by liliana borcea - university of michiganborcea/publications/france_course.pdf · 2 liliana borcea...

WAVE PROPAGATION AND IMAGING IN RANDOMWAVEGUIDES

by

Liliana Borcea

Contents

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Set-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53. Ideal waveguides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94. Waveguides with random boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135. Waveguides with random internal inhomogeneities. . . . . . . . . . . . . . . 296. Net scattering effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317. Model of the array data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428. Time reversal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439. Imaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110. Numerical results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2 LILIANA BORCEA

1. Introduction

Array imaging is an important technology with a wide range of applications inunderwater acoustics, seismology, non-destructive evaluation of materials, medical ul-trasound, and elsewhere. It is concerned with locating remote, compactly supportedsources and/or scatterers from measurements at arrays of sensors. The sensors are de-vices that transform one form of energy into another. Depending on the applicationthey may be antennas that convert electromagnetic waves to/from electric signals,hydrophones that convert changes in water pressure to electrical signals, ultrasonictransducers that transmit and receive ultrasound waves, and so on. When the sen-sors are located sufficiently close together, they behave as a collective entity, calledthe array. The sensors in passive arrays are receivers of the waves generated by un-known remote sources. Active arrays have sensors that play the dual role of sourcesand receivers. The sources emit waves that propagate through the medium and arescattered back to the array, where they are captured by the receivers.

The recordings at the receivers are called the array data. The coherent imagingprocess seeks to transform the data into an imaging function that peaks in the supportof the unknown sources or scatterers. The key data processing step in the imageformation is the synchronization of the received signals using a mathematical modelof wave back-propagation from the array to a search (imaging) location ~rs. Theexpectation is that when ~rs lies in the support of the unknown sources or scatterers,the recordings are synchronized and add up over the sensors to give a peak value ofthe imaging function. Indeed, this happens if we have an accurate model of wavepropagation between the array and the imaging region.

Image formation is somewhat similar to the time reversal process. Time reversalis an experiment that uses an active array to first receive the waves from a remotesource, and then re-emit the time-reversed recordings back in the medium. Thewave equation is time reversible if the medium is non-dissipative, so the waves areexpected to propagate back to the source and focus near it. The focusing resolutionin the range direction is mostly affected by the bandwidth of the emitted signals.The focusing in cross-range (i.e., in the plane orthogonal to the range) depends onmany factors, such as how large the array is, how far the source is, and how muchscattering occurs between the source and receiver. The focusing is never perfect,because the array cannot capture the whole wave field emitted from the source. Thus,the time reversal process is not exactly like solving the wave equation backward intime to recover the initial source. The expectation is that the larger the array is,the better the focus, because more of the waves are captured and turned back. Butno matter how large the array is, even if it surrounds the source, the evanescentwaves cannot reach it, and the focusing resolution is limited by diffraction. Forexample, the resolution of refocusing of time harmonic waves cannot exceed the Abbediffraction limit of λ/2, where λ is the wavelength [9]. In most applications the array

WAVE PROPAGATION AND IMAGING IN RANDOM WAVEGUIDES 3

is supported in a set of small diameter (aperture) |A| with respect to the distance Lof propagation, and the expected resolution of refocusing in cross-range is given bythe Rayleigh formula λL/|A| if there is no scattering in the medium. The interestingproperty of time reversal, known as super-resolution, is that scattering may improvethe cross-range focusing. This is demonstrated and explained in [12, 13, 20], and itis analyzed theoretically and numerically in [3, 2, 25]. The latter studies are basedon random models of the scattering medium, and introduce the important concept ofstatistical stability. Stability means that the focusing is robust, it is independent ofthe realization of the random medium, it is observable and reproducible. Stability isguaranteed in general only for sources emitting broad-band signals, as pointed out in[3], although there are regimes where it holds for narrowband signals as long as thesources and arrays have sufficiently large support [25].

The super-resolution and robustness of time reversal are due entirely to the wavespropagating back in exactly the same medium they came from. The observer hasaccess to the vicinity of the source and can see the focus there, whereas in imagingthe access is limited to the array. Moreover, the medium is not known in detail inmany imaging applications. For example, in underwater acoustics, the smooth partof the wave speed is known, but there are small scale fluctuations due to internalwaves [14] that cannot be known, and are not even of interest in imaging. It isthe uncertainty about such small scale inhomogeneities that we model with randomspatial processes, and thus speak of imaging in random media.

The array data processing for image formation may appear to mimic the back-propagation of the waves in the time reversal process, but there is a fundamentaldifference. While in time reversal the waves go back in the same medium, the back-propagation in imaging is done mathematically, using a fictitious medium model thatincorporates the large scale features of the wave speed, but not its fluctuations, whichare unknown. This difference is essential in regimes with strong cumulative scatteringof the waves by the random inhomogeneities. The longer the waves travel in therandom medium, the stronger the scattering, which leads to loss of coherence ofthe wave field. The coherent waves are useful in imaging because we can relatethe locations of the sought-after sources or scatterers to their arrival time. Multiplescattering by the inhomogeneities transfers energy to the incoherent part of the waves,the random fluctuations which are unwanted reverberations. We may think of thereverberations as noise, but we should remember that they are not ordinary additive,uncorrelated noise. They have complicated statistical structure, with correlationsacross the array and in the bandwidth, and are difficult to mitigate. Our goal is toquantify the deterioration of resolution and robustness of images in random media,and to propose efficient methods for mitigating cumulative scattering effects.

4 LILIANA BORCEA

The data processing in image formation must take into account the presence ofscattering boundaries, because they create multiple traveling paths of the waves re-ceived at the array. This happens in waveguides, where boundaries trap the wavesand guide the energy in a preferred direction, the waveguide axis, along which themedium is unbounded. To analyze the wave field in waveguides, we may decompose itmathematically in an infinite, countable set of monochromatic waves called waveguidemodes. In ideal waveguides, with flat boundaries and wave-speed that is constant orvaries smoothly in the waveguide cross-section, the modes are uncoupled. Finitelymany of them propagate along the waveguide axis at different speeds, and we mayassociate them to plane waves with different angles of incidence at the boundary. Theslower modes correspond to nearly normal incidence. They bounce off the boundariesmany times, thus taking a long path from the source to the array. The faster modescorrespond to small grazing angles at the boundary, and shorter paths to the ar-ray. The remaining infinitely many modes are evanescent, with amplitudes decayingexponentially with the distance along the waveguide axis.

We are concerned with waveguides that are randomly perturbed versions of theideal ones. They may have boundaries with small random fluctuations, and/or inter-nal inhomogeneities that cause small fluctuations of the wave speed. We present arigorous asymptotic theory of wave propagation in such waveguides, where the asymp-totics is in the small parameter that measures the magnitude of the fluctuations. Weshow that when the waves travel sufficiently far from the source, the small fluctua-tions cause significant cumulative scattering. It amounts to coupling of the waveguidemodes, and gradual loss of coherence of their amplitudes. We give a detailed analysisof mode coupling and quantify the net scattering via three important length scales:the scattering mean free path, the transport mean free path and the equipartition dis-tance. The mode dependent scattering mean free path is the distance over which themode loses its coherence, meaning that its random fluctuations dominate its statisti-cal mean. The transport mean free path is classically defined as the distance beyondwhich the waves forget their initial direction [26]. Because the modes are associatedto directions of incidence at the boundary, we define the mode dependent transportmean free path as the length scale over which the mode exchanges energy with theother modes. The equipartition distance is the longest of the three length scales, andit gives the distance over which scattering distributes the energy uniformly amongthe modes, independent of the initial source excitation.

The three length scales are important for understanding the limitations of imaging.For example, coherent imaging cannot work for arrays that are farther from the sourcethan the scattering mean free path of all the modes, because there is no coherenceleft in the data. The recordings are just random medium reverberation. But ifthe array is not beyond the equipartition distance, we can still track how energy istransported in the waveguide by the modes, and thus localize sources or scatterers with


ε

array range ZAε2

z

Figure 1. Schematic of the problem setup. The source emits a signal in awaveguide and the wave field is recorded at a remote array. The waveguidehas fluctuating boundaries around the values y = 0 and y = D and themedium may have fluctuating wave speed.

a more involved, incoherent parameter estimation approach. Beyond the equipartitiondistance imaging is impossible, because the waves lose all the information about theirinitial state, due to scattering.

We present an explicit analysis of imaging in random waveguides and contrast itwith time reversal. For simplicity, we consider only the problem of imaging sourceswith passive arrays. Extensions to imaging with active arrays are straightforward ifwe use the Born approximation at the scatterers. Our study is for scalar (acoustic)waves in two dimensions. The three-dimensional waveguides are very similar to thetwo-dimensional ones if they have a bounded cross-section. We refer to [4] for a studyof wave propagation in three-dimensional waveguides with unbounded cross-sections,in a set-up motivated by applications in underwater acoustics. The waves are trappedthere by top and bottom boundaries, but the medium is unbounded in the remainingtwo directions.

The outline is as follows: We begin in section 2 with the set-up of the problem.The analysis of the wave field in ideal waveguides is in section 3, and in randomwaveguides in sections 4 and 5. The results are summarized in section 6. The modelof the data measured at the array is in section 7. We use it in sections 8 and 9 to studytime reversal and imaging. The analysis is illustrated with numerical simulations insection 10.

2. Set-up

Consider a two dimensional waveguide with range axis denoted by z ∈ R andtransverse coordinate (cross-range) y belonging to a bounded interval, the waveguidecross-section. In ideal waveguides y ∈ (0,D). In perturbed waveguides the top andbottom boundaries Y(z) and Yb(z) may fluctuate as illustrated in Figure 1. Theacoustic pressure field p(t, y, z) satisfies the wave equation

(2.1)(∂2y + ∂2

z − c−2∂2t

)p(t, y, z) = F (t, y)δ(z),

where F models the source distribution at the origin of range, and c is the wave speed.Equation (2.1) holds for y ∈ (Yb(z),Y(z)), z ∈ R and t > 0. It is complemented with

6 LILIANA BORCEA

the pressure-release boundary conditions

(2.2) p(t, y = Yb(z), z) = p(t, y = Y(z), z) = 0,

and the initial condition

(2.3) p(t, y, z) = 0, t 0.

Ideally, the wave speed c is constant or smoothly varying in the cross-section. Butin media with numerous small, and typically weak inhomogeneities, c has rapid fluc-tuations. We neglect large scale variations of c for simplicity, but take into accountsmall scale fluctuations.

We study wave propagation in perturbed waveguides with either randomly fluctu-ating boundaries (section 4), or randomly fluctuating wave speed (section 5) aroundthe mean constant value co. We separate the fluctuations in order to compare theirnet scattering effects, but the results extend easily to waveguides with both inte-rior inhomogeneities and perturbed boundaries. The analysis is asymptotic, in thelimit ε → 0, where ε is the small parameter that quantifies the magnitude of thefluctuations. In the waveguides with fluctuating boundaries we have (see Figure 1),

(2.4)|Yb(z)|

D= O(ε) and/or

|Y(z)−D|D

= O(ε),

and similarly, in the waveguides with internal inhomogeneities,

(2.5)|c(y, z)− co|

co= O(ε).

In either case, the fluctuations occur on a length scale `, called correlation length,which is of the same order as the central wavelength λo.

To define λo let us denote by

(2.6) F (ω, y) =∫ ∞−∞

dt F (t, y)eiωt

the Fourier transform of the signal emitted from the source point y. We assume thatit is supported in the frequency interval centered at ωo, of length 2πB, where B is thebandwidth. The central wavelength λo is defined by

(2.7) λo =2πcoωo

.

We use it throughout as the reference, order one length scale. See section 2.1 formore details on the source, and an example used in the analysis of time reversal andimaging.

Because the fluctuations are weak, the net scattering effects are negligible unlessthe waves travel a long distance in the waveguide. We will see that for both types offluctuations, which we model with mean zero random processes, this means scalingthe range as z = O(λo/ε2). This is why the array is at range ZA/ε2 in Figure 1.


Before considering the problem in random waveguides, it is useful to describe thewave field po(t, y, z) in ideal waveguides. It satisfies the wave equation

(2.8)(∂2y + ∂2

z − c−2o ∂2

t

)po(t, y, z) = F (t, y)δ(z), z ∈ R, y ∈ (0,D),

with boundary conditions

(2.9) po(t, 0, z) = po(t,D, z) = 0,

and initial condition

(2.10) po(t, y, z) = 0, t 0.

The solution po(t, y, z) can be modeled as a superposition of monochromatic waves,called waveguide modes. The mode decomposition is based on the eigenfunctions ofthe Sturm-Liouville operator

(2.11) Ly = ∂2y +

(ω

co

)2

defined on the vector space C20 (0,D) of functions that are twice continuously differ-

entiable in (0,D), and vanish at y = 0 and y = D. There are infinitely many modes.Finitely many propagate to the right and left of the source, and we can associatethem to plane waves that strike the boundaries at different angles of incidence. Theiramplitudes are completely determined by the source excitation and do not changewith range. Since problem (2.8)-(2.10) is separable, the modes are not coupled.

The wave field p(t, y, z) in the randomly perturbed waveguides can also be decom-posed into propagating and evanescent modes, as shown in sections 4 and 5. Thekey point in the decomposition is that the perturbation effects are mapped only tothe mode amplitudes, which are coupled and vary with range because of scattering.The analysis in 4 and 5 gives a detailed characterization of the random mode ampli-tudes. We are especially interested in their first and second moments, which allowus to quantify the range scales over which the waves lose coherence and decorrelatedue to scattering. We summarize and compare the net scattering effects of randomboundaries and interior random inhomogeneities in section 6. These are the mainresults of the analysis, and we use them in sections 8 and 9 to analyze time reversaland imaging.

2.1. Narrowband and broadband sources. — We describe here a source thatemits the same signal from all the points in its support. We use it in the analysis ofsections 8 and 9.

The source F (t, y) has the separable form

(2.12) F (t, y) = ϕ(t)1

∆yρ

(y − y?

∆y

),

8 LILIANA BORCEA

with non-negative source density ρ compactly supported in the interval of length ∆y

centered at y? ∈ (0,D). We assume that[y? − ∆y

2, y? +

∆y

2

]⊂ (0,D),

and normalize ρ by ∫ D

0

dy

∆yρ

(y − y?

∆y

)= 1.

If ∆y D we have a point-like source at y?.The source emits the signal

(2.13) ϕ(t) = e−iωotf(Bt),

defined by the function f of dimensionless arguments, with Fourier transform f sup-ported in the interval [−π, π]. The temporal duration of f(Bt) and therefore of ϕ(t)is of order 1/B. The Fourier transform of the emitted signal is given by

(2.14) ϕ(ω) =∫ ∞−∞

dt ϕ(t)eiωt =∫ ∞−∞

dt f(Bt)ei(ω−ωo)t =1Bf

(ω − ωoB

),

and it is supported at frequencies ω ∈ [ωo − πB, ωo + πB].To study wave propagation to an array at long range ZA/ε2 from the source, we

observe the waves on a time scale

(2.15) t =T

ε2.

We distinguish between two regimes: narrowband and broadband. In the narrowbandregime the bandwidth B is related to the central frequency by

(2.16) B = ε2ΩB,ΩBωo

= O(1),

and the emitted signal

(2.17) ϕ

(t =

T

ε2

)= e−iωoT/ε

2f (ΩBT )

has large temporal support, of the order of the travel time from the source to thearray. In the broadband regime

B = εqΩB, q < 2,

and the signal

(2.18) ϕ

(t =

T

ε2

)= e−iωoT/ε

2f(ε2−qΩBT

)has temporal support that is smaller by a factor of ε2−q than the travel time. We callsuch a signal a pulse.


Obviously, the larger B is, the smaller the pulse duration. For our analysis, itsuffices to consider q = 1, so we define the broadband regime by

(2.19) B = εΩB,ΩBωo

= O(1).

3. Ideal waveguides

A waveguide mode is a monochromatic wave P (ω, y, z)e−iωt with P (ω, y, z) solvingthe Helmholtz equation

(3.1)(∂2y + ∂2

z + k2)P (ω, y, z) = 0

for y ∈ (0,D) and z ∈ R, with homogeneous Dirichlet boundary conditions at y = 0and y = D. Here k is the wavenumber, defined by

k =ω

co=

2πλ.

