evolution of local spectra in smoothly varying nonhomogeneous environments—local canonization and...

Evolution of local spectra in smoothly varying nonhomogeneous environments--Local canonization and marching algorithms

Ben Zion Steinberg Department of Interdisciplinary Studies, Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel

(Received 4 June 1992; revised 30 November 1992; accepted 12 December 1992)

By applying the windowed Fourier transform directly to the Helmholtz wave equation a new formulation that governs the evolution of the local spectra of wave fields in a general nonhomogeneous environment is derived. By further invoking the so-called locally homogeneous approximation, a simplified evolution equation, termed as the locally homogeneous wave equation is developed, together with an upper bound on the error associated with the approximation. It is shown how simple analytical solutions of the new wave equation in a general smoothly varying nonhomogeneous environment can be obtained using well-known analytical techniques, and how the marching methodology connects these new solutions to the original problem described by the Helmholtz equation.

PACS numbers: 43.30.Bp, 43.30.Dr, 43.20.Bi

INTRODUCTION

Propagation models for wave phenomena governed by linear differential equations have been traditionally based on the expansion of the radiation field by sets of basis functions like spatial Fourier harmonies, modes, and Green's functions. Essential to the utility of this approach is the requirement that the evolution of the basis functions through the propagation environment constitutes a simplified problem with an exact or approximate closed- form solution. The simplified problem can formally be obtained by applying a properly defined transformation to the original wave equation. Thus, for example, the evolution of a spatial Fourier harmonic through a homogeneous medium is governed by a reduced wave equation, obtained by applying the Fourier transform to the Helmholtz equation, with a simple and well-known solution--the plane wave. Similarly, modal representations of wave fields in layered media are governed by reduced wave equations of the Sturm-Liouville type obtained by applying a Fourier transform along the coordinate perpendicular to the medium variability. However, since global basis functions like plane waves occupy the entire domain and point source excitation radiates to all directions, their evolution through a nonhomogeneous environment constitutes a problem that may become at least as difficult to solve as that of the propagation of the total field. From the mathematical point of view, the difficulty stems from the inability to get, with the traditional global techniques like the Fourier transform, a simplified differential equation that governs the evolution of the basis function through the nonhomogeneous medium.

The recognition of the propagator global nature as the difficultie's physical genesis led researchers to develop and investigate propagation models based on the expansion of the total field by sets of spatially confined wave objects like complex source-generated beams and Gaussian beams. 1-5 In analogy with global representations, two essential steps

are involved in the computation of the propagating field. The first is the expansion of the initial data as a weighted sum of the spatially confined basis functions, i.e., construc- tion of a spectrum. The second is the evolution of the beam basis functions through the propagation medium. The first is based on well-established mathematical tools like the

complex Huygens principle used for the expansion of a point source excitation, 6'7 or like the windowed Fourier transform (WFT) used for initial plane data. 8 The second step, however, still raises difficulties, mainly due to the lack of a simple equation governing the evolution of the basis functions through the nonhomogeneous medium. The only technique available is to track Gaussian beams locally along curved ray trajectories. 2'3 This can be done as long as the beamwidth and wavelength are small compare to the inhomogeneity lengthscale. It is well known, however, that every beam-type propagator is bound to diverge and to loose its localization properties after a certain propagation distance--the diffraction length. Thus, ocean acoustic applications where nonhomogeneous environments and arbitrary large propagation distances are considered may render these schemes incompatible. Moreover, it has been observed that ray trajectories in such environments may exhibit chaotic behavior, 9-11 thus increasing the scheme implementation complexity. Finally, to the best of the au- thors knowledge, a systematic framework for assessing the error associated with the aforementioned technique has not been developed.

Motivated by the observation that the spatial confinement of beam-type propagators can be kept as long as the propagation distance is not large, a framework that com- bines the methodology of marching a radiation field with the windowed transformation approach has been recently suggested. 12 It considers the evolution of the WFT of the field along the propagation path, instead of the evol. ution of the associated bases. Since the scheme plays an important role in the present work, we shall briefly summarize its essentials. The WFT U(•,•,Az) of a two-dimensional field

2566 J. Acoust. Soc. Am. 93 (5), May 1993 0001-4966/93/052566-15506.00 @ 1993 Acoustical Society of America 2566

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•-30

The phase-space propagaLor

T ø ,,o I i I i i

3O 1

(a)

_

0.2 -

_

-0.2 -

_

-0.6 -

_

(•/x.•,O=(o..o.4) windows width •=2X

sLep size bz=2X

-02

_

- -0.2

_

- -0.6

_

L I I I I I I I I I I I -1 30 -20 -10 O 10 20 30

•/x

(b)

FIG. 1. The phase-space short-range propagator in a homogeneous medium. (a) A contour plot in the (•0,•0) plane for a given point in the (.•,•) plane. The outer contour shows 5% of the highest magnitude. (b) A physical domain interpretation.

u (x,/Xz), along its transverse coordinate x [see Eq. (9) for definition] is related to the WFT U(,•,•,0) of the field u(x,O) via

=ff (1)

Following the interpretation of U(,•,•, z) as the description of the field u(x, z) over a space of locations (,•) and directions (•)mthe phase space representation--one can further interpret the phase-space short-range propagator B(,•,•; ,•0,•0,•z) as a measure of the degree of coupling of the (-•0,•0) phase-space constituent on the plane z=0, to the (,•,•) phase-space constituent at the plane z=/Xz (see Fig. 1 ). Using asymptotic analysis of the phase-space propagator, it has been shown that this coupling is strong if the phase-space points (-•0,•0) and (,•,•) belong to the same straight ray, and it decreases exponentially in those phase- space points that do not form together a ray trajectory.

Accordingly, for a given (,•,•), B is centered in the (-•0,•0) plane around the "skeletal points:" those points in phase space connected to (,•,•) by a straight ray trajectory. The phase-space short-range propagator for a homogeneous medium is shown in Fig. 1 (a) and its localization around the skeletal point, schematized in terms of a ray in the physical space, is shown in Fig. 1 (b). Although the propagator in Ref. 12 has been derived only for a homogeneous medium, it was argued that, due to the spatial and directional confinement, it can also be used in smoothly varying nonhomogeneous environments provided that the window width and the step size are small compare to the medium variability. This condition was termed as the locally homogeneous approximation. Numerical simulations, performed for a homogeneous medium and for a smooth layered medium, have shown how the phase-space representations kept their local nature when marched over ranges of hun- dreds of wavelengths.

A few important aspects, however, were not addressed directly. Since the analytical framework used for comput- ing the propagator was that of a homogeneous medium, estimates on the error associated with the locally homogeneous approximation in a smooth environment could not be derived directly from the formulation. The role of the three lengthscales--wavelength, window width, and inhomogeneity, as well as the regime of validity of the locally homogeneous approximation and its connection to the step size were discussed only heuristically. Finally, and most important, although it was pointed out that the approach may be suitable to handle propagation in range-dependent media, the arguments were primarily qualitative and no specific analytical or computational framework was developed to handle such environments.

The present work approaches these difficulties from a wider point of view. The evolution of the local spectra U(•,•,z) in a general nonhomogeneous medium is addressed on the fundamental level of an exact governing equation, derived directly from the Helmholtz wave equation. Although this general evolution equation is difficult to solve, it provides a "safe ground" from which approximate, and much simplified, differential equations can be formu- lated. Thus, for the first time, a simple approximate equation that governs the propagation of windowed spectra through a general, smoothly varying nonhomogeneous medium is derived and an upper bound for the associated error is obtained. This equation is termed here as the locally homogeneous evolution equation and it is shown how it generalizes the concept of "local homogeneity" discussed heuristically in Ref. 12. The role of the three lengthscales--wavelength, inhomogeneity, and window width is investigated in detail and it is further shown how the error upper bound can be controlled by the width of the window.

