the mutual coherence function in a scattering channel— a two-scale solution

The mutual coherence function in a scattering channel-- A two-scale solution

Shimshen Frankenthal a) Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

(Received 15 August 1986; accepted for publication 2 May 1988)

By using the two-scale embedding procedure, an approximate expression for the mutual coherence function in a refractive and scattering channel is derived. The solution reduces to the known limiting monochromatic solutions, and provides a good approximation for the bichromatic coherence in a homogeneous scattering medium. Simple expressions are derived for both the coherence bandwidth and coherence time (Doppler spreading) in a quadratic channel with a quadratic scatterer. The possibility of inversion is explored.

PACS numbers: 43.30.Ft

INTRODUCTION

The design of optimally coded communication links • requires knowledge of the decorrelation, in frequency and in time, that a signal experiences in its transit through a scattering channel. The basic measures of these effects are the coherence and Doppler bandwidths, and the attendant characteristic times. Scattering will destroy the correlation between two pure tones if their frequency separation exceeds the coherent bandwidth. The inverse quantityrathe multipath- spread time--describes the distortion of an erstwhile sharp pulse. The Doppler bandwidth measures the broadening of a single pure tone (spectral line) as a result of time variations in the scattering process. Its inverse, i.e., the coherence time, indicates how long a pure tone remains coherent. These quantities often sufiSce to characterize grossly the effect of the medium, and can serve to define the frequency-time "cells" that must be allocated to a single element of the signal. However, the design of signal waveforms requires the full functional dependence of the signal correlation on separations in both the frequency and the observation time.

For any applications that require coherent data processing, the information described above is contained in the gen- eralized second-order mutual correlation function (MCF), which also takes into account the separation between receiver locations. Once the equation that governs this quantity is solved, one can also consider using frequency and time cor- relations as a diagnostic tool for studying the properties of the medium. The mutual correlation function has been treat-

ed extensively in nonrefracting media, and for simple excitations such as plane or spherical waves; see, e.g., Ishimaru 2 and references cited therein. Explicit solutions have also been obtained with other excitations when the medium is a

"quadratic scatterer," i.e., has a structure function that depends quadratically on the separation. 3-5

In the present article, the M CF equation is extended to include the effects of a refractive channel. We then proceed to present a general approximate solution of this equation, which accounts for the effect of the channel as well as the

Permanent address: Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Ramat Aviv, Israel.

effect of an arbitrary scatterer. The solution is obtained by the two-scale embedding technique, which has been success- fully applied in the recent few years to the treatment of coherence propagation problems in both deterministic media, 6 where it permits dealing with caustics, and stochastic media, 7-9 where it has been used to determine the range and frequency dependence of the scintillation index for arbitrary scatterer properties. The present study considers only cases in which the quadratic-channel approximation sufiSces to describe the refractive effects. 6'•ø However, the approach can be extended (by a three-scale embedding) to treat situations where this approximation breaks down (e.g., near caustics). It can also be extended to situations where the paraxial approximation is inappropriate, and its becomes necessary to apply the parabolic approximation along large- angle rays. • •

In Sec. I, we formulate the equation for the MCF in a scattering refractive channel, and recast it in a form that is amenable to treatment by the two-scale procedure. Section II outlines the procedure and introduces the (approximate) general solution for the MCF. This solution is then used to treat the nonrefractive regime: The quadratic scatterer results are compared with the exact solutions that are available in this case, and thus serve to assess the errors that are introduced by the approximations involved in the two-scale solution. We then treat the case of a quadratic channel, quoting explicit results for the quadratic scatterer. Finally, we consider the inversion of the solution, and potential diagnostic applications that are implied by this possibility.

I. FORMULATION OF THE COHERENCE EQUATION

The MCF equation in a nonrefractive medium is well known, see Ref. 2. We redefive this equation here partly in order to show the genesis of the refractive term, but primar- ily in order to introduce the transformations that render this equation amenable to treatment by the two-scale procedure.

The complex amplitude p of the pressure p = Re p exp-j(kr q-rot) is taken to be governed by the parabolic wave equation

ø• -J 2k o• 2 j-•- ktr ( r, ,k,t) = O, (1) 104 J. Acoust. Sec. Am. 85 (1), January 1989 0001-4966/89/010104-10500.80 ¸ 1988 Acoustical Society of America 104

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.138.73.68 On: Sun, 21 Dec 2014 04:58:33

where r denotes the range, u the transverse position in a range plane, k is the wavenumber corresponding to the frequency co at the reference sound speed Co, and Atr is the measure of the total deviation of the sound speed c from its reference value

,

k = W/Co, Atr = Co 2 / c• - 1. (2) Following the standard procedure, :'l: we write Eq. ( 1 ) for •b• and its conjugate forthS, premultiply these equations by•b• and •b•, respectively, and ensemble-average their sum (the subscripts 1 and 2 denote two operating wavenumbers and transverse locations in a range plane). The result is the equation for the mutual coherence function, which can be written in a symbolic form thus (the symbol ( ) denotes ensemble- averaging)

•9 F (D + R -4- S)F, 3r (3)

•= (•b( r,u,,k•,t, )b* ( r, u2,k2,t2) ). This merely states that the range variation of the coherence reflects the combined effects of diffraction (D), refraction (R), and scattering ($). We now consider each of these op- erators.