At the central frequency ωo we call it ko.We decompose the solution po(t, y, z) of (2.8)-(2.10) in modes using the eigenfunc-

tions φj(y) of the operator Ly defined in (2.11). This operator is self-adjoint, withreal and simple eigenvalues λj(ω), and the set φjj≥1 of orthonormal eigenfunctionsis complete in L2(0,D). See for example [27]. The eigenfunctions are independent ofthe frequency because the wave speed is constant. They would depend on ω other-wise, but the analysis below would remain the same in all essential aspects. We haveexplicitly

(3.2) Lyφj(y) = λj(ω)φj(y), j = 1, 2, . . . ,

where

(3.3) φj(y) =

√2D

sin(πjy

D

),

and

(3.4) λj(ω) =( π

D

)2[(

kD

π

)2

− j2

].

Because only the first N(ω) eigenvalues are positive, where

(3.5) N(ω) =⌊kD

π

⌋and b c denotes the integer part, we write

(3.6) λj(ω) =β2j (ω) if j = 1, . . . , N(ω)−β2

j (ω) if j > N(ω).

10 LILIANA BORCEA

The solution of (2.8)-(2.10) is given by

(3.7) po(t, y, z) =∫ ∞−∞

dω

2πpo(ω, y, z)e−iωt,

with

po(ω, y, z) = 1(0,∞)(z)

N(ω)∑j=1

φj(y)u+j,o(ω, z) +

∞∑j=N(ω)+1

φj(y) v+j,o(ω, z)

+1(−∞,0)(z)

N(ω)∑j=1

φj(y)u−j,o(ω, z) +∞∑

j=N(ω)+1

φj(y) v−j,o(ω, z)

.(3.8)

The indicator function

1(a,b)(z) =

1 if z ∈ (a, b)0 otherwise

allows us to write po to the right and left of the source. The first N(ω) components u±j,oof the solution are propagating waves satisfying one dimensional Helmholtz equationswith wavenumber βj(ω)

(3.9)[∂2z + β2

j (ω)]u±j,o(ω, z) = 0, z 6= 0,

outgoing (radiation) conditions at |z| → ∞, and source jump conditions at z = 0

u+j,o(ω, 0+)− u−j,o(ω, 0−) = 0,

∂zu+j,o(ω, 0+)− ∂zu−j,o(ω, 0−) = Fj(ω) :=

∫ D

0

dy φj(y)F (ω, y).(3.10)

Explicitly, they are given by right and left going waves

(3.11) u+j,o(ω, z) = aj,o(ω)eiβj(ω)z, u−j,o(ω, z) = bj,o(ω)e−iβj(ω)z,

with amplitudes

(3.12) aj,o(ω) = bj,o(ω) =Fj(ω)

2iβj(ω), j = 1, . . . , N(ω).

The components v±j,o(ω, z) solve the equations

(3.13)[∂2z − β2

j (ω)]v±j,o(ω, z) = 0, z 6= 0,

with decay condition lim|z|→∞ v±j,o(ω, z) = 0, and source jump conditions similar to(3.10). They are the evanescent waves given by

(3.14) v±j,o(ω, z) = ej,o(ω)e−βj(ω)|z|, ej,o = − Fj(ω)2βj(ω)

, j > N(ω).

We assume throughout that none of the mode wavenumbers βj are zero in thebandwidth, meaning that there are no standing waves. We also suppose that N(ω)


does not vary in the bandwidth, and we drop its dependence on ω. This amounts tohaving

(3.15)kD

π= N + ϑ(ω), ϑ(ω) ∈ (0, 1), ∀ω ∈ [ωo − πB, ωo + πB].

The wave field to the right of the source becomes

(3.16) po(ω, y, z) =N∑j=1

aj,o(ω)φj(y)eiβj(ω)z +∞∑

j=N+1

ej,o(ω)φj(y)e−βj(ω)z, z > 0,

with negligible evanescent part at long ranges.

3.1. Plane waves interpretation. — Recalling definition (3.3) of φj , we can writethe propagating part of (3.16) as a superposition of plane waves

±aj,o(ω)i√

2Dexp

[i

(±πj

D, βj

)· (y, z)

], j = 1, . . . , N.

We need both signs in the phases so that when we add the waves together, we satisfythe boundary conditions. The waves travel in the direction of the slowness vectors

K±j =(±πj

D, βj

)and strike the boundaries, where they are reflected.

Suppose that the waveguide supports many propagating modes, that is N 1.Because the wavenumbers decrease with the index j, we see that the higher indexedmodes have slowness vector that is almost orthogonal to the boundaries. Indeed, forthe last mode we have

K±N =(±πN

D, βN

)with

πN

D≈ k, βN =

π

D

√2ϑN ≈ k

√2ϑN k,

so the slowness vector is almost parallel to the y axis. These waves strike the boundarymany times and thus take a very long path from the source to the array. The modepropagates slowly at group speed

1β′N (ω)

= coβN (ω)k

≈ co

√2ϑ(ω)N

co.

The fastest mode is indexed by 1. It gives the slowness vector

K±1 =(± π

D, β1

)≈(± k

N, k

)

12 LILIANA BORCEA

that is almost parallel to the range axis. This mode barely sees the boundaries, andtravels quickly to the array with speed

1β′1(ω)

= coβ1(ω)k≈ co.

3.2. Long range pulse propagation. — Neglecting the evanescent part in (3.16)and using equation (3.12), we obtain that the pressure field measured at the rangeZA/ε

2 of the array is

(3.17) po

(t, y,

ZAε2

)≈

N∑j=1

φj(y)∫ ∞−∞

dω

2πFj(ω)

2iβj(ω)eiβj(ω)ZA/ε

2−iωt.

We describe it here for the separable source density (2.12), emitting the signal ϕ(t)given by (2.13), and for the time t = T/ε2. Since

(3.18) Fj(ω) =∫ D

0

dy φj(y)∫ ∞−∞

dt F (t, y)eiωt,

we obtain that

po

(t =

T

ε2, y,

ZAε2

)≈

N∑j=1

φj(y) fεj(T )∫ D

0

dy′

∆yρ

(y′ − y?

∆y

)φj(y′),(3.19)

where we denote by fεj(T ) the signals propagated by the modes,

(3.20) fεj(T ) =∫ ∞−∞

dω

2πBf

(ω − ωoB

)ei[βj(ω)ZA−ωT ]/ε2

2iβj(ω).

In the narrowband regime defined by (2.16), we can change variables as

ω = ωo + ε2w,

and obtain that

fεj(T ) =ei[βj(ωo)ZA−ωoT ]/ε2

2iβj(ωo)

∫ ∞−∞

dw

2πΩBf

(w

ΩB

)eiβ′j(ωo)wZA−iwT +O(ε2)

=ei[βj(ωo)ZA−ωoT ]/ε2

2iβj(ωo)f[ΩB(T − β′j(ωo)ZA

)]+O(ε2).(3.21)

Thus, the modes propagate the emitted signal at the group velocity

(3.22)1

β′j(ωo)= co

βj(ωo)ko

≈ co

√1− j2

N2,

which varies with the mode index. The first arrival is for mode j = 1, with speed1

β′1(ωo)≈ co.


The other modes follow, with the latter ones traveling at much smaller speed thanco. Since ϕ has long temporal support, of the same order as the travel time, it is notpossible to distinguish the arrival of the modes. The array records the superpositionof signals that are mixed together.

In the broadband regime defined by (2.19) we let

ω = ωo + εw,

and obtain from (3.20) that

fεj(T ) =ei[βj(ωo)ZA−ωoT ]/ε2

2iβj(ωo)

∫ ∞−∞

dw

2πΩBf

(w

ΩB

)eiwε (β′j(ωo)ZA−T)+i

β′′j (ωo)w2ZA2

=ei[βj(ωo)ZA−ωoT ]/ε2

2iβj(ωo)f ? dj

[ΩB(T − β′j(ωo)ZA

)ε

],(3.23)

where the star denotes convolution and dj is the dispersion kernel

(3.24) dj(ΩBt) =∫ ∞−∞

dw

2πΩBeiβ′′j (ωo)w2ZA/2−iwt.

The modes travel with speed 1/β′j(ωo) as before, but now the array receives a trainof pulses(1), which are well separated because their temporal support is small withrespect to the travel time. The pulse shapes are affected by dispersion, due to thequadratic phase term proportional to

|β′′j (ωo)| =(πj/D)2

c2oβ3j (ωo)

.

It increases monotonically with the mode index, so we can expect that the late arrivalshave significant pulse shape alteration due to dispersion.

4. Waveguides with random boundaries

The asymptotic analysis presented here is essentially the same if one or both bound-aries of the waveguide have fluctuations. For simplicity, we assume that the topboundary Y(z) fluctuates about the mean value D, but set the bottom boundary to

Yb(z) = 0.

Waveguides with random Yb(z) and Y(z) are studied in detail in [1] for both Dirichletand Neumann boundary conditions. See also the studies [17, 18] for random waveg-uides with unbounded cross-section that support radiative modes in addition to thepropagating and evanescent ones.

(1)See the top left picture in Figure 4 for an illustration of a train of pulses received at the array.

14 LILIANA BORCEA

4.1. Model of the fluctuations. — We model the fluctuations of the top bound-ary with a random process µ(z)

(4.1) Y(z) = D [1 + εµ(z)] , ε 1.

We assume that µ(z) is bounded, stationary, with mean zero

(4.2) E[µ(z)] = 0

and mixing, i.e., with enough decorrelation(2). Its covariance

(4.3) Rµ(z) = E[µ(z + s)µ(s)]

is integrable, and we normalize it by

(4.4) Rµ(0) = 1.

We use the dimensionless parameter ε 1 to model the small amplitude of thefluctuations.

Our method of solution is based on a coordinate change that straightens the bound-ary and maps the fluctuations inside the domain. It requires that µ(z) be twice differ-entiable, with almost surely bounded derivatives. This implies that Rµ(z) has at leastfour derivatives. If µ(z) is not differentiable, another approach based on conformalmappings may be used, as explained in [18]. In the end, the results should not bethat different.

The covariance Rµ(z) is an even function, with maximum at z = 0. It decays tozero as |z| → ∞, on a length scale called the correlation length `. It is the range offsetover which the fluctuations become statistically decorrelated. We define it using anauxiliary even, integrable and four times differentiable function R of dimensionlessargument, satisfying R(0) = −R′′(0) = 1. We let

(4.5) Rµ(z) = R(z`

),

and therefore

(4.6) ` =

√−Rµ(0)R′′µ(0)

=

√− 1R′′µ(0)

.

An alternative definition of ` is given by the integral of Rµ(z). It agrees with (4.6)up to an order one constant, because equation (4.5) gives∫ ∞

−∞dzRµ(z) = `

∫ ∞−∞

duR(u) = `O(1).

As an illustration, consider the Gaussian

Rµ(z) = e−z2

2`2 .

(2)Explicitly, µ(z) is a ϕ-mixing process, with ϕ ∈ L1/2(R+), as stated in [21, Section 4.6.2].


Its integral is ∫ ∞−∞

dzRµ(z) =√

2π`

and the fluctuations are uncorrelated for range offsets exceeding 3`, because

Rµ(z) ≈ 0 if |z| ≥ 3`.

In the analysis we suppose that the boundary fluctuates only in the range intervalz ∈ [0, Lε],

(4.7) Y(z) = D, z ∈ (−∞, 0) ∪ (Lε,∞),

so we can specify outgoing boundary conditions on the Fourier coefficients p(ω, y, z)of the wave field. The confinement of the fluctuations to z ≤ Lε can be motivated bythe hyperbolicity of the time-domain problem. If we wish to study wave propagationto a maximum range Lε that is large and depends on ε in our case, we observe thefield p(t, y, z) in a time window supported at t . Lε/co. In this window the waves arenot affected by the medium beyond a range Lε & Lε, so we may set the fluctuationto zero for z > Lε. Here the symbols . (and &) mean less or equal (and larger orequal) up to an order one constant.

The truncation of the fluctuations at negative ranges can be justified under the for-ward scattering approximation, which says that when the correlation length is largeenough, we can neglect the backscattered field. In this approximation the medium onthe left of the source has little influence on the wave field at positive ranges, so we candiscard the fluctuations at z < 0. We refer to [16] for a study of wave propagation inrandom waveguides with forward and backward wave coupling. The analysis is morecomplicated, and it accounts for important phenomena, such as enhanced backscat-tering, that cannot be analyzed under the forward scattering approximation. Theresults relevant to imaging are not that different, so we will consider regimes wherethe forward scattering approximation applies.

4.2. Change of coordinates. — To analyze the wave field in the random waveg-uide, we reformulate the problem with a change of coordinates that flatten the bound-ary

(4.8) y = Y(z)Y

D, Y ∈ [0,D].

We could use many other coordinate changes, but we prefer (4.8) because it is simple.Clearly, if the method is any good, the result should be independent of the choicethat we make. This turns out to be the case, as shown in detail in [1].

Let the wave field in the new coordinate system be P (t, Y, z). It is defined by

(4.9) P (t, Y, z) = p

(t,Y(z)

Y

D, z

), z ∈ R, Y ∈ [0,D],

16 LILIANA BORCEA

and it satisfies the boundary conditions

(4.10) P (t, 0, z) = P (t,D, z) = 0.

In the original coordinates we have

(4.11) p(t, y, z) = P

(t,yD

Y(z), z

), z ∈ R, y ∈ [0,Y(z)].

After Fourier transforming in t and using the chain rule, we obtain that the Fouriercoefficients of (4.9) satisfy

(4.12) LεP (ω, Y, z) = 0,

for z ∈ (0, Lε) and Y ∈ (0,D). The partial differential operator Lε is given by

Lε = ∂2z +

[D2 + (Y Y ′(z))2

]Y2(z)

∂2Y −

2Y Y ′(z)Y(z)

∂2Y z

+Y

[2(Y ′(z)Y(z)

)2

− Y′′(z)Y(z)

]∂Y + k2.(4.13)

Its dependence on the small parameter ε follows from the definition (4.1) of Y(z), andwe can write it as a perturbation series of operators

(4.14) Lε = Lo + εL1 + ε2L2 + ε3L3 + . . .

The leading order operator corresponds to the ideal waveguide

(4.15) Lo = ∂2z + ∂2

Y + k2.

The next two operators are given by

L1 + εL2 = −2Y µ′(z)[1− εµ(z)]∂2Y z +

[−2µ(z) + 3εµ2(z) + ε(Y µ′(z))2

]∂2Y

+[−µ′′(z) + εµ(z)µ′′(z) + 2ε(µ′(z))2

]Y ∂Y ,(4.16)

and the remainder in (4.14) is an operator with O(ε3) random coefficients that dependon Y and z,

(4.17) ε3L3 + . . . = O(ε3)∂2Y z +O(ε3)∂2

Y +O(ε3)∂Y .

We do not write it explicitly, because it is negligible in our asymptotic analysis.The boundary is flat for z ∈ (−∞, 0)∪ (Lε,∞) by assumption (4.7), so the change

of variables (4.8) is trivial in this range domain

(4.18) Y = y, z ∈ (−∞, 0) ∪ (Lε,∞),

and P satisfies the same equation as in the ideal waveguide

(4.19) LoP (ω, Y, z) = 0, z ∈ (−∞, 0) ∪ (Lε,∞).


4.3. Mode decomposition. — We ensure that the boundary conditions (4.13)are satisfied exactly by decomposing P (ω, Y, z) in the L2(0,D) basis of orthonormaleigenfunctions φj(Y )j≥1,

(4.20) P (ω, Y, z) =N∑j=1

φj(Y )uj(ω, z) +∞∑

j=N+1

φj(Y )vj(ω, z).

This looks similar to the decomposition (3.8) in ideal waveguides, but the problem isno longer separable, and the propagating and evanescent components uj and vj arerandom functions accounting for scattering at the random boundary.

