I. INTRODUCTION. J. Acoust. Soc. Am. 113 (1), January /2003/113(1)/223/22/$ Acoustical Society of America

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1 Spectra and moda formuations for the Dopper-shifted fied scattered by an object moving in a stratified medium Yi-san Lai and Nichoas C. Makris a) Massachusetts Institute of Technoogy, Cambridge, Massachusetts 239 Received 5 Apri 2; revised 3 March 22; accepted 2 June 22 Spectra and norma mode formuations for the three-dimensiona fied scattered by an object moving in a stratified medium are derived using fu-fied wave theory. The derivations are based on Green s theorem for the time-domain scaar wave equation and account for Dopper effects induced by target motion as we as source and receiver motion. The formuations are vaid when mutipe scattering between the object and waveguide boundaries can be negected, and the scattered fied can be expressed as a inear function of the object s pane wave scattering function. The advantage of the spectra formuation is that it incorporates the entire wave number spectrum, incuding evanescent waves, and therefore can potentiay be used at much coser ranges to the target than the moda formuation. The norma mode formuation is more computationay efficient but is imited to onger ranges. For a monochromatic source that excites N incident modes in the waveguide, there wi be roughy N 2 distinct harmonic components in the scattered fied. The Dopper shifts in the scattered fied are highy dependent upon the waveguide environment, target shape, and measurement geometry. The Dopper effects are iustrated through a number of canonica exampes. 23 Acoustica Society of America. DOI:.2/ PACS numbers: 43.3.Gv, 43.3.Vh, 43.3.Es DLB I. INTRODUCTION a Eectronic mai: makris@mit.edu Standard active sonar and radar systems estimate the instantaneous veocity of a moving target in free space by resoving Dopper shifts in the frequency spectrum of scattered waves. To obtain a components of the veocity vector, a mutistatic measurement geometry may be necessary. This type of active scenario is we suited to the veocity estimation of a distant body because the frequency spectrum of the source is known and controabe and so can be taiored to the resoution constraints of the probem at hand. In passive sonar and radar, however, veocity estimation by Dopper shift anaysis is often ess reiabe because the distant object must itsef radiate enough power to be detected. Additionay, the frequency spectrum of this radiation must be known, and have sufficienty narrow bandwidth and stabiity for Dopper shifts to be extracted robusty. The probem of using active sonar to estimate the veocity of an underwater target moving in an ocean waveguide has compications not found in the free-space anaogue. This is because propagation and scattering effects in a waveguide are typicay not separabe as they are in the far fied freespace scenario. Aso, mutipe frequency components are typicay present in the fied scattered from an object moving in an ocean waveguide even if the active source of radiation is harmonic. An accurate physica mode for the fied scattered from an object moving in a stratified ocean waveguide must then be derived before techniques can be deveoped to estimate the submerged object s veocity. It is the goa of this paper to derive such a mode and to investigate the Dopper effects induced by motion of a source, target, and receiver in a stratified ocean waveguide. Incusion of source and receiver motion is aso necessary because the source and receivers are typicay mounted on research vesses that move with speeds simiar to that of the target, and so induce their own Dopper effects that must be differentiated from those induced by the target. Dopper effects induced by the motion of a radiating source that is passivey measured at a moving receiver in free space have been extensivey studied in acoustics.,2 Dopper effects for the corresponding passive probem of a moving source and a moving receiver submerged in a stratified ocean waveguide have aso been studied in the iterature. 3,4 Mutimoda propagation and dispersion make the Dopper effects far more compicated in a waveguide than in free space. For exampe, the fied radiated by a time harmonic source moving in an ocean waveguide can be received with mutipe frequency components because of mutimoda propagation. A number of modes exist for three-dimensiona scattering from targets submerged in a stratified medium, as described in Ref. 5. A particuary convenient and widey used approach is the singe-scatter theorey deveoped in Refs The major advantage of this approach is that the scattered fied is expressed in terms of the target s free-space pane wave scattering function. This theory is vaid when the propagation medium is horizontay stratified and rangeindependent; 2 the object is contained within an isoveocity ayer; 3 mutipe scattering between the object and waveguide boundaries make negigibe contribution at the receiver; and 4 the range from the object to source and receiver is sufficienty arge that the scattered fied can be expressed as a inear function of the object s pane wave scattering function. This theory, however, assumes that the source, receiver, and target are not moving so that Dopper effects must be negigibe. In this paper, the singe scatter theory is generaized to J. Acoust. Soc. Am. 3 (), January /23/3()/223/22/$9. 23 Acoustica Society of America 223