The diffraction operator is

o = T ', . (4) In the monochromatic limit k• = k2 = k, the use of the sum and difference variables

I1=• (U 1 +112) , q = Ul -- U2, (5) reduces this operator to the simple form (j/k) (•3/c9u) (c• / o•q) that is essential to the simplification of the ensuing mani- pulations. In the case at hand, a more complicated transformation is required in order to achieve this simplification. We first define the mean wavenumber k, and the measure 12 of the wavenumber (or frequency) separation, as follows:

k = 4k•k2 , 12 = 2 k• -- k 2 ( + ) ( + )

k• = k( 1 -- 12/2)-2, k 2 = k( 1 + 1]/2) -2, (6)

noting that, when the separation parameter fl is small, the mean (or carrier) wavenumber k is indeed the algebraic average of kl and k2. We next define the modified transverse position variables, and their sum and difference, and record, for future reference, the relations between the "modified" and the original (geometric) sum and difference variables of Eqs. (5), as follows:

Ulm = x/(k•/k)u•, U2r n = x/(k2/k)u 2, '

Um =• (Ulrn "•-U2rn), qm =Ulm --U2rn,

U = Urn -- (fl/4)qm, U m = y[U + (fl/4)q],

q= --flU.,+qm, qm=Y(flu+q),

y = ( 1 -- 122/4) --•.

(7)

The use of these definitions in Eq. (4) reduces the diffraction operator to the form

D-- j 0 o9 (8) ko•mtgqm

which is considerably simpler than the corresponding forms used in conventional treatments of the bichromatic problem. 3 The penalty for this is a more cumbersome form of the scattering operator, which can nonetheless be dealt with by the two-scale procedure.

The refraction coefficient R reflects the dependence on the depth (uy) of the average refractivity At, and can be obtained by inspection of the parabolic Eq. (1) for the pressure. It reads

R= (j/2)[k•t(u,y) -- k2At(u2y) ], At(uy) =•(uy/lr), (9)

where lr is the characteristic scale of the depth variation of At, the refractive scale (or convergence length). In the monochromatic limit, the quadratic approximation is used to replace the operator R by the lowest-order term of its Taylor- series expansion, 0.5jkAt(•)(%,) qy (where At(k) is the kth derivative oftt). Upon performing this expansion in terms of the modified variables defined in Eqs. (7), and retaining the lowest-order terms, we find

R = (j/2)kAt(•)(Umy)qmy +jkflg(Umy), (10)

g(Umy ) = t•(U•y ) -- « U•y•/(U•y ). • this expression is exact, and In a quadratic channel •-• uy

the term g - 0. The scattering operator S manifests the second-order

statistics of the fluctuations of the refractivity measure tt r. The simplified version that we shall use here is acceptable when (a) the ranges of interest are much larger than the range correlation length (see Refs. 2 and 12); and (b) the frequency is sufficiently large so that the "quasi-isotropic" regime prevails, i.e., Ia/k! • ,• 1, where • and Ia are the vertical and horizontal correlation lengths associated with the refractivity fluctuations. • The relevant statistics are em- bodied in the integral (over all range separations) of the correlation function B,, of the refractivity fluctuations

- f - A(q) = dqzB,,(q,qz)=A(O)A(q),

](0) = (11)

which depends only on the transverse separation and the attendant correlation lengths li (where i = x,y or h,v). In the definition of •(0), (At2) is the mean-square refractivity fluctuation, and the integral scale lz arises as a result of the integration over range-separations. It is convenient to express S thus

S= y2k 2F(q) + • (k, - k2)2](0), F(q) = r• .4(0)[ 1 - A(q) ],

y2k 2F(q) = ( 1/lf)f(qi/li ), i= x,y, or h,v, l/ i= • •2k 2(At2)lz '

(12)

The quantity l s in Eqs. (12) is the extinction length: It reflects the k • dependence of the first term in the expression for $, and is proportional to the strength of the scattering, which

105 J. Acoust. Soc. Am., Vol. 85, No. 1, January 1989 Shimshon Frankenthal: Scattering coherence function 105


is determined by (/.•2) and may vary with the depth uy. In isotropic media, the function F depends only on the magnitude of its argument, and vanishes if this argument is zero. To account for the motion (at the transverse velocity W) of the "frozen" scattering structure, this argument is set to q- Wta, where ta = tl -- t2 is the separation between the observation times. In terms of the modified variables defined

in Eq. (7), we now have

S = •2k 2F(qm -- •'•u m -- Wta ) + « Yqk 2•,2•(0) ß (13) In the quasi-isotropic regime of stratified nonisotropic media, we shall assume that F depends on the magnitudes of Cartesian components of q, and vanishes when both components are zero.

The second term in the expressions for S, Eqs. (12) and ( 13 ), depends only on the frequency separation. This term is readily disposed of by extracting a range-dependent exponential coefficient from the coherence function--see Ref. 2--thus

•' = • exp -- -• y4k 2fi2•(0)r. (14) In the regime of interest, this range-dependent exponential does not play a dom,•nant role in limiting the two-frequency coherence, so that F is the main quantity of interest. Upon substituting Eqs. ( 8 ), (10), and ( 13 ) i•Eqs. ( 3 ), and taking account of Eq. (14), the equation for F becomes

8 1 8 8 j k /.t{•)(Umy ) 37r- T

--jkO, g(u•y) + y2k 2F(q• -- O,u• -- Wtd) ) A

X r' m (r, Um,qm;•,td ): O, A A

Fm (O,Um,qm;•,td ) = Fsm (Um,qm). (15)

A A

Here, we have used the notation F m for the coherence F, expressed j,n terms of the modified variables Um and qm' Likewise, Fsm is the source condition (at the plane r = 0) on the coherence: It is obtained from the source pressure, and expressed in terms of Um and qm- We have supp•ssed the mean wavenumber k from the list of arguments OfFm. Also, since fi and ta play the role of parameters in the differential equation (15), we shall frequently suppress them in the discussion of the solution of that equation.