The scope of the evolution equation approach, however, is far beyond that of addressing the above aspects. On a somewhat abstract level of description, the WFT projects the Helmholtz wave equation, that "lives" in the (x, z) space, onto an extended domain of mixed location direction and range coordinates (•,•, z). Due to the overcom-

2567 J. Acoust. Soc. Am., Vol. 93, No. 5, May 1993 Ben Zion Steinberg: Evolution of local spectra 2567


pleteness property of the WFT, the latter space is much "bigger" than the former, •3 thus allowing for added de- grees of freedom in seeking new fundamental solutions. The unique solution of the original physical problem, which must (a) satisfy prescribed boundary conditions and (b) lie within the range of the map defined by the WFT, is then described by an appropriately defined summation of the new fundamental solutions. While condition (a) above is quite straightforward to meet, it is shown that condition (b) can be met only by incorporating the marching methodology.

How do the ideas above actually materialize? On the more "algorithmic" level of description, the strength of the new formulation lies in the specific structure of the locally homogeneous evolution equation. Let us recall, for a mo- ment, the structure of the Helmholtz wave equation [see Eq. (2)]. The inhomogeneity of the medium, expressed by n(x, z), does not commute with the differential operators associated with the equation. From the mathematical point of view, this can be regarded as an essential source of difficulty in solving the wave equation in nonhomogeneous environments. In the locally homogeneous evolution equation, defined in the extended (•,•, z) domain, a general inhomogeneity becomes "parametrized" in at least one direction. That is: it commutes with the differential operators along, at least, one coordinate (the coordinate along which the WFT is performed). This fact allows for particularly simple solution techniques, that are completely analogous to well-known techniques used traditionally only in one- dimensional (or layered) wave equations. This procedure is termed here as "local canonization." For example, new fundamental solutions like windowed plane waves that generalize the conventional plane waves in a nonhomogeneous medium are derived. The WKB method, applied traditionally in layered media, can now be applied in general inhomogeneity, and the turning point condition of the conventional WKB implementation that has a "plane," or global, sense (x or z independent), becomes now a local turning condition--x and z dependent, signifying a local turning event of a windowed plane wave. Similarly, a solution of a plane wave at a turning point in a layered medium, expressed in terms of an Airy function, •6 is generalized to a solution at a local turning event in a general inhomogeneity. It is anticipated that the last result is of fundamental importance for marching/propagating fields in range-dependent environments. (We repeat for emphasis that although the locally homogeneous evolution equation is approximate, an explicit, simple, upper bound for the associated error is provided and it is shown that one can always tune the window width as to render the associated error acceptably small.) However, while these new solutions satisfy the new wave equation in the extended domain, they in fact express wave constituents that do not belong to the function space M(R 2) defined as the range of the map established by the WFT. That is, they cannot be expressed as WFT of legitimate, finite energy fields. Nev- ertheless, it is shown here that by a properly defined summation of the new fundamental solutions, one may construct a solution so that its "distance" in function space

from M(R 2) has an upper bound proportional to the propagation distance and to the ratio between the window width and the inhomogeneity. This fact motivates the use of the new approach in conjunction with the marching methodology.

The structure of the paper is as follows. In Sec. I, after the introduction of some basic notations and a brief dis-

cussion of the WFT, the new evolution equation is derived and the locally homogeneous approximation is introduced and discussed. The procedure of local canonization, together with analytic solutions of the locally homogeneous evolution equation in layered media, in general smooth inhomogeneity and near local turning events, are presented in Sec. II. This section explores also the use of the marching approach in conjunction with the new solutions and shows how it connects to the unique solution. In Sec. III we compare the formulation developed here to related schemes such as the phase-space Gaussian beam summation method (GBSM), 2-5'8 the parabolic equation approach, TM and its generalization, •5 and point out the connection between the evolution equation approach and the GBSM. Section IV discusses some specialty aspects con- cerning the choice of the window to be employed in the WFT. Numerical demonstrations are provided in Sec. V and concluding remarks in Sec. VI.

I. FORMULATION

In this section an exact integrodifferential evolution equation for the WFT of the field is derived and reduced to a simplified form using the so-called "locally homogeneous" assumption. The conditions under which this assumption is valid and the properties of the new evolution equation are discussed.

Our starting point is the two-dimensional Helmholtz equation in a nonhomogeneous medium (medium density assumed constant),

[c92+c92 kg n2(x,z)]u(x,z) O, Z X + = (2)

where n (x, z) = Vo/V (x, z) is the index of refraction, defined as the ratio between a reference wave speed Vo and its local value v(x, z). Here, ko=w/v 0 is the wave number and co is the radian frequency of the time-harmonic field. To facilitate compact representation of the forthcoming derivations we shall rewrite n2(x, z) in the form,

n2(x, z) = 1 +Ix(x, z). (3)

An initial field Uo(X)=U(x,O) is assumed to propagate from the z=0 plane into the z > 0 half-space. Accordingly x and z are termed as the cross range and range directions, respectively, and the initial field is said to propagate in the down-range direction.

Since the refraction index generally depends on z, bi- furcation and back refraction are likely to occur. Thus, the down-range radiation condition is not conserved along the propagation path. Clearly, any specific solution of Eq. (2) must be accompanied with auxiliary radiation conditions.

The field Uo(X) and its spectrum •0(•) are related via the Fourier transform pair,



U-o(•) =; Uo(x)e -iø•gx dx, (4) Uo(X) =•-• ffo(•)e i•øgx d•. (5)

It is well known that in a homogeneous medium (/.t=0) the field radiating into the z > 0 half-space is given by the summation,

U(X, z)=•-• •'o(•)e icø(gx+v) d•, (6) where

•=v• -1 x/1-(Vo•) 2, Re •>0, Im •>0. (7) The synthesizing integral in Eq. (6) describes the field as a superposition of plane waves, propagating along the direction

0 (•) = COS-- 1 ( U0 •) ( 8 )

relative to the z axis, with excitation amplitudes given by the initial spectrum fro (•)-

A. The windowed Fourier transform of the field

Here we define the windowed Fourier transform

(WFT) of the field and briefly discuss some general fea- tures important for subsequent derivations.

Following Ref. 12, the WFT of the field u(x, z) along its cross-range coordinate is written as

U(.ff,•-, z) =; u(x, z)w*(x--.ff)e -kø•x dx --2½r ff(•, z)•*(•--•)e i•ø(g-•)x d•, (9)

where w(x) is a square integrable spatial window function centered more or less around the origin and the tilde denotes the conventional Fourier transform as defined in Eq. (4). The field u(x, z) can be reconstructed from its WFT via,

oo f f z)w(x_.)eiogX ' u(x, z) --2•rN 2 (lO)

where N is the "L 2 norm" of the window, defined by

N 2-- f Iw(x) 12dx. (11) Generally one uses a spatially confined window [i.e., w(x) is either of compact support or it tapers fast to zero so it can be "practically" set to zero outside a given interval]. Thus, Eq. (9) can be interpreted as a local measure of the spectral content of u around •. Similarly, Eq. (10) can be interpreted as a description of the field over a space of locations • and directions Vo• (see Refs. 8 and 12). Thus, the name phase-space representation is often used. A survey of the WFT can be found in Ref. 13. Its application to phase-space beam-type representations of radiation from apertures has been considered in Ref. 8. The evolution of

the WFT in a smoothly varying layered media has been pursued in the context of marching algorithms in Ref. 12, together with a detailed discussion of the motivation for this approach and its physical interpretation.