2 incude the effects of source, receiver, and target motion. Anaytica expressions are obtained for the fied scattered to a moving receiver from a moving target in a stratified ocean waveguide by a moving source. The formuations are fuy bistatic, and a the motions are assumed to be horizonta with constant veocities. Both the expressions for a simpe harmonic source and a source with arbitrary time dependence are derived in this paper. Spectra and moda representations of the scattered fied are derived from first principes using the time-domain formuation of Green s theorem. The spectra representation makes fewer assumptions and is more accurate than the norma mode representation at coser ranges, but the norma mode formuation provides a compeing physica interpretation and can be used at onger ranges without significant oss of accuracy. The singe scatter theory of Refs. 6 and 7 then becomes a specia case of the present more genera theory when the source, receiver, and target are at rest. The four isted restrictions of the stationary singe scatter theory aso appy to the generaized theory deveoped in this paper. It is noteworthy that when the target, source, or receiver are moving, the scattered fied no onger obeys reciprocity, as is evident in our present formuation. The concept of a timereversa mirror 9 therefore is not directy appicabe under motion of the target, source, or receiver. This is true in both free space and in a stratified medium. A simpe and intuitive technique for deriving the fied radiated from a moving source measured at a moving receiver using deta functions is aso presented for both spectra and moda formuations. The spectra representation is identica to the resut of Ref. 4. The norma mode representation makes more accurate approximations than those used in Ref. 3. The resuting expressions are used in the scattering probem to describe the incident fied from the moving source at the moving target. II. ANALYTIC FORMULATION Anaytica expressions for the fied scattered from a moving source by a moving object measured at a moving receiver are derived from first principes using the timedomain scaar wave equation and the corresponding timedomain formuation of Green s theorem. Some of the basic approximations and techniques used in Refs. 6 and 7 to sove the stationary scattering probem are aso appied here. The major difference, however, is that we must sove the probem with the time-domain scaar wave equation instead of the Hemhotz equation to account for motion of the source, receiver, and target. The time-domain scaar wave equation for the tota fied T with a source function q(r,t )is 2 T r,t 2 T r,t c 2 2 qr t,t. The Green function for the time-domain scaar wave equation satisfies 2 Gr,tr,t 2 Gr,tr,t c 2 2 rr t tt. 2 By appying Green s theorem, the tota fied T can be expressed as 2 T r,t t dt dv Gr,tr,t qr,t t dt ds "Gr,tr,t T T Gr,tr,t c 2 dv Gr,tr,t t T Gr,tr,t T t t, which differs from Eq of Ref. 2 ony by a 4 factor due to differing choices for the deta function normaization. The first integra represents the incident fied i induced by the source, and the second integra represents the scattered fied s. The third integra accounts for the transient response. For exampe, given a time harmonic source turned on at t, this integra vanishes after the source has been operating for a time duration t arge compared to the source period. The first two integras then represent the steady state response, and the tota fied is the summation of the first two integras T r,t i r,t s r,t 4 with incident fied i r,t t dt dv Gr,tr,t qr,t and scattered fied s r,t t dt ds "Gr,tr,t T T Gr,tr,t. Foowing the type of abbreviating convention adopted in Refs. 6 and 7, we wi drop the first term in Eq. 6 in the derivation to avoid cumbersome and uninformative agebra. The derivation with both terms proceeds in exacty the same manner and eads to exacty the same expression for the scattered fied. This expression is in terms of the object s pane wave scattering function for an object with arbitrary boundary conditions. 5 7 The scattered fied from a rigid surface with unspecified shape is s (r,t) t dt ds " T (r,t ) G(r,tr,t ) t dt ds " i r,t s r,t Gr,tr,t. For a steady wave probem, this eads to Eq. 37 of Ref. 6 directy J. Acoust. Soc. Am., Vo. 3, No., January 23 Y.-s. Lai and N. C. Makris: Dopper fied scattered by an object

3 s r,t t dt ds " i r,t s r,t Gr,tr,t, 2 where the surface integra is carried out on the surface of the scatterer. The incident fied induced by a simpe-harmonic source at frequency moving with horizonta veocity v and received at a point r on the surface of an object moving with horizonta veocity v is obtained from Eq. A as i r,t d 2 2 i gz,z ; i "v FIG.. Measurement geometry for a submerged object in a horizontay stratified waveguide ensonified by a point source. The coordinate system is centered at the centroid of the object with positive z pointing down. Each ayer i is characterized by sound speed c i, density i, and attenuation a i. For economy, the notation of Ref. 7 is used here and in the remainder of this artice. Figure 2 of Ref. 7 shows the geometry of spatia and wave number coordinates. For exampe, the object centroid at the initia ocation of the object is at the center of a coordinate systems, as shown in Fig.. Source coordinates are denoted by (x,y,z ), receiver coordinates by (x,y,z), and coordinates on the surface of the target by (x,y,z ) where the positive z axis points downward and is norma to the interfaces between horizonta strata. Spatia cyindrica (,,z) and spherica (r,,) systems are defined by xr sin cos, yr sin sin, z r cos, and 2 x 2 y 2. Wave number coordinates for the incident ( ix, iy, i ) and scattered fied ( x, y,) aso originate at the target center and are reated to poar and azimutha propagation anges by 2 x 2 y 2, where x k sin cos, y k sin sin, z k cos, k 2 2 c The superscript is used to denote the initia positions of the source, target, and receiver, for exampe, x(t) t x. A. Spectra representation of the Dopper-shifted fied scattered from a moving target by a simpe-harmonic source in a stratified waveguide A spectra representation for the fied from a moving source, scattered by a moving target at a moving receiver, is now derived. The source is taken to be a simpe-harmonic one with frequency, and the motions of the source, target, and receiver are a horizonta with constant veocity. In order to cacuate the scattered fied, Eq. 7 is appied where the incident fied at a point r on the surface of the target depends on time t. The scattered fied at the receiver ocation r at time t can then be cacuated by e i i " e i i "v v t. 3 With the decomposition proposed in Eq. 6 of Ref. 7, the depth-dependent Green function defined in Eq. A7 becomes gz,z ; i Az ; i e i i i z Bz ; i e i i i z 4 with the shifted frequency of the incident fied i i "v. The ocation of a point on the surface of the target is r r v t 5 6 with r as its initia ocation at t and v as its horizonta veocity. The incident fied in Eq. 3 then becomes i r,t d 2 2 i e i i " e i i "v v t Az ; i e i i " i i z Bz ; i e i i " i i z. 7 The spectra representation of Green s function for the Hemhotz equation in a stratified waveguide is Grr ; d 2 gz,z 2 ;e i". 8 Simiary, the depth-dependent Green function in Eq. 8 is decomposed as gz,z ;Cz;e i i z Dz;e i i z. The motion of the receiver is expressed as rr vt, 9 2 where r is its initia ocation at time t and v is its horizonta veocity. The Green function for the time-domain scaar wave equation from the surface of the target r at time t to the receiver ocation r at time t then becomes J. Acoust. Soc. Am., Vo. 3, No., January 23 Y.-s. Lai and N. C. Makris: Dopper fied scattered by an object 225

