An energy-conserving parabolic equation incorporating range refraction

$Page 1: An energy-conserving parabolic equation incorporating range refraction$
An energy-conserving parabolic equation incorporating range refraction

I.W. Schurman, W. L. Siegmann, and M.J. Jacobson Department of Mathematical Sciences, Rensselaer Polytechnic Institute• Troy, New York 12180-3590

(Received 28 December 1989; accepted for publication 27 August 1990)

A new parabolic equation (PE) is presented that is independent of ko and capable of handling relatively large range variations in the index of refraction. This equation is similar to, and ostensibly simpler than, an earlier range refraction PE (RAREPE). The modified range refraction parabolic equation (MOREPE) is obtained by a transformation approach, and operator and multiscale formalisms are described to validate the equation. Principal properties of MOREPE are developed, including energy conservation and possession of the correct (Helmholtz) rays in the high-frequency, small-angle limit. Exact solutions with range variation in sound speed are presented to illustrate differences between standard PE (SPE) and MOREPE. Propagation examples in range-independent environments demonstrate close agreement between MOREPE and SPE, while examples with strong range dependence exhibit significant differences between the two equations in their predictions of acoustic intensity. Analytical and numerical comparisons of solutions to the one-way Helmholtz equation (HE1), MOREPE, and SPE demonstrate the increased accuracy of MOREPE over SPE in range-dependent environments.

PACS numbers: 43.30.Bp

INTRODUCTION

The parabolic equation method for the prediction of low-frequency acoustic propagation was first introduced by Tappert I in the early 1970s. Since that time, the technique has flourished, and, today, a wider variety of parabolic equations (PEs) is available. Different physical situations may require different PEs. Regardless of the choice, the essence of the technique remains the same; the elliptic Helmholtz equation (HE) is replaced with a one-way parabolic approximation, and the acoustic field is generated as the solution of an initial value problem. The usefulness of the method re- suits because the approximate solution retains many full- wave effects of the exact HE solution, while being more efficient to implement computationally. Thus solutions to many problems of interest can be found quickly and accurately.

Perhaps the most widely implemented parabolic equation is a two-dimensional version of Tappert's original equation. i Often referred to as the "standard parabolic equation" (SPE), this equation neglects azimuthal variations in the field, propagation at wide angles, and significant variations in sound speed. As a result of the inherent limitations of SPE, a number of PEs have been developed to correct one or more of its deficiencies in the last decade. A family of wide-angle PEs, with various implementations, has been formulated (e.g., Refs. 2-7). These equations are well suited to problems in which energy propagates at steep angles, such as might occur with significant bottom interaction. Also, implementations of three-dimensional PEs have been reported (Refs. 8-10, for example). Other researchers have developed PEs that incorporate effects such as density variations 11'12 and currents. 13'14 Recently, a "PE-type" wave equation was designed I-s so that its rays match exactly the rays of HE in a range-independent environment.

The aforementioned PEs (and others not referenced here) contribute to the wide class of available parabolic and parabolic-type equations. Each equation is germane to a propagation problem for which SPE is not well suited but reduces to SPE in an appropriate limit. It has been demon- strated 16 recently that a higher-order PE is relatively insensi- tive to the choice of reference wave number ko and is quite accurate for problems involving strong depth variation in sound speed. Some PEs, however, can be sensitive to the choice of ko, especially in range-dependent environments. I I Further, most PEs allow only weak range variation in the index of refraction. Such an assumption can be invalid in an ocean volume where sound speed varies strongly with respect to spatial variables such as might occur in fronts or eddies, or in the presence of other processes near boundaries. Tappert and Lee 17 recently presented a range refraction parabolic equation (RAREPE) designed for range-dependent problems. However, range-dependent examples pub- lished in Ref. 17 do not confirm that RAREPE is, in fact, more accurate than SPE. Hence, the most compelling issue currently facing PE modeling is improving the accuracy of PE predictions for range-dependent problems.

In this paper, attention is focused on range dependence arising from variations in sound speed. A new parabolic equation is presented that is capable of handling relatively large variations in the refractive index n and that does not require a choice of ko. Our modified refraction parabolic equation (MOREPE) is similar to, but simpler than, RAREPE, and is obtained by three different derivations which are summarized in Sec. I. Like RAREPE and SPE,

MOREPE is valid for narrow-angle propagation. In Sec. II, MOREPE is shown to reduce to SPE when the refractive

index differs little from unity, to conserve energy in range, and to match the rays of the far-field HE within the small-

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angle approximation. Exact solutions of MOREPE and SPE in the special case where sound speed varies only in range are provided to illustrate different features of intensity predictions from the two equations, and an asymptotic analysis of the one-way HE (HE1) validates the range capabilities of MOREPE. In addition, a slow-variation equation is obtained from the transformation and multiscale derivations of

MOREPE, and this successor equation could have useful computational advantages. Minor modifications to an exist- ing implicit-finite difference (IFD) code ms allow a numerical implementation of MOREPE. In Sec. III, an example with range dependence absent demonstrates the correspon- dence of results from MOREPE and those of SPE. Propaga- tion examples with only range variation in sound speed af- firm predictions made from the analytical solutions of MOREPE and SPE. Finally, numerical comparisons of exact solutions of HE1, MOREPE, and SPE in a particular range-dependent environment illustrate the improved accuracy of MOREPE.