Let us consider now several characteristics of the

WFT that will be referred to in the forthcoming analysis. A straightforward application of the Cauchy-Schwartz inequality to Eq. (9) yields

I l <N. Nu(z), (12)

where Nu is the L 2 norm of the field at a given range coordinate,

N2u(Z) = f lu(x,z)12 dx. (13)

Hence, for finite energy fields, the WFT is always finite. It is also straightforward to show that U(•,•, z) at a given z is square integrable,

12dd 2 2 =•NN u . (14)

Thus, for a given z, Eq. (9) defines a map from the vector space L2(R) (assuming a finite energy field) to the vector space L2(R2). Since the latter is "bigger" than the former, the range of this map is the space M(R 2) C L2(R2), which is much smaller than L2(R 2) and the inversion formula (10) is not unique. •3 We shall return to discuss this point further. An important identity that will be used later is obtained by operating with Eq. (9) on both sides of Eq. (10),

U(•o,•o, z) B(•,•; •o,•o,0)d•o d•o,

(15)

where, up to a multiplicative constant, B(•,•; •o,•o,0) is a WFT of a shifted and modulated window,

B(X,g; Xo,go,0)

f -- 2•-N2 w*(x--•)w(x--•o)e -iø•(•-•ø)x dx

= 2-• d•ø(xø•ø-g) 5'(•-•) X •(•-•-o)e iø•(x-xo)g d•. (16)

The reason for the additional variable with the value of

zero in the arguments of B will become clear later. Al- though Eq. (15) reconstructs U(Z,•,z) from itself, the kernel B(•,•; •o,•o,0) is not singular. It is given by a WFT of the window--a function in L2(R)--hence it is always finite [see Eq. (12)]. This reconstruction is not unique. There are many kernels that can replace B in Eq. (15) and still reconstruct U correctly. An obvious example is the singular distribution tS(•--•o)tS(•--•o). However, among all possible kernels, B in Eq. (16) is the only one within the range of the map defined by Eq. (9), i.e., it is the only one that can be expressed as a WFT of a finite energy function.

2569 d. Acoust. Soc. Am., Vol. 93, No. 5, May 1993 Ben Zion Steinberg: Evolution of local spectra 2569


B. An exact integrodifferential evolution equation using Gaussian windows

An exact equation that describes the evolution of the WFT of the field along the propagation path can be obtained by operating with Eq. (9) on Eq. (2). Although the following derivations can in principle be applied with any kind of window, we shall restrict the analysis to WFT obtained by using Gaussian windows only. This choice is made for two reasons. The Gaussians localize best in phase space and they convey analytically tractable results.

In keeping with the notations in Refs. 8 and 12 we shall use the window,

W( X ) =e-a•( a/2 )X2-- e- (X//l )2/2A2 (17)

•(•) = (2•r/3/to)1/2e-tø(/•/2)•2 ( 18 ) where

l•= l/a (19)

and for which

N2= (z'/wa) 1/2. (20)

Here a is a real positive parameter that determines the window width and

A= (2•a/•v) -1/2 (21)

is the window width in units of wavelength. Also B(g,•; g0,•0,0) is the two-dimensional-modulated Gauss- ian

exp --w 2 - - 2 (g--go)

-i• (•-•o)(•+•o) ß (22) The specific choice of the window width (namely, the parameter a) will be discussed in detail in Sec. IV. It can be shown that, with (17) the WFT of a function f(x) de- noted F(•,•), and the WFT of its derivative f' (x) de- noted Fø(•,•), are related via (see Appendix A)

Fø(:•,•) = DF (2,•) (23) with

D=iaO•--wa2+iw•. (24)

[This identity is valid if and only if F(•,•)•M(R2).] Ac- cordingly, by operating with Eq. (9) on the Helmholtz wave equation (2) we get

(0• 2 + D. D + k•) U(2,•, z)

-- -kg ; !•(x, z)u(x, z)w*(x-:•)e -iø'•' dx, (25) where the notation D. D is used instead of D 2 to avoid the problem of operator ordering for the noncommuting terms 0• and • (see Appendix A). Expressing u(x, z) in the right-hand side of Eq. (25) as an inverse WFT [Eq. (10) ], we get

(0• 2 + D. D+ k•) U(.•,•, z)

; C(2,•; .•'o,•'-o, z)U(•o,•-o, z)d:•o d•-o , (26)

where

C(•,•; •0,•0, z)=kg 2rrN 2 l•(X, z)w(x-:• o) X w*(x--•)e -i'ø(•-•ø)• dx, (27)

and where/• is defined in Eq. (3). Equation (26) is the exact integrodifferential evolution equation of the local spectra U(2,•,z). In its present form, it is more difficult to solve than the original Helmholtz equation. Nevertheless, it provides a systematic framework for various approximations by which it can be reduced to much simpler forms, allowing for straightforward, physically meaningful solutions.

C. The locally homogeneous approximation

The integral operator in the right-hand side of Eq. (26) represents coupling of phase-space constituents due to medium inhomogeneity. This coupling decreases as the distance between •70 and • increases. Moreover, Gaussian windows render the kernel C, as well as the integrand of Eq. (27), exponentially small when Ix-l or Ix-01 is large compared to the typical width of tlie window. Similar considerations can be applied to the right-hand side of Eq. (25). This suggests the following expansion for continuously varying media:

•(X,Z)=•(•',Z)-•- Z •m("•,Z)( x--"•')m, (28) m=l

where

•m(.•, Z)= (1/m!)O x•/• (x, z)I•=•. (29)

Substituting into Eq. (25), using Eq. (3), and utilizing the identity x,• (i/w) O• we get

[0 •2 + D. D+ ko 2 n2 (.•, z)lU(•,•,z)

=--kg Z •m('• z) 0•--• U(.•,•-•z). (30) m=l

[The same result can be obtained by substituting Eq. (28) into Eqs. (26) and (27) and utilizing Eq. ( 15).] The inhomogeneity of the medium is taken into effect by the term n2(•, z) in the left side, and by the operators on the right side. The point is that, while the former is just a multiplicative function depending solely on the medium properties, the magnitude of the latter can be controlled by the width of the window. An easy calculation shows that (see Ap- pendix B)

I[ (i/('ø)O•--'•]mu(•,• z) I <Nu(z)F1/2(m+«)

X (lo•t) -(1/2)(m+1/2), (31)



where Nu is defined in Eq. (13) and F(x) is the gamma function. From Eq. (17), (we) -1/2 is the width of the window. The typical length scale of the inhomogeneity in the g direction, given by Lx(g, z)--n(g, z)/I cg•n(g, z) I , is approximately I/•l(g, z) l -1. Therefore, in a continuously varying medium the terms associated with the inhomogeneity on the right side of Eq. (30) can be made as small as one wishes, and the series can be truncated. From Eq. (31 ), the associated error is bounded by the energy of the field and by an inverse power of (we)1/2. The locally homogeneous approximation is obtained by choosing a window such that

I (coa)- 1/2 ß/• (•, z) I ,•1, (32) so that the series can be neglected altogether, with an error proportional to (toe) - 3/4. The resulting equation,

[O 2+D. D+k• n2(g, z) ] U(g,•, z)=0. (33) z

is referred to as the locally homogeneous evolution equation. It should be emphasized that its validity depends on

the inhomogeneity length scale in the x (cross-range) direction, as well as on the width of the window. Since at this point there is no limit on how narrow the window can be, one can always tune e as to render the associated error acceptable. The fundamental advantage of the procedure outlined above, and of the resulting evolution equation, stems from the fact that the cross-range dependence of the inhomogeneity, namely the g dependence of n2(•, z), has been "parametrized." That is, it commutes with the operators associated with the evolution equation. As we shall see in the next section, this allows for particularly simple solution procedures.