4 Gr,tr,t d e itt 2 d 2 e i" e i"vt e i"v t 2 Cz;e i" i z Dz;e i" i z. 2 Inserting Eqs. 7 and 2 into Eq. 2 eads to the scattered fied s r,t t dt 2 ds " 2 d 2 i e i i " e i i "v v t Az ; i e i i " i i z Bz ; i e i i " i i z s r,t d e itt d 2 e i" e i"vt e i"v t 2 Cz;e i" i z Dz;e i" i z, 22 where "zkr(,;,) and,;,cos cos sin sin cos 23 is the cosine of the ange between the propagation direction, and fied coordinate direction, where the anges, may be compex. Substituting this anguar representation into Eq. 22 yieds s r,t t dt 2 ds " d 2 2 i e i i " e i i "v v t Az ; i e ik i r i i, i ;, Bz ; i e ik i r i i, i ;, s r,t d e itt d 2 e i" e i"vt e i"v t 2 Cz;e ikr,;, Dz;e ikr,;,, 24 where ( i, i ) is the propagation direction of the incident pane waves and (, ) is the direction of r, the initia ocation of a point on the target with respect to the initia position of the target centroid which is the origin of a coordinates. For ow Mach number motion, the scattered fied on the surface of the object in Eq. 24 is approximatey ˆ sr,t ˆ sr ; i e i i t 25 for a given incident pane wave. The wave number vectors for the downgoing and upgoing waves are defined as When the Mach number of the target motion is sma, the scattered fieds on the surface of the moving target ˆ s(r,k i ; i ) and ˆ s(r,k i ; i ), which are induced by downgoing and upgoing incident pane waves with unit ampitudes, can be approximated as the scattered fieds at the initia ocations of the target mutipied by a phase shift factor e i i "v t that accounts for the rigid body transation of the centroid. The scattered fied on the object then becomes s r,t d 2 2 i e i i " e i i "v v t Az ; i ˆ sr,k i ; i k i i i î z, 26 Bz ; i ˆ sr,k i ; i. 28 k i i i î z. 27 Introducing Eq. 28 into Eq. 24, then eads to the scattered fied 226 J. Acoust. Soc. Am., Vo. 3, No., January 23 Y.-s. Lai and N. C. Makris: Dopper fied scattered by an object

5 s r,t t dt 2 ds " d 2 2 i e i i " e i i "v v t Az ; i e ik i r i i, i ;, ˆ sr,k i ; i Bz ; i e ik i r i i, i ;, ˆ sr,k i ; i d e itt d 2 e i" e i"vt e i"v t 2 Cz;e ikr,;, Dz;e ikr,,,. 29 For sufficienty ong time duration t, the integra over t introduces the deta function ( i "(v v )"v ) to the integrand. Integrating over then eads to s r,t d 2 2 i d 2 e i" i " e i i "v v "v vt ds Az ; i e ik i r i i, i ;, ˆ sr,k i ; i Bz ; i e ik i r i i, i ;, ˆ sr,k i ; i Cz; s e ik s r s,;, Dz; s e ik s r s,;, 3 where the Dopper shifted frequency of the scattered fied is s i "v v "v. 3 It is important to note that the time dependence has been factored from the surface integra in going from Eq. 29 to Eq. 3 foowing our approximation for the assumed ow Mach number motion. This means, for exampe, that the object s orientation with respect to the incoming and outgoing waves is not significanty atered for a time period arge enough compared to the source period for the source to be considered harmonic. This is discussed in more detai, for exampe, in Sec. II C and Appendix B. We can then express the scattered fied in the waveguide in terms of the pane-wave scattering function S(,; i, i ;) of the object. With the aid of Eq. C9, Eq. 3 becomes s r,t d 2 d 2 i k s e i" i " e i i "v v "v vt Fz,z ;, i ; s, i, 32 which is an expression for the fied scattered by a moving target with arbitrary shape, where Fz,z ;, i ; s, i Az ; i Cz; s S s,; i i, i ; s Az ; i Dz; s S s,; i i, i ; s Bz ; i Cz; s S s,; i i, i ; s Bz ; i Dz; s S s,; i i, i ; s. 33 The formuation is fuy bistatic and incorporates horizonta veocities of the source, target, and receiver. The source is assumed to be a simpe-harmonic one radiating at frequency, but the received time series wi contain mutipe frequency components due to Dopper effects. The Dopper frequency shifts are indicated in the argument of the compex exponentia function of Eq. 32. When the source, target, and receiver are at rest, a incident frequencies i and scattered frequencies s are equa to the source frequency. In this case Eq. 32 reduces to Eq. 8 of Ref. 7 mutipied by exp(it) where reciprocity for harmonic waves d Gr r ;d Gr r ; 34 was invoked for the incident fied and the medium densities d and d in the ayers of the source and target depth were assumed identica. In Eq. 33, a coefficients A s and B s of the incident fied are evauated at the incident frequency i, and a the coefficients of the scattered fied C s and D s are evauated at the scattered frequency s. The wave number normaization k and the pane-wave scatter function S are evauated J. Acoust. Soc. Am., Vo. 3, No., January 23 Y.-s. Lai and N. C. Makris: Dopper fied scattered by an object 227