I. DERIVATION OF MOREPE

A. Transformation formalism

We begin with the two-dimensional HE in cylindrical coordinates r and z, for the spatially varying part of the acoustic pressure p ( r,z ) :

p,.,. + (1/r)pr +Pzz + k2p -0. (1)

A point source of frequency co in rad s-• is assumed so that the full acoustic pressure is pe- io,,. The range- and depth- dependent wave number k(r,z) is defined by k = kon(r,z), where n = Co/C(r,z) is the index of refraction, ko = co/Co, c(r,z) is the sound speed, and Co is some reference sound speed. It is convenient to factor out cylindrical spreading via the transformation

u(r,z) p(r,z) = •. (2)

Applying Eq. (2) and the far-field approximation kor>> 1, Eq. ( 1 ) becomes

U,.,. -+- Uzz -[- k 2u -- 0. (3)

Next, we suppose that u can be written as the product of a slowly varying function •(r,z), which modulates a rapidly varying function containing the phase function tt (r,z)'

u(r,z) = q)e"'. (4)

For example, if/• has the simple form/• = kor, inserting Eq. (4) into Eq. ( 3 ) produces SPE upon the usual neglect of the second radial derivative of

2ikodP,. + dpzz + k • ( n 2 -- 1 )dO - O. ( 5 )

It is seen that SPE is explicitly dependent on the reference wave number ko, while HE [Eq. (1)] is not. As a result, different choices of k o yield solutions of SPE that are not mathematically equivalent.

In order to construct a PE that is independent of ko, we choose tt in the integral form

tt = k ( s,z ) ds. (6)

With this, Eq. (4) can be regarded as an extension of the generic parabolic approximation transformation discussed by Pierce. • In that development the reference parameter ko appears as a function of range in lieu of the range- and depth- dependent wave number k ofEq. (6). Applying the transformation in Eq. (4) combined with the definition of tt in Eq. (6), and again neglecting •,.r, yields

2ik•,. + •zz + 2ilUz •z + ( ik,. + ilUzz -/Zz 2 )• = o. (7)

Equation (7) is the analog of SPE in the sense that the solution • is an envelope function that modulates the rapidly varying function e i". We remark that partial derivatives of the function k are normally quite small; however, the possi- bility of their becoming significant is allowed here and, further, effects of k derivatives may accumulate with range. Consequently, retention of terms containing these derivatives is justified by the objective of incorporating range refraction effects. A second transformation,

4p = uø(r,z)e- i,, (8)

effectively the inverse of Eq. (4), yields, with no further approximations, MOREPE:

2iku• ø + u•ø• + ( ik, + 2k • ) u ø = O. (9) The function u ø is a one-way approximation to the solution of the elliptic far-field HE Eq. (3). Unlike most PEs, MOREPE has nonconstant coefficients of the solution and

its derivatives, displays range refraction via the term k•, and does not contain the constant reference wave number ko.

B. Operator formalism

Although the transformation derivation of the preced- ing subsection is perhaps the simplest one for obtaining MOREPE, it is somewhat nontraditional. We now provide another derivation using a more familiar formalism.

The far-field HE Eq. (3) is rewritten in an operator form:

(c•. + k•Q2)u-O, (10)

where the pseudodifferehtial operator Q is defined by 2 9 I/9 Q- [n 2+(1/ko)o9;] -, (11)

with positive square root implied. Formal factorization of Eq. ( 1 0) gives

(c•,. - ikoQ) (c•,. + ikoQ)u + iko(c•,.,Q}u- O, (12)

where {c•,.,Q} represents the commutator of 09,. and Q. Fol- lowing Tappert, •7 we assume that the dependence of n on spatial variables is sufficiently weak (although not negligi- ble) that commutation is valid. It follows that

(c•,. -- ikoQ)u_•O (13)

for the outgoing wave. The narrow-angle approximation 20•2 I k ,, I '• 1 in Eq. ( 1 1 ) means Q_• n, and Eq. ( 1 3 ) becomes

(c•,. -- ik)u_•O. (14)

135 J. Acoust. Soc. Am., Vol. 89, No. 1, January 1991 Schurman eta/.: Energy-conserving PE 135


From Eq. (14), we see that the operator (O•r --/k) is, in some sense, small. To use this fact, we express the second range-derivative operator as shown below, and expand the product, keeping track of operator order:

r32 = (c• r -- ik + ik) 2,

= (c•,.- ik) 2 + ik(c• r - ik) + (c•,.- ik)ik- k 2, = (r3,.- ik) 2 q- 2ik& + ik,. + k 2. (15)

Neglecting (o•r -- ik) 2 in favor of remaining, presumably larger, terms of Eq. (15), and inserting the result into Eq. (3), produces

2iku + U zz + ( ikr + 2k 2) ,. u =0. (16)

Equation (16) is exactly MOREPE, as derived in Sec. I A.

C. Multiscale formalism

Both the transformation and operator formalisms above produce the same equation, MOREPE, and this result differs from the previous range refraction equation RAREPE. Moreover, it can be shown that there is no transformation linear in u that .maps either equation into the other. Also, the approximate commutation of operators in the argument leading to Eq. (14) is necessary in order to achieve a one-way equation, but the neglect involved is not easily quantified. Consequently, we believe it is appropriate to further validate MOREPE by applying asymptotic methods. An asymptotic derivation of MOREPE can be found in the Appendix.

II. PROPERTIES OF MOREPE

We have mentioned already two principal properties of MOREPE: one, that it is independent of the choice of ko, and, two, that it possesses the capability to treat range refraction. A related fact is that the range refraction influence appears explicitly and in a simple form. We now indicate five additional properties.