II. SOLUTIONS OF THE LOCALLY HOMOGENEOUS EVOLUTION EQUATION

A. General considerations

Any meaningful solution of Eq. (33), attempting to describe a solution of the physical problem defined by Eq. (2) and by the auxiliary boundary conditions must admit the following properties:

U(g,•-,O) -- f Uo(X)W*(X-- g)e -iø•x dx-- Uo(g,•) (34)

and

U(g,•-, z)•M(R2). (35)

Equation (34) is simply a boundary condition. A brief discussion of condition (35) is warranted. It states that the solution must be within the range of the map defined in Eq. (9), that is, at any z, it can be expressed as a WFT of a finite energy field. This is not only a "physical" condition. The procedure leading from the Helmholtz equation to Eq. (26) is not valid if (35) is not admitted [see Eq. (23) and Appendix A]. However, since Eq. (33) is linear and homogeneous one may attempt to synthesize, or at least approximate, a solution that belongs to M(R 2) by a properly defined summation over fundamental solutions not neces-

sarily belonging to it. This added degree of freedom, not available with conventional Fourier analysis, is a direct result of the overcompleteness property of the representation in Eq. (10). It will be used throughout the rest of this paper. [For example, it is recognized that the harmonic functions e ikl•+ik2• do not admit condition (35), nevertheless, since they constitute a basis of L2(R2), any U(g,•-, z) •M(R 2) C L2(R 2) can be synthesized, at a given z, by a proper summation over kl, k2.]

B. Layered media

The methodology of solution articulated above is particularly simple in layered media, for which n2(•,z)-•n2(•). It is straightforward to show that the windowed plane wave, defined by

W• (.•,•, z)=•*(•--•)exp[iw .•(•--•) +iw•(.•,•)z]

-- (2•r/•/o)1/2exp [- o (/•/2) (•_•)2

+ico •(•-•)+iw•(•,•)z], (36)

where • (•,•) generalizes (7),

•(•,•)=v• -l•/n2(•)-(vo•) 2, Re •0, Im •0, (37)

and for which

DW•(•,•, z)=ico• W•(•,•, z), (38) is an exact solution of Eq. (33). A brief discussion of this new solution is warranted. Unlike the conventional plane wave, that constitutes a fundamental solution of the Helm- holtz equation in a homogeneous medium and propagates along a constant direction [see Eqs. (6)-(8)], the propagation direction of W• is generally not a constant. The new solution W• describes a wave radiating at the angle

0 (•,•)-- COS-- 1 [/20•(•,•) ] (39)

relative to the z axis. This direction depends on the local properties of the medium, taken into effect by n(g). More- over, the window function •* (•-- •) localizes •, which "parametrizes" the solution, to the neighborhood of •. Thus the radiation pattern of W• is confined essentially around the angle 0(g,•) relative to the z axis. We shall refer to W• as the fundamental solution of the locally homogeneous evolution equation in layered media. It is clear that any summation of W• over • is also a solution. Fol- lowing the discussion after Eq. (35), we shall look for the specific sum that satisfies conditions (34) and (35).

One may check by inspection that the solution defined by the • superposition,

z)

= 2-N z),/g, (40) becomes identical with the reconstructing kernel in Eq. ( 15 ) when z vanishes [see definition in Eq. (16) ]. It follows therefore that



(41)

is an exact solution of the locally homogeneous evolution equation in layered media that satisfies the boundary condition in Eq. (34).

Condition (35) is more difficult to satisfy. In fact, it is not strictly admitted by the solution defined in Eq. (41). Nevertheless, one may hope to satisfy it approximately. It is evident from the superposition integral (41 ), that (35) is admitted if and only if B(•,•; •0,•-0, z) •M(R2). It can be shown that for short ranges z-• Az, such that

X0v0G(•,•0) It*•(•) I (•øa)-•x2• 1 (42) and if the width of the window is in the order of (or larger than) a wavelength, B can be approximated by •M(R2), with an error bounded by (see Appendix C for details),

7r 1/2

IB-l02 I/•l(•) I (røa)-l/2' (43) which is proportional to the range Az and can be controlled by the ratio between the window width (roa)- •/2 and the inhomogeneity length scale (/•i-1). When restricted to initial radiation angles such that (o0•-0) 2 < n2(,•), Eqs. (42)- (43), which are the algorithmic manifestation of condition (35), provide the motivation for the marching methodology advised by Eq. (1). We shall therefore refer to B in (40) as the short-range propagator of the locally homogeneous approximation in layered media. Being aware to the interpretation of the WFT as a phase-space representation, the name phase-space short-range propagator will also be used. It gives a measure of the degree of coupling of the phase-space (location-direction) component (g,•) at the plane z--ZXz to the phase-space component (g0,•0) at the plane z=0. Clearly, in a homogeneous medium Lu• (g) =0] it constitutes, together with Eq. (41 ), an exact solution of the problem. Equations (40) and (41 ) are the phase-space short-range propagator and the marching algorithm that have been obtained for a homogeneous medium in Ref. 12 using different considerations.

Remarks:

(1) Note that the error estimates (42) and (43) are not valid if the window width is small compare to a wavelength. The limit of a very narrow window awaits further study.

(2) The • integration in Eq. (40) has the form of a FFT integral, with g0 in the dual domain. The computation of the short range propagator is therefore easily implemented.

C. Smooth nonhomogeneous media and local canonization

Fundamental solutions for the locally homogeneous evolution equation (33) in general inhomogeneous, or range-dependent, environments can be sought after by substituting the ansatz

= (2rrfl/w) 1/2exp [ --co (fl/2) (•-- •) 2

+ i•o•(•- •) 1,• (•, z,•) (44)

into Eq. (33). Since no derivatives with respect to 2 or • are associated with the operator D, Eq. (38) still holds. The expression in (44) is thus an exact solution of the locally homogeneous evolution equation, provided that A (•, z,•) satisfies the one-dimensional wave equation,

[Oz2-q- 092•2(.•, Z,•')]A(.•', z,•'):0, (45) where •(g, z,•) is a generalization of (37),

•(•', Z,•):V• -1 •/n2 (,•, J)- (o0•) 2. (46)

Clearly, • and • are parameters. Here lies one of the fundamental advantages of the present approach. A plethora of asymptotic and numerical techniques, developed to handle one-dimensional wave equations like (45) in general and to handle propagation in layered media in particular, are now available (WKB, Bremmer series, the method of canonical mapping and modal analysis, to name a few•6). These solution techniques, when applied to Eq. (45) become "parametrized" by the transverse location coordinate •. This is to contrast with their traditional implementations in layered media where each fundamental solution is independent of at least one location coordinate and represent a global plane, or modal, wave. Thus, Eqs. (44) and (45), together with the conditions in Eqs. (34) and (35) and with the superposition of fundamental solutions like Eqs. (40) and (41 ), provide the systematic framework for applying a hierarchy of asymptotic techniques in a truly local manner (i.e., not in a plane wave or layered sense). This procedure, referred to as "local canonization," will be demonstrated in the next sections.

1. WKB solutions and Bremmer series

A large co asymptotic solution to Eq. (45) may be constructed by using the WKB method (Ref. 16, Chap. 3.5c). The result can be written as

A (.•', z,•) = [•(.•,0,•)/•(.•, z,•)]1/2. (A+ei•o•+A-e-i•o•),

(47)

where A + and A- are arbitrary amplitudes and

i0 •= •(•7, z',•)dz'. (48)

Together with (44), we get two fundamental solutions describing down-range (+z) and up-range (--z) propagating windowed plane waves,

W•: (.•',• Z): [•(.•',0,•)/•(.•, Z,•) ]1/2. X exp [kog(•--•) + iro•b]. (49)

This solution has a local turning point zt(Z,•) along the z axis, given by

•(•', Zt,•) :0. (50)



Unlike the conventional global (x-independent) turning point of the WKB solution, this turning point is of local (• dependent) nature. It gives both the range and cross-range coordinates at which the windowed plane wave undergoes a "turning around" or back refraction event. At this point (or set of points) the down-range and up-range fundamental solutions strongly couple and the asymptotic expression in (49) fails. Nevertheless, it provides the information about local turning around events for which more robust asymptotic techniques (canonical mapping, matched as- ymptotics) should be applied (see next section).