6 at the scattered frequency s as we. The equivaent eevation anges i of the incident pane waves are evauated at the incident frequency i, and the equivaent eevation anges of the scattered pane waves are evauated at the scattered frequency s. B. Spectra representation of the Dopper-shifted fied scattered from a moving target by a source with arbitrary time dependence in a stratified waveguide The Dopper-shifted scattered fied induced by a source with arbitrary time dependence q(t) and frequency spectrum Q() can be obtained in the receiver s frame of reference from Eq. 32 by Fourier synthesis as s r,t dq 2 2 d 2 d 2 i k s e i" i " e i i "v v "v vt Fz,z ;, i ; s, i. 35 A direct impementation of Eq. 35, however, wi be inefficient because the four-dimensiona wave number integras are couped with time in the argument of the compex exponentia function and so need to be evauated at each individua time instant. Simiar difficuties for the passive probem of modeing propagation from a moving source to a moving receiver are discussed in Ref. 4 by Schmidt and Kuperman. They note that by transforming the Dopper-shifted fied from the source frequency to a representation in terms of the receiver frequency, the wave number and frequency integrations can be integrated independenty. 4 The frequency spectrum of the scattered fied in the receiver s frame of reference is obtained by appying a Fourier transform to Eq. 35, s r, dte it s r,t 36 where is the frequency in the receiver s frame of reference. Integrating over t introduces the deta function ( i "(v v )"(v v)) in the integrand. Upon integrating over, the frequency spectrum of the Doppershifted scattered fied in the receiver s frame of reference then becomes s r, d 2 d 2 i k s ei" i " Q i "v v "v v Fz,z ;, i ; s, i, 37 where the shifted frequencies i and s in terms of are equa to and i "vv i "v s "v Equation 37 can be impemented efficienty and directy without the need for time domain processing. C. Norma mode representation of the Dopper-shifted fied scattered from a moving target by a simpe-harmonic source in a stratified waveguide At sufficienty ong source and receiver ranges from the target, the scattered fied can be we represented as a sum of norma modes. The moda representation for the scattered fied with a simpe-harmonic source is derived in this section. Green s function for the time-domain wave equation in a waveguide can be written as an inverse Fourier transform of the moda form of Green s function for the Hemhotz equation, Gr,tr,t d Grr ;e itt 2 d e itt 2 id 4 m u m z;u m z ; H m, 4 where u m (z;) and m () are the ampitude function and horizonta wave number of the mth mode at frequency.we assume m (), and the asymptotic form of the zeroth-order Hanke function of the first kind is used H m 2 m ei m /4. For a moving target, the horizonta position vector is v t cos i x sin i y 4 v t cos i x v t sin i y, 42 where is its initia position at t and v is its horizonta veocity. Simiary, the horizonta position vector of the receiver is vt cos i x sin i y vt cos i x vt sin i y, 43 where is its initia position at t and v is its horizonta veocity. For the bistatic configuration used in the scattering probem, the horizonta range to a point on the target is much smaer than the range to the receiver so that. For ow Mach number motions of the target as in typica sonar scenarios, the dispacements v t of a target point and vt of the receiver are aso much smaer than so that the azimutha ange of the vector vt is approximatey equa to the azimutha ange of the vector even for a time duration t so much arger than the source period that the source can be considered harmonic. An approximation for can then be made that 228 J. Acoust. Soc. Am., Vo. 3, No., January 23 Y.-s. Lai and N. C. Makris: Dopper fied scattered by an object