A. Reduction to SPE

First, we show that MOREPE reduces to SPE in the limiting case n_• 1. In nondimensional form, the SPE solution gb is related to the far-field HE solution p by

poc [½(?,2)/x/•7]e '•'/f, (17)

and the MOREPE solution u {(" is similarly related by

p oc uIø)(?,•)/x/•/e. ( 18 )

Examination ofEq. (A2) shows that } - n in this case, since the appropriate choice for K is ko. Comparing Eqs. (17) and (18), we see that we could hope to retrieve SPE from MOREPE by the transformation

u (ø) -- ½e '•'/f, ( 19 ) where a constant factor has been suppressed for convenience. In a weakly range-dependent environment, the index of refraction differs little from unity, so we write

n- 1 + e0(?,2), (20)

where l el ,• 1 and 0 is assumed to be of order unity. This

corresponds to •5 -- 0 and/c 2 -- • in Eq. (A3). Applying Eq. (19) to Eq. (A 17) and using Eq. (20), we obtain

2igb• + gb• + 20gb -- O(e). (21 )

Noting that 2 0 = e-•(1/2 -- 1 ) + O(e) and ignoring O(e) terms, Eq. (21 ) becomes, upon dimensionalizing,

2 1/2 2iko• + fizz + k o ( - 1 ) ½ - 0, (22)

which is precisely SPE.

B. Conservation of energy

MOREPE is shown to conserve with range the quantity œk lu(r,z)12dz, and therefore to conserve energy, in the sense that power flow is independent of range. Relevant defini- tions and a discussion of this property for parabolic approximations can be found in Ref. 11.

Beginning with Eq. (16), we multiply by u* (asterisk denotes complex conjugate) and subtract the product of u and the conjugate of Eq. (16) to obtain

2i •3 (kuu*) + UzzU* Uz*zU O. (23) cgr

Next, we assume a pressure-release surface and a completely attenuating layer at some depth zB below the bottom; i.e.,

u(r,O) = u(r,z• ) = 0. (24)

Integrating Eq. (23) over the entire depth, and using uu* -- l u[ 2 and Eq. (24), we obtain

a (f k(r,z)lul2dz)=O. (25) cgr

C. MOREPE rays

In this section, we derive the rays for MOREPE and show that, within the small-angle approximation, they match those of HE. In particular, the MOREPE rays contain the correct first-order depth- and range-refraction terms.

We begin by expressing the MOREPE solution as

u( r,z) -- A ( r,z)e '•'"•( '"z), (26)

where A is a real amplitude and • is a real phase. Inserting Eq. (26) into Eq. (16) and separating real and imaginary parts produce

c2 (nA 2) + c2 8--7 •zz (•zA 2) = 0 (27) and

- 2n•b,..A + ( 1/k • )Az: -- •A + 2n2A -- O. (28) Equation (27) is the transport equation for MOREPE. Use of the high-frequency approximation Azz/A •k 2 in Eq. (28) produces the eikonal equation

•,. q- (1/2n)•z 2 -- n=F(•b,.,•bz,r,z) -- O, (29) where the definition of F in Eq. (29) follows Ref. 15. Equa- tion (29) can be written as a system of five ordinary differential equations by parametrizing along characteristics •9'2ø (rays); i.e., r = r(s) and z = z(s). The resulting system is

_

dr --= Fq• r = 1 (30) ds '

136 J. Acoust. Soc. Am., Vol. 89, No. 1, January 1991 Schurman ot a/.' Energy-conserving PE 136


and

dZ = F,• = l qb z, (31) ds n

d• = •zFqbz "JI- •rFqbr = •2 '-JI- •r, ds

d•z ds 1 •3•2 ) •= -4zF•-Fz=n• l+2n 2 ,

(32)

(33)

•=n,. 1 +••2 (34) ds 2n 2 '

Equation (30) shows that the parameter s can be replaced by r. Differentiating Eq. (31 ) with respect to r and using Eq. (33) give

d2z_c•logn [1 1 (dz) 2 dzc•logn (35) d--•- - az -5- Z dr Or '

Equation (35) defines the MOREPE rays. For the purpose of comparison, the eikonal, transport,

and ray equations for the far-field HE, MOREPE, and SPE are written next. For HE and SPE, an argument similar to that above (see also Ref. 20) can be used to find the relevant equations. In Eqs. (36)-(38), (a) corresponds to the eikonal equation, (b) to the transport equation, and (c) to the ray equation:

HE:

•3• + •3• 2 -- n 2, (36a)

,:9 (4,,A 2) + o • o•-• •zz (•A 2) _ 0, (36b)

d2z_[ (dz•2](.c•logn dzc•logn.) dr 2 - l + \dr/ I c•z dr c•r ' MOREPE:

(36c)

•,. + (1/2n)• 2 -n, (37a)

c•r •zz (qbzA -) -- O, (37b) d 2z

dP log n &

SPE:

i 2 He 4•+•4•=•+ •,

dz c) log n (37c) ß

dr

(38a)

8 (A 2) + o• Or •zz (c)zA 2) _ O, (38b) d 2z c• log n = rt 2 . ( 38c) dr 2 cgz

Comparison of the ray equations for HE [Eq. (36c)] and MOREPE [Eq. (37c) ] shows that the MOREPE rays agree, to within the narrrow-angle assumption (dz/dr)2 ,• 1, with those of HE. Particularly important is that the MOR- EPE rays contain the correct first-order range-refraction behavior, as seen by the last term in Eq. (37c). This arises from the fact that MOREPE contains nonconstant coefficients. In

fact, it has been shown •5 that there can exist no constant- coefficient parabolic equation that contains this range-re-

fraction term, because F(q•r,q•z,r,g) is linear in •r for any such equation. The SPE rays contain none of the correct terms, as can be seen by comparison of Eqs. (38c) and (36c).