We assume at present that no back refraction takes place. Also, auxiliary radiation conditions are specified for the field in the initial (z=0) plane,

u(x,O) (x) + (x)

and for the associated local spectra,

Vo(X,) = Vo + + v;

(51)

(52)

where as previously the + signs correspond to down-range (+z) and up-range (--z) propagation, respectively. Fol- lowing the procedure of the last section we define the short-range propagators

B (x,g; x0,g0,az)

•(•._•o)e i,o Xo(•o- g) W• (•,•,Az) d•, (53)

where W + is defined in Eqs. (48) and (49). Evidently both B + and B- become identical to Eq. (16) when Az vanishes. (It is clear that B depends now on the propagation distance Az as well as on the absolute range z. For simplicity of notation the absolute range is suppressed here and henceforth.) The local spectra at a short range step Az > 0 is given now by

u(x,g, az) = u + (x,g, az) + u- az),

where

(54)

B + (•,•; •o,•o, Az) U•: (•o,•o) d•o d•o. (55)

Equations (53) and (54), together with (49), constitute a local WKB solution of the locally homogeneous evolution equation in range-dependent environments. By virtue of (53) and (15), the initial conditions (52) or (34) are automatically satisfied. As with the layered media case, however, (35) can be satisfied only approximately. By following essentially the same procedures outlined in Appen- dix C one may show that the range-step conditions in Eqs. (42) and (43) still hold, with /.tl Az/•' replaced by

Vo2(d/d•)• I•,aZ,•o [see Eq. (48)]. As with the previous case, the computation of the short-range propagator is easily implemented by a FFT algorithm.

(a) Brernrner series: continuous internal reflections. It is well known that weak down-range up-range propagation coupling in range-dependent environments takes place even when the WKB turning point condition •--0 is not

met. The backshading of radiation away from the turning point is usually to-1 smaller in magnitude relative to that occurring at the turning point. Nevertheless, it might become important, especially in cases for which the •--0 condition is never met (like a range-dependent medium with a monotonically increasing index of refraction). A wave solution describing this phenomenon, expressed as an infinite series of correction terms for the (uncoupled) WKB solutions far from the turning point, was developed by Bremmer. 17 Clearly, the same procedure can be applied to Eq. (45) and its uncoupled solutions (47)-(48), with ,•,• as parameters. The two-way propagating solutions of (45) are expressed by (47), where the amplitudes A + -•A + (,•, z,•) are given by the sum,

A+(•, z,•)= • A•(.•, z,•), (56) n=0

and are coupled by,

1 ;o: Oz'bt(•' z') e•:2i,or)(z, ) (X, VO2C2 (X, z',g) XA • (•, z',•)dz', (57) n--1

and where the zero-order terms are determined by the initial radiation condition. For an initial condition of down-

range propagation one has A0 + -- 1 and A•- =0. From (57), A• gives rise to a series of elements A/+, i--1,3,5,... and Af, j--2,4,6,.... Upon substituting these series into Eqs. (47) and (44), a corresponding sequence of solutions W•: for the locally homogeneous evolution equation is generated. In analogy with the traditional implementations of the Bremmer series, the result can be interpreted as an infinite sequence of continuous internal reflections of windowed plane waves.

2. Solutions at local turning points

The well-established method of canonical mapping (or comparison equation method) can be used to solve Eq. (45) in the neighborhood of the local turning point zt(g,•) defined by (50). A detailed presentation of the approach and its applications can be found in Ref. 16, Chap. 3.5b, where it is termed as the comparison equation method. We shall apply this procedure to solve Eq. (45) in the presence of a local turning point, and consequently, to get solutions of the locally homogeneous evolution equation that support local back refractions and two-way couplings. Assum- ing that •2(•, z,•) >0 for z<zt(:•,•) and •2(•, z,•) <0 for z> zt(:ff,•), an asymptotic (large co) solution in the presence of a turning point is given by

A (.g,z,•') = D[ (I)(z, zt) ] 1/6 [•(.•, Z,•) ] --1/2

X hi-[-- [•co• (z, Zt) ] 2/3), ( 58 )

where Ai(-) is the Airy function, D is a normalization amplitude given by

[ •(-•,0,•) ] 1/2 D- (59)

[ (I) (0, zt) ] 1/6Ai{-- [•to(I)(0, zt) ]2/3} '



and q) is the phase accumulated from z to z,

f zt( :•,g ) •(z, zt) = •(:•, z',•)dz', Im •(z, zt) >•0,

(60)

with [•(z, zt)]2/3>O for z>zt(:•,•) and [•(z, zt)]2/3 <O for z < zt(•,•). Thus incident and reflected windowed plane waves of the form of (47) are propagating at z<zt(:•,•) and an evanescent windowed plane wave is present at z> zt(:•,•). (This is checked easily by employing the large argument approximations of the Airy function.) From (58) it follows that A (•,0,•) = 1. Thus, following again the procedure outlined in the previous sections, we define a propagator by the • superposition (40),

B(Z'•; Zø'•ø' z) = 2• •(•-- •ø) eio •ø(•ø- •) X W•(:•,•, z)d• (61)

with the fundamental solution W• defined by (44) and (47) far from the local turning point, and defined by (44) and (58) close to the local turning point. This propagator reduces to the reconstructing kernel given by (16) when z-}0, thus a superposition like (41 ) generates a solution to the locally homogeneous evolution equation that satisfies the initial condition (34). As with the solutions discussed previously, since B•M(R2), condition (35) can be satisfied only approximately. Short-range propagation renders the error upper-bound proportional to the step size and to the ratio between the window width and the inhomogeneity lengthscale (see Appendix D for details). This motivates again the use of B as a short-range propagator in range-dependent environments and the use of (41) with the corresponding propagator as a marching algorithm.

Remarks:

(1) As with the layered medium case, the estimates are valid if the window width is in the order of (or larger than) a wavelength. The very narrow window case awaits further study.

(2) Note that each of the summed fundamental solutions in Eq. (61) has an associated turning point, whose location relative to the specific z coordinate varies with •--the summation variable. Thus, the phase-space short- range propagator is synthesized simultaneously by fundamental solutions located before, at, and after their respec- tive turning events. However, since the integrand in (61 ) is localized around •=•0 (by the spectral window), the nature of B is dominated by those fundamental solutions with • close to •0.

(3) As with the layered medium case, the synthesizing integral of the phase-space short-range propagator in the presence of a turning point can be implemented by a FFT algorithm.