7 is ap- vt v t vt v t cos. Simiary, the azimutha ange of the vector proximated as because. This eads to 44a and the incident fied are now expressed as a inear superposition of equivaent pane waves in the ayer of the target via Gr,tr,t d e itt 2 vt cos v t cos. 44b Then since we have cos vt cos v t cos. 44c Green s function for the time-domain scaar wave equation from a point on the surface of the target r at retarded time t to the receiver ocation r at time t then can be approximated as Gr,tr,t d e itt 2 id 8 ei/4 m e i m cos u m z;u m z ; m e i m v cost e i m v cos t. 45 As in Eqs. 4 and 42 of Ref. 6, the Green function and m A m r ;e ikr m, ;, B m r ;e ikr m, ;, e i m v cos t e i m v cos t 46 i r,t v v G cos A r ; e ik r, ;, B r ; e ik r, ;, e i v cos t, 47 where Eq. 47 is derived in from Eq. B, and is the Dopper-shifted frequency of the th mode as defined in Eq. B. Substituting Eqs. 46 and 47 into Eq. 2 eads to s r,t t dt 2 ds v /v G cos A r ; e ik r, ;, B r ; e ik r, ;, e i v cos t s r,t d e itt m A m r ;e ikr m, ;, B m r ;e ikr m, ;, e i m v cost e i m v cos t. 48 For ow Mach number motion, the scattered fied on the surface of the object in Eq. 48 is approximatey ˆ s(r,t )ˆ s(r ; )e i t for a given incident pane wave. We define the wave number vectors for the downgoing and upgoing waves for the th mode as k î î z, 49 k î î z. 5 The scattered fied on the surface of the target given in Eq. 48 can then be represented in terms of downgoing and upgoing pane incident waves with unit ampitudes ˆ s(r,k ; ) and ˆ s(r,k ; ), respectivey, via J. Acoust. Soc. Am., Vo. 3, No., January 23 Y.-s. Lai and N. C. Makris: Dopper fied scattered by an object 229

8 s r,t e i t v v G cos A r ; ˆ sr,k ; B r ; ˆ sr,k ;. 5 Just as in the derivation for the spectra representation, in Sec. II A, approximations are made for ˆ s(r,k ; ) and ˆ s(r,k ; ) that account for rigid body transation. Equation 5 then becomes s r,t v v G cos e i v cos t A r ; ˆ sr,k ; B r ; ˆ sr,k ;. When this is inserted into Eq. 48 the scattered fied takes the form 52 s r,t t dt 2 ds " v /v G cos ei v cos t A r ; e ik r, ;, ˆ sr,k ; B r ; e ik r, ;, ˆ sr,k ; d e itt m A m r ;e ikr m, ;, B m r ;e ikr m, ;, e i m v cos t e i m v cos t. 53 For sufficienty ong time duration t, integration over t introduces the deta function ( ( )v cos( ) m ()v cos( )) to the integrand. In order to integrate over, we need to sove the transcendenta equation for the argument h() of the function dh d d m d v cos v v G m cos, 55 h v cos m v cos. 54 Equation 54 can be soved numericay. However, an approximation that can be evauated anayticay is desired. The derivative of h() with respect to is where v m G () is the group veocity of the mth mode at frequencey. For ow Mach number motions, Eq. 55 is cose to unity, so that h() is neary inear around the roots of Eq. 54. Using the Newton Raphson method with the frequency as an initia guess, the first iteration yieds a reasonaby accurate soution of Eq. 54 as m, h h v cos m v cos v v G m cos, 56 where v m G ( ) is the group veocity of the mth mode at frequency. Here m, is the douby Dopper-shifted frequency with respect to the th incident mode and the mth outgoing mode. Using the property of the function for any functions f, h Eq. 9.6 in Ref. 2 it must hod that 23 J. Acoust. Soc. Am., Vo. 3, No., January 23 Y.-s. Lai and N. C. Makris: Dopper fied scattered by an object

9 f f hd, 57 dh/d* where * is a zero of h, i.e., h(*). Integrating over gives s r,t m ds v /v G cos v /v G m m, cos A r ; e ik r, ;, ˆ sr,k ; B r ; e ik r, ;, ˆ sr,k ; A m r ; m, e ik m, r m, ;, B m r ;e ik m, r m,,, e i m, m m, v cos t. 58 With the aid of Eq. C9, the scattered fied can finay be written in terms of the scattering function of the target as s r,t4 m v v G cos v k m, v G m m, cos A r ; A m r ; m, S m m,, ;, ; m, A r ; B m r ; m, S m m,, ;, ; m, B r ; A m r ; m, S m m,, ;, ; m, B r ; B m r ; m, S m m,, ;, ; m, e i m, m m, v cos t. 59 When there is no motion of the source, target, or receiver, a the incident and scattered frequencies are evauated at the source frequency, and Eq. 59 eads to the specia case Eq. 5 of Ref. 6 mutipied by exp(it). If the number of modes that truncates the moda summation excited at the source frequency is N, this same number can be used to truncate the incident and outgoing moda summations for ow Mach number motions. The tota number of discrete frequency components wi then be roughy N 2 due to the couping between incident and scattered modes at the target. D. Norma mode representation of the Dopper-shifted fied scattered from a moving target by a source with arbitrary time dependence in a stratified waveguide For a source with arbitrary time dependence q(t) and frequency spectrum Q(), the norma mode representation of the Dopper-shifted scattered fied can be formuated by Fourier synthesis as s r,t d Qs r,t, 6 2 where s (r,t) is given in Eq. 59. Equation 6, however, is computationay inefficient because the moda summation needs to be evauated at every time instant. Just as in the spectra representation of the scattered fied from a source with arbitrary time dependence, transformation to the frequency spectrum in the receiver s frame of reference can speed up the computation significanty. Appying a Fourier transform to Eq. 6 is not desired because both shifted frequencies and m, of the incident and scattered fied are approximations obtained by the Newton Raphson method in terms of the source frequency. A derivation for the shifted frequencies in terms of the receiving frequency based on those approximated vaues wi give inaccurate and compicated resuts. Therefore, the frequency spectrum in the receiver s frame of reference needs to be derived from intermediate expressions for the incident fied and scattered fied before the approximations by the Newton Raphson method are made. The derivation is engthy and is given in Appendix D. With the aid of Eq. C9, the scattered fied of Eq. D4 is expressed in terms of the scattering function of the target as J. Acoust. Soc. Am., Vo. 3, No., January 23 Y.-s. Lai and N. C. Makris: Dopper fied scattered by an object 23