The relevance of MOREPE as an approximation to HE can be established by further examination of the HE eikonal and transport Eqs. (36a) and (36b). Since every parabolic approximation is a one-way equation, it is natural to factor the HE eikonal equation in order to compare with approximate eikonal equations. It follows that Eq. (36a) becomes

•,.- n[1 -- (1/n2)qb•2] '/2, (39)

where the positive square root is assumed. An expansion of Eq. (39) for •3• 2 ,• n 2 gives, to first order,

•3,. = n -- (1/2n)•3•. (40)

Equation (40) is equivalent to Eq. (29), the eikonal equation for MOREPE. We note that, for HE, dz/dr = c•/c•r, •s and with Eq. (36a) and using • ,• n 2, it follows that dz/dr -•C•z/n. Hence, the validity condition of Eq. (40) is tanta- mount to the small-angle approximation (dz/dr)2,•l. Moreover, use of •r --• n in Eq. (36b) yields exactly the transport equation for MOREPE, Eq. (37b). Therefore, the eikonal and transport equations for HE reduce to those of MOREPE upon making only the small-angle approximation. In contrast, the additional assumptions that n is ap- proximately one and slowly varing in range and depth are required to obtain the corresponding equations for SPE, as can be seen by comparing Eqs. (37a) and (37b), and Eqs. (38a) and (38b).

We conclude that, from a high-frequency or ray-theore- tic point of view, MOREPE is a natural choice for small- angle propagation. We have also affirmed the refraction properties of our equation.

D. Exact solutions when c=c(r)

We now provide exact solutions of MOREPE and SPE in the special case where sound speed varies only with range. While such sound-speed structures are not often realistic, they serve to illustrate distinctions between MOREPE and SPE because these exact solutions permit predictions of characteristics that can be tested numericallyß In addition, they provide a means for accuracy verification of numerical implementations.

We assume a pressure-release surface and rigid bottom, so that boundary conditions for the MOREPE and SPE solutions u(r,z) and ½(r,z), respectively, are

u(r,O) -- Uz (r, za ) -- O, (41a)

and

½(r, 0) - ½z (r,z,) - 0, (4lb)

where za is the bottom depth. Formally separating variables in Eqs. (16) and (22), applying a general initial condition to u at initial range ro, and using the dimensional form of Eq. (19), which relates u and •p, to thereby determine the initial condition for •p, we find

137 J. Acoust. Soc. Am., Vol. 89, No. 1, January 1991 Schurman eta/.' Energy-conserving PE 137


• e•Yr. k(

and

•!, ( r,z )

1

o• - iy•,,• •/2•aSsi n X E Ane (AnZ) (42) n=O

( fr, r )] exp -- ikoro + i [ k o ( n • - 1 )/2 ] ds

X E "tne-i•2•r-r"•/2•"sin('•'n z)' (43) n=O

where the eigenvalues are

A,, = [ (2n q- 1)rc/2z• ], n = 0,1,2,..., (44) and we have assumed k (ro) = ko. The coefficients A,, can be found by integrating the initial condition and using orthogo- nality of eigenfunctions. Adopting a conventional interpre- tation of acoustic intensity, TM and anticipating numerical comparisons, we write

and

c(r) 2

--iyrr"[X•C•S•/2ø•]CtSsin(,•,nZ) , (45)

iA 2 ro)/2w S - •c,,•- in(AnZ) 2

, (46)

where we have used k(r) -- w/c(r) and ko -- W/Co. It is evi- dent that Eqs. (45) and (46) are equivalent in the isospeed case c- Co. In general, however, the intensity of the MOREPE solution contains the function c(r) in lieu of the arbitrary reference sound speed co, which suggests that both amplitude and phase differences can occur between the intensity predictions of the two equations when sound speed varies with range. Analysis of a solution with only a few modes, for example, shows that MOREPE shifts the intensity curve upward and toward the source when c increases with range. Similarly, a sound speed that decreases with range causes a decreased amplitude and outward shift of the MOREPE curve. These types of predictions are, in fact, substantiated in the range-dependent numerical examples of Sec. III.

To validate the form of the exact MOREPE solution

shown in Eq. (42), we provide a high-frequency asymptotic analysis of the far-field HE in the case c - c(r). Separating variables in Eq. (3) and applying boundary conditions anal- ogous to those used for the MOREPE and SPE solutions, one obtains for the far-field Helmholtz solution U

U(r,z) -- • B,, W,, (r)sin(A,,z), (47)

where A,, is given by Eq. (44) and the range function W,, satisfies

W,'; + [k2(r) --A ,2,] W,, --0. (48) While exact solutions to Eq. (48) cannot be found, it is illu- minating to perform an asymptotic WKB analysis of Eq. (48) and compare the result with the exact range functions of MOREPE [ Eq. (42) ] and SPE [ Eq. (43) ].

Following Ref. 22, we assume W,, has the high-frequency asymptotic form

g,,(r)

•-.exp[wOo(r) + 01(t' ) -3 I- O-)--1•92(t ') -3 I- '''], O.)'• OO, (49)

where k(r) = w/c(r). Insertion of Eq. (49) into Eq. (48) yields a sequence of ordinary differential equations for the functions 0i. Including terms through 02, it can be shown that the outgoing contribution (corresponding to e -• •ø' time dependence) to W,, (r) has the form

W,, (r)--• 1 exp i k(s) " x/k(r) . 2k(s)

q- 1 (3k'2(s) ,

4k 2 2k(s) •-k"(s)) ds}, (5O)

Comparison of Eq. (50) and the range function in Eq. (42) reveals that the MOREPE range function differs from the asymptotic form of the range function for the far-field (one- way) HE 1 by only the last phase term in Eq. (50). Further- more, this additional term has no bearing on intensity in a modal solution, since it does not involve A,,. By contrast, $PE has none of the correct terms, as can be seen by comparing Eq. (50) with the range function in Eq. (43). Conse- quently, our high-frequency analysis validates the range behavior of the MOREPE solution and suggests its improved accuracy over the $PE solution.