II. THE SOLUTION OF THE COHERENCE EQUATION

A solution of Eqs. (15) may be obtained by the technique of multiple-scale embedding, which has been discussed in a number of previous publications. 6'8 In the case at hand, we have in effect relinquished any attempt to treat the coherence in the vicinity of caustics, by adopting, from the very beginning, the quadratic approximation to the refractive coefficient R (in order to treat caustics, an additional term in the series expansion of this coefficient would have to be retained). Thus it suffices to employ a two-scale embedding, in order to deal with the presence of both the transverse separation q and location u as arguments of the scattering function Fin Eqs. (15). The process is then identical to that of Ref. 8 (which deals with the fourth moment of a plane

wave in a scattering nonrefractive medium), except for mi- nor complications caused by the presence of the refractive term. We shall therefore content ourselves here with an out- line of the procedure.

( 1 ) Recast Eqs. (15) in a dimension-free form, using the following quantities:

= r -- Um -- qm r=-- u=•, q----,

lf' luq luq

luq =• ]•= luq ' lr ' luq

(16)

where •i will later be taken to be small compared to 1. (2) Fourier transform the fi dependence to an •1 depend-

ence.

(3) Embed the resulting equation in a higher-dimen- sioned space, which also includes

•i = •i•i, Oi = 6iqi, i = x,y, (17) as additional arguments of the (transformed) coherence function.

(4) Retransform the ql dependence to a v dependence, and transform the q dependence to a½r dependence.

This results in an integrodifferential equation, which governs a multiply transformed and embedded version of the coherence. This equation contains an approximate represen- tation of refractive effects (it is exact only for a quadratic channel, in which there are no caustics), but it is still exact insofar as scattering is concerned. When the solution of this equation is expanded in an e series, its lowest-order term is found to satisfy a first-order partial differential equation, and may therefore be obtained by the method of characteris- tics. Upon subjecting this lowest-order solution to the inverse sequence of transforms and embedding outlined above, we finally obtain an explicit approximate expression for the coherence in terms of the modified coordinates Um and qm'

The solution is expressed in terms of properties of the rays of geometrical acoustics. The relevant ray functions are the (transverse) position V(•,v,tr) and direction (slope) S(•,v,tr), at the range r-•, of the ray that crosses the observation range r at the position v in the direction {r. These functions are defined by the ray equations and their "initial" conditions (at the observation range), namely:

-_&, = o,,

(18)

•Sy _ 1 i[•(l)(Vy), Sy(O) & (•,v•,a•) = a•, c9• 2 where we note that each Cartesian component ( i = x or y) of V and S depends only on the corresponding component of v and {r. In addition, we need the functions Vio (•,oi,rri) and Vi• (•,oi,tri), which are, respectively, the sensitivities of the components Vi of the ray position at the range r-• to the components of the ray position oi and direction tri at the observation range r. These sensitivities may be obtained by partial differentiation of the ray-position function V (r,v,tr), or by a numerical solution of the ray-sensitivity equations



0Vv

V• (0) - 1, = o,

1 /z(•)(F•)F• ' $•(0)=0, 2

1 p(2)( Vy ) Vy•, S• (0) = 1 2 '

(19a)

which can be integrated simultaneously with Eqs. (18). For the horizontal (x) components, we always have

V• = 1, V• = --•, (19b)

since Vx = vx - a• •, Eqs. (19) are always valid in a horizontally stratified medium.

In terms of the functions defined above, the approximate solution of Eqs. (15) may be expressed as follows:

^ 4fff ^ F m (r, Um,qm ) -- (7;) 4 dv d dq•n Fsm [V(r,v,e),qL ]exp[ -jk(qm'• + q•,.S(r,v,e) ]

Xexp[jkn•(r, vy,•ry) ] f d• exp[jk•'(Um - v)]exp[ - Zk2Q(r,v,•,qm,•) ], •r $( r, vy,ay ) = a• g[ vy (•,vy,ay ) ],

Q(r,v,•,qm,•) = d•F[q•mCo(•,v•,a•) + •iC•(•,v•,a•) - •C(•,v,,a•) - Witd],

(20)

where all integrals extend from - oe to oe, unless otherwise indicated. In this expression, r is the range; u,• and qm are, respectively, the modified average location and separation of two points of observations; k is the mean (or carrier) wavenumber; f• measures the frequency separation; and td measures the time separation. The quantity Fs,• (Urn ,qm ) is the coherence in the source plane, in terms of the modified coordinates defined in Eqs. (7).