The propagating components satisfy the differential equations[∂2z + β2

j (ω)]uj(ω, z) = Fj(ω)δ(z)

− 1(0,Lε)(z) εN∑l=1

[Qεjl(µ(z)) ∂z +Mε

jl(µ(z))]ul(ω, z)

− 1(0,Lε)(z) ε∞∑

l=N+1


jl(µ(z))]vl(ω, z) +O(ε3) ,(4.21)

for j = 1, . . . , N , with outgoing conditions at |z| → ∞, Similarly, the evanescentcomponents satisfy[∂2z − β2

j (ω)]vj(ω, z) = Fj(ω)δ(z)

− 1(0,Lε)(z) εN∑l=1


jl(µ(z))]ul(ω, z)

− 1(0,Lε)(z) ε∞∑

l=N+1


jl(µ(z))]vl(ω, z) +O(ε3)(4.22)

for j > N , with decay conditions lim|z|→∞ vj(ω, z) = 0. The matrices Qεjl and Mεjl

couple the propagating and evanescent components, and are given by

(4.23) Qεjl(µ(z)) = −2µ′(z)[1− εµ(z)]∫ D

0

dy y φj(y)φ′l(y),

and

Mεjl(µ(z)) =[−2µ(z) + 3εµ2(z)]

∫ D

0

dy φj(y)φ′′l (y)

+ ε[µ′(z)]2∫ D

0

dy y2φj(y)φ′′l (y)

+[−µ′′(z) + εµ(z)µ′′(z) + 2ε(µ′(z))2

] ∫ D

0

dy y φj(y)φ′l(y).(4.24)

18 LILIANA BORCEA

The characteristic function 1[0,Lε](z) in equations (4.21)-(4.22) indicates that the fluc-tuations are supported in the range interval z ∈ [0, Lε].

We begin with the study of the propagating modes. The evanescent ones areanalyzed in section 4.6.

4.4. The forward and backward going waves. — Note that equations (4.21)are the same as in the ideal waveguide in the range interval z ∈ (−∞, 0) ∪ (Lε,∞).Therefore, we write directly the solution as a backward going wave to the left of thesource

(4.25) uj(ω, z) = b−j (ω)e−iβj(ω)z, z < 0,

and a forward going wave beyond the range of the fluctuations

(4.26) uj(ω, z) = a+j (ω)eiβj(ω)z, z > Lε.

Here b−j (ω), a+j (ω) are constant mode amplitudes that remain to be determined from

the continuity conditions at z = Lε

a+j (ω) = uj(ω, Lε)e−iβj(ω)Lε

iβj(ω)a+j (ω) = ∂zuj(ω, Lε)e−iβj(ω)Lε ,(4.27)

and the source jump conditions at z = 0

uj(ω, 0+)− b−j (ω) = 0,

∂zuj(ω, 0+) + iβj(ω)b−j (ω) = Fj(ω).(4.28)

The solution of (4.21) in the range interval z ∈ (0, Lε) is a superposition of forwardand backward going waves, with amplitudes aj and bj ,

(4.29) uj(ω, z) = aj(ω, z)eiβj(ω)z + bj(ω, z)e−iβj(ω)z, j = 1, . . . , N.

Because the fluctuations end at z = Lε, there should be no backward going wavethere, so we ask that

(4.30) bj(ω, Lε) = 0,

and rewrite the continuity conditions (4.27) as

aj(ω, Lε) = a+j (ω),

eiβj(ω)Lε∂zaj(ω, Lε) + e−iβj(ω)Lε∂zbj(ω, Lε) = 0.(4.31)


The source jump conditions at z = 0 are

aj(ω, 0+) + bj(ω, 0+) = b−j (ω),

iβj(ω) [aj(ω, 0+)− bj(ω, 0+)] = Fj(ω)− iβj(ω)b−j (ω)

− [∂zaj(ω, 0+) + ∂zbj(ω, 0+)] .(4.32)

Moreover, substituting (4.29) in (4.21) we obtain

iβj(ω)[eiβj(ω)z∂zaj(ω, z)− e−iβj(ω)z∂zbj(ω, z)

]=

− ∂z[eiβj(ω)z∂zaj(ω, z) + e−iβj(ω)z∂zbj(ω, z)

]− ε

N∑l=1


jl(µ(z))] (al(ω, z)eiβl(ω)z + bl(ω, z)e−iβl(ω)z

)− ε

∞∑l=N+1


jl(µ(z))]vl(ω, z) +O(ε3), z ∈ (0, Lε).(4.33)

Equations (4.30)-(4.33) do not define uniquely the amplitudes aj and bj . We needan extra equation to close the system. To derive it, we note that because we have aperturbation problem, the solution should behave like in the ideal waveguide in theproximity of the source. This happens when we ask that

(4.34) eiβj(ω)z∂zaj(ω, z) + e−iβj(ω)z∂zbj(ω, z) = 0, ∀ z ∈ (0, Lε).

This equation is consistent with (4.31), and when combined with (4.33) it gives firstorder evolution equations for aj and bj . These equations, with the initial conditions

(4.35) aj(ω, 0+) =Fj(ω)

2iβj(ω)= aj,o(ω)

derived from (4.32), and the outgoing conditions (4.30), determine aj(ω, z) andbj(ω, z). The waves on the left of the source are determined by

(4.36) b−j (ω) =Fj(ω)

2iβj(ω)+ bj(ω, 0+),

and the waves at ranges z > Lε follow from the continuity conditions (4.27).We write next the infinite system of differential equations that describe the evolu-

tion and coupling of the modes.

20 LILIANA BORCEA

4.5. Mode coupling. — The forward going wave amplitudes satisfy

∂zaj(ω, z) = iε

N∑l=1

Cεjl(ω, µ(z)) al(ω, z)ei[βl(ω)−βj(ω)]z

+ iε

N∑l=1

Cεjl(ω, µ(z)) bl(ω, z)e−i[βl(ω)+βj(ω)]z

+iε

2βj(ω)

∞∑l=N+1

e−iβj(ω)z[Qεjl(µ(z)) ∂z +Mε

jl(µ(z))]vl(ω, z) +O(ε3)(4.37)

for z ∈ (0, Lε), with initial condition (4.35). Here the bar denotes complex conjuga-tion. Similarly, the backward going wave amplitudes satisfy

∂z bj(ω, z) = −iεN∑l=1

Cεjl(ω, µ(z)) al(ω, z)ei[βl(ω)+βj(ω)]z

− iεN∑l=1

Cεjl(ω, µ(z)) bl(ω, z)e−i[βl(ω)−βj(ω)]z

− iε

2βj(ω)

∞∑l=N+1

eiβj(ω)z[Qεjl(µ(z))∂z +Mε

jl(µ(z))]vl(ω, z) +O(ε3) ,(4.38)

for z ∈ (0, Lε), with end condition (4.30). The amplitudes are coupled directly by thecoefficients

(4.39) Cεjl(ω, µ(z)) =1

2βj(ω)[Mεjl(µ(z)) + iβl(ω)Qεjl(µ(z))

]j, l = 1, . . . , N,

with Qεjl and Mεjl defined in (4.23)-(4.24), and indirectly through the evanescent

modes. We show next that the evanescent components vj(ω, z)j>N can be writ-ten in terms of the propagating amplitudes, and thus obtain a closed system foraj(ω, z), bj(ω, z)j=1,...,N .

4.6. Analysis of the evanescent modes. — When substituting (4.29) in (4.22),we obtain that vj(ω, z)j>N satisfy the equations[∂2z − β2

j (ω)]vj(ω, z) = Fj(ω)δ(z)− 2εwεj (ω, z)

− 1(0,Lε)(z) ε∞∑

l=N+1


jl(µ(z))]vl(ω, z) +O(ε3) ,(4.40)

with decay conditions

(4.41) lim|z|→∞

vj(ω, z) = 0,


where we let

wεj (ω, z) = 1(0,Lε)(z)2βj(ω)N∑l=1

[Cεjl(ω, µ(z)) al(ω, z)eiβl(ω)z

+Cεjl(ω, µ(z)) bl(ω, z)e−iβl(ω)z].

To solve these equations, we begin by inverting the leading differential operator,using the Green’s function

Gj(ω, z) =e−βj(ω)|z|

2βj(ω),

satisfying (∂2z − β2

j (ω))Gj(ω, z) = −δ(z), and lim

|z|→∞Gj(ω, z) = 0.

We obtain a system of integral equations for the infinite vector v = (vN+1, . . .),

[(I− εH) v]j (ω, z) = − Fj(ω)e−βj(ω)|z|

2βj(ω, z)+ ε

∫ ∞−∞

ds e−βj(ω)|s|wεj (ω, z + s),

where I is the identity and H is the linear integral operator

[Hv]j (ω, z) =∞∑

l=N+1

∫ ∞−∞

ds e−βj(ω)|s|Hεjl(ω, z + s)vl(ω, z + s),

with kernel

Hεjl(ω, z) =1(0,Lε)(z)2βj(ω)

(Qεjl(µ(z)) ∂z +Mε

jl(µ(z)))

+O(ε2).

It is shown in [1, Lemma 3.1] that H is a bounded operator on the space of squaresummable sequences of L2(R) functions with weights, equipped with the norm

‖v‖ =

∞∑j=N+1

(j‖vj‖L2(R)

)2 12

.

Thus, we can invert the operator I− εH with the Neumann series

(I− εH)−1 = I + εH + ε2H2 + . . . ,

and write explicitly the evanescent waves in terms of the propagating ones.

22 LILIANA BORCEA

The solution is

vj(ω, z) = − Fj(ω)2βj(ω, z)

e−βj(ω)|z|

+ ε

N∑l=1

al(ω, z)eiβl(ω)z

∫ ∞−∞

ds e−βj(ω)|s|+iβl(ω)sCjl(ω, µ(z + s))

+ ε

N∑l=1

bl(ω, z)e−iβl(ω)z

∫ ∞−∞

ds e−βj(ω)|s|−iβl(ω)sCjl(ω, µ(z + s)) +O(ε2),(4.42)

because equations (4.37)-(4.38) give

al(ω, z + s) = al(ω, z) +O(ε), bl(ω, z + s) = bl(ω, z) +O(ε)

for s = O(1), where the decaying exponential in (4.42) is not negligible. We neglectedthe term [

H

(FN+1e

−βN+1|z|

2βN+1, . . .

)]j

(ω, z) =∞∑

l=N+1

Fl(ω)2βl(ω)

×∫ ∞−∞

ds e−βj(ω)|s|Hεjl(ω, z + s)e−βl(ω)|s+z|,

because it is smaller than the residual at long ranges, due to the exponential decay inz. We also let Cjl(ω, µ(z)) be the leading (order one) part of the coupling matricesdefined in (4.39), which we rewrite as

(4.43) Cεjl(ω, µ(z)) = Cjl(ω, µ(z)) + εcjl(ω, µ(z)).

Note that the first term in (4.42) is precisely the solution in the ideal waveguide.It is exponentially small at long ranges, so we neglect it in the calculations below.

4.7. Closed system for the propagating modes. — The substitution of (4.42)in equations (4.37)-(4.38) gives a closed system of equations for the amplitudes of thepropagating modes. We write it in compact form as

(4.44) ∂z

[a(ω, z)b(ω, z)

]=[εΥ(ω, µ(z), z) + ε2γ(ω, µ(z), z) +O(ε3)

] [ a(ω, z)b(ω, z)

],

for the vectors a and b with components aj and bj , for j = 1, . . . , N , and matrices Υand γ with block structure

(4.45) Υ(ω, µ(z), z) =

[Υ(a)(ω, µ(z), z) Υ(b)(ω, (µ(z), z)

Υ(b)(ω, µ(z), z) Υ(a)(ω, µ(z), z)

],

and

(4.46) γ(ω, µ(z), z) =

[γ(a)(ω, µ(z), z) γ(b)(ω, µ(z), z)γ(b)(ω, µ(z), z) γ(a)(ω, µ(z), z)

].


The leading order N ×N coupling matrices Υ(a,b) have the entries

(4.47) Υ(a)jl (ω, µ(z), z) = iCjl(ω, µ(z))ei[βl(ω)−βj(ω)]z,

and

(4.48) Υ(b)jl (ω, µ(z), z) = iCjl(ω, µ(z))e−i[βl(ω)+βj(ω)]z,

and the second order coupling matrices are given by

γ(a)jl (ω, µ(z), z) = iei[βl(ω)−βj(ω)]z

cjl(ω, µ(z)) +

12βj(ω)

∑l′>N

∫ ∞−∞ds e−βl′ |s|+iβl(ω)s

× [βl′(ω)Qjl′(ω, µ(z))sgn(s) +Mjl′(ω, µ(z))]Cl′l(ω, µ(z + s)),(4.49)

and

γ(b)jl (ω, µ(z), z) = −ie−i[βl(ω)+βj(ω)]z

cjl(ω, µ(z)) +

12βj(ω)

∑l′>N

∫ ∞−∞ds e−βl′ |s|−iβl(ω)s

× [βl′(ω)Qjl′(ω, µ(z))sgn(s) +Mjl′(ω, µ(z))]Cl′l(ω, µ(z + s)).(4.50)

Here we recalled definition (4.43), and wrote similarly the matrices Qε and Mε definedin (4.23)-(4.24)

(4.51) Qε(µ(z)) = Q(µ(z)) +O(ε) and Mε(µ(z)) = M(µ(z)) +O(ε).

Note that the entries in Υ are linear in the random process µ, and so their expec-tation is zero,

(4.52) E[Υ(a)jl (ω, µ(z), z)

]= E

[Υ(b)jl (ω, µ(z), z)

]= 0, j, l = 1, . . . , N.

The entries in γ are quadratic in µ(z), and their expectation is not zero.

4.8. The propagator. — Because a(ω, z) satisfies the initial conditions (4.35) atz = 0, and b(ω, z) satisfies the end conditions (4.30), it is convenient to write thesolution of (4.44) in terms of the 2N × 2N propagator matrix P(ω, z). It satisfies theevolution equation

(4.53) ∂zP(ω, z) =[εΥ(ω, µ(z), z) + ε2γ(ω, µ(z), z) +O(ε3)

]P(ω, z),

for z > 0, and the initial condition

(4.54) P(ω, 0) = I2N ,

where I2N is the 2N × 2N identity matrix. The solution of (4.44) is given by

(4.55)[

a(ω, z)b(ω, z)

]= P(ω, z)

[a(ω, 0+)b(ω, 0+)

],

24 LILIANA BORCEA

with a(ω, 0+) defined in (4.35), and b(ω, 0+) determined from the boundary identity

(4.56)[

a(ω, Lε)0

]= P(ω, Lε)

[a(ω, 0+)b(ω, 0+)

].

The special structure of the coupling matrices in (4.53) implies that if (a,b) is asolution of equations (4.44), then so is (b,a). Thus, we must have

(4.57)[

b(ω, z)a(ω, z)

]= P(ω, z)

[b(ω, 0+)a(ω, 0+)

],

in conjunction with (4.55), which implies that the propagator has the block form

(4.58) P(ω, z) =

[P(a)(ω, z) P(b)(ω, z)

P(b)(ω, z) P(a)(ω, z)

].

The blocks P(a,b)(ω, z) ∈ CN×N satisfy the initial conditions

(4.59) P(a)(ω, 0) = I and P(b)(ω, 0) = 0,

where I and 0 are the N ×N identity and zero matrices, respectively.

4.9. The long range scaling. — Because the right hand side in the evolutionequations (4.53) is of order ε, we expect that there is no net scattering effect overdistances of order one, meaning that

limε→0

P(ω, z) = I2N .

If we considered ranges of order ε−1, we would have an order one right hand sidein the scaled equations (4.53). Still, there would be no net scattering effect in thelimit ε→ 0, because the expectation of the leading term is zero. This is a stochasticaveraging result given for example in [15, Section 6.4]. It takes longer ranges, of orderε−2 to see net scattering effects, so we let z = Z/ε2, with scaled range Z of order one.This means in particular that the limit range of the fluctuations is Lε = L/ε2.