10 s r,4 m Q m k m v v G m m cos v v G,m cos A r ;,m A m r ; m S m m, ;,m, ; m A r ;,m B m r ; m S m m, ;,m, ; m B r ;,m A m r ; m S m m, ;,m, ; m B r ;,m B m r, m S m m, ;,m, ; m, 6 where the frequency of the source spectrum is m,m,m v cos. 62 Equation 6 can be impemented efficienty and directy without the need for time domain processing. with carrier frequency f c 2 Hz, t s and. s. Its frequency spectrum is Q f e /22 2 f f c 2 e i2 ft. 64 III. ILLUSTRATIVE EXAMPLES Equation 6 is impemented with a modified version of the norma mode code KRAKENC. 3 The formuation is fuy bistatic and incorporates the motion of source, target, and receiver. For simpicity, ony monostatic configurations are iustrated. These have the strongest Dopper frequency shifts when ony the target is in motion and the source and receiver are at rest. The source function to be used in a exampes is a Gaussian moduated wave form The ampitude and phase of the time series of the source demoduated by the carrier frequency f c 2 Hz is shown in Figs. 2a and 2b. The frequency spectrum of the source is shown in Fig. 2c. qt 2 ett o 2 /2 2 e i2 f c tt o 63 FIG. 2. Pot a and b show the ampitude and phase of the source function, demoduated by the 2 Hz carrier frequency, versus time. Pot c shows the magnitude of the frequency spectrum of the source. FIG. 3. The magnitude of the free space pane-wave scattering function S(9,; i 9, i ) for a a pressure-reease sphere with ka2 at 2 Hz and b a pressure-reease circuar disk with ka2 at 2 Hz. The incident wave is parae to the disk s surface norma, i.e., at broadside to the disk. 232 J. Acoust. Soc. Am., Vo. 3, No., January 23 Y.-s. Lai and N. C. Makris: Dopper fied scattered by an object

11 FIG. 4. Measurement geometry for object scattering in a Pekeris waveguide. The source and receiver are coocated at a depth of 2 m, and the centroid of the target is at a depth of 5 m. A time series iustrations in this paper foow the same convention used in Figs. 2a and 2b. They show the magnitude and phase of the signas demoduated by the carrier frequency at 2 Hz. The phase is ony shown at times when the signa ampitude is not negigiby sma. A horizonta axes of time series pots in this section are abeed with reduced time, which is the actua time minus the round-trip horizonta range divided by the sound speed. Two types of targets are used as to iustrate scattering characteristics, incuding a pressure-reease sphere and a perfecty refecting circuar disk which both have ka2 at 2 FIG. 6. The scattered fied and its scaed phase rate for the Gaussian moduated source. The bottom type is sit. Source and receiver are coocated at 2-m depth with 5-m target depth. The horizonta range of the target is 2 m from the source. The object is a pressure-reease sphere of ka2 at 2 Hz. The dashed curves are for a stationary target. The soid curves are for a target moving toward the source at m/s. Pots a and b show the ampitude and phase of the time series demoduated by the 2 Hz carrier frequency. Pot c shows the scaed phase rate of the demoduated time series Eq. 65. Pot d shows the frequency spectra. FIG. 5. The scattered fied and its scaed phase rate for the Gaussian moduated source in free space. The object is a pressure-reease sphere of ka 2 at 2 Hz. The dashed curves are for a stationary target. The soid curves are for a target moving toward the source at m/s. Pots a and b show the ampitude and phase of the time series demoduated by the 2 Hz carrier frequency. Pot c shows the scaed phase rate of the demoduated time series from Eq. 65. Pot d shows the frequency spectra. Hz, where a is the radius of the sphere and disk. The free space pane wave scattering function of the sphere is given in Eq. A2 of Ref. 7. The scatter function of the disk is given in Ref. 4. Figures 3a and b show the magnitude of the scatter functions versus scattering ange for the sphere and the disk, respectivey. The incident wave is parae to the disk s surface norma, i.e., at broadside to the disk. Before iustrating the probem in a waveguide, exampes of object scattering in free space are shown for comparison. The measurement geometry is the same as that shown in Fig. 4 but without the waveguide boundaries. A monostatic sonar with coocated point source and receiver senses a pressure-reease sphere with ka2 at f c 2 Hz. The sonar and target are in water with a sound speed of 5 m/s, and they are initiay separated by 2 m in the horizonta and 3 m in the vertica. Equation C8 is used to perform the simuations. The dashed curves in Figs. 5a and 5b show the ampitude and phase of the demoduated time series of the scattered signas from a stationary target. The dashed curve in Fig. 5d shows its frequency spectrum. Since free space is nondispersive, and the scatter function is neary constant over the frequency band of the source, the received wave form appears effectivey as a J. Acoust. Soc. Am., Vo. 3, No., January 23 Y.-s. Lai and N. C. Makris: Dopper fied scattered by an object 233