E. Relation between MOREPE and RAREPE

MOREPE and RAREPE [ Eq. (22) of Ref. 17 ] are rewritten here for convenience as

2iku,. + U zz + ( ik,. + 2k 2)u- 0 (16)

and

2 i k ø2/ r q- ø2/ zz -- ( k z / k ) ø2/z

+ where the MOREPE solution u(r,z) is related to acoustic pressure by Eq. (2) and the RAREPE solution •2/(r,z) satisfies

p•qr/xf•. (52)

Examining Eqs. (52) and (2), it is natural to apply

@ = x•-u (53)

to Eq. (51 ) and compare the result with MOREPE. Per- forming the transformation in Eq. (53), we obtain with no approximations

2iku ,. + Uzz + ( ik,. + 2k 2 __ k z 2/4k 2 )/,/: 0. (54)

Equation (54) differs from MOREPE by the additional term - k 2 u/4k 2 The discrepancy between MOREPE and z

RAREPE might be attributed to the differing approaches in derivation of the two equations. MOREPE is derived in Sec. I by three procedures, and RAREPE is derived by a different method in Reft 17.

138 d. Acoust. Soc. Am., Vol. 89, No. 1, January 1991 Schurman eta/.' Energy-conserving PE 138


It is shown in Ref. 17 that the finite difference scheme

used to implement RAREPE conserves energy. It can be shown, by an argument similar to that in Sec. II B, that the actual partial differential equation conserves energy as well. It is also shown in Ref. 17 that RAREPE has the same rays as are found here for MOREPE. We have extended the ray analysis here to show the reduction of the HE transport and eikonal equations to those of MOREPE in the narrow-angle limit. In addition, we have analytically validated MOREPE for narrow-angle, range-dependent propagation. We offer MOREPE as a simpler equation that visibly displays range refraction in a single term and that is validated, both by three derivations and by analytical comparisons with the far-field HE.

III. NUMERICAL ILLUSTRATIONS

In this section, we illustrate the capabilities of MOREPE, mainly in strongly range-dependent environments. Also, very close agreement is shown between MOREPE and SPE in a range-independent medium.

One numerical implementation of our equation can be achieved in the following way. A generic PE has the form

u,. = a(r,z)u + [3(r,Z)Uzz + y(r,Z)Uz, (55)

containing three coefficient functions. For SPE, a,/3, and ?' are given by

at = iko(n 2 - 1 )/2, /3= i/2ko, ?' - 0. (56) The implicit-finite difference (IFD) scheme la is one way to solve Eqs. ( 55 ) and (56). For MOREPE, the coefficients in Eq. (55) are

at = (i/k) -- (k,./2k), /3 = i/2k, ?' = 0. (57)

Only minor modifications are needed in IFD to use it for solving MOREPE. Moreover, the alternative slow-field version of MOREPE, Eq. (7), could be implemented via the following equations:

2 -zz) k,. 2ik 2k ' 2k k

(58)

where ttz -- œ"kz (s,z)ds and similarly for tt•z. This equation could be advantageous in removing fast variation and, therefore, could be more computationally efficient.

To test MOREPE, we performed a number of relative- intensity computations with the modified IFD algorithm suggested by Eqs. (55) and (57). In each example we assume a pressure-release surface and a flat, artificially absorb- ing bottom at 100 m in depth. The field is modeled at initial range by a Gaussian distribution. TM In each of the following examples, a choice of co (or, equivalently, ko) is required for SPE. The IFD algorithm provides a default value for co, chosen as the depth average of c in the first sound-speed profile. This option was exercised in most of the examples below.

We turn first to an example without range variation, where we would expect to see close agreement between MOREPE and SPE. In Fig. 1, a source of frequency f = co/ 2rr = 50 Hz is placed at middepth zs - 50 m, and the signal is propagated to a receiver at depth z/e = 25 m. There is a

z LIJ I-- Z

-.4O

-5O

-6O

MOREPE

SPE

C=1450 ' C=1500 '

F=50Hz I

I 2 3

RANGE (KM)

FIG. 1. Relative intensity I versus range r for MOREPE and SPE; f= 50 Hz, zs = 50 m, z/• = 25 m; c = c(z), with g_: 0.5 s -•.

constant sound-speed gradient gz -- 0.5 s-•, so that the isospeed curves are horizontal lines as sketched. This large gradient was chosen to challenge the agreement between the two propagation models. The depth and range increments Az and Ar for this example were both chosen to be 0.5 m and co = 1475 m s-i. The solid curve represents relative intensity versus range for the MOREPE solution, while the dashed curve is that for SPE. As expected, the solutions of MOREPE and SPE agree quite closely in this environment without range variation; differences between the two curves are less than 1 dB. Other calculations we performed in range- independent situations consistently echo this close agreement. We reiterate that MOREPE circumvents the problem of choosing the precise value of ko.