We denote the propagator in this long range scaling by

(4.60) Pε(ω,Z) = P(ω,

Z

ε2

),

and obtain from (4.53) that it satisfies the equations(4.61)

∂ZPε(ω,Z) =[

1εΥ(ω, µ

(Z

ε2

),Z

ε2

)+ γ

(ω, µ

(Z

ε2

),Z

ε2

)+O(ε)

]Pε(ω,Z),

with initial conditions

(4.62) Pε(ω, 0) = I2N .


Recall the block structure (4.58) of the generator. The off-diagonal blocksPε(b)(ω,Z) satisfy homogeneous initial conditions. They couple the forward andbackward propagating waves, and we show in section 4.11 that they are negligible inthe ε→ 0 limit if the correlation length is larger than the wavelength. We recall firstthe theorem that gives the limit of Pε(ω,Z).

4.10. The diffusion approximation. — The system (4.61) belongs to the generalclass of problems that can be analyzed in the limit ε→ 0 with the diffusion approx-imation limit theorem. This theorem is stated in [15, Section 6.5], and we refer to[23, 24] for its proof. We summarize it briefly.

Consider a system of stochastic differential equations of the form

(4.63) ∂ZX ε(Z) =1εF (0)

(X ε(Z), µ

(Z

ε2

),Z

ε2

)+ F (1)

(X ε(Z), µ

(Z

ε2

),Z

ε2

),

for Z > 0 and vector or matrix valued X ε(Z) with real entries, satisfying the initialconditions

(4.64) X ε(0) = Xo.

The functions F (0,1)(X , µ, τ) are at most linearly growing and smooth in X , and thedependence in τ is periodic, or almost periodic. Moreover, F (0)(X , µ, τ) is centered,meaning that for any fixed X and τ ,

E[F (0)(X , µ, τ)] = 0.

All these assumptions are satisfied in our case: The right hand side in (4.61) is linearin Pε, the matrices Υ and γ are periodic in the last argument, and E[Υ] = 0.

The diffusion approximation theorem states that as ε → 0, X ε(z) converges indistribution to the diffusion Markov process X (z) with generator G acting on smoothfunctions ϕ(X ) as

Gϕ(X ) = limT→∞

1T

∫ T

0

dτ

∫ ∞0

dz E[F (0)(X , µ(0), τ) · ∇X

[F (0)(X , µ(z), τ + z) · ∇Xϕ(X )

]]+

1T

∫ T

0

dτ E[F (1)(X , µ(0), τ) · ∇Xϕ(X )

].(4.65)

We can apply it to our problem by letting X ε be the matrix in R4N×4N obtained byconcatenating the absolute values and phases of the entries in the propagator Pε.

4.11. The forward scattering approximation. — When we use the diffusionlimit theorem described above, we obtain that the limit entries of Pε(b)(z) are coupledto the limit entries of Pε(a)(z) by coefficients that are proportional to

Rµ(βj + βl) = 2∫ ∞

0

dzRµ(z) cos[(βj + βl)z] ,

26 LILIANA BORCEA

for j, l = 1, . . . , N . Here Rµ is the power spectral density of the process µ, the Fouriertransform of the covarianceRµ. It is evaluated at the sum of the wavenumbers becausethe phase factors present in the matrices Υ(b) and γ(b) are (βj + βl)z. The phases inthe matrices Υ(a) and γ(a) are (βj − βl)z, so the limit entries of Pε(a)(z) are coupledto each other through Rµ(βj − βl), for j, l = 1, . . . , N .

The forward scattering approximation is based on the assumption that the powerspectral density decays fast enough to get

(4.66) Rµ(βj + βl) ≈ 0 , for j, l = 1, . . . , N.

If this is true, Pε(a) and Pε(b) decouple in the limit. Equivalently, the forward andbackward going wave amplitudes are asymptotically decoupled, and due to the bound-ary conditions b(ω, Lε) = 0, we may neglect the backward going waves.

To illustrate the assumption (4.66), consider the Gaussian covariance

(4.67) Rµ(z) = e−z2

2`2 .

The wavenumbers decrease monotonically in the index j, so assumption (4.66) holdsif the power spectral density

Rµ(β) =√

2π`e−β2`2

2 .

satisfiesRµ(β) ≈ 0, ∀β ≥ 2βN (ω).

Explicitly, we need that

(4.68) 2βN (ω) =2πD

√2ϑ(ω)N ≈ 2k

√2ϑ(ω)N

≥ 3`

where we recalled definition (3.16) of N and ϑ(ω). Thus, the forward scatteringapproximation is valid when the correlation length is larger than the wavelength

(4.69)k`

2π=`

λ&√N.

As before, the symbol & stands for larger or equal up to a multiplicative constant oforder one.

4.12. The asymptotic limit. — Gathering the results, we can approximate theacoustic pressure field as

(4.70) p

(t, y,

Z

ε2

)≈∫ ∞−∞

dω

2π

N∑j=1

φj(y)aεj(ω,Z)eiβj(ω) Zε2−iωt,


with mode amplitudes aεj(ω,Z)j=1,...,N converging in distribution as ε → 0 to acomplex valued diffusion Markov process

(4.71) aεj(ω,Z) αj(ω,Z)√βj(ω)

.

We write the limit process as

(4.72) αj(ω,Z) = P1/2j (ω,Z)eiθj(ω,Z),

with phases θj(ω, z), and scaled powers

(4.73) Pj(ω,Z) = |αj(ω,Z)|2, j = 1, . . . , N.

The scaling by√βj in (4.71) symmetrizes the constant coefficient matrices that ap-

pear in the infinitesimal generator G given by

(4.74) G = GP + Gθ.

The first term is a partial differential operator in the powers

(4.75) GP =N∑

j, l = 1

j 6= l

Γ(c)jl (ω)

[PlPj

(∂

∂Pj− ∂

∂Pl

)∂

∂Pj+ (Pl − Pj)

∂

∂Pj

],

and the second term is a partial differential operator in the phases

Gθ =14

N∑j, l = 1

j 6= l

Γ(c)jl (ω)

[PjPl

∂2

∂θ2l

+PlPj

∂2

∂θ2j

+ 2∂2

∂θj∂θl

]+

12

N∑j,l=1

Γjl(ω)∂2

∂θj∂θl

+12

N∑j, l = 1

j 6= l

Γ(s)jl (ω)

∂

∂θj+

N∑j=1

κj(ω)∂

∂θj.(4.76)

We denote henceforth multiple sums by multiple summation indexes. That is to sayN∑

j,l=1

≡N∑j=1

N∑l=1

.

Now let us describe the matrices of coefficients in the generator, which depend onthe covariance Rµ of the random process µ. The matrix Γ(c)(ω) is symmetric, withrows summing to zero

(4.77) Γ(c)jj (ω) = −

∑l 6=j

Γ(c)jl (ω) < 0,

and non-negative off-diagonal entries

(4.78) Γ(c)jl (ω) =

D2

4βj(ω)βl(ω)[φ′j(D)φ′l(D)

]2 Rµ[βj(ω)− βl(ω)], j 6= l.

28 LILIANA BORCEA

This is because Rµ(β) ≥ 0 for any β ∈ R, by Bochner’s theorem. The matrix Γ(ω) issimilar, but all its entries are positive and given by

(4.79) Γjl(ω) =D2


]2 Rµ(0) > 0.

The matrix Γ(s)(ω) has the off-diagonal entries

(4.80) Γ(s)jl (ω) =

D2


]2∫ ∞0

dz sin [(βj(ω)− βl(ω))z]Rµ(z)

which may be positive or negative, and its rows sum to zero

(4.81) Γ(s)jj (ω) = −

∑l 6=j

Γ(s)jl (ω).

All the terms in the generator except for the last one in (4.76) are due to the leadingorder term in the evolution equation for the amplitudes aεj , which is not affected bythe evanescent modes. The second order term in the evolution equations gives

(4.82) limT→∞

∫ T

0

dτ E[γaj,l(ω, µ(0), τ)

]= iκj(ω)δjl.

The matrix is diagonal because of the phase (βj − βl)τ , and after a straightforwardcalculation we obtain that the result is imaginary. Given the expression (4.49) of γ(a)

we see that

(4.83) κj(ω) = κ(a)j (ω) + κ

(e)j (ω),

with the first part due to the direct coupling of the propagating modes, and the seconddue to the coupling via the evanescent modes. We have

κ(a)j (ω) = E [cjj(ω, µ(0))]

= − 12βj(ω)

3(πj

D

)2

+1`2

∫ D

0

dy[y2φ′j(y) + yφj(y)

]φ′j(y)

,(4.84)

and κ(e)j follows similarly.

The asymptotic limit described in this section allows us to analyze imaging andtime reversal with remote arrays. Given the generator, we can compute the statis-tical moments of the wave field in order to understand the loss of coherence due toscattering, and the propagation of energy. Before we do so, we show next that theresult in waveguides with randomly fluctuating wave speed is very similar. In fact,the generator of the limit diffusion process is exactly the same. However, the coeffi-cients Γ(c)

jl , Γ(o)jl , Γ(s)

jl and κj are not the same. Thus, random boundaries and randominterior inhomogeneities have different scattering effects, as explained in section 6.


5. Waveguides with random internal inhomogeneities

Consider two dimensional random waveguides with interior inhomogeneities, butflat boundaries. We model the inhomogeneities as random fluctuations of the soundspeed, so that

(5.1)c2o

c2(y, z)= 1 + 1(0,Lε)(z) εν(y, z),

for y ∈ (0,D) and z ∈ R. The mean zero stationary random function ν(y, z) isbounded and mixing, with integrable covariance

(5.2) Rν(y, z) = E [ν(y + y′, z + z′)ν(y′, z′)] .

We normalize the covariance by Rν(0) = 1, and consider weak fluctuations withamplitude scaled by ε 1. The fluctuations are isotropic, with correlation length `,and we motivate their confinement to the range interval (0, Lε) with Lε = L/ε2 asbefore.

The Fourier transform p(ω, y, z) of the pressure field satisfies a similar equation to(4.12) for z ∈ (0, Lε),

(5.3) Lεp(ω, y, z) = 0, y ∈ (0,D),

with homogeneous Dirichlet boundary conditions

(5.4) p(ω, 0, z) = p(ω,D, z) = 0.

The partial differential operator Lε is the same to leading order, but it has a differentO(ε) part,

(5.5) Lε = ∂2z + ∂2

y + k2 + 1(0,Lε)(z) εk2ν(y, z).

The analysis is exactly as before. We decompose p in the waveguide modes, solvefor the evanescent part in terms of the propagating amplitudes, and obtain a closedsystem of equations that looks the same as (4.44). The entries in the matrices Υ(a),(b)

are of the form (4.47)-(4.48), but the random matrices Cjl have a different expression

(5.6) Cjl(ω, z) =k2

2βj(ω)βl(ω)

∫ D

0

dy ν(y, z)φj(y)φl(y).

The second order matrices γ(a),(b) are due only to the coupling via the evanescentmodes. The first is given by

γ(a)jl (ω, ν, z) = iei[βl(ω)−βj(ω)]z

∑l′>N

∫ ∞−∞ds e−βl′ |s|+iβl(ω)sCjl′(ω, z)Cll′(ω, z + s)

2βj(ω)βl(ω)βl′(ω),

and the second by

γ(b)jl (ω, ν, z) = −ie−i[βl(ω)−βj(ω)]z

∑l′>N

∫ ∞−∞ds e−βl′ |s|−iβl(ω)sCjl′(ω, z)Cll′(ω, z + s)

2βj(ω)βl(ω)βl′(ω).

30 LILIANA BORCEA

5.1. The asymptotic limit. — Since the coupled system of equations has the sameform as (4.44), we can write directly the asymptotic approximation of the acousticpressure

(5.7) p

(t, y,

Z

ε2

)≈∫ ∞−∞

dω

2π

N∑j=1

φj(y)aεj(ω,Z)eiβj(ω) Zε2−iωt.

Here we used the forward scattering approximation justified as before, based on theassumption that the correlation length is large with respect to the wavelength. Themode amplitudes aεj(ω,Z)j=1,...,N converge in distribution as ε → 0 to a complexvalued diffusion Markov process

(5.8) aεj(ω,Z) αj(ω,Z)√βj(ω)

, αj(ω,Z) = P1/2j (ω,Z)eiθj(ω,Z).

The infinitesimal generator is given by (4.74)-(4.76), but the matrices of coefficientsare different.

The symmetric matrix Γ(c)(ω) has the off-diagonal entries

Γ(c)jl (ω) =

k4

4βj(ω)βl(ω)

∫ ∞−∞dz cos[(βj(ω)− βl(ω))z]

×∫ D

0

dy

∫ D

0

dy′Rν(y − y′, z)φj(y)φl(y)φj(y′)φl(y′),(5.9)

and its rows sum to zero. Thus, the diagonal entries are given by

(5.10) Γ(c)jj (ω) = −

∑l 6=j

Γ(c)jl (ω) < 0.

Because Γ(c)jl are proportional to the power spectral densities of the stationary random

processes Cjl defined in (5.6), they are non-negative by Bochner’s theorem. Thesymmetric matrix Γ(ω) has the entries

Γjl(ω) =k4

4βj(ω)βl(ω)

∫ ∞−∞dz

∫ D

0

dy

∫ D

0

dy′Rν(y − y′, z)φ2j (y)φ2

l (y′),(5.11)

for all l, j = 1, . . . , N . Its diagonal Γjj is positive, because it is proportional to thepower spectral density of Cjj , evaluated at zero.

The matrix Γ(s) has the off-diagonal entries

Γ(s)jl (ω) =

k4

2βj(ω)βl(ω)

∫ ∞0

dz sin[(βj(ω)− βl(ω))z]

×∫ D

0

dy

∫ D

0

dy′Rν(y − y′, z)φj(y)φl(y)φj(y′)φl(y′),(5.12)


that may be positive or negative, and its rows sum to zero

(5.13) Γ(s)jj (ω) = −

∑l 6=j

Γ(s)jl (ω) .

Finally, the coefficient κj in the last term of the generator Gθ is due only to theevanescent modes,

(5.14) κj(ω) = κ(e)j (ω).

Its expression follows from the expectation of γ(a)(ω, ν, τ) averaged over the phase τ ,as in (4.82).

6. Net scattering effects

In this section we describe the statistics of the moment amplitudes, with emphasison the first and second moments. They allow us to understand the loss of coherenceand the mode decorrelation due to scattering. We also compare the results in the twotypes of waveguides: with boundary fluctuations and with interior inhomogeneities.But first, we make some general remarks.

Remark 1. — The coefficients of the partial derivatives in the powers Pj in thegenerator G depend only on Pjj=1,...,N . This means that the scaled mode powersβj |aεj |2j=1,...,N converge in distribution as ε → 0 to the diffusion Markov process|αj |2 = Pjj=1,...,N with generator GP given in (4.75).

Remark 2. — The evanescent waves influence only the coefficient κj in the lastterm of Gθ. Since they do not appear in GP , they do not change the energy of thepropagating modes in the limit ε → 0. Moreover, since κj is in the diagonal part ofGθ, they do not affect the mode coupling as ε→ 0. The only effect of the evanescentmodes is a mode and frequency dependent phase modulation.

Remark 3. — The symmetry of the matrix Γ(c) of coefficients in GP gives that

(6.1) GP( N∑j=1

Pj)

=N∑

j, l = 1

j 6= l

Γ(c)jl (ω)(Pl − Pj) = 0.

Thus, the energy of the limit diffusion process is conserved. The process αjj=1,...,N

is supported on the sphere in CN centered at zero and of radius

(6.2) rP (ω) =N∑j=1

|αj(ω, 0)|2 =N∑j=1

βj(ω)|aj,o(ω)|2 =N∑j=1

|Fj(ω)|2

4βj(ω).

Here we used the initial conditions (4.35). We will see later that if we let Z → ∞,the process becomes uniformly distributed over this sphere. This is the equipartitionlimit where the waves forget their initial state, and imaging becomes impossible.