In the monochromatic limit f• = 0, we may drop the subscript rn since the modified coordinates then correspond to the geometric coordinates. If we then perform the q' integration, the source coherence is then replaced by its directional counterpart (the Fourier transform of the q' dependence to a • dependence), and Eqs. ( 18 )-(20) revert directly to the result of Ref. 13. We observe that this is all that would

be necessary in order to calculate the (monochromatic) coherence time and Doppler spread in the frozen regime. The

development outlined above was required in order to deal with the bichromatic problem. Equations (20) may be recast in several alternative forms, in which the source coherence is expressed in terms of the modified coordinates Um and qm, the geometric coordinates u and q, or is replaced by its directional counterpart. However, in all these forms, the computation of the MCF requires tracing a threefold family of rays, parametrized by v, •, and •l, from the observation range (where • = 0) to the source plane (where • = r), and weighting the source coherence by phase and attenuation factors such as those given in Eqs. (20). We must trace enough rays to be able to perform the integration indicated in Eqs. (20) to the desired level of accuracy.

A particularly useful version of Eqs. (20) is the superposition integral, which weighs the coherence of the source by a Green's function, thus

F (r,u m ,qm ) = du•n dq• n Fsm (U•,,q•n) Gm or F (r,u,q) du' dq' F• (u',q') G, ,:•

Gm =Y-:• =-• dv d(r 6[V(r,v,•) - u•, ]exp[ -jk(qm'(r + q•,.S(r,v,•)] (

Xexp[jk•(r, vy,ay) ] f d•l exp[jk•l'(Um - ¾)]exp[ - •2k2Q(r,¾,o',qm,•l) ],

where •b and Q, which are discussed further below, are •defined in Eqs. (20). The 6 function in the integrand of Gm signifies that, is to be eliminated in favor ofuL. By integrating this 6 function, we also introduce the Jacobian factor I% I-'.

The integrand in Eqs. (20) contains two ray integrals. One is the deterministic phase factor 0• (r,v• ,% ) that is asso-

107 J. Acoust. Soc. Am., Vol. 85, No. 1, January 1989

(21)

ciated with the propagation of signals at two different fre- quencies through the channel. This factor vanishes when the

2

channel is either uniform (p = 0) or quadratic (p-uy ), and contributes a pure phase in a linear channel. In other channels, a deterministic multipath can exist (several eigen- rays that connect the source and the receiver), and then the sum of several phase terms can produce a change of the mag-

Shimshon Frankenthal' Scattering coherence function 107


nitude of the coherence even in the absence of scattering. Note that, in the absence of scattering, the •1 and v integrations indicated in Eqs. (20) can be carried out directly: This results in the replacement of v by Um in the remainder of the integration.

The other ray integral is the scattering exponent Q, which accounts for the variation of the magnitude of the coherence with range along a single ray. In the examples studied in this article, this integral alone is responsible for the limitation on the coherence bandwidth. We recall at this

point that, in the absence of scattering, the ray-sensitivity functions V,-v and Via serve only to trace the geometrical spreading of the rays. In the case at hand, these functions also serve to trace scattering, since they appear as arguments of the scattering function, and thus affect the scattering exponent: the combination qim V•.v + 'rh Vi,• traces, to lowest order, the separation between two rays that are initiated, respectively, with (v,•) and (v + qm ,• + •1 )'

For the sake of compactness, the entire solution was written in vector notation, taking advantage of the fact that the process of tracing the rays and their sensitivities, and the computation of the phase factors in the integrand of Eqs. (20), can be carried out separately for the horizontal and the vertical components. The two components "couple" only in the scattering exponent. In a nonisotropic medium, where the horizontal correlation length is much larger than the vertical, one can argue that the horizontal terms in the argument of F may be neglected, to lowest order, because the decorrelating effect of the vertical terms becomes pro- nounced at much shorter ranges, or at lower frequency separations (in the monochromatic case, one can rigorously jus- tify disregarding the horizontal scattering terms in studying the horizontally averaged coherence). The effect of the horizontal terms may be computed by a higher-order iteration.

An example for such an iterative process is offered by Beran and McCoy's TM treatment of the coupling between vertical excursions of the ray and horizontal decorrelation. This was done for monochromatic signals, using a two-scale expansion in range in which the ratio of the correlation lengths served as a small parameter. The treatment employed a very simple zero-order solution, which holds only if the decorrelating effect of vertical scattering can be ignored on the scale of one convergence cycle. Instead, a zero-order solution that accounts for the effects of vertical scattering can be obtained from Eqs. (20), and employed in a bichromatic version of the analysis of Ref. 14. However, such an analysis is beyond the scope of this article.

III. NONREFRACTIVE MEDIA

This section considers nonrefractive, transversely isotropic scattering media, and in particular the quadratic scatterer limit, for which an exact solution of Eqs. (15) is available. 3 This can help establish bounds on the range of usefulness of the approximations involved in the two-scale procedure.

In the case at hand, the ray equations (18) yield

V(•,v,•) = v- •, Vxv = 1, Vy• = 1, (22) = = - = -

Using this in Eqs. (21 ), and performing the •r integration, we obtain the Green's function

G• (u•,q•,u•,q;.)

__ k 4 1 ; dv exp [jk•'(qm q•, ) ] 2= 4 •

X; d•l exp[jk•l'(Um -- v)]exp( -- y2k2Q) •0 r Q= d•F[qm -- fly-- (•1 -- fl&)•-- Wtd], (23)

•= (v- u•.)/r,

in terms of the modified coordinates.