In order to illustrate distinctions between MOREPE

and SPE, we consider other types of test problems with sound speed varying only with range. While such sound- speed structures are not necessarily realistic, they allow comparison with the qualitative predictions of the exact solutions discussed in Sec. II D. In the next three computational examples, the sound speed is constant on either side of a region over which it varies linearly between two constant values of 1500 and 1540 m s- i. In each case, co is taken to be the initial sound speed, Ar = 1 m, and Az = 5 m. Isospeed curves are vertical lines, as suggested in the small sketch in each of Figs. 2-4. In order to underscore differences between the two numerical calculations over short ranges, a large horizontal sound-speed gradient g,. and short transition region are taken.

In Fig. 2, source and receiver depths are 10 and 25 m, respectively. The intensity curves from MOREPE and SPE agree identically up to the start of the transition at 500 m. However, the subsequent increase in sound speed causes a

139 J. Acoust. Soc. Am., Vol. 89, No. 1, January 1991 Schurman et aL' Energy-conserving PE 139


I I

-50' MOREPE SPE

600 m -60 z, Zs 1500 1540

z 200 m

[ F=25Hz I .

Z -- I

• -80 I I

-lOO

0 2 4 6 8 10

RANGE (KM)

FIG. 2. Relative intensity I versus range r for MOREPE and SPE; f= 25 Hz, zs = 10 m, z• = 25 m; c = c(r), with g,. = 0.2 s-'

-5O

-6O

-7O

-80

-90 -

-100

MOREPE

SPE

600 m

I !J]Jl l '•r zs 1500 1540 z,

I F = 50 Hz

i .

, I , I , I , I • I >r 0 2 4 6 8 10

RANGE (KM)

FIG. 4. Relative intensity I versus range r; parameters as in Fig. 2, except f= 50 Hz.

shift upward and toward the source of the MOREPE curve relative to the SPE curve. This is precisely the behavior predicted by the exact solutions. Typical discrepancies of about 3 dB occur past the transition.

We now consider the effect of reversing the horizontal

I

.5o •

-6o

-7o

-8o

MOREPE

SPE

600 m

Zs I 1540 I [ 1500 ZR 5oo ;, z

F = 25 Hz" '!

-90 , I , I • r 0 2 4 6 8 10

RANGE (KM)

FIG. 3. Relative intensity I versus range r for parameters as in Fig. 2, except z• = 50m, g,. = -- 0.2 s-'.

140 J. Acoust. Soc. Am., Vol. 89, No. 1, January 1991

sound-speed gradient, i.e., the initial speed is 1540 m s- ' and the final speed is 1500 m s -'. In Fig. 3, all other parameters are identical to those in Fig. 2 except Co = 1540 m s-• and, for variety, we have elected to increase the receiver depth to 50 m. In this case, MOREPE shifts local intensity peaks outward in range and downward, opposite to the situation observed in the previous example. Again, this qualitative behavior is predicted by the analytical solutions for the special case where sound speed varies only with range and decreases with range. We note that in all calculations we performed, reversal of the horizontal sound-speed gradient across a transition region resulted in a reversal of the relative config- urations of the MOREPE and SPE curves, regardless of source and receiver depths. We conclude that MOREPE handles range refraction in the expected manner.

In Fig. 4, another comparison of MOREPE and SPE is performed when c is the function of range specified for Fig. 2. In this example, f-- 50 Hz, while all other parameters are as in Fig. 2. Decibel differences of about 3 dB still occur, but noticeable shifts in position begin at a slightly greater range. As before, these shifts are consistent with the analytical predictions of the exact solutions.

To test MOREPE in a range- and depth-dependent environment, we constructed a sound-speed model in which the depth gradient is linearly modulated in range across a transition region. In a numerical example, an initial gradient gz of 0.5 s- 1 is graduated to a final value of -- 0.5 s- 1 and the transition region is 200 m long, as shown in Fig. 5 (a). Range and depth increments are both 1 m, and Co was selected to be 1500 m s- 1. Isospeed curves for this example are depicted in Fig. 5 (b). Relative-intensity curves are shown in Fig. 6 with source and receiver depths of 10 and 20 m, respectively. The

Schurman et al.: Energy-conserving PE 140


o

100 m

SOUND SPEED (m/s)

1550 1500 1450

c (r,z)

r = 500 m r= 700 m

(a)

100 m --

500 m 700 m

(b)

FIG. 5. Sound speed c = c(r,z)' (a) gradient g transition fi'om -- 0.5 s • to 0.5 s- •' (b) isospeed curves.

source frequency is 25 Hz. One feature of the MOREPE relative-intensity curve is that, beyond about 4 km, its pat- tern appears to be displaced downward and outward in range from that of the SPE curve. Since sound speed is both depth- and range dependent, analysis based on the exact so-

.50 J

-6O

n-'

-90

-lOO

MOREPE

SPE

c = c (r,z)

I F = 25 Hz I

, i I , I • I i I • I • r 0 2 4 6 8 10

FIG. 6. Relative intensity/versus range r; f= 25 Hz, zs = 10 m, z, = 20 m; c = c(r,z) from Fig. 5.

lutions of Sec. IID does not apply here. Nevertheless, at any fixed depth, sound speed does increase, so that the behavior of the intensity predictions is plausible in view of our exact solutions. In fact, this type of generalization held in all of a number of computational test examples that we examined. With this very strong c(r,z) variation, maximum differences of about 5 dB occur.