32 LILIANA BORCEA

6.1. Single frequency moments. — The results stated in section 4.12 give aftera direct calculation that uses the expression of the generator G, that

(6.3) E[aεj(ω,Z)

] ε→0−→ E [αj(ω,Z)]√βj(ω)

= aj,o(ω)e

"Γ(c)jj

(ω)−Γjj(ω)

2

#Z+i

"Γ(s)jj

(ω)

2 +κj(ω)

#Z

,

where aj,o(ω) is the amplitude in the ideal waveguide, given by (3.12). Thus, themode amplitudes of the coherent (mean) pressure field E[p] are not the same as inthe ideal waveguide. They decay exponentially with Z, because

Γ(c)jj − Γjj < 0.

The mode powers satisfy

(6.4) E[|aεj(ω,Z)|2

] ε→0−→ E [Pj(ω,Z)]βj(ω)

=〈Pj〉 (ω,Z)βj(ω)

,

where 〈Pj〉 is the solution of the initial value problem

∂Z 〈Pj〉 (ω,Z) =N∑l=1

Γ(c)jl (ω) [〈Pl〉 (ω,Z)− 〈Pj〉 (ω,Z)] , Z > 0,

〈Pj〉 (ω, 0) = βj(ω)|aj,o(ω)|2, j = 1, . . . , N.

Because the rows of Γ(c) sum to zero, we can write the system in vector form

∂Z

〈P1〉 (ω,Z)...

〈PN 〉 (ω,Z)

= Γ(c)(ω)

〈P1〉 (ω,Z)...

〈PN 〉 (ω,Z)

and obtain that

(6.5)

〈P1〉 (ω,Z)...

〈PN 〉 (ω,Z)

= eΓ(c)(ω)Z

β1(ω)|a1,o(ω)|2...

βN (ω)|aN,o(ω)|2

.

We already know that Γ(c)jl ≥ 0, for j 6= l. If we assume in addition that they

are not zero, meaning that the power spectral density of the fluctuations evaluatedat βj − βl does not vanish, we can use the Perron-Frobenius theorem to describe thematrix exponential in (6.5). Because Γ(c)

jl is symmetric, it has real eigenvalues denotedby Λj(ω) and orthonormal eigenvectors uj , for j = 1, . . . , N . The Perron-Frobeniustheorem states that the largest eigenvalue is simple

ΛN (ω) ≤ . . . ≤ Λ2(ω) < Λ1(ω),


and the components of the leading eigenvector u1 have the same sign. In our case

Λ1(ω) = 0 and u1 =1√N

1...1

.

Thus, the solution of (6.5) is given by

(6.6)

〈P1〉 (ω,Z)...

〈PN 〉 (ω,Z)

=N∑j=1

eΛj(ω)Z ujuTj

β1(ω)|a1,o(ω)|2...

βN (ω)|aN,o(ω)|2

,

and as Z grows, all the terms in the sum, except the first one decay exponentially tozero. More explicitly,

(6.7) supj=1,...N

∣∣∣∣〈Pj〉 (ω,Z)− rP (ω)N

∣∣∣∣ ≤ O (e−|Λ2(ω)|Z),

so the mode powers converge to the uniform distribution on the sphere of radius rP (ω)defined in Remark 3.

Because E[αj ] decays exponentially with Z, its signal to noise ratio (SNR)

(6.8) SNRj(ω) =|E[αj(ω,Z)]|√

〈Pj〉 (ω,Z)− |E[αj(ω,Z)]|2∼ e−Z/Sj(ω)

decays on the scale

(6.9) Sj(ω) =2

Γjj(ω)− Γ(c)jj (ω)

.

We call Sj(ω) the scattering mean free path. It is the range scale over which the modeloses its coherence, meaning that its mean is dominated by the random fluctuations.

To quantify the fluctuations of the mode powers, we need the fourth order moments

(6.10) 〈Pjl〉 (ω, z) = E [Pj(ω,Z)Pl(ω,Z)] .

They satisfy the differential equations

(6.11) ∂Z 〈Pjj〉 (ω, z) = −2Γ(c)jj (ω) 〈Pjj〉 (ω, z) + 4

N∑n=1

Γ(c)jn (ω) 〈Pjn〉 (ω, z),

for j = l = 1, . . . , N and

∂Z 〈Pjl〉 (ω, z) = −2Γ(c)jl (ω) 〈Pjl〉 (ω, z)

+N∑n=1

[Γ(c)jn (ω) 〈Pnl〉 (ω, z) + Γ(c)

ln (ω) 〈Pjn〉 (ω, z)],(6.12)

34 LILIANA BORCEA

for j 6= l, and the initial conditions

(6.13) 〈Pjl〉 (ω, 0) = βj(ω)βl(ω)|aj,o(ω)al,o(ω)|2, j, l = 1, . . . , N.

This linear system can be put in matrix form, and analyzed using the Perron-Frobenius theorem, as before. Again, the simple leading eigenvalue is zero, so as Zgrows, the solution tends to the stationary one, lying in the span of the leading eigen-vector. The components of this normalized eigenvector are (1 + δjl)/

√N(N + 1), so

we obtain that

(6.14) 〈Pjl〉 (ω,Z) Z→∞−→

r2P

(ω)

N(N+1) , if j 6= l,2r2P

(ω)

N(N+1) , if j = l.

The variance of the mode powers follows from (6.14) and (6.7)

(6.15) var(Pj) = 〈Pjj〉 (ω,Z)− [〈Pj〉 (ω,Z)]2 Z→∞−→r2P

(ω)(N − 1)N2(N + 1)

≈r2P

(ω)N2

,

where the approximation is for N 1. Thus, the standard deviation [var(Pj)]1/2 ofthe powers is approximately the same as their mean 〈Pj〉. The fluctuations do notoverwhelm the coherent (mean) part, but they are significant. The mode powers arenot deterministic. In fact, it is shown in [15, Section 20.3.4] that in the limit Z →∞and for N 1, they are approximately exponential random variables, with meanrP(ω)/N .

We also get from (6.14) and (6.7) that the covariance satisfies

(6.16) cov(Pj ,Pl) = 〈Pjl〉 (ω,Z)− 〈Pj〉 (ω,Z) 〈Pl〉 (ω,Z) Z→∞−→−r2P

(ω)N2(N + 1)

for j 6= l. The correlation of the mode powers is

(6.17) corr(Pj ,Pl) =cov(Pj ,Pl)√

var(Pj)var(Pl)Z→∞−→ −1

N − 1,

and if N 1, we see that the modes are essentially uncorrelated at long ranges.

Remark 4. — We can also state the results in terms of the limit propagatorP(α)(ω,Z), the matrix in CN×N that maps the initial conditions

(6.18) αj(ω, 0) =√βj(ω)aj,o(ω) =

Fj(ω)2i√βj(ω)

to the range dependent limit amplitudes

(6.19) αj(ω,Z) =N∑l=1

P(α)jl (ω,Z)αl(ω, 0), Z > 0.


We have the convergence in distribution

(6.20) Pε(a)(ω,Z) ε→0 B−

12 (ω)Pα(ω,Z)B

12 (ω),

where B is the diagonal matrix

(6.21) B(ω) = diag (β1(ω), . . . , βN (ω)) .

6.2. Multi frequency moments. — Because the source emits a signal of band-width B, we need multi frequency moments of the propagator Pε,(a)(ω,Z), to describethe wave field in the limit ε→ 0. The calculation of these moments amounts to using(4.61) to derive evolution equations for the products

m∏j=1

Pε,(a)(ωj , Z), for m > 1,

and then applying the diffusion approximation theorem to get their limit as ε → 0.We refer to [15, Chapters 6,8,20] for details of the calculation.

For the analysis of imaging and time reversal, we need the auto-correlation ofPε,(a)(ω,Z) at two different frequencies, given in [15, Proposition 20.7]. The propa-gator decorrelates at different frequencies, meaning that

(6.22) E[Pε,(a)jl (ω,Z)Pε,(a)

j′l′ (ω′, Z)]ε→0−→ E

[P(α)jl (ω,Z)

]E[P(α)j′l′(ω

′, Z)], ω 6= ω′.

The mean propagator is

(6.23) E[P(α)jl (ω,Z)

]= δjl e

− ZSj(ω) +i

ZAζj(ω) ,

where we let

(6.24) ζj(ω) =

[Γ(s)jj (ω)

2+ κj(ω)

]−1

.

There is correlation over frequency offsets of order ε2, and the second moments are

E[Pε,(a)jl (ω,Z)Pε,(a)

j′l′ (ω − ε2h, Z)]ε→0−→ δjj′δll′

βl(ω)βj(ω)

W(l)j (ω, h, Z)e−ihβ

′j(ω)Z

+(1− δjj′)δjlδj′l′eKjj′ (ω)Z .(6.25)

Here we introduced the coefficients

Kjj′(ω) =Γ(c)jj (ω) + Γ(c)

j′j′(ω)− Γjj(ω)− Γj′j′(ω) + 2Γjj′(ω)2

+ i

[1

ζj(ω)− 1ζj′(ω)

],

(6.26)

and we let

(6.27) W(l)j (ω, h, Z) =

∫ ∞−∞

dτW(l)j (ω, τ, Z)eihτ

36 LILIANA BORCEA

be the Fourier transform of the mean Wigner distribution W(l)j (ω, τ, Z). It solves the

transport equations

[∂Z + β′j(ω)∂τ

]W(l)j (ω, τ, Z) =

N∑n = 1

n 6= j

Γ(c)jn (ω)

[W(l)n (ω, τ, Z)−W(l)

j (ω, τ, Z)]

(6.28)

for Z > 0, with initial conditions

(6.29) W(l)j (ω, τ, 0) = δjlδ(τ).

To compare with the results of the previous section, let us rewrite the secondmoments in terms of the amplitudes

E[aεj(ω,Z)aj′ε(ω − ε2h, Z)

] ε→0−→ δjj′N∑l=1

W(l)j (ω, h, Z)e−ihβ

′j(ω)Z βl(ω)

βj(ω)|al,o(ω)|2

+(1− δjj′)eKjj′ (ω)Zaj,o(ω)aj′,o(ω).(6.30)

We note that when j 6= j′ the right hand side decays exponentially with Z, becausethe real part of Kjj′ is negative. The covariance is

cov[aεj , a

εj′]

= E[aεj(ω,Z)aj′ε(ω − ε2h, Z)

]− E

[aεj(ω,Z)

]E [aj′ε(ω,Z)]

ε→0−→ E[αj(ω,Z)]E[αj′(ω,Z)]√βj(ω)βj′(ω)

[e2Γjj′ (ω)Z − 1

],(6.31)

with expectations given by (6.3), and the correlation of aεj and aεj′ decays exponentiallywith Z, meaning that the mode amplitudes decorrelate at long ranges. We alreadysaw in the previous section that the mode powers decorrelate in the limit Z →∞.

In the case j = j′, and for a frequency offset h = 0, we recover the mean powerresult (6.4), with

(6.32) 〈Pj〉 (ω,Z) =N∑l=1

W(l)j (ω, 0, Z)|αl(ω, 0)|2,

and αl(ω, 0) given by (6.18). For a non-zero frequency offset, W(l)j (ω, h, Z) satisfies

the linear system of differential equations

(6.33)[∂Z − ihβ′j(ω)

]W(l)j (ω, h, Z) =

N∑n=1

Γ(c)jn (ω)W(l)

n (ω, h, Z), Z > 0,

with initial conditions

(6.34) W(l)j (ω, h, 0) = δjl.


The solution is given by the exponential of the matrix ihB′ + Γ(c),

(6.35) W(l)j (ω, h, Z) = ej · exp

[ihB′ + Γ(c)

]Z

el,

where B′ is the diagonal matrix with entries β′j and ejj=1,...,N is the canonical basis

in RN . We can write W(l)j as the sum of the coherent part, which decays exponentially

with Z, and a remainder

(6.36) W(l)j (ω, h, Z) = δjle

ihβ′l(ω)Z+Γ(c)ll (ω)Z + W

(l)j (ω, h, Z).

The remainder satisfies the homogeneous initial conditions

(6.37) W(l)j (ω, h, 0) = 0,

and evolution equations driven by the coherent part

[∂Z − ihβ′j(ω)

]W

(l)j (ω, h, Z) =

N∑n=1

Γ(c)jn (ω)W (l)

n (ω, h, Z)

+Γ(c)jl (ω)eihβ

′l(ω)Z+Γ

(c)ll (ω)Z .(6.38)

Equivalently, in the time domain, the Wigner transform is given by

(6.39) W(l)j (ω, τ, Z) = δjle

Γ(c)ll (ω)Zδ (τ − β′l(ω)Z) +W

(l)j (ω, τ, Z),

where the coherent terms propagates the singularity along the characteristic τ = β′lZ,but its weight decays exponentially in Z. The remainder W (l)

j is a continuous densitythat does not decay exponentially in range, so using equation (6.30) we obtain that

(6.40) limε→0

E[aεj(ω,Z)ajε(ω − ε2h, Z)

]≈ e−ihβ

′j(ω)Z

N∑l=1

W(l)j (ω, h, Z)

βl(ω)|al,o(ω)|2

βj(ω)

for large Z.

6.3. Pulse propagation. — We described in section 3.2 the signal received at thearray at range ZA/ε2, in homogeneous waveguides. Now we look at the signal receivedin random waveguides.

We obtain from equation (4.70) that

p

(t, y,

ZAε2

)≈

N∑j,l=1

φj(y)∫ D

0

dy′

∆yφl(y′)ρ

(y′ − y?

∆y

)

×∫ ∞−∞

dω

2πBf

(ω − ωoB

)Pε,(a)jl (ω,ZA)eiβj(ω)

ZAε2−iωt,(6.41)

where we used the initial conditions (4.35) and the source model (2.12). We analyzethe signal (6.41) in the narrowband and broadband regimes defined by equations(2.16) and (2.19).

38 LILIANA BORCEA

In the narrowband case we let

ω = ωo + ε2w

and evaluate equation (6.42) at time t = T/ε2. We obtain that

(6.42) p

(t =

T

ε2, y,

ZAε2

)≈

N∑j,l=1

φj(y) fεjl(T )∫ D

0

dy′

∆yφl(y′)ρ

(y′ − y?

∆y

),

where the signals

(6.43) fεjl(T ) =ei[βj(ωo)ZA−ωoT ]/ε2

2iβj(ωo)Fεjl(T )

have the random factor

(6.44) Fεjl(T ) =∫ ∞−∞

dw

2πΩBf

(w

ΩB

)Pε,(a)jl (ωo + ε2w,ZA)eiw[β′j(ωo)ZA−T ].

If the kernel Pε,(a) where the identity matrix, we would recover the result in the idealwaveguide. In our case the kernel is random, with moments described by equations(6.23) and (6.25). The mean of Fεjl satisfies

(6.45) E[Fεjl(T )

] ε→0−→ δjl f[ΩB(T − β′j(ωo)ZA

)]e− ZASj(ωo) +i

ZAζj(ωo) ,

so the coherent signal E[p] is similar to that in the ideal waveguide, except that thecontribution of each mode is exponentially damped on the range scale of the scatteringmean free path Sj(ωo). There is also a mode dependent phase modulation. It isobvious from the second moments (6.25) that the standard deviation of p exceedsits mean for large ZA, meaning that the random fluctuations of the narrowbandsignals received at the array overwhelm its coherent part. The array data is basicallyincoherent at long ranges.

In the broadband regime we change variables in (6.41) as

ω = ωo + εw,

and obtain

(6.46) p

(t =

T

ε2, y,

ZAε2

)≈

N∑j,l=1

φj(y) fεjl(T )∫ D

0

dy′

∆yφl(y′)ρ

(y′ − y?

∆y

).

Here fεjl(T ) has the same form as in (6.43), but the random factor is

Fεjl(T ) =∫ ∞−∞

dw

2πΩBf

(w

ΩB

)dj

(w

ΩB

)Pε,(a)jl (ωo + εw,ZA)eiw[β′j(ωo)ZA−T ]/ε,

(6.47)


with dispersion kernel dj defined in equation (3.24). We have a fast phase in (6.47),which vanishes when T equals the travel time of the j-th mode

(6.48) T = Tj = β′j(ωo)ZA.