12 FIG. 7. Same as Fig. 6 except bottom type is sand. scaed and time-shifted version of the transmitted signa, with negigibe spectra distortion. The time series after demoduation and the spectrum of the fied scattered from a sphere moving at m/s toward the source are shown as the soid curves in Figs. 5a, 5b and 5d. It can be seen that the free space Dopper-shifted spectrum can be very cosey approximated by a transated version of the stationary spectrum since negigibe distortion is introduced by the Dopper shift, dynamica factors described in Appendix C, and scatter function over the frequency band of the source. The phase ange versus time in Fig. 5b shows that the phase is neary a constant versus time for a stationary object. For the moving target, the phase ange is decreasing with respect to time at a constant rate, which represents a singe frequency shift induced by the target motion. The frequency shift is ineary porportiona to the radia component of target veocity in free space when the scattering funcion of the target does not vary significanty versus frequency within the band of the source. Active sonar and radar systems in free space typicay take the scaed phase rate t c dt 4 f c dt 65 as an estimate of the target s radia veocity where (t) is the phase ange of the sonar return after demoduation by the carrier frequency. The dashed curve and the soid curve in Fig. 5c show that the scaed phase rate (t) matches the FIG. 8. Same as Fig. 6 except bottom type is imestone. true vaue of the radia veocity of the targets, m/s and m/s for the exampes shown. In a iustrative exampes of this section, a water coumn of m depth is used to simuate a typica continenta shef environment. The density of the water is kg/m 3, the sound speed is 5 m/s, and the attenuation is 6. 5 db/. The simuations are performed over different seabed types to iustrate the dependence of the Dopper effects on bottom properties. A seabeds are modeed as hafspaces. The source and the receiver are coocated at a depth of 2 m without motion, and the centroid of the target is at a depth of 5 m. First, we show how different bottom types affect the Dopper shifts. Sand, sit, and imestone are used as the homogeneous materia of the bottom haf-space. The density, sound speed, and attenuation are taken to be 9 kg/m 3, 7 m/s, and.8 db/ for sand, 4 kg/m 3, 52 m/s, and.3 db/ for sit. The density, compressiona speed and shear speed of imestone are 22 kg/m 3, 25 m/s, and 8 m/s, respectivey. The attenuation coefficients are. and.2 db/ for compression and shear, respectivey. A sit bottom is used for the simuations in Fig. 6. A pressure-reease sphere with ka2 at 2 Hz is used as the target. The dashed curves in Figs. 6a and 6b show the ampitude and phase of the demoduated time series of the scattered signas from a stationary target, and the dashed curve in Fig. 6d shows its frequency spectrum. Both the ampitude of the time series and frequency spectrum appear to be Gaussian, which indicates that the dispersion due to mutipath effects in the waveguide is weak for this type of 234 J. Acoust. Soc. Am., Vo. 3, No., January 23 Y.-s. Lai and N. C. Makris: Dopper fied scattered by an object

13 FIG. 9. Same as Fig. 6 except the bottom type is sand and target veocity is changed. The target is moving toward the source at 5 m/s. FIG.. Same as Fig. 6 except the bottom type is sand and target moves away from source at m/s. bottom. The scattered fied is dominated by the owest order mode. The soid curves in Figs. 6a and 6b show the received time series scattered by a sphere moving toward the source at m/s, and the soid curve in Fig. 6d shows its frequency spectrum. The shape of the time series sti ooks Gaussian, and the arriva time is sighty earier than the stationary case due to the shortening of the horizonta distance. The frequency shifts due to Dopper effects can be observed in the soid curve in Fig. 6d. The spectrum aso ooks Gaussian but is shifted with the frequency shift of the first mode, which is cose to the frequency shift of the scattered fied in free space. Simiar to the exampes for free space, the phase versus time shown in Fig. 6b is neary a constant for the stationary target, and the phase is changing at neary a constant rate for the moving target. Appying Eq. 65, the scaed phase rate (t) is cacuated for both the stationary and the moving targets and is potted as the dashed and the soid curves in Fig. 6c, respectivey. Because the sonar return is not significanty distorted by the mutimoda dispersion and Dopper effects, the scaed phase rate is cose to the target s true radia veocity for both the stationary and the moving target. This indicates that the scaed phase rate (t) in Eq. 65 can be used to estimate the radia veocity of targets for this particuar scenario of a weaky dispersive waveguide. Figure 7 shows demoduated time series and frequency spectra for a sand bottom. The same spherica scatterer is used as in Fig. 6. The dashed curves in Fig. 7a and 7b show the ampitude and phase of the demoduated time series when the sphere is stationary. We see that the time series has not ony the arriva from the first mode but aso the ate arrivas from the higher order modes with sower group veocities. This indicates that the dispersion is much stronger for a sand bottom than a sit bottom. The received signas from a moving sphere soid curves in Fig. 7a show that not ony the first arriva is earier than in the stationary case but the contributions of the higher order modes are aso different. From Fig. 7d, we can see that the spectrum of the stationary case dashed curve is distorted due to mutimoda effects. The shifted spectrum soid curve is aso distorted and is not simpy a transated version of the stationary spectrum dashed curve. This is because the ower order modes have arger frequency shifts than the higher order modes so that energy is nonuniformy shifted across frequency. The phase of the demoduated time series in Fig. 7b shows that the phase ange versus time for the stationary target dashed curve varies sowy but is no onger neary a constant ike in free space and for a sit bottom. This is because of the higher order modes introduce different phase changes. The phase change versus time soid curve is not changing at a constant rate as in free space or for a sit bottom. The higher order modes introduce mutipe Dopper shifts and ater the rate of phase change. Figure 7c shows the scaed phase rate (t) of the demoduated time series cacuated by Eq. 65. The dashed curve is for the stationary target and the soid curve is for the moving target. The strong ate arrivas in the received fied shown in Fig. 7a introduce significant distortion of the phase ange in Fig. 7b and make the scaed J. Acoust. Soc. Am., Vo. 3, No., January 23 Y.-s. Lai and N. C. Makris: Dopper fied scattered by an object 235