To demonstrate the relative accuracy of MOREPE, we next compare exact solutions of MOREPE, SPE, and HE1 in a range-dependent environment. In this example, two homogeneous regions of differing sound speeds are adjoined by a transition interval over which k •-(r) varies linearly with range from an initial value k • to a final value k •'

k•, r•r I

k2(r) = ••,d- rl(r-- r•), rl<r<r2 (59) • r•<r,

where r/= (k 22 - k •2 )/(r: - r• ). A pressure-release surface and fiat rigid bottom are assumed, so that the exact solutions of Sec. IID apply. All three solutions are taken as identical in the first region (r<r•), up to the start of the transition region (r- r•). This initial field is chosen to match the far-field behavior of the exact solution for a point source disturbance at depths Zs in a homogeneous environment (see, for example, Ref. 24). With k :(r) given by Eq. ( 59 ), the HE 1 range œq. (48) admits a solution :• that can be written in the form of Hankel functions H <• and H 1/3 1/3 ß

Corresponding to the outgoing HE1 solution, we have

W n Bx[-•,,H (') (203n/2/31rl[) (60) -- 1/3

where Vn = k 2 __ A 2 and B is a constant. For our parameter n

values, the Hankel function argument in Eq. (60) is in excess of about 50 for all propagating modes, and we therefore em- ploy an asymptotic form ofH <• Consequently, the MOR- 1/3 ß

EPE, SPE, and far-field HE1 intensities in the transition region are, respectively,

lul = _ 1 N - iA 2n[k(r) -- k,]/I 2 k(r) • Ane nlsin(JtnZ) , (61) n•O

- 1 --i'•"r--r')/2k'sin(/[nZ) (62) '• A,,e , N 2

IUI 2= • B.v•-•/'•e:i[k-'(r•-•"'"-731nlsin(A.z) , rt=0

(63) where

- !/2 2 !/4 B,, =k,

Xexp[ -- 2/(k• (64) and N d- 1 is the number of propagating modes. Beyond the transition region, the MOREPE, SPE, and HE1 intensities are found from Eqs. (45)-(47), the latter using the simple solution of Eq. (48). Each solution is then matched appro- priately to its transition-region counterpart at r- r•_.

A three-mode solution to each equation is generated for source frequencies of 150 and 300 Hz. With this number of modes, propagation is easily shown to qualify as narrow angle. Parameter choices are r I • 500 m, r•_ = 700 m, zs = 10 m, z• -- 50 m, and z• -- 100 m. The sound speed is initially 1540 m s-! and decreases across the transition region to a

141 J. Acoust. Soc. Am., Vol. 89, No. 1, January 1991 Schurman et aL: Energy-conserving PE 141


final value of 1500 m s- •. Figure 7 shows two sets of relative- intensity curves corresponding to the two source frequencies, with the 300-Hz curves shown heavier. As expected, MOREPE tracks the HE1 solution quite closely in each case, while the SPE curves exhibit positioning errors. In the 150-Hz case, MOREPE is in error by only a fraction of a decibel, and at 300 Hz the MOREPE and HE1 curves are indistinguishable. The SPE curve is visibly distinct in each case. The improvement in accuracy between MOREPE and HE 1 as frequency increases is explained by the asymptotic analysis of Sec. IID. For a fixed number of modes, each MOREPE mode matches the corresponding HE1 mode more closely as frequency increases. Conversely, at a fixed frequency, the MOREPE solution more closely resembles the HE1 solution as the number of propagated modes decreases. Since higher modes propagate at steeper angles, this also illustrates the narrow-angle nature of the MOREPE solution. These characteristics of MOREPE, HE1, and SPE held for a number of different exact-solution comparisons that we performed. We conclude from this example and the supporting analyses that MOREPE is more accurate for range-dependent problems than is SPE.

Finally, we mention that it is possible to transform Tappert and Lee's •7 RAREPE into one that is similar to MOREPE by the procedure shown in Sec. IIE. Numerical comparisons of results from these two equations led to agreement within minor decibel differences. We emphasize, however, the relative simplicity and accuracy verification of MOREPE.

I

-50 F=IõOHz F=3OOHz • MOREPE • MOREPE, HE!

..... SPiC 600 m

1540 1500 z R

-60 OO•

(/3 z LLJ I'- --70 Z

-80

--

-90 O

..... HEI ..... SPE

j .,? ',iI ',.\11 til

1

4 6 8 IO

RANGE (KM)

•r

FIG. 7. Relative intensity I versus range r for exact MOREPE, SPE, and HE1 solutions; f= 150 and 300 Hz, z• = 10 m, z/• = 50 m; c• = 1540 m S I - ,c,1500ms •.

_

IV. SUMMARY

A new energy-conserving parabolic equation which possesses range-refraction capability and ko independence is derived. Transformation, operator, and multiscale formalisms validate the modified range refraction parabolic equation (MOREPE). MOREPE is shown to reduce to the standard

PE in weakly range dependent environments, but can supply significant corrections when the index of refraction varies strongly with range. The transport, eikonal, and ray equations for MOREPE are shown to be equivalent to those for the Helmholtz equations within the small-angle approximation. Exact solutions provide a mechanism for accuracy verification and illustrate distinction features between predictions of MOREPE and SPE. Occurrence of these types of distinctions is substantiated by numerical examples that incorporate strong range dependence. Asymptotic analysis of an exact HE solution justifies MOREPE's range behavior and suggests an improvement in accuracy over SPE. Com- parisons of exact solutions to all three equations verify this analysis and demonstrate MOREPE's ability to correctly handle narrow-angle range-dependent propagation. MOREPE is similar to an earlier range refraction PE, but it is simpler in form and in numerical implementation, and is validated here by analytical and numerical comparisons with HE1. In addition, our derivations produce a successor range-refraction PE for which a numerical implementation could be more efficient because the computation of fast variation is not required. Although MOREPE is limited to narrow-angle propagation, we suggest that it provides an important step in the development of range-dependent PE modeling.