Let us observe the signal in a time window centered at Tj and of width of order ε,similar to that of the emitted pulse. Explicitly, let

(6.49) T = Tj + εs,

and obtain from (6.47) that

Fεjl(Tj + εs) =∫ ∞−∞

dw

2πΩBf

(w

ΩB

)dj

(w

ΩB

)Pε,(a)jl (ωo + εw,ZA)e−iws.(6.50)

We can compute the mean and variance of this signal as we did above, but note thefollowing. Because the propagator decorrelates rapidly, over frequency offsets of orderε2, the right hand side in (6.50) is essentially the superposition of uncorrelated randomvariables, and we expect it to be close to its mean by the law of large numbers. Werefer to [15, Section 20.4.3] for details on the proof and cite the result from there.

As ε→ 0, we have the convergence in distribution

Fεjl(Tj + εs) δjl eΓ(c)jj

(ωo)ZA2 +i

ZAζj(ωo) +iBj(ZA)

f ? dj (ΩB s) ,(6.51)

where Bj is Brownian motion with variance

(6.52) E [Bj(ZA)] = Γjj(ωo)ZA.

This is a pulse stabilization result, because it says that when we observe the wavearound the expected travel time Tj , in a time window comparable to the emitted pulsewidth, it is deterministic except for the Brownian motion in the phase. Furthermore,the pulse shape which is affected by dispersion is the same as in the ideal waveguide.The essential difference is that the pulse amplitude decays exponentially in ZA, onthe scale

SPj =2

|Γ(c)jj (ωo)|

.

As the waves travel in the waveguide, the energy is transferred from the coherentpart which arrives first, to the incoherent part that arrives after it over a longer timeinterval than the pulse width. See Figure 4 for an illustration.

6.4. Comparative study of the net scattering effects. — We have now seenthat the coherent field decays exponentially with range on the mode dependent scaleSj(ωo), the scattering mean free path. Definition (6.9) shows that 1/Sj is given bythe sum of two terms

(6.53)1

SPj (ωo):= −

Γ(c)jj (ωo)

2

40 LILIANA BORCEA

and

(6.54)1

Sθj (ωo):=

Γjj(ωo)2

.

The first term (6.53) is defined by the power diffusion coefficients in the generatorGP , and we denote by SPj the range scale over which the modes exchange energy. Weknow that each mode is associated to a direction of propagation of a plane wave, sowe call Sθj the transport mean free path. Classically, this is defined as the distancebeyond which the wave loses its initial direction due to scattering [26].

The second term (6.54) is defined by the phase diffusion coefficients in the generatorGθ. They are also the variances of the Brownian motions Bj appearing in the phaseof the coherent front (6.51). We denote by Sθj the range scale over which the randomphase of the mode amplitudes become significant, thus giving the exponential dampingof their expectation. For example, when taking the expectation of (6.51), we get

E[eiBj(ZA)

]=∫ ∞∞

du√2πΓjj(ωo)ZA

exp[iu− u2

2Γjj(ωo)ZA

]= e−Γjj(ωo)ZA/2.

We wish to compare the scattering and transport mean free paths in waveguideswith random boundaries vs. waveguides with internal inhomogeneities. These scalesare different because the diffusion matrices Γ and Γ(c) have different expressions, givenby (4.78)-(4.79) and (5.9)-(5.11), respectively. The details of the estimation can befound in [1]. We recall here the results for the scaling regime

(6.55)√N .

`

λo N.

The first inequality is the assumption (4.69) made in the forward scattering approxi-mation. The second inequality says that the wavelength is large in comparison withthe waveguide depth D, so that there are many propagating modes.

In both types of waveguides Sj decreases monotonically with the mode index. Thisis intuitive, because the high order modes bounce more often at the boundary andthus take long paths from the source to the array. The cumulative scattering effectsbuild up over the longer paths, and the damping rate of the expectation of the modeamplitudes is larger.

In the waveguides with random boundary, and for the first modes, we have

(6.56) SP1 (ωo) ∼ S1(ω) ∼ D

√`

λo& D√N,

where the symbol ∼ indicates that the scales are of the same order. For the lastmodes the scattering mean free path satisfies

(6.57) SN (ωo) ∼ DλoN2`

. DN−5/2,


but the transport mean free path has a different behavior. It is given by

(6.58) SPN (ωo) ∼ D`

N3λo& SN (ωo),

when` ∼√Nλo λo,

and it is much longer than the scattering mean free path in high frequency regimes

(6.59) SPN (ωo) SN (ωo), `√Nλo.

This is particular to the waveguides with random boundaries, and it can be understoodas follows. When ` λo, the high order modes strike the boundary many timesover a range of the order of the correlation length, at almost the same angle ofincidence. At each strike, the amplitude gains a random phase, and because the phasesare statistically correlated they accumulate, thus giving a large diffusion coefficientΓjj , for j ∼ N . The deviation of the direction of the waves due to the boundaryfluctuations is typically small from one strike to another, and there is less energyexchanged between the high order modes. This is why SPj Sj for j ∼ N .

In the waveguides with interior inhomogeneities, and for the first modes we have

(6.60) SP1 (ωo) ∼ S1(ωo) ∼ DλoN`

. DN−3/2,

and for the last modes

(6.61) SPN (ωo) ∼ SN (ωo) ∼ DλoN2`

. DN−5/2.

Thus, we see that the first modes are much more affected by the interior inhomo-geneities than the random boundaries. This is important for imaging, because it saysthat when the boundary scattering effects dominate, we can use first mode pass filtersto obtain reliable results up to long ranges, satisfying

D ZA . D√N.

Such filters are useless in waveguides with interior inhomogeneities, because even thefirst modes lose their coherence at ranges

ZA . DN−3/2 D.

The high order modes have the same scattering mean free path in both types ofwaveguides. However, the transport mean free path is different. The main mechanismfor the loss of coherence in the waveguides with random inhomogeneities is due tothe exchange of energy between the modes. The scattering and transport mean freepaths are similar for all the modes in such waveguides.

It remains to compare the equipartition distance

(6.62) SE(ωo) :=1

|Λ2(ω)|,

42 LILIANA BORCEA

defined in terms of the second eigenvalue Λ2 of the matrix Γ(c). As shown in equation(6.7), it is the range scale over which the power becomes uniformly distributed overthe modes, independent of the initial excitation. This means that with arrays atranges ZA > SE it is not possible to determine the cross-range location of the source.

We do not have a theoretical estimate of the equipartition distance, but we calcu-lated it numerically in [1] for the case of Gaussian autocorrelations Rµ and Rν . Theresult is that in waveguides with random boundaries

(6.63) SE(ωo) ∼ S1(ωo),

meaning that once all the modes have lost coherence, it is no longer possible todetermine the cross-range of the source, no matter what imaging method we use.

In waveguides with random inhomogeneities we get

(6.64) SE(ωo) S1(ωo).

This shows that once the modes lose their coherence, it is useful to use incoherentimaging methods to estimate source locations at ranges ZA ∈ (S1,SE).

7. Model of the array data

We denote by pA(t, y) the pressure wave measured at the sensors of the arraysupported in the interval

A = [ym, yM ] ⊂ [0,D],

at range

zA =ZAε2.

The length |A| = yM − ym of the interval is called the aperture. The array has fullaperture when |A| = D, and partial aperture when |A| < D. We assume for conve-nience that the sensors are spaced sufficiently close together to make the continuumarray approximation. This means that when computing sums over the locations ofthe sensors we can replace them by integrals over y ∈ A.

The data is

(7.1) pA(t, y) = 1A(y)χ(ε2t

T

)p

(t, y,

ZAε2

),

with p given by (4.70). The recordings are over a finite time interval modeled with thewindow function χ of dimensionless arguments and support of order one. The scalingby T/ε2 of the argument of χ reflects that the sensors record over a sufficiently longinterval so that at least some of the modes reach the array. We allow an arbitraryshape of the window, and we normalize it by

(7.2) χ(0) = 1.


The Fourier transform of the data is

(7.3) pA(ω, y) = 1A(y)∫ ∞−∞

du

2πχ(u) p

(ω − ε2u

T, y,

ZAε2

),

and from (4.70), with the mode amplitudes written in terms of the propagator,

(7.4) p

(ω, y,

ZAε2

)≈

N∑j,l=1

φj(y)Pε,(a)jl (ω,ZA) al,o(ω)eiβj(ω)

ZAε2 .

The initial amplitudes are

(7.5) al,o(ω) =1

2iβl(ω)

∫ D

0

dy′F (ω, y′)φl(y′),

and with the source model (2.12) we have

(7.6) F (ω, y) =1

∆yρ

(y − y?

∆y

)1Bf

(ω − ωoB

).

Gathering the results,

pA(ω, y) ≈ 1A(y)∫ D

0

dy′

∆yρ

(y′ − y?

∆y

) N∑j,l=1

φl(y′)φj(y)2iβl(ωo)

∫ ∞−∞

du

2πχ(u) e−iβ

′j(ωo)

uZAT

× 1Bf

(ω − ωoB

− ε2u

BT

)Pε,(a)jl

(ω − ε2u

T, ZA

)eiβj(ω)

ZAε2 ,(7.7)

where the approximate sign means equality up to a residual term that vanishes in thelimit ε → 0. Since we consider bandwidths that are much smaller than ωo, we canapproximate the wavenumber in the denominator by its value at the central frequency.

If we had an ideal waveguide, the propagator would be just the identity matrix,and (7.4) would simplify as

(7.8) po

(ω, y,

ZAε2

)≈

N∑j=1

φj(y)aj,o(ω)eiβj(ω)ZAε2 .

Here the approximation is because we neglect the evanescent waves.

8. Time reversal

Time reversal is an experiment that uses an active array of sensors to record thedata pA and then re-emit its time reversed version in the same medium. The waves areexpected to refocus back at the source by the time reversibility of the wave equation.Here we study mathematically the time reversal process in order to understand therole played by cumulative scattering in random waveguides.

44 LILIANA BORCEA

The time reversed recordings are

(8.1) pTRA (t, y) = pA

(T

ε2− t, y

),

and in the Fourier domain

(8.2) pTRA (ω, y) = pA(ω, y)eiω

Tε2 ,

where we recall that the bar denotes complex conjugation. The array emits the signalpTRA and we observe the resulting wave field denoted by O(t, y, Z/ε2). Its Fourier

transform is similar to (7.4), with the source F (ω, y) replaced by (8.2). The wavesare moving backward, so we should be careful with the propagator.

Let us denote by Pε,(a)(ω,Zo, Z) the propagator between the ranges Zo/ε2 andZ/ε2, with Z > Zo. Up to now Zo was zero, and we suppressed it in the arguments.We have

aε(ω,Z) = Pε,(a)(ω,Z, Zo)aε(ω,Zo),

and because the energy is conserved in the limit ε→ 0, the propagator must satisfy

[Pε,(a)]∗Pε,(a) ≈ I.

Here the index ? denotes the conjugate transpose, and the approximation means thatthere is a residual that tends to zero as ε→ 0. Therefore we obtain that

aε(ω,Zo) ≈[Pε,(a)(ω,Z2, Z1)

]?aε(ω,Z).

We need the backward going waves. Recalling the structure (4.58) of Pε, in particularits second block on the diagonal, we see that the backpropagation should be donewith the complex conjugate of the matrix above, which is the transpose of Pε,(a).

It remains to calculate the amplitudes of the backward going emerging wave at thearray. We decompose the wave in forward and backward going parts as in section4.4, and then impose the source condition that is at now at the array range ZA/ε2

instead of zero. We find the amplitudes

bj(ω,ZA) =eiβj(ω)

ZAε2

2iβj(ω)

∫ D

0

dy pTRA (ω, y)φj(y),

and their propagation to range Z < ZA is

bj(ω,Z) =N∑l=1

Pε,(a)lj (ω,ZA, Z)bl(ω,ZA).

The observed wave field is given by

(8.3) O(t, y,

Z

ε2

)≈

N∑j=1

φj(y)∫ ∞−∞

dω

2πbj(ω,Z)e−iβj(ω) Z

ε2−iωt,

and more explicitly


O(t, y,

Z

ε2

)≈∫ D

0

dy′

∆yρ

(y′ − y?

∆y

) N∑j,l,j′,l′=1

Mj′lφl′(y′)φj(y)

4βl(ωo)βl′(ωo)

×∫ ∞−∞

du

2πχ (u) eiuβ

′j′ (ωo)

ZAT

∫ ∞−∞

dω

2πBf

(ω − ωoB

− ε2u

BT

)eiω( T

ε2−t)

× Pε,(a)lj (ω,ZA − Z)Pε,(a)

j′l′

(ω − ε2u

T, ZA

)ei[βl(ω)−βj′ (ω)]

ZAε2−iβj(ω) Z

ε2 .(8.4)

Here we denote by Mjl the matrix that couples the modes for arrays with partialapertures

(8.5) Mjl =∫ D

0

dy 1A(y)φj(y)φl(y).

It equals the identity for arrays with full aperture.The time reversal function is defined by (8.4) observed at time t = T/ε2 and at

the range of the source,

(8.6) J TR(y) = O(T

ε2, y, 0

).

We write it as

(8.7) J TR(y) ≈∫ D

0

dy′

∆yρ

(y′ − y?

∆y

)ΨTR(y′, y),

in terms of the point spread kernel ΨTR. The model of the kernel is

ΨTR(y′, y) =N∑

j,l,j′,l′=1


4βl(ωo)βl′(ωo)

∫ ∞−∞

du

2πχ (u) eiuβ

′j′ (ωo)

ZAT

×∫ ∞−∞

dω

2πBf

(ω − ωoB

− ε2u

BT

)ei[βl(ω)−βj′ (ω)]

ZAε2

× Pε,(a)lj (ω,ZA)Pε,(a)

j′l′

(ω − ε2u

T, ZA

).(8.8)

Ideally, it should be the Dirac distribution δ(y′ − y), but we cannot achieve it. Ourgoal is to understand the behavior of the kernel, and quantify its focusing aroundthe expected peak at y′ = y. We present below the resolution analysis of ΨTR forbroadband sources. The narrowband case is treated in detail in [15, Section 20.9].

8.1. Resolution analysis for broadband signals. — In the broadband case, wecan change variables in the frequency integral in (8.8)

ω = ωo + εw, |w| ≤ πΩB,

and assuming a differentiable f , we obtain the point spread function

46 LILIANA BORCEA

ΨTR(y′, y) ≈N∑

j,l,j′,l′=1


4βl(ωo)βl′(ωo)ei[βl(ωo)−βj′ (ωo)]

ZAε2

×∫ ∞−∞

dw

2πΩBf

(w

ΩB

)dj

(w

ΩB

)dj′

(w

ΩB

)eiw[β′l(ωo)−β′

j′ (ωo)]ZAε

×∫ ∞−∞

du

2πχ (u) eiβ

′j′ (ωo)

uZAT Pε,(a)

lj (ωo + εw,ZA)Pε,(a)j′l′

(ωo + εw − ε2u

T, ZA

),(8.9)

where dj is the dispersion kernel (3.24). To analyze resolution, we need the meanand variance of ΨTR. The variance is small, of order ε2, because the decorrelationfrequency is of order ε2. Thus, the point spread function is close to its mean in theneighborhood of its peak,

ΨTR(y′, y) ≈ E[ΨTR(y′, y)].

We say that the time reversal process is statistically stable, because its point spreadfunction is essentially deterministic.

The cross-range resolution ry is the offset |y′−y| that bounds the essential supportof E[ΨTR]. That is to say, E[ΨTR(y, y′)] is large when |y′ − y| ≤ ry. In practice, bylarge we mean that E[ΨTR] is greater than the standard deviation of the fluctuationsby a user defined threshold factor.

Intuitively, we expect that the best point spread function is given by full aperturearrays. This is because such arrays capture all the propagating waves that arrivein the time measurement window. We will see that this is indeed the case in idealwaveguides. Surprisingly, in random waveguides we can achieve the same resolutioneven with small apertures. This is the super resolution property of time reversal.