In the monochromatic limit f• = 0, the v terms in the argument of F disappear, and the subscript m may be omit- ted. The v integration yields a delta function (2•)2•[ (k/ r) (q - q') - •1 ], and the •1 integration yields

G = (•r•) • .exp u - ).(q - ) exp( - k 2r) r

X d• F q -- (q -- q')---• -- Wtd , (24)

which is the well-known solution for the coherence in this

case; see Ref. 2, Eqs. (20)-(63). As a further check, 'consider plane-wave propagation,

where we have the source coherence

•'sm (Urn ,qm ) = 1. (25a)

The q•,integration produces (2•r) 26 [ k/r(v - u•,) ] in the integrand. By integrating u•,, setting g = q•, - fly - Wtd, •1 = -- •lr, and taking note of Eq. (7), we obtain

A

F (r,u,q) = k 2

X exp -- •rr •1'(• -- q + Wtd )

Xexp( -- y2k2r) d•F(•l• + V), (25b)

where the coherence depends, as it must in this case, only on the geometric separation variable q.

Indeed, it is possible to recast the general expression for the coherence, Eqs. (23), in terms of the geometrical variables, and simultaneously simplify the scattering exponent drastically, by using the transformation

• = qm -- I•V, fi = -- (•1 -- II&)r, fi = (v -- u•, )/r, (26)

and making use of Eqs. (7). The expression for the Green's function becomes



( r (2•r): (r•):

Xexp --j-• (•! q- •-- )'(•--q) X exp[ -- •k :r•(•!,g) ],

( ' = - q )+ 1+

X (u -- u').(q -- q')], (27) Q(•,V) = d• F(q• -- g -- Wta ),

where the scattering exponent depends linearly on the range r and on the function Q of the integration variables q and •.

To see how Eqs. (27) reverts to the monochromatic limit, note that the kernel of the (q,V) integral tends to 6 (• + • + q') 6 (• - q) as • • 0. The integration then re- places the argument of F by that shown in Eq. (24). More formally, we "undo" Eqs. (26) by defining

• + • = q' + •r•/k q, • = q + •r•/k •, so that the (q,V) integral in Eqs. (27) becomes

• dq d{ exp( -jq.{)exp( - •k2r) d•

XF q--(q--q')•+ W [{ + (•-- })•]--wtd ß If both q and q' are of the order of the correlation length 1•, then as far as scattering is concerned, propagation is "quasi- monochromatic" as long as the condition

2 2 • • •r/kl • • 1 (28)

is satisfied. However, if q = q'= 0, as in the calculation of the two-frequency coherence of a point source (below), the surviving terms in the argument of F are of order a•. With F(q) • q", for example, the coherent bandwidth is then de- temined by the parameter •k 2 (r•/kl • ) n. A

To illustrate the application of Eqs. (27), we obtain G fora quadratic scatterer, where F = ( 1/•k 2) (1/1•) (q/lc)2

,) • •k 2 exp -- ½0 G = (•2 r P(p)

( , r ) Xexp -- 3P(p) lfi• [•2 +•'2 +•'q'(l +jp2)] , • = q -- Wta, •' = q'-- Wta, (29)

) = -g p2 + p4, p: = nr/t/ 2c. A

For plane-wave propagation, Fs (u,q)= 1, the MCF becomes

^ 1 exp( F (r,u,q;ll,td) = P1 (P-----• P1 (P ) = 1 --j2p 2 + « p4.

lr ) Pi(P) lf12c (q -- Wtd)2 ' (30)

For an omnidirectional point source of pressure, whose coherence is 6(u - Uo)6(q),, the MCF is readily obtained by setting u' = uo,q' = 0 in G of Eqs. (29).

The two-frequency coherence at a single receiver is ob-

tained by setting q = td = 0 above. Thus only the factors P and Pl in the last two equations determine the magnitude of the coherence [the distance u - Uo in gbo in Eqs. (29), defined as per Eqs. (27), contributes only a phase]. The frequency separation measure fl can affect these factors only via the parameter

p2= 11r2/klfl 2 = kll {gt2) lzr2/412 (31) c c,

where we have neglected the y2 term in the expression for the extinction length If; see Eqs. (12). In the case of a plane wave, we obtain the coherence bandwidth by setting 2p 2 = 1:

kd,½o h = 2kl](p2=0.5) = (41•2/{la2)lz)(1/r2). (32) This is independent of the mean (or carder) wavenumber k, but decreases as r -2. By comparison, the coherence bandwidth imposed by the exponential coefficient in Eq. (14) decreases with r- •/2, and is therefore irrelevant at sufficiently large ranges.

Note that the bichromatic coherence retains the Gaus-

sian dependence on the quantity q - Wtd, which obtains in the monochromatic limit, and also the ratio W between the spatial coherence length and the coherence time. However, the spatial coherence between two frequency components has a characteristic scale lq (11) (given below) that varies with the frequency separation. The width of a Doppler- spread spectral line scales as

,•7•,= W/lq(O), lq(O)= [21c/(kt21•)'/2](k2r)-'/2, lq(11) = lq(O)P•(p). (33)

We observe here that, with both a plane wave and a point source, the expression for the ta dependence of the coherence in the monochromatic limit can be obtained for an arbi-

trary scatterer. Equations (24), (25), or (27) will all yield the dependence exp[ - k 2rF(Wta ) ]. Thus the coherence time (or the Doppler linewidth) will always depend on the quantity kP/2. However, the direct proporfionality dis- played in Eqs. (33) above holds only in the quadratic scattering regime.