III. COMPARISON WITH RELATED SCHEMES

As pointed out in the introduction, propagation models based on the expansion of the total field by sets of spatially confined wave objects constitute a subject of an ongoing research for more than a decade. The Gaussian

beam summation methods (GBSM) 2-6 attracted much at- tention since its initial proposal in Refs. 2 and 3. More recent publications have demonstrated the intimate rela- tions between the GBSM and phase-space transformations like the WFT or the Gabor representations. Phase-space transformations can be conveniently used to match a col- lection of Gaussian beams to an initial plane data 8 (the matching problem was left open in Refs. 2 and 3). More specifically, the two-dimensional field u(x, z) that radiates to the z>•0 half-space due to an initial field distribution u(x,O) =u0(x) at z--0 is given by the phase-space superposition, 8

u(x,z)= Uo(:•,•)b(x,z ; :•,•)d:•d•, (62)

where U0(:•,•) is the WFT of the initial plane data Uo(X) obtained via Eq. (9) and b(x, z; :•,•) defines a beam in the physical domain (x, z), whose axis originates in the z=0 plane at x--• with initial direction •--sin-• (v•) relative to the z axis. It has been shown in Ref. 8 that with a

properly chosen window U0(:•, •) favors a priori the beams propagating along the preferred direction of radiation of the initial field in the neighborhood of :•, thereby establish- ing effective localization in the (:•,•) plane around skeletal lines that coincide with the geometrical description of the initial field Uo(X). The fundamental difficulty in the scheme advised by Eq. (62) is the propagation of the beam field b (x, z; :•,•) in the configurational domain (x, z). The problems are articulated in the introduction and the discussion will not be repeated here.

The evolution equation approach in the present work offers an alternative route. The local spectra U(•,•, z) is propagated and not the beams. Two advantages are offered by this approach. The first, discussed in detail in Ref. 12, is the ability to conserve localization over large propagation distances. The second, shown in the present work, is the existence of a simplified differential equation that governs the evolution of the local spectra in a general smoothly varying nonhomogeneous medium--the locally homogeneous evolution equation. The equation is approximate, but an upper bound on the error is derived, and it is shown that the error can be controlled by controlling the width of the window. The windowed plane waves derived here [see Eqs. (36) and (44) and discussions thereafter] are the fundamental solutions of the new equation. They are used to construct a local spectral representation of the field in a general smoothly varying medium. Unlike the GBSM, this representation is based completely on linear computations and does not require any geometric ray calculations, which sometimes tend to become chaotic. Of course, these advantages come at the expense of employing a marching algorithm.

It is natural to ask whether the evolution equation approach leads to a representation that can be connected to the phase-space GBSM presented in Eq. (62). The answer is yes. This connection can be established by repeating the same procedure used in Ref. 12 to develop a marching algorithm in phase space: restrict the propagation distance in Eq. (62) to short ranges, and operate with the WFT on



both sides of that equation. The final result is exactly the one given in Eq. (1) with the alternative definition of B,

f O(x,5z ; --•7)e -i•'•x dx. (63)

Thus the short-range propagator developed in the present work can be presented also by the procedure in Ref. 12, namely, as the WFT of a configurational domain short- range propagated beam. This procedure is, in some sense, the simplest and most straightforward way to derive a marching algorithm in the WFT phase space. Its main disadvantages are that it depends on one's ability to derive the configurational domain beam b (x, z; •7,•) and it lacks a systematic framework for assessing the level of accuracy of the approximations made. Indeed, the propagator in Ref. 12 was developed only for a homogeneous medium, and then applied to a smooth layered medium using heu- ristic considerations.

Yet another methodology of solution that is well known to the community of the underwater acoustics is the marching of the radiation field in the configurational and spectral domains. These approaches are governed by the paraxially valid parabolic equation (PE) TM and its wide angle generalizations like the factorized Helmholtz equation developed by Fishman and McCoy. 15 Both are one- way wave theories. They are based on the assumption of a perfectly layered medium, or, what amounts to the same thing, the assumption that the up-range and down-range propagations are uncoupled. Furthermore, the schemes are global in the cross-range direction exactly in the sense discussed in the introduction, thus the short-range propagator may become difficult to evaluate even for a relatively simple cross-range variability. To contrast, the results presented in Sec. II C suggest that the evolution equation approach is potentially suitable to handle range-dependent environments, and the results in Sec. IIB shows that analytic solutions of the locally homogeneous evolution equation in a layered medium can be obtained in a straightforward manner.

IV. ON THE CHOICE OF THE WINDOW

The choice of the optimal window width in terms of computational efficiency and accuracy is of great practical importance. Besides validating the various approximations made here, to which explicit conditions on the window width are given, one may look for the width that would allow simple and accurate asymptotic evaluations of the propagator B, or look for the width that maximizes localization of the integrand in the marching scheme ( 1 )mthus reducing to a minimum the domain of integration needed for the accomplishment of a step. The first aspect-- asymptotic evaluation of the propagator B--was addressed explicitly in Ref. 12 and the results can be directly applied to the evaluation of the propagator developed here for layered media, or for range-dependent media far from a turning event. To facilitate asymptotic evaluation the window width should be in the order of, or larger than a wave-

length (see Ref. 12 for details). An added "bonus" of this condition is the localization property that connects us to the second aspect. For such widths and for step sizes not exceeding several wavelengths, the propagator is rendered highly localized in phase space around the "skeletal points" (see discussion in the introduction and Fig. 1) with an exponential (Gaussian, in fact) decay. Since the skeletal points are given by simple geometrical considerations, and further the width of the propagator in the •0 and •0 directions are given by simple algebraic expressions, one can easily use this information to substantially cut down the area of integration in Eq. (1). We repeat for emphasis that since the generic forms of the expressions for B in our analysis in Secs. IIB and II C 1 are the same as those in Ref. 12, the aforementioned procedure can be directly applied here. This, however, is not valid when the propagator approaches a local turning event. Unfortu- nately, simple asymptotic expressions for evaluating B near the turning point are not available. Furthermore, it will be shown in the numerical example that at the turning point localization of B is seriously degraded.

Another possibility to cut down the integration area is to look for the conditions that maximize localization of the

phase-space distribution of the field itself, U(•,•, z). It has been shown in Ref. 8 that the window that minimizes the

effective area of U in phase space depends strongly on the specific field under consideration. Thus, it certainly would change during the evolution, as the structure of the field itself changes during propagation. Consequently, this possibility is of less interest to us.

From the discussion above, a global optimal choice of the window cannot be given. It is, certainly, medium dependent as some of the validity conditions depend upon p• and coa. Furthermore, under extreme circumstances some conditions may even contradict. For example, the locally homogeneous condition in (32) requires the window to be narrow, whereas the localization of B and the validity of the error estimates (Appendixes C and D) require a window wider than a wavelength. In a fast varying medium these conditions may not overlap. However, it follows from the above that when a smoothly varying medium is considered (on the scale of)t0), a "rule of thumb" can be used: a window width of one or two wavelengths would keep the locally homogeneous approximation valid, and at the same time would yield a well-localized propagator that allows for a profound confinement of the integration area in (1). This will be demonstrated in the numerical section.

V. INTERPRETATIONS AND EXAMPLES

Before indulging in numerical examples, it is instruct- ful to recall again the physical interpretation of the marching scheme advised by Eq. (1). It is seen from the WFT inversion formula (10) that the field at any z is described by a summation of spatially confined windows shifted by • and modulated by •. The weights in this two-fold summation are the WFT--the local spectra of the field. By asso- ciating the direction

0(•, z,•)=sin- • [v(•, z)•] (64)



with the (•,•) point in phase space, one may interpret U(•,•, z) as a local measure of the directional properties of the field at the plane z-a phase-space representation, and the inversion formula (10) as a phase-space superposition. It is clear from Eq. (64) that the direction of radiation associated with a given value of • may vary with the coordinates (•, z). Thus, a point in phase space (•,•) repre- senting a moderate off-axis radiation direction at z--0, may represent a wave constituent that approaches a turning point at a different range z. From (64), the phase-space points corresponding to propagation perpendicular to the range direction (waves reaching a turning event) are given by the constraint

=u--l( "•' Z)' (65)

which is equivalent to (50) with • replaced by •. Further- more, regions in phase space for which

> U--1 ("•' Z) (66)

can be interpreted as a local evanescent spectrum domain. While a clear location-direction sense can be assigned to points for which I •l < v- 1 (•, z) (the propagation domain), the local evanescent spectrum domain eliminates any sense of directionality.