To highlight the effect of scattering on the time reversal process, we compareE[ΨTR] with the point spread function ΨTR

o in ideal waveguides. It is obtained from(8.9) by replacing the propagators with the identity matrix. Explicitly, we have

ΨTRo (y′, y) ≈

N∑j,j′=1

Mj′jφj′(y′)φj(y)

4βj(ωo)βj′(ωo)χ

(β′j′(ωo)ZA

T

)ei[βj(ωo)−βj′ (ωo)]

ZAε2

×∫ ∞−∞

dw

2πΩBf

(w

ΩB

)dj

(w

ΩB

)dj′

(w

ΩB

)eiw[β′j(ωo)−β′

j′ (ωo)]ZAε .(8.10)


The expressions (8.4) and (8.10) simplify in the full aperture case, where Mjl = δjl.We have

ΨTRF (y′, y) ≈

N∑j,l,l′=1

φl′(y′)φj(y)4βl(ωo)βl′(ωo)

∫ ∞−∞

dw

2πΩBf

(w

ΩB

)

×∫ ∞−∞

du

2πχ (u) eiβ

′l(ωo)

uZAT Pε,(a)

lj (ωo + εw,ZA)Pε,(a)ll′

(ωo + εw − ε2u

T, ZA

)(8.11)

in random waveguides, and

ΨTRo,F(y′, y) ≈ f(0)

4

N∑j=1

φj(y′)φj(y)β2j (ωo)

χ

(β′j(ωo)ZA

T

).(8.12)

in ideal waveguides. The index F reminds us that we have full aperture arrays.

8.1.1. The mean point spread function. — We use the second moments (6.25) toanalyze the expectation of ΨTR. We obtain after straightforward calculations that itsis given by the sum of two terms

(8.13) E[ΨTR(y′, y)

]≈ 〈Ψ1〉 (y′, y) + 〈Ψ2〉 (y′, y).

The first term comes from the second moment E[Pε,(a)lj Pε,(a)

lj ], given by the Wignertransform,

〈Ψ1〉 (y′, y) =f(0)

4

N∑j,l=1

Mllφj(y′)φj(y)β2l (ωo)

∫ ∞−∞

du

2πχ (u)W(j)

l

(ωo,

u

T, ZA

).(8.14)

The dispersion kernels disappear because |dj |2 = 1. The second term is due to the

moments E[Pε,(a)jj Pε,(a)

j′j′ ],

〈Ψ2〉 (y′, y) =N∑

j, j′ = 1

j 6= j′

Mj′jφj′(y′)φj(y)

4βj(ωo)βj′(ωo)ei[βj(ωo)−βj′ (ωo)]

ZAε2−Kjj′ (ωo)ZA

× [f ? dj ? dj′ ]( [β′j′(ωo)− β′j(ωo)]ZA

ε

)χ

(β′j′ZA

T

),(8.15)

and it does not contribute in (8.13), because the pulse has compact support.Recalling the expression (6.36) of the Wigner transform, with continuous density

W(j)l , we obtain that∫ ∞

−∞

du

2πχ (u) W(j)

l

(ω,u

T, ZA

)= δjl χ

(β′j(ωo)ZA

T

)e− 2ZASPj

(ωo)

+∫ ∞−∞

dτ χ( τT

)W

(j)l (ωo, τ, ZA) .(8.16)

48 LILIANA BORCEA

Here we used the definition of the transport mean free path SPj and rewrote theintegral over u as a convolution of the Wigner density and the time reversed windowχ. The expectation of the point spread function becomes

E[ΨTR(y′, y)

]≈ f(0)

4

N∑j=1

Mjjφj(y′)φj(y)β2j (ωo)

e− 2ZASPj

(ωo)χ

(β′j(ωo)ZA

T

)+

f(0)4

N∑j,l=1

Mllφj(y′)φj(y)β2l (ωo)

∫ ∞−∞

dτ χ( τT

)W

(j)l (ωo, τ, ZA) .(8.17)

The first term sums the product of the eigenfunctions with positive weights that decaywith the mode index. The weights are positive because

(8.18) Mjj =∫ D

0

dy 1A(y)φ2j (y) > 0,

and they decay because the transport mean free path SPj decreases monotonicallywith the index j. All the weights decrease exponentially in ZA, so at long rangesZA > SP1 , the mean point spread function is given by the second term in (8.17).

It is impossible to calculate analytically the integral of the Wigner density over therecording window. But if we let ZA grow, the density becomes independent of j andl, and (8.17) simplifies to

(8.19) E[ΨTR(y′, y)

]∼

NT∑j=1

φj(y)φj′(y), ZA > SE(ωo).

Here we let NT ≤ N be the number of modes for which W(j)l peaks in the recording

window, and use ∼ to denote approximate, up to a multiplicative constant. We referto [15, Section 20.6.2] for the analysis of W (l)

j at large ranges. It is shown there that

W(l)j (ωo, τ, ZA) ZAS

E

∼ 1N√

2πσ2ZAexp

[− (τ − 〈β′〉ZA)2

2σ2ZA

]where 〈β′〉 is the average group speed

〈β′〉 =1N

N∑j=1

β′j(ωo),

and σ2 is similar to

σ2 ∼ 2SE(ωo)N

N∑j=1

[β′j(ωo)− 〈β′〉

]2.

Thus, all the modes contribute in the sum if

T > 〈β′〉ZA + 3√σ2ZA.


Figure 2. Plot of the function sinc(t) for t ∈ [−6π, 6π]. The argument tis scaled by π in the abscissa.

To see what this means, let us suppose that NT = N 1, and use the definition(3.3) of the eigenfunctions and the approximation

π

D≈ koN.

We can estimate the sum in the right hand side of (8.19) asN∑j=1

φj(y)φj′(y) =2D

N∑j=1

sin(πjy

D

)sin(πjy′

D

)

≈ koN

N∑j=1

sin(kojy

N

)sin(kojy

′

N

)

≈ ko∫ 1

0

du sin(kouy) sin(kouy′)

=ko2sinc [ko(y − y′)]− sinc [ko(y + y′)](8.20)

The sinc function is equal to one when its argument is zero and it decays away fromit, as illustrated in Figure 2. Because y and y′ are positive, we can neglect the secondterm in (8.20) and conclude from (8.19) that

(8.21) E[ΨTR(y′, y)

]∼ ko

2sinc [ko(y − y′)] , ZA > SE .

The kernel is large for

(8.22) |y − y′| ≤ ry :=π

ko=λo2,

and the resolution is the ideal Abbe diffraction limit independent of the aperture ofthe array!

50 LILIANA BORCEA

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y’−y0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y’−y0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y’−y

Figure 3. The point spread functions ΨTRo (y, y′) (in blue) and ΨTR(y, y′)

(in red). From left to right: full aperture and NT = N ; full aperture andNT = N/2; 5% aperture and NT = N/2. Each plot is normalized to havea maximum equal to one.

8.1.2. The point spread function in ideal waveguides. — Because of the large phasesin the integral of the expression (8.10) of the point spread function, we see that onlythe terms j′ = j contribute. We obtain that

(8.23) ΨTRo (y′, y) ≈ f(0)

4

N∑j=1

Mjjφj(y)φj(y′)β2j (ωo)

χ

(β′j(ωo)ZA

T

),

and even simpler,

(8.24) ΨTRo (y′, y) ∼

NT∑j=1

Mjjφj(y)φj(y′)β2j (ωo)

,

where NT is the number of modes that arrive in the recording window.To illustrate the difference between ΨTR

o and E[ΨTR], we plot in Figure 3 the righthand sides of equations (8.24) and (8.19), for a waveguide with N = 40 propagatingmodes, and an array with aperture

A = [0, |A|],

so that

Mjj =∫ |A|

0

dy φ2j (y) =

|A|D

[1− sinc

(2πj|A|

D

)].

The left plot is for NT = N and for |A| = D. We see that both functions peak aty = y′, but ΨTR

o has large sidelobes. This is because of the weights β−2j in (8.24) that

are very large when j = N . The middle plot is again for a full aperture, but withNT = N/2. Here the weights are comparable for all the recorded modes, and thepoint spread functions look similar. The resolution is about half of that in the leftplot. The right plot is for NT = N/2 and an aperture equal to 5% of the waveguidedepth, |A| = 0.05D. We note that the point spread function ΨTR

o is affected by thepartial aperture, it has large sidelobes.


9. Imaging

The goal of the imaging process is to transform the data pA(t, y) into a functionJ (y, Z) that has two properties:

– J (y, Z) is large when (y, Z) is in the support of the unknown source, and smallotherwise. This allows us to estimate the location of the source from the peaksof J (y, Z) in the search domain.

– J (y, Z) is statistically stable in the vicinity of its peaks

E[J (y, Z)] ≈ J (y, Z).

This gives robust estimates of the source location.

A typical coherent imaging method forms J (y, Z) by superposing the recordings pAover the array aperture, after synchronizing them relative to the search points (y, Z).The synchronization uses a mathematical model of wave propagation to relate thesearch points to arrival times of the waves at the array. In waveguides the synchro-nization is tricky, because the modes travel at different group speeds and the signal isaffected by dispersion. The best way to do it, is to backpropagate the time reversedrecordings in our model of the waveguide. The synthetic backpropagation is done an-alytically or numerically, and it is mathematically equivalent to that in time reversalif we have perfect knowledge of the waveguide. This is not possible in random waveg-uides. Thus, we backpropagate in a surrogate waveguide with straight boundariesand constant wave speed co.

We analyze coherent imaging in section 9.1. We show that it can work for broad-band sources if we complement it with a low order mode pass filter. That is to say, ifwe keep only the modes with scattering mean free path Sj > ZA. This is possible inwaveguides with random boundaries, but not in those with random inhomogeneities,where Sj is very small for all the modes. The array data in such waveguides is basicallyincoherent.

A natural idea for imaging at long ranges is to work with cross-correlations of thedata. It has been exploited successfully in a variety of contexts, and we describe insection 9.2 how it can be implemented for our problem.

To illustrate the points, we refer to numerical results described in in section 10.

9.1. Coherent imaging. — Recall from section 8 the amplitudes of the emergingbackward wave at the array

bj(ω,ZA) =eiβj(ω)

ZAε2

2iβj(ω)

∫ D

0

dy pTRA (ω, y)φj(y).

52 LILIANA BORCEA

We use them as initial conditions for the wave equation in the surrogate ideal waveg-uide, and obtain the following model of the backpropagated wave field

(9.1) O(t, y,

Z

ε2

)≈

N∑j=1

φj(y)∫ ∞−∞

dω

2πbj(ω,ZA)e−iβj(ω) Z

ε2−iωt.

Similar to what we did for time reversal, we define the imaging function by

J (y, Z) = O(t =

T

ε2, y,

Z

ε2

).(9.2)

If the coherent imaging method is to be useful, we expect it to work best in broad-band regimes. Recall that the waves decorrelate over frequency offsets of order ε2, sothe broadband should help with the statistical stability. We consider henceforth thebroadband regime defined by (2.19). We also take for simplicity an array with fullaperture, where the coupling matrix M equals the identity.

We obtain from definitions (9.1)-(9.2) and the expression of pTRA that

JF (y, Z) =∫ D

0

dy′

∆yρ

(y′ − y?

∆y

)ΨF(y, y′, Z),(9.3)

where the index F reminds us that we have full aperture and ΨF is the point spreadfunction

ΨF(y, y′, Z) =N∑

j,l=1

φl(y′)φj(y)4βj(ωo)βl(ωo)

e−iβj(ωo) Zε2

×∫ ∞−∞

dw

2πΩBf

(w

ΩB

)e−iwβ

′j(ωo)Zε −iw

2β′′j (ωo)Z2

×∫ ∞−∞

du

2πχ (u) eiuβ

′j(ωo)

ZAT Pε,(a)

jl

(ωo + εw − ε2u

T, ZA

).(9.4)

We analyze next its mean and variance.

9.1.1. The mean point spread function for full aperture arrays. — We take the ex-pectation of (9.4) and use the moment formula (6.23) to obtain

E [ΨF(y, y′, Z)] ≈N∑j=1

φj(y′)φj(y)4β2

j (ωo)e−iβj(ωo) Z

ε2− ZASj(ωo)−i

ZAζj(ωo)χ

(β′j(ωo)ZA

T

)

×∫ ∞−∞

dw

2πΩBf

(w

ΩB

)e−iwβ

′j(ωo)Zε −iw

2β′′j (ωo)Z2 .(9.5)

Range focusing: Because the signal f has O(1) support, it is clear that (9.9) islarge only when Z = O(ε). From this argument we expect the range resolution

(9.6) z =Z

ε2∼ coεΩB

=coB.


But in (9.9) we also sum over terms that oscillate rapidly due to the large phaseβjZ/ε

2. If enough modes contribute to the sum, the range resolution improves toZ = O(ε2), i.e.,

(9.7) z =Z

ε2∼ λo.

The recording window captures NT ≤ N modes, but the contribution of each modeis weighted by exp(−ZA/Sj). Suppose that ZA < Sj for j = 1, . . . , NS ≤ N . Then,the number of modes that contribute to the sum in (9.9) is

(9.8) N = minNT , NS.

We need a large enough N to get the improved range resolution (9.7).Cross-range focusing: Let us evaluate the mean point spread function at the

range Z = 0 of the source, and obtain

E [ΨF(y, y′, 0)] ≈ f(0)4

NT∑j=1

φj(y′)φj(y)β2j (ωo)

exp− ZASj(ωo)

− i ZAζj(ωo)

.(9.9)

This is similar to the time reversal point spread function (8.12) in ideal waveguides,but the mode contributions have phase modulations on the range scale ζj , and expo-nentially decaying weights on the scale Sj . In regimes with

ζj(ωo) ∼ Sj(ωo) > ZA, j = 1, . . . ,N,

the behavior of (9.9) is similar to that of ΨTRo,F with NT = N. See the middle plot in

Figure (3) for an example of the point spread function. The phase modulation playsa role when

ζj(ωo) < ZA < Sj(ωo), j = 1, . . . ,N.

The point spread function may not be focused at y = y′ unless we weight optimallythe modes. That is to say, instead of backpropagating like in (9.1), we can multiplyeach mode amplitude bj by a weight to be determined with optimization, given afigure of merit that measures the quality of the image.

9.1.2. Statistical stability of the point spread function. — The mean point spreadfunction is focused at y = y′ and Z = 0, but the imaging method is useful only if theSNR at the peak is large, where

SNR =|E [ΨF (y, y, 0)]|

E [|ΨF(y, y, 0)|2]− |E [ΨF(y, y, 0)]|21/2

.

54 LILIANA BORCEA

The variance is

var [ΨF(y, y, 0)] ≈N∑

j,l,j′,l′=1

φj(y)φl(y)φj′(y)φl′(y)16βj(ωo)βl(ωo)βj′(ωo)βl′(ωo)

∫ ∞−∞

du

2π

∫ ∞−∞

du′

2πχ(u′)χ (u)

× ei[uβ′j(ωo)−u′β′

j′ (ωo)]ZAT

∫ ∞−∞

dw

2πΩB

∫ ∞−∞

dw′

2πΩBf

(w

ΩB

)f

(w′

ΩB

)×

E[Pε,(a)jl

(ωo + εw′ − ε2u′

T,ZA

)Pε,(a)j′l′

(ωo + εw − ε2u

T, ZA

)]−δjlδj′l′e

−ZA»

1Sj′ (ωo) + 1

Sj(ωo)

–+iZA

»1

ζj′ (ωo)−

1ζj(ωo)

–,(9.10)

where we used the expectations (6.23). We know that the propagator decorrelates forfrequency offsets that exceed O(ε2). Thus, only the frequencies satisfying |w −w′| =O(ε) contribute in (9.10), and we let

w − w′ = εw, |w| ≤ Ω = O(1).