Finally, consider briefly the simplest anisotropic model for a quadratic scatterer, F = (1/If) (q2•/1 • + qy2/l 2 v ). Since the transverse coordinates are still uncoupled, the algebra is almost identical to that which produced, say, Eqs. (30) for plane-wave propagation. The eventual result is that the two- frequency coherence is given by [Pl (Ph)Pl (Pv) ] 1/2, where ,o h and Pv are defined as in Eqs. (29), with lc replaced by either I h or I v . In Eq. (32) for the coherent bandwidth, 1 •2 is replaced by [ 1/1 • + 1/12 v ] -i. Thus the shorter of two very disparate correlation lengths dominates the determination of the coherent bandwidth: To lowest order, the term that involves the longer correlation length may be ignored.

Similar reasoning does not apply to the coherence time' The ratio of the horizontal and vertical components of the transverse translatory motion may be of the same order as the ratio of the correlation lengths.

By way of assessment, we note that the various decorrelation phenomena and parameters that have been obtained here reflect our conventional understanding of these effects, which is based, by and large, on the quadratic scatterer as- sumption. We also note that, for plane-wave propagation

109 J. Acoust. Soc. Am., Vol. 85, No. 1, January 1989 Shimshon Frankenthal' Scattering coherence function 109


through a quadratic scatterer, the p dependence of the poly- nomial P•, Eqs. (30), reproduces the lowest two terms of the cos (2j) 1/2p dependence that is predicted by the exact solution, which is available in this case. 3 Thus, over the narrow range of frequency spreads encountered in practical applications, the approximations introduced by the two-scale procedure do not lead to serious errors in the results. The advan-

tage of this approximation is that it provides an explicit expression for the coherence in an arbitrary scattering medium.

IV. THE QUADRATICALLY REFRACTING CHANNEL

Uy2/l 2 The quadratic channel/• (uy) = • merits special attention, for several reasons. In the vicinity of the axis of the deep ocean channel, it provides a good approximation to physical reality. Also, the integrodifferential equation that arises in the course of the two-scale procedure accounts exactly for the refractive effects in this case. Finally, it allows a tractable analytical evaluation of the integral expression for the coherence, and thus provides a benchmark for testing numerical procedures.

In this case, the horizontal projections of the rays re- main straight lines. These projections and the attendant sensitivities are given by Eqs. (22), and the simplifications that arise from this can be handled precisely as in the preceding section, i.e., by the use of Eqs. (26). As for the vertical projections, Eqs. (18) and (19) yield sinusoidal rays

Vy = Oy COS •'- lray sin •, Vyo = cos •, Sy = (vy/lr)sin • + try COS •, (34) Vyo = -- l• sin g, •= • /l•,

which focus periodically at the ranges nl•. Noting that the "vertical" terms in the argument of F assume the form

(q•y - livy )cos • - l r ( •y -- •tYy )sin • = v-y cos • + •/y sin •, ( 35 )

we switch to the new variables of integrations •/y and •y, and also eliminate try in favor of

u•,y = vy cos 7 - lray sin 7, 7-- r/lr. (36) This yields a simplified superposition-integral expression, akin to Eqs. (27),

2 rrl r k f driy 2•r lr•

k x exp -j lr?l• + - q3 ) - )),

__ ,)2 •b• =fl(u• u• + (fl/4)(qx _q•)2

+ ( 1 d- 1'•2/4) ( Ux -- u•, ) (q• -- q•, ),

[ C( Uy + Uy ) -- 2UyUy ]

+ (•/4)[•(q• + q3 2) - 2qyq• ]

+ ( 1 + a2/4) [ (•uy -- u• )qy + (•u 3 -- uy )q• ],

Q = d• F • • + • - W, ta,•y sin • r

- ) +•ycos•-- Wyta ,

•sin?, ?•cos?,

(37)

which provides the coherence in terms of the geometrical coordinates u and q.

Equations (37) display fully both the vertical and the horizontal terms as arguments of the scattering function F. However, for the monochromatic regime, Ref. 7 shows that we can bypass the need to deal with the horizontal terms, provided that we rest•rict our attention to the horizontally averaged• coherence F•. This quantity is obtained by integrating F over u•, and setting qx = 0 (which implies averaging the transformed coherence over all horizontal direc- tions). The above argument does not extend rigorously to the bichromatic case. However, a simplification is possible when the conditions

Iv 'glh (38a)

(strongly disparate correlation lengths), and

a• ----rfl/(kl• ) ,g 1 (38b)

are satisfied.

To show this, we proceed to construct the expression for •x by setting q• = 0, and integrating H• in Eqs. (37) over u•. This leaves the (•7• ,•x ) integral, and also the phase exponent exp( :t,2, -- Jqx ) , where

•, l/2q• q•, = ( k /4fir) .

Next, following the procedure used earlier to reduce Eqs. (27) to the monochromatic limit, define 0• and b• so that the phase exponents of the integrals in H• and Hy are of order 1. With the definition

2 2 tzo• lrll/kl o

the dimensionless arguments of F read

Ct h [ 2q• (•/r) + • + (• -- • ) (•/r) ], (1/1o sin 7) [q} sin • + qy sin (7 -- •e) ]

+ av [0y sin •e + by sin (7- •) ].