ACKNOWLEDGMENTS

This work was supported by Code 1125OA, Office of Naval Research. This article is taken in part from a thesis to be submitted by I. W. Schurman in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematical Sciences at Rensselaer Poly- technic Institute, Troy, NY 12180-3590.

APPENDIX: MULTISCALE DERIVATION OF MOREPE

In this section, variables with carets denote nondimensional quantities, while dimensional physical variables are unadorned. We scale the independent variables r and z by

? = eKr, • = xi•Kz, (A 1 )

as suggested originally by Tappert I and used by others, where • is a small dimensionless parameter and K represents the scale of the dimensional wave number k; i.e.,

k(r,z) = K•c(?,•). (A2) The order-one :6 function •c (?,•) will be assumed to have the form

•c(?,•) -- 1 + •1(•') JI- •'•2(•',•), (A3) where the component functions of •c are defined as follows:

•c.(?)----av• [•c -- 1], •----maxlav • [•c -- 1]l (A4) and

142 J. Acoust. Soc. Am., Vol. 89, No. 1, January 1991 Schurman eta/.' Energy-conserving PE 142


•:(?,•) =•- av• •, •=maxl•- av• • I, (A5) where av• implies a depth average. The parameter 6 represents the maximum of the accumulated depth-averaged range variation of•:, and could be a small quantity. However, we do not require it to be small, but instead permit it to be as large as O( 1 ). Doing this, while requiring 6 to be small, is one way to account for significant range variation in the refractive index. In the following development we will assume that the order relation

0(6) < 0(6)<0( 1 ) (A6)

holds.

The scaled version of the far-field HE is

62U?? -•- 6U}} -•- • 2U = 0. (A7) We apply the method of multiple scales 2v'2• by seeking a solution to Eq. (A7) in the form

(A8) u(?,2) - U[•,2,-?(?,2);e],

where

-7--- 1 • /c(},2)d} (A9)

represents a fast variable. This quantity is one way to incorporate the predominantly fast change of the acoustic field, and the remaining variation in U incorporates the slower modulation. We note that when the nondimensional func-

tion •c in -7- is approximated by one, which is appropriate for weakly range-dependent environments, -7- assumes the simple form -7' - ?/6, and SPE can be shown TM to result.

With our choice of -?, application of Eq. (A8) in Eq. (A7) gives

e:u. + e(e. + + + + 2•U•, + •U,) + •:(U** + U) =0, (A10)

where we have used the fact that ?•. = •/6 and &•.•. = •./6. In Eq. (A 10), depth derivatives of & are order 1, as can be seen from Eqs. (A9) and (A3) together. In addition, •. is O(8) because of Eq. (A3), and this term is retained in Eq. (A 10) by virtue of the order relation, Eq. (A6). We next assume an asymptotic expansion of U in the form

U• U (ø) + eU (I) + '-', e•0. (A11)

Inserting Eq. (A 11 ) into Eq. (A 10) produces a sequence of equations, the first two of which are

U•O• U(O• • + = 0 (A12)

and

•2(U(i) (i) •+U )

- • + •. • +-zz +2•U•

+ U © • +'7'•U• ). (A13)

We note that Eqs. (A 12) and (A 13 ) reduce, in the absence of depth variation, to a corresponding procedure for deriving a WKB approximation. 27 Also, we have assumed •5 > O(6), so that the inclusion of O(•5) terms with those of O( 1 ) is justified, but the cases of •5 as small as 6, or zero, can be treated if desired.

Since the assumed time dependence is e- i•,,, the outgoing solution of Eq. (A 12) is

U (ø) - •p(?,2)ei( (A14)

The solution ofEq. ( A 10) will remain bounded for all -7- only if the right side of Eq. (A13 ) sums to zero. 26 Applying this condition and using Eq. (A 14) gives

2i•:q•. + q• + 2i•-•q• + (i•:•. + i•-• -- • )q• = 0. (A15)

Equation (A15) is a nondimensional version of the slow field, Eq. (7), obtained by transformation arguments. It could provide a basis for numerical implementation that is an alternative to MOREPE, as discussed in Sec. III. To obtain the nondimensional version of MOREPE, we perform the transformation

•p -- ulø)(•',2)e '•' in Eq. (A 15 ), which yields

:?.. (o) . (o) 2[c 2/6 U (0) 2,,,,7. + + (i[c• + ) --O.

(A16)

(A17)

Equation (A 17) can be matched with Eq. (9), the dimensional form of MOREPE, by using the scaling relations Eqs. (A1) and (A2).

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2j. F. Claerbout, Fundamentals of Geophysical Data Processing (Blackwell, Oxford, 1985), p. 206.

3L. E. Estes and G. Fain, "Numerical technique for computing the wide angle acoustic field in an ocean with range-dependent velocity profile," J. Acoust. Soc. Am. 46, 38-43 (1977).

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7A. Bamberger, B. Engquist, L. Halpern, and P. Joly, "Higher order parax- ial wave equation approximations in heterogeneous media," SIAM J. Appl. Math. 48, 129-154 (1988).

8R. N. Baer, "Propagation through a three-dimensional eddy including effects on an array," J. Acoust. Soc. Am. 69, 70-75 ( 1981 ).

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144 J. Acoust. Soc. Am., Vol. 89, No. 1, January 1991 Schurman eta/.: Energy-conserving PE 144


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An energy-conserving parabolic equation incorporating range refraction