Using the second moments (6.25), and the expression (6.36) of the Wigner transform,we obtain that the variance is given by the sum of two terms

var [ΨF(y, y, 0)] ≈ T1 + T2.(9.11)

The first term is determined by the continuous density of the Wigner transform,

T1 =εΩ‖f‖2

16πΩB

N∑j,l=1

φ2j (y)φ2

l (y)β2j (ωo)β2

l (ωo)

∫ ∞−∞

χ2( τT

)W

(l)j (ωo, τ, ZA)

× sinc

Ω[β′j(ωo)ZA − τ

].(9.12)

Here we let‖f‖2 =

∫ ∞−∞

du

2π|f(u)|2,

and obtained the right hand side in (9.12) after standard manipulations. The secondterm is due to the diagonal (coherent) terms that decay exponentially with range,

T2 =εΩ‖f‖2

32πΩB

N∑j,l=1

φ2j (y)φ2

l (y)β2j (ωo)β2

l (ωo)χ

(β′j(ωo)ZA

T

)χ

(β′l(ωo)ZA

T

)

×

(1− δj,l)e− ZASj′ (ωo)−

ZASj(ωo) +i

»ZA

ζj′ (ωo)−

ZAζj(ωo)

– [eΓjl(ωo)ZA − 1

]+δj,l

[e− 2ZASPj

(ωo) − e−2ZASj(ωo)

].(9.13)

The second term dominates the first at shorter ranges, but as ZA increases, the firstterm becomes the larger one. In any case, we can see that the variance is proportional


to εNT , if NT N . The integral of the Wigner transform is close to the identitymatrix for smaller ranges, but it becomes a full matrix with entries of order 1/N aswe approach the equipartition regime. Only the modes that arrive in the support ofthe measurement window contribute to the sums in (9.12) and (9.13), and we haveβj ∼ k if NT N . The variance is larger for NT ∼ N .

The peak value of the mean point spread function is of the order of the first termin (9.9), and the SNR estimate is

(9.14) SNR ∼ ε−1/2N√NT

e− ZAS1(ωo) ,

where we assumedSj(ωo) ∼ S1(ωo), j = 1, . . . ,N.

It shows that the imaging method is useless if ZA > S1. Coherent imaging worksonly when ZA < Sj for j = 1, . . . , NS , and NS sufficiently large. The numberN = minNT , NS of coherent modes that contribute in the point spread function de-termines its resolution. The SNR decreases with NT , so we should have NT ≈ N. Thequality of the image may be improved by weighting the contribution of the modes.The weights can be determined by optimizing a measure of quality of the image.

9.2. Imaging with cross-correlations. — The idea of imaging with cross-correlations of the data has been used successfully in the coherent interferometric(CINT) imaging approach introduced and analyzed in [7, 8, 5]. The point is thatwhen net scattering is significant, it is better to work with the square of the wavefield, i.e., with the energy resolved locally in time and over directions of arrival.Because the second moments of the mode amplitudes do not tend to zero as the rangeincreases, we can obtain statistically stable imaging methods when we backpropagatethe cross-correlations of the recordings pA(t, y).

Let us write the data projected on the span of the eigenfunctions

Dj(ω) =∫ D

0

dy pA(ω, y)φj(y),

for frequencies ω = ωo + εw and |w| ≤ πΩB. We assume for simplicity that the arrayhas full aperture, and obtain from (7.7) that

Dj(ω) ≈∫ D

0

dy

∆yρ

(y − y?

∆y

)1εΩB

f

(w

ΩB

)dj

(w

ΩB

)eiβj(ωo)

ZAε2

+iwβ′j(ωo)ZAε

×∫ ∞−∞

du

2πχ(u) e−iβ

′j(ωo)

uZAT

N∑l=1

φl(y)2iβl(ωo)

Pε,(a)jl

(ωo + εw − ε2u

T, ZA

).(9.15)

56 LILIANA BORCEA

We wish to image by backpropagating the mode dependent energy resolved locally intime

(9.16) Ej(τ) =∫ ∞−∞

dω

2π

∫ ∞−∞

d(εw)2π

Dj

(ω + ε2 w

2

)Dj

(ω − ε2 w

2

)e−i ewτ .

Here we scale the offset w by ε because we know that the propagator decorrelatesover frequency intervals exceeding ε2. We saw that the modes decorrelate due toscattering, so we consider one mode at a time. This implies that we work with wavestraveling in the same direction.

To understand how to extract the source location from (9.16), we calculate itsexpectation. We rewrite (9.16) as

Ej(τ) ≈∫ D

0

dy

∆yρ

(y − y?

∆y

)∫ D

0

dy′

∆yρ

(y′ − y?

∆y

)Ψj(y, y′, τ),(9.17)

with kernel Ψj following obviously from (9.15). Its expectation is given by

E [Ψj(y, y′, τ)] ≈ ‖f‖2

4ΩBχ2

(β′j(ωo)ZA

T

) N∑l=1

φl(y)φl(y′)β2l (ωo)

W(l)j (ωo, τ, ZA) ,(9.18)

where we used the second moments (6.25).Because we are in the broadband regime and the waves decorrelate over frequency

offsets of order ε2, the variance of Ej(τ) is small in the vicinity of its peak. Thus, wecan analyze the imaging method by looking at the expectation of the kernel.

9.2.1. Range estimation. — The information about the range of the source is encodedin the τ peak location of the right hand side in (9.18). The problem is that we donot know the transport speed of W(l)

j . It is not the same as the mode group speedβ′j(ωo), unless ZA is smaller than the transport mean free path SPj . We are interestedin longer ranges, where coherent imaging does not work. Recall from the discussion insection 8.1.1 that in the very long range limit, beyond the equipartition distance, thetransport speed is 〈β′〉, the average of the mode group speeds. But we already knowthat we cannot image beyond the equipartition distance, because the waves forget theinitial conditions. The interesting regime for imaging is

SPj (ωo) < ZA < SE(ωo),

where the speed needs to be estimated from the transport equations (6.28).The matrix Γ(c) of coefficients in the transport equations is determined by the power

spectral density of the autocorrelation of the fluctuations, evaluated at differences ofthe wave numbers. If we know the autocorrelation, i.e., Γ(c), we can relate the τ peaklocation of Ej(τ) to the range of the source. If we do not know it, we can try toestimate both the source range and the autocorrelation of the fluctuations. This wasproposed and analyzed in [22, 6] for a point source, i.e., ∆y D, where (9.16) and


(9.19) give

E [Ej(τ)] ≈ E [Ψj(y?, y?, τ)]

≈ ‖f‖2

4ΩBχ2

(β′j(ωo)ZA

T

) N∑l=1

φ2l (y

?)β2l (ωo)

W(l)j (ωo, τ, ZA) .(9.19)

We do not know the weights φ2l (y

?), because we do not know y?. However, we areinterested only in the τ peak location of this expression, which is not really affectedby the weights. The estimation in [22, 6] computes

(9.20) Ej(t) =Ej[t+ β′j(ωo)(ZA − Z)

]maxτ |Ej(τ)|

,

and compares it with

(9.21) Ej,M (t) =Ej,M

[t+ β′j(ωo)(ZA − Z)

]maxτ |Ej,M (τ)|

,

for a search range offset Z of the source. The offset by the deterministic travel timeβ′j(ZA − Z) allows us to observe the deviation from zero of the t peak location ofEj(t). The search model Ej,M (τ) is given by

(9.22) Ej,M (τ) =N∑l=1

1β2l (ωo)

W(l)j,M (ωo, τ, ZA) ,

and the Wigner transformW(l)j,M solves the transport equations (6.28), with the model

matrix of coefficients Γ(c)M .

We compute Γ(c)M using definition (4.78) for waveguides with random boundaries

and (5.9) for waveguides with internal inhomogeneities. These definitions require amodel of the correlation Rµ(z) or Rν(y, z) of the fluctuations. If we know the model,we should use it. If not, it turns out that the estimation is quite robust with respectto the model. In [22, 6] we considered waveguides with interior inhomogeneities andused the correlation

Rν(y, z) = R(y`,z

`

),

with different choices of R. The correlation length ` is determined at the same timeas the range of the source. The results show that the range estimation is independentof the particular choice of R. We refer to section 10 for an illustration with numericalsimulations. More detailed simulations are in [22, 6].

9.2.2. Cross-range estimation. — Once we have estimated the range of the sourceand the model Γ(c)

M , we can seek the cross-range. Take for example the integral overτ in (9.17) and use that

(9.23) W(l)j (ωo, 0, ZA) =

∫ ∞−∞

dτW(l)j (ωo, τ, ZA) = ej · eΓ(c)(ωo)ZAel,

58 LILIANA BORCEA

to obtain

(9.24)∫dτ E[Ψj(y, y′, τ)] ∼

N∑l=1


ej · eΓ(c)(ωo)ZAel, j = 1, . . . , NT .

Here we let ∼ denoting approximate up to a multiplicative constant, as before, andwe take into account the finite size of the recording window by limiting the modeindex j ≤ NT .

We know that (9.23) becomes independent of l when ZA approaches the equipar-tition distance, so we can write

(9.25)∫dτ E[Ej(τ)] ∼

N∑l=1


, ZA & SE .

This is like the time reversal point spread function ΨTRo,F in ideal waveguides, and it

peaks at y ≈ y′. Therefore, we get that

(9.26)∫dτ E[Ej(τ)] ∼

∫ D

0

∫ D

0

dy

∆yρ

(y − y?

∆y

)∫ D

0

dy′

∆yρ

(y′ − y?

∆y

)ΨTRo,F(y, y′),

and if we have a point-like source at y?,

(9.27)∫dτ E[Ej(τ)] ∼ ΨTR

o,F(y?, y?), ∆y D.

Obviously, this is independent on where y? lies in (0,D), so we cannot estimate thesource location. As we said before, in the equipartition regime the waves forget theirinitial conditions and imaging is no longer possible.

If ZA < SE , we can estimate the source location from the kernel (9.24), by com-paring it with its mathematical model

N∑l=1

φ2l (y)

β2l (ωo)

ej · eΓ(c)M (ωo)ZAel, j = 1, . . . , NT ,

with Γ(c)M computed previously. We refer to [22, 6] for more details of the cross-range

estimation.

10. Numerical results

We recall a few a few numerical simulations from [6], for two dimensional waveg-uides with interior inhomogeneities. The wave field p(t, y, z) is simulated by solvingthe wave equation in a medium with wave speed fluctuations modeled as in (5.1), andwith Gaussian correlation

Rν(y, z) = e−y2+z2

2`2 .


Figure 4. The acoustic pressure computed at the receiver location y =D/2 for (clockwise from top left): ε = 0, ε = 1%, ε = 2% and ε = 3%.

Figure 5. The coherent imaging function JF (y, z) for ε = 1%.

We generate the process ν(y, z) numerically using random Fourier series [10]. Thecorrelation length is ` = λo/2 and the perturbation parameter ε ranges between(3)

0%− 3%.The simulations are for a point source at y? = D/2, emitting a pulse with extra

wide bandwidth of 1.5 − 4.5kHz. It is larger than the bandwidth considered in ourmodel (2.19), but the analysis can be extended easily to this case.

The central wavelength in the simulations is λo = 0.5m, calculated at the constantsund speed co = 1.5km/s in water. The depth of the waveguide is D = 20λo, so thatN(ωo) = 40. The array is at range zA = 500λo.

We plot in Figure 4 the computed signal at the receiver location y = D/2, for threedifferent strengths ε of the fluctuations. We see a clean train of pulses in the idealwaveguide (ε = 0). The scattering effects are significant for ε > 0, as predicted bythe theory, and the signal looks almost like noise when ε ≥ 2%.

(3)If we look carefully at the scaling in the model of the fluctuations in [6], we see that the increase

of ε may be related to an increase in the net scattering effects in our model.

60 LILIANA BORCEA

Figure 6. Coherent function JF (y, z) for ε = 2% and two realizations ofthe fluctuations. The true source location is shown with a white circle.

Range in λc

j ∈ S

440 460 480 500 520 540

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Range in λc

j ∈ S

440 460 480 500 520 540

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Figure 7. Computed Ej(t) and model EM (t) plotted as a function of ZA−t/β′j in the abscissa and j in the ordinate. The model is computed withthe true autocorrelation and source range location. Because the transportspeed varies with the mode index, the peak locations change with j.

In Figures 5 and 6 we plot the coherent imaging function (9.3) for ε = 1% and2%, respectively. The source is in the middle of the domain and the method locatesit when ε = 1%, because the net scattering effects are not so strong. However, wesee from the results in 6 that when the scattering is stronger, the image changesunpredictably with the realization of the fluctuations of the wave speed. This is anillustration of the lack of statistical stability of the imaging function (9.4).

To illustrate the ideas of the incoherent range estimation, we plot in Figures 7 and8 the calculated Ej(t) and model Ej,M (t). The model uses a Gaussian autocorrelationwith correlation length that is determined in the range estimation. The calculatedEj(t) is noisy due to scattering and numerical error (left plot in Figure 7). But still,it compares well with the model computed with the correct correlation length andsource range (right plot in Figure 7). If we have the wrong range and/or the wrongcorrelation, the model changes as shown in Figure 8. The location of the peaks isdifferent.

We refer to [6] for many more simulations and detailed range and cross-rangeestimation results.


Range in λc

j ∈ S

440 460 480 500 520 540

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8

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Range in λc

j ∈ S

440 460 480 500 520 540

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Figure 8. Model EM (t) as a function of ZA − t/β′j in the abscissa and jin the ordinate. In the left picture we evaluate the model at the searchrange ZA − 20λo and in the right picture we use a correlation length thatis half the true one.

Acknowledgments

These lecture notes describe results obtained in collaboration with: Ricardo Alonsoand Josselin Garnier for wave propagation in waveguides with random boundaries [1],Leila Issa for the incoherent imaging algorithm based on a model of transport of energyin waveguides [22, 6], and Chrysoula Tsogka for the numerical simulations of wavepropagation in [6]. We also use extensively the studies [19, 11, 15].

The work of L. Borcea was partially supported by the AFSOR Grant FA9550-12-1-0117, the ONR Grant N00014-12-1-0256, and by the NSF Grants DMS-0907746,DMS-0934594.

References

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[2] G. Bal and L. Ryzhik. Time reversal and refocusing in random media. SIAM Journalon Applied Mathematics, 63(5):1475–1498, 2003.

[3] P. Blomgren, G. Papanicolaou, and H. Zhao. Super-resolution in time-reversal acoustics.Journal of the Acoustical Society of America, 111:230, 2002.

[4] L. Borcea and J. Garnier. Paraxial coupling of propagating modes in three-dimensionalwaveguides with random boundaries. arXiv preprint arXiv:1211.0468, 2012.

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[11] L. B. Dozier and F. D. Tappert. Statistics of normal mode amplitudes in a randomocean. Journal of the Acoustical Society of America, 63:533–547, 1978.

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[18] C. Gomez. Wave propagation in shallow-water acoustic waveguides with rough bound-aries. arXiv preprint arXiv:0911.5646 [math AP], 2012.

[19] W. Kohler and G. Papanicolaou. Wave Propagation and Underwater Acoustics, J. B.Keller and J. S. Papadakis, eds., volume 70 of Lecture Notes in Physics, chapter Wavepropagation in randomly inhomogeneous ocean. Springer Verlag, Berlin, 1977.

[20] W.A. Kuperman, W.S Hodkiss, H.C. Song, T. Akal, C. Ferla, and D.R. Jackson. Ex-perimental demonstration of an acoustic time-reversal mirror. Journal of the AcousticalSociety of America, 103:25–40, 1998.

[21] H. J. Kushner. Approximation and weak convergence methods for random processes.MIT Press, Cambridge, 1984.

[22] Issa L. Source localization in cluttered acoustic waveguides. PhD thesis, Rice University,2010.

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[25] G. Papanicolaou, L. Ryzhik, and K. Sølna. Self-averaging from lateral diversity in theito-schrodinger equation. Multiscale Modeling & Simulation, 6(2):468–492, 2007.

[26] M.C.W. van Rossum and Th.M. Nieuwenhuizen. Multiple scattering of classicalwaves:microscopy, mesoscopy, and diffusion. Rev. Mod. Phys., 71:313–371, 1999.

[27] J. Weidmann. Spectral theory of ordinary differential operators. Lecture Notes in Math-ematics 1258. Springer-Verlag, Heidelberg, 1987.

Liliana Borcea, Computational and Applied Mathematics, Rice University, MS 134, Houston, TX

77005-1892 • E-mail : [email protected]

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