When Iv •lh, the inequality ah ,gay prevails at ranges

r<lr(lh/lo) 2. With the usual forms of F, we may then expect to obtain the dominant behavior by setting the horizontal terms above to 0, and retaining qy and q}, of order Iv, as long as ah ,g 1. We find

110 J. Acoust. Sec. Am., Vol. 85, No. 1, January 1989 Shimshen Frankenthal: Scattering coherence function 110


^ X exp( -- •ak 2Q• ),

dg F( -- Wxta,•b sin g + •y cos g -- Wyta ), (38c)

where Hy is defined in Eqs. (37). The monochromatic regime discussed in Ref. 15 is reached when 12 is such that both ah and av are ,• 1. The thrust of the preceding discussion is that the departure from this regime is dominated by the vertical term in the argument of F, except at very large ranges where the cumulative effect of the horizontal terms becomes

significant. To provide the simplest illustration of the behavior of

the two-frequency correlation, consider again an omnidirectional point source, which has the coherence lo6 (u - Uo) 6 (q), in the presence of a quadratic scatterer. Since we set q = ta = 0 in order to obtain the two-frequency correlation, we need not resort to the horizontal averaging process in this case: It suffices to invoke an ,•av, and Eqs. (37) then yields

nitely spread in the point source limit), and provides the limiting behavior in regions of the order of the Fresnel radius kb 2 about the focal planes. At the focal planes, the bichromatic-averaged coherence reduces to

- k 1 1 exp •?(1 +y2f•2) , r = 7/2•r br x/•'?(1 + Ten 2) where the parameter $ measures the scattering intensity. In the absence of scattering, this expression reduces to $ (u•y), implying that the beam focusses to a horizontal line image of the source.

For the sake of completeness, we note that the time coherence of the radiation field of a point source, per se, is independent of the channel parameters. The expression for this quantity, in the presence of arbitrary scattering, is

•(r,u,O;td)- k2 Iø exp[--k2rF(Wtd)]. (41) 4rr 2 rlr sin ?

2 2

With a biquadratic scatterer 1/13(q2•/l• + q;/l o ) this produces the Doppler spread

•)[ = [ (/•2)1 z ]1/2( W2•/i• + Wy2/12• )krl/2. (42)

A • -- 1/2 r(r,u,0;fl) = Io k2 1 Pe 4• rlr sin ?

X exp[j/cfl( (u,,--Uo,,)2 : u_x 2 si.•n_•?/] sin • cos • / j' fl 112

Pe = 1 +j-• + n---•-' u•, =uy--Uo, cos•,

212 sin • '

(39)

= ' (72 -- sin 2 7)1/2. 4/2

The failure of these expressions near the focal planes sin 7 - 0 is discussed below. Elsewhere, the source-receiver displacements contribute only a phase term, which is linear in the wavenumber k corresponding to the carrier frequency. The coherence magnitude is determined by the remaining factor of Eqs. (39). The coherence bandwidth

kd, co h -• 2kfl 1 = 412 C.OS ? 1 (40) (/•2) lz l 2 s•n Y

is independent of the carrier frequency, and limited only by the vertical correlation length 1o.

The singular behavior of Eqs. (38) and (39) near the focal plane is "endemic" to the point source, and manifests itself even in the monochromatic limit, where the intensity is uniform across the transverse planes, and varies as (r sin r) - ', independently of the strength of the scattering. To resolve this behavior, we replace the omnidirectional point source by a beamed point source, which has a vertical coherence ( 1/b)exp [ 2 2) , - (q•/b ] or a Gaussian angular spread in the vertical direction that is parametrized by 0•, = 2/kb. This restores the dependence of the solution on the vertical distance u•x from the beam axis (which is infi-

V. THE INVERSE PROBLEM

The explicit expressions obtained here for the dependence of the coherence on receiver and frequency separations suggest that it is possible to infer information about the statistics of the scattering process from measurements of the spatial (or the directional) bichromatic correlation. In particular, in the regime fir/kl ], ,• 1, we may expect to accom- plish this for the dependence of F on vertical separations.

To illustrate, consider a simultaneous measurement of the bichromatic coherence in an arbitrarily scattering quadratic channel ensonified by an omnidirectional point source. Using Eqs. (37) with qx = 0, and invoking the regime fir/kl • ,• 1 with its attendant simplifications, we find

^ k 2 1

F(r,u,%;fi) - Io •-•-• H(v, qy )exp(jk•b),

H(¾,qy) lff 2•r

X exp[ --jv(•lyJ + •)(•y -- qy ) ]

X exp[ -- y2k 2Qx (r, Oy,1)-y ) 1,

Q• (r,•ly,•x ) = d• F(O,• b sin • + •y cos •),

k v=•, ?=sinT, ?=cosT,

•lr?

--(1 +--•)(Uoy--•uy)qy.

(43)

It is assumed that the deterministic refractive parameters of the channel are known, as are the locations of the transmitter and the receivers. For simplicity, we also assume (although this is not essential) that the phase factorjk•p can be disposed



of by suitable data processing. The question is, then, whether the integral H(v, qy ) in Eqs. (43) above can be inverted to yield the scattering factor Qx (r,•/,•) defined above, and hence also the scattering function F.