The phase-space short-range propagator B constitutes a measure of coupling of a local wave constituent at (•0, z) propagating in the direction O(.•'o,Z,•'o) [see (64)], to a local wave constituent at (•, z+ Az) propagating in the direction 0(•, z+ Az,•). The point of maximal coupling in the (•0,•0) plane, for a given value of (•,•), has been

eo/Xo •,oo.o _•,o.o _•,o.o _•o.o Zo.o o.o •o;o •o;o •o;o •o;o 1. i i i I i i i i i i f '1

0.6

0.4

,-

0.2

_

• 0.0 - _

--02 -

--0.4

-0.6 (•/)•.•l) = (0..0.4) window width

step size Az=O.l• -0.8 range z=10•

_•.o -100.0 -60.0 -60.0 -40.0 -20.0 0.0 20 0 40 0 60 0 80 0

eo/•o (a)

eo/• _,oo.o _•,o.o -•,.o _•?.o _•?.o o.o •o;o •o;o •o;o •o;o 1.0 f f • • I • i • • I' I

0.8

0.6

0.4

0.2

;

0.0 - _

-0.2 -

_

-0.4 -

_

-o.6 - (•/•,w•) = (0.,0.4) window width A=2•

_

step size Az=0.1• -0.8 range z=15•

-100.0 -80.0 -60.0 -40.0 -20.0 0.0 20 0 40.0 80.0 80 0

100.0 1.0

_

- 0.8

_

- 0.6

_

- 0.4

_

- 0.2

0.0

-0.2

_

- -0.4

_

- -0.6

_

- -0.8

_

-1.0 100.0

100.0 1.0

_

- 0.8

_

0.6

0.4

- 0.2

_

-0.0 _

- -0.2

_

- -0.4

_

- -0.6

_

- -0.8

_

-1.0 100.0

0.6

0.4

0.2

eo/Xo 1 •100'0 -87.0 -67.0 --40.0 --20.0 0.0 200 400 600 80 0 100

_

0.8 -

_

0.0

-0.2

(•/)•,•o•) = (0.,0.4) window width

step size Az=O.l?• range

-0.6

-0.8

-100.0 -80.0 -80.0 --40.0 -20.0 0.0 20 0 40 0 60 0 60 0

•ø/Xo

0.6

0.4

0.2

(c)

_

_

_

_

-' -0.6

_

- -0.8

_

-1.0 100.0

.•o/Xo ,o ,o

0.8 • 0.8 0.6 0.6

0.4 0.4

0.2 0.2

0.0 - _

-0.2 -

(•/•.•0=(o..o.4) window width A=2A.

step size Az=O.1)• range z= 19.4•,

-100.0 -80.0 -60.0 -40.0 -20.0 0.0 20 0 40 0 60.0 80.0

(d) •o/Xo

-0.2

_

- -0.4

_

- -0.8

-0.8

_

-1.0 100.0

FIG. 2. The phase-space short-range propagator in the nonhomogeneous range-dependent environment defined by Eq. (67). (a) The propagator at range 10;to; (b) the propagator at range 15;to; (c) the propagator at range 18;to; and (d) the propagator at range 19.4;t o.



termed in Ref. 12 as the "skeletal point." As pointed out in previous works and as schematized in Fig. 1, usually one can use ray-type intuition to predict where this coupling is strong: the propagator is localized in phase space around those regions that together form ray trajectories. The skeletal point in the (•0,•0) plane, therefore, is given by the initial location-direction coordinates of a ray leaving the plane z and reaching the plane z + Az at location • with the direction associated with •. The local evanescent spectrum domain, however, is an exception to this generality, since no sense of direction can be assigned to points satisfying (66). We turn now to demonstrate these inferences through numerical examples.

A. Numerical examples

Equation (61 ) implemented by a FFT algorithm with the fundamental solutions in (44), ( 49 ) and (44), ( 58 ), has been used to compute the phase space short range propagator in the range dependent environment defined by the index of refraction,

n2(x, z) = 1 + nld2/(d 2 q-x 2) --c2z 2 (67)

with d= 10A0, c=0.05/A0, nl=0.1. With these values we get max z> 1-6,5 x 10-3 To validate the locally homogeneous approximation and the error estimates in Eqs. (30)-(32) and Appendixes C and D, we choose to use a window width of 2A0, thus I (cøa)-i/2 ß /•l (•, z) I < 1.3 X 10 -2. The range step is 0.1A 0. Figure 2 shows the phase-space short-range propagator in the (•0,•0) plane for the point (•,•)= (0,0.4) at a sequence of ranges. With the chosen numerical values, this point cor- responds to off-axis directions of 26 ø , 37 ø , and 48 ø , at ranges of 10A0, 15A0, and 18A0, respectively, and it reaches the local evanescent spectrum domain at z= 19.4A0. Figure 2 (a) and 2 (b) shows plots of the propagator at the ranges of 10A0 and 15A 0. The contours depict the region in phase space in the plane z that "effectively" contributes to the location-direction coordinate (0,0.4) in the plane z+ ZXz. The outer contour, depicting 5% of the maximal magnitude, represents at its upmost extent radiation below 73 ø . Localization around the skeletal point is clear. Figure 2 (c) shows the propagator at the range of 18A0. Radiation with v0•0 at and below 0.43 is still well localized in phase space. The "steeper" regions of the propagator, however, have penetrated the local evanescent spectrum domain and localization has been seriously degraded. Finally, Fig. 2(d) shows the propagator at the range of the turning point associated with (•,•) = (0,0.4). The contributing regions in phase space cannot be localized for these parameters.

VI. CONCLUSIONS

By applying the windowed Fourier transform to the Helmholtz equation, a new wave equation governing the evolution of the local spectrum of a field in general nonhomogeneous environments was developed. By further invoking the so called locally homogeneous approximation the equation is reduced to a much simplified form, with an associated error that can be controlled by the width of the

window employed in the WFT. The strength of the new reduced wave equation is that the inhomogeneity in the direction along which the WFT is applied became "parametrized." That is, it commutes with the rest of the terms in that equation. Thus, analytical techniques developed to handle propagation in homogeneous or layered media can now be generalized to solve the new wave equation in general smoothly varying nonhomogeneous environments. The unique solution of the "original" physical problem, defined as the solution that (a) satisfies prescribed boundary conditions and (b) lies within the range of the map established by the WFT, can be synthesized by using an appropriately defined summation over fundamental solutions of the new wave equation, in conjunction with the marching approach. Numerical examples demonstrating solutions in the windowed Fourier transform domain for a

range-dependent environment and in the presence of a turning event were presented. It is anticipated that the proposed scheme can serve as a framework for marching fields in a range-dependent inhomogeneity and for tracking the back shading of radiation typical to such environments. The incorporation of such a framework in a marching algorithm is currently under investigation.

ACKNOWLEDGMENTS

This work was supported by the Office of Naval Re- search under contract No. N00014-89-J- 1666. The author

is also grateful to the reviwer for his helpful comments.

