω1 + ω2
2.
We have from (A.20) that
E
[Tj(ω +
ω
2, X1, X
′1, Z
)Tl(ω − ω
2,X2,X′
2, Z
)]≈ βj(ω)
2πZsinc−
12
{(1 + i)Z
�βj(ω)
[ω∂ωβj(ω)
Sj(ω)
] 12
}
× exp
⎧⎨⎩−ω2(∂ω
1√Sj(ω)
)2Z ++
i|[(X1 −X′1)− (X2 −X′
2)]βj(ω) + ω[(X1−X′
1)+(X2−X′2)]
2β′j(ω)|2
2Zωβ′j(ω)
+βj(ω)[|X1 −X2|2 + |X′
1 −X′2|2]
�ωβ′j(ω)(1 + i)
[ωβ′
j(ω)
Sj(ω)
] 12
cot
⎡⎣ (1 + i)Z
βj(ω)�
[ωβ′
j(ω)
Sj(ω)
] 12
⎤⎦−
2βj(ω)(X1 −X2)(X′1 −X′
2)
�ωβ′j(ω)(1 + i)
[ωβ′
j(ω)
Sj(ω)
] 12
sin−1
⎡⎣ (1 + i)Z
βj(ω)�
[ωβ′
j(ω)
Sj(ω)
] 12
⎤⎦⎫⎬⎭ .(A.23)
A.3. Frequency decorrelation. To study the decorrelation over frequency off-sets, let X1 = X2 and X ′
1 = X ′2 in (A.23):
E
[Tj(ω +
ω
2, X,X ′, Z
)Tj(ω − ω
2, X,X ′, Z
)]
≈ βj(ω)
2πZsinc−
12
{(1 + i)Z
βj(ω)
[ω∂ωβj(ω)
Sj(ω)
] 12
}
× exp
{−[ω∂ωS
− 12
j (ω)]2Z +
iωβ′j(ω)(X −X ′)2
2Z
}.(A.24)
We have two factors that decay exponentially in ω. The first is the sinc, decaying atthe rate
(A.25) |ω| � Ωd,j(ω) =Sj(ω)β
2j (ω)
2
Z2|β′j(ω)|
=βj(ω)
|β′j(ω)|
Sj(ω)βj(ω) 2
γ2S21 (ω)
,
and the second is the Gaussian with standard deviation
and using (2.13), (2.14), (6.3), and the high-frequency assumption N � 1, we have
(A.28) Ωd,j(ω) ≈ω 2β1(ω)
16γ2S1(ω)
[(N + α(ω)− 1
2
)2 − (j − 12
)2]5/2N5(j − 1/2)4
.
Here
(A.29) 2β1S1(ω)
≈ σ2
32N
(π
D
)3
≈ σ2( k)3
32N4=π3σ2( /λ)3
4N4,
and /λ = O(1), because the scaled correlation length is similar to the wavelength λ.Moreover, the rate (A.26) is given by
(A.30) Ωj(ω) ≈ω
4√2γ
[(N + α(ω)− 1
2
)2 − (j − 12
)2]3/2N3(j − 1/2)2
,
and it is larger than Ωd,j(ω).Thus, we call Ωd,j(ω) the mode-dependent decoherence frequency, the frequency
scale over which the second moments decay. Note that when the frequency offsetssatisfy |ω| � Ωd,j the moment formula (A.24) simplifies to expression (6.28) in Propo-sition 6.3, because
exp
[− ω2
2Ω2j(ω)
]≈ 1,
when
|ω| � Ωd,j � Ωj(ω).
Appendix B. The fourth moments. We denote the moments by(B.1)
MjlJL := E
[Tj(ω1, X1, X
′1, Z)Tl(ω2, X2, X ′
2, Z)TJ (ω3, Y1, Y′1 , Z)TL(ω4, Y2, Y ′
2 , Z)]
and obtain from (5.21) that they satisfy the partial differential equation
∂ZMjlJL =
[i
2βj∂2X1
− i
2βl∂2X2
+i
2βJ∂2Y1
− i
2βL∂2Y2
]MjlJL
+
[−(
1√Sj
− 1√Sl
)2
−(
1√SJ
− 1√SL
)2
− 2Co(X1 −X2)√SjSl
− 2Co(Y1 − Y2)√SJSL
]MjlJL
+2
Ro(0)
[Ro(X1 − Y2)√
SjSL
− Ro(X1 − Y1)√SjSJ
− Ro(X2 − Y2)√SlSL
+Ro(X2 − Y1)√
SlSJ
]MjlJL(B.2)
for Z > 0, with the initial condition
(B.3) MjlJL = δ(X1 −X ′1)δ(X2 −X ′
2)δ(Y1 − Y ′1)δ(Y2 − Y ′
2) at Z = 0.
Let us consider the case j = l, J = L, ω1 = ω2 = ω, and ω3 = ω4 = ω′. Thesemoments MjjJJ are needed in section 8 to show the statistical stability of the timereversal function in the case of an array that spans the entire depth of the waveguide.
Here we can see that the fourth-order moment is not equal to the product of thesecond-order moments.
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