The answer is affirmative, if the two-frequency coherence decreases sufficiently rapidly with gl, to the point where terms in •a [which appear in the coefficient of l/; see Eqs. (12) ], can be ignored. The inverted integral reads

exp(-•ak•Q•)=••/y• 2 dv dqy X exp[jv(•ly• + •y•) (•y -- qy ) ]m( Y•qy )

(44)

as may be verified by substitution. In particular, setting •y = O, we find

d• F( 0,•?y sin •) = J(•/),

1 In 1 dv

x(faq exp(--jv•yqy )H(v, qy )), (45) and this may be inverted to obtain the form ofFexplicitly; as may be verified by expanding both F and J in power series.

We observe that the inversion entails two Fourier trans-

forms. The first is quite familiar. It transforms the dependence of the integral H on the vertical separation qy to a directional dependence. Note, in passing, that if one measures the directional dependence of the coherence [the Four- ier transform of Eqs. (43 ) ], the data must be retransformed to a qy dependence in order to permit the removal of the qy- dependent phase factor that multiplies the i•ntegral H. An inversion procedure that works directly on F can probably be worked out, but it would be equivalent to the procedures indicated above.

The second transform operates on the variable v, which is inversely proportional to the frequency-separation measure gl. Intuitively, one would expect to transform the separation itself, and obtain the spread-time spectrum of the signal as a means for determining the scattering function. At this time, we cannot offer an intuitive physical interpretation to the mathematical fact that 1/gl is the variable to be used.

VI. SUMMARY AND DISCUSSION

The method of two-scale embedding has been employed in this article to derive an approximate solution of the equation that governs the mutual coherence function in a scattering refractive channel. This extends the previous theory, in that: (1) A general (though approximate) expression for the mutual coherence function, with arbitrary ensonifica- tion, is provided; and (2) the effects of refraction (i.e., an inhomogeneous deterministic background) are incorporat- ed, to the level of the quadratic approximation. The imple- mentation requires tracing the rays and their sensitivities: The latter are needed to track the effect of scattering, as well as the usual effect of geometrical ray spreading.

In the monochromatic limit, the resulting equation exactly reproduces the existing solutions for nonrefractive and quadratic channels. For bichromatic propagation through a nonrefractive, quadratically scattering medium, the solution constitutes a good approximation to the known exact solutions, over the narrow range of frequency separations that are envisaged in practical cases.

The theory contains a term that accounts for Doppler line spreading due to a frozen translatory scattering structure. It can be expected to apply, say, in the presence of a moving eddy, but may not be applicable to scattering by internal waves (where the time dependence enters through a dispersion relation, rather than by the "frozen-in" model). In channels for which the ray sensitivity is independent of the ray parameters, it is noted that the Doppler broadening of a pure tone from a point source is independent of the refractivity profile, and simply reflects the character of the scattering function. It scales with kr 1/2, but is proportional to it only in a quadratic scatterer; see comments ff [Eqs. ( 33 ) ] and preceding Eq. (41 ).

In a quadratic channel, the theory accounts exactly for the effect of refraction, but only approximately (the limitation of the two-scale procedure) for the effect of scattering, on the bichromatic coherence. In anisotropic media with widely disparate horizontal and vertical correlation scales, a simplification can be gained by considering the horizontally averaged coherence. In the regime glr/kl • ,• 1, the horizontal scattering terms can then be disregarded. This is tanta- mount to the intuitive argument that the effect of vertical scattering becomes manifest much more rapidly, and thus plays a dominant role in setting the coherent bandwidth. In the computation of the two-frequency correlation of the field of a point source, horizontal averaging is unnecessary. Ana- lytical scaling laws for the coherent bandwidth are obtained for the case of a quadratic scatterer.

A simple expression is obtained for the vertical variation of the MCF, when a quadratic (or uniform) channel, with arbitrary scattering but very disparate correlation lengths, is ensonified by a point source. This expression contains a phase factor, which reflects the transmitter and receiver locations, and a transformlike integral involving the unknown statistics of the vertical scattering process. It is found that this integral is invertible. In principle, this shows that mea- sured data can be processed so as to determine the scattering function of the medium. For this application, it is easy to simulate point-source propagation with a physical source. It suffices that the linear source aperture be smaller than the smallest correlation length to be detected. By comparison, if we were to employ the time coherence of the point-source field for measuring the scattering function (see above), the source aperture would have to be smaller than the distance ta traversed during a time-resolution element, which is likely to be more difficult to achieve.

The expression that is presented here, Eqs. (20), permits the calculation of the full MCF, or any special feature thereof, for an arbitrary refractive and scattering channel, with an accuracy consistent with the various approximations cited in the course of the discussion. Several possible avenues may lead to simplified versions of this expression,

112 J. Acoust. Sec. Am., Vol. 85, No. 1, January 1989 Shimshen Frankenthal: Scattering coherence function 112


which would reduce the computational effort involved. These possibilities are currently being explored.

ACKNOWLEDGMENTS

The author wishes to express his gratitude for the hospi- tality of the Department of Ocean Engineering at the Massa- chusetts Institute of Technology, Cambridge, MA, where he was visiting when this work was performed. This work was supported, in part, by the Charles Stark Draper Laborato- ries of Cambridge, MA, under Contract No. DL-H-261815, and in part by the O/rice of Naval Research, under Contract No. N00014-77-C-0226.

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the mutual coherence function in a scattering channel— a two-scale solution

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