APPENDIX A: DERIVATION OF SOME PROPERTIES OF THE WINDOWED FOURIER TRANSFORM

(I) Let F(•,•) and Fø(•,•) be the WFT of f(x) and f' (x), respectively (the prime denotes a derivative with respect to the argument). The function f and all its derivatives vanish at infinity. Then,

Fz> (•,•) = DF (•7,•), (Ala)

where the operator D is given by

D=O•+iro•. (Alb)

Prooff By definition,

Fø(•,•) = • f' (x)w*(x--•)e -i•ø• dx. (A2) Integrating by parts and recalling that the integrand vanishes at infinity we get from Eq. (A2),

Fø(•,•) =-- • f(X)Ox[W*(X--•)e-i•ø•X]dx. (A3) Since 3xw* (x -- •) = -- 3•w* (x -- •) we get

F z> (:•,•) = (0•+ ico•) F (•7, •), (A4)

which was to be proved. (II) If the WFT of f(x) is computed with the Gauss-

ian window,

w(x ) =e -•ø( a/2 )x2 ( A5 ) then the operator D can be written in the form,



D=iaO•--wa•+iw•. (A6)

Proof.' Substituting Eq. (A5) into Eq. (A3) we get

Fø(X,•) - f f(x) [wa(x--)7) q-iw•] X exp[ --w(a/2) (x--•7) 2-- iw• x]dx,

but wx=iS•, thus we get

Fø ( •,•) = ( iat3•--coa:•-3- iw•) F ( •,•),

(A7)

(A8)

which was to be proved.

Since 3• and • do not commute, a question of operator ordering may arise whenever the operator D is applied twice to find the transform of the second derivative of the

function f However, by repeating the procedure outlined here, one may show that applying the operator in Eq. (A8) twice does give the right result as long as the operations are ordered the same way in the first and second D.

Finally, we repeat for emphasis that the forms of D in Eq. (Alb) and Eq. (A8) are equivalent only when operating on two-dimensional distributions that are obtained by the WFT of "legitimate" functions (L 2, vanishing at infinity) and when Gaussian windows are employed.

APPENDIX B: PROOF OF INEQUALITY (31)

With Eqs. (9) and (17), the left side of Eq. (31 ) can be written as

rø(a/2)(x--•)2--kø•Xu(x, Z)

Utilizing the Cauchy-Schwartz inequality, we get

I•<IlI2 ,

where

(B1)

(B2)

I•= f l u(x, z) l 2 dx=N2u(Z), (B3)

12 2-- f (x--'ff)2me-wa(x-:•)2 dx F(m+ 1/2)

(03(Z) m+ 1/2

1.3-5-.. (2m-- 1)

2m( oa ) re+l/2 (B4)

Combining the last results together, we get the inequality (31).

APPENDIX C: APPROXIMATE ADMISSIBILITY (43)

By a straightforward manipulation one may rewrite Eq. (40) in the form,

B(•,•; •0,•0, z)

= 2-•-• w*(x--•)e-'•ø•"F(x' œ; •o,•-o, z)dx, (c1)

where

F(x, X; •o,•o, Z)

f - _ = •(•-•o)exp[-iro •o(•-•o) +iro•(•,•)z

+ iw x•']d•' (C2)

and where •(•,•) is defined in Eq. (37). Apart from the • dependence ofFwhich comes from •(•,•), Eq. (C1) looks exactly like a WFT of a function of x defined by Eq. (C2). Since the integrand of Eq. (C1) localizes • around x, we can expand [note that • in (C2) localizes further to •_• •0],

•(•,•) •_•(x,•) +a- (x-•), (C3a)

where

a = 2v•(•,•0) (C3b) and where •1 is defined in Eq. (29). The typical width of the Gaussian window in (C 1 ) is given by (wa) - 1/2. Thus this approximation is valid for I wa (wa) - mz I • 1, which yields

T/'Z

We define now a new function B(•,•; •0,•0, z) as,

B(•,g; •0,g0, z)

(C4)

= 2'-•-• w*(x-:•)e- (x; •0,•0, z)dx, (C5) where F a is given by (C2) with • (•,•) replaced by • (x,•). Clearly, •M(R 2) and becomes identical with B as z vanishes. Under condition (C4), the difference between B and B is given approximately by

I

- < l ß (c6) where

I/el = w*(x-X)exp[ia x(•-•) +ia•(x,•)z]

X (1-e-'•øa(x-•)Z)dx I

<2 f Iw(x-) l , Isin[wa(x-)z/2] Idx. (C7)



Substituting the window [Eq. ( 17)] and utilizing Eq. (C4) we finally get

(0 2

I B- za(•(0ot ) -1/2 'TF 1/2 I

l Itl()l (c8) Thus, the upper bound on the error is proportional to z and can be controlled by controlling the width of the window relative to the inhomogeneity. Note, however, that we have assumed in (C3a) that the window •(•--•0) localizes • to the neighborhood of •0- Thus, the spatial domain window cannot be made as narrow as one wishes. The spectral width of the window should be below l/v0. This means that the width of the window in the spatial domain should be in the order of or larger than a wavelength.

APPENDIX D: ADMISSIBILITY CONDITION NEAR A TURNING EVENT

We shall follow essentially the same steps outlined in Appendix C. One can rewrite Eq. (61 ) in the form,

; z)

= w*(x-•) e-iø•xFt(x, •; •o,•o, z)dx, (D1)

where

Ft ( x, .•; .•o,•o, Z )

f - _ = •(•--•0)exp [--i(0 •70 (•-- •0)

+i(0 x• ]A (•?, z,• )d• (D2)

and where A (•7, z,•) is defined in Eq. ( 58 ). As with (C 1 ) and (C2), apart from the • dependence of F t, which comes from A (•, z,•), Eq. (D1) looks exactly like a WFT of a function defined by (D2). The Gaussian window in the integrand of (D1) localizes œ around x. Thus, provided that A (x, z,•) changes slowly in x compare to the window, we can expand A (•, z,•) around A (x, z,•). This yields

A(•, z,•)•_A(x,z,•)+a' (x--•7), (D3)

where [note that • in (D2) localizes • further to •0],

a= --8-•t (•, z,•0) (D3a)

with the following condition [recall that the Gaussian width is ((0a) - 1/2].

I ((0a) -1/2 al < 1. (D3b) With (D3) in (D2), B is expressed as •eM(R 2) plus a correction term proportional to a. Due to the (0 dependence of the arguments of Ai [see (58)-(59)], we assume that the variations of A (x, z,•) come essentially from the Airy functions. Moreover, for small step size, located in proximity with the turning point, the magnitude of the

remaining terms in (58)-(59) is approximately unity. Thus,

Ai [1),•2/3 (z, Zt) ] a_• --c9• Ai [1),(i>2/3 (0 ' Zt ) ] , (D4)

where, for brevity II (3 2/3 =- •(0) . From this point on it is meant that zt [see definition in (50)] is evaluated with

Using the identity q>(z, zt) =q>(0, z t) --q>(O, z) and assuming again a small step size with z•--zt, we may approximate (D4) further as

2 llO• t (•)(O,z)C•)(O, zt)_l/3 Ai'[•dp2/3(O, zt)]) a•_• Ai [1),(i>2/3 (0 ' Zt ) ] , (D5)

where a prime denotes a derivative with respect to the argument. Performing the derivative with respect to x and keeping only leading orders of the large parameter 11, we get

a •_•QI•2• (0, Z)a•(I)2/3 (0, (D6)

with

d Ai' (y)

Q=•-1/3(0' it) dy Ai(y) ' Y=1•2/3(0' it)' (D7) Under condition (D3b) the difference between B and •[•M(R2)] is identical to that given in (C6) with

= la[' a-3/2l•-•le-(ø/2a)(g-•)2 (D8)

(note that a is independent of x). Carrying on with (C6) yields

I B-•I •<(2rr3/2)-l(01al ((00•) --1/2. (D9) Substituting (D6), interpreting • (0, z) as an effective step size, and •2/3(0, gt) as an effective measure of the inhomogeneity in the x direction (it is proportional to •l), we get again to the conclusion that the error is proportional to the step size and to the ratio between the window width and the inhomogeneity lengthscale. However, for reasons identical to those articulated at the end of Appendix C, the results here are valid for window widths in the order of or

larger than a wavelength.

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