14 FIG.. Same as Fig. 6 except the bottom type is sand and the target is at 3 m horizonta range from source/receiver. FIG. 2. Same as Fig. 6 except the bottom type is sand and the target is a perfecty refective disk with ka2 at 2 Hz. phase rates in Fig. 7c inconsistent with the target s true radia veocities. Even if the target is not moving at a, a rapid change occurs in the scaed phase rate when a strong ate arriva corresponding to a higher order mode with sower group veocity arrives. The difference between the scaed phase rate (t) and the target s true radia veocity can be greater than m/s when a strong ate arriva is received. This exampe shows that when the sonar return is significanty distorted by mutimoda effects, the scaed phase rate (t) in Eq. 65 cannot be used to reiaby estimate the target s radia component veocity. Limestone bottoms typicay have reativey ow attenuation, support many higher order modes and so ead to highy dispersive shaow water propagation. As shown in the received fied scattered by a stationary sphere dashed curves and by a sphere moving at m/s toward the source soid curves in Fig. 8a, severa ate arrivas are present with ong time deays induced by the higher order modes. The highy distorted spectra for a stationary sphere and a sphere moving toward the source at m/s are shown in Fig. 8d. Again, the Dopper-shifted frequency spectrum soid curve is not simpy a transated version of the stationary spectrum dashed curve. Figure 8b shows that the phase changes significanty versus time due to the mutimoda effects even if the target is not moving. Whie the target is moving, the phase change is compicated due to the mutipe Dopper shifts. The scaed phase rate (t) of the demoduated time series in Eq. 65 for both the stationary and moving target are shown as the dashed curve and the soid curve in Fig. 8c. As in the waveguide with a sand bottom, rapid changes of scaed phase rate occur making it differ by more than m/s from the true vaue of the target s radia veocity. A these resuts indicate that the Dopper shifts in the scattered fied are highy dependent on the ocean environment. Since Dopper effects are a function of target veocity, target veocity may be estimated by measurements of Dopper shifted fieds given a known source function and waveguide environment. The sensitivity of the Dopper shifted fied to variations in target veocity then becomes an important factor. To investigate this issue, consider again the case of a sand bottom with a spherica target as in Fig. 7, but now with the target moving toward the source at 5 m/s rather than m/s. Figure 9 shows the time series and spectrum of the resuting scattered fied, where the soid curve in Fig. 9d is the Dopper shifted spectrum. As expected, the dispersive effect in the time series and the frequency shift in the spectrum is smaer for reduced target speed. The phase of the demoduated time series for the target moving at 5 m/s aso changes sower than when the target is moving at m/s as shown in Fig. 9b. These effects are significant since the reduction in time spread of the higher order modes is on the order of tenths of a second and the frequency spectrum is significanty atered over the entire bandwidth of the signa. When the target is moving away from the source, the Dopper frequency shifts are negative. To iustrate this, Fig. shows the time series and spectra for the scattered fied from a sphere moving away from the source at m/s, where the bottom type is sand as in Fig. 7. The first arriva for the 236 J. Acoust. Soc. Am., Vo. 3, No., January 23 Y.-s. Lai and N. C. Makris: Dopper fied scattered by an object

15 FIG. 3. Same as Fig. 2 except the bottom type is imestone. moving target in Fig. a arrives sighty ater than in the stationary case because the target is moving away from the source. The negative frequency shift is significant, and on the order of the signa bandwidth, as is evident in the spectrum in Fig. d. The negative frequency shift is aso shown as the positive rate of phase change of the soid curve in Fig. b. On the other hand, the rate of phase change is negative for a target moving toward the source and receiver. The next exampe iustrates Dopper effects at greater target ranges. Using sand as the bottom type and the sphere at 3 m initia range from the source, the time series dashed curves in Fig. a is dispersed ess than the dashed curves in Fig. 7a where the horizonta range is 2 m. The scattered fied from a target moving toward the source at m/s soid curves in Fig. a is aso dispersed ess than those in Fig. 7a. This indicates that Dopper effects are highy dependent on the measured geometry. A perfecty refecting circuar disk facing the source with the same radius as the sphere of ka2 at 2 Hz is used to iustrate variations in the scattered fied for fat versus rounded targets. A sand bottom as used in Fig. 7 is aso used in Fig. 2. Figures 2a, 2b, and 2d show the scattered fied from a stationary disk and a disk moving toward the source at m/s. With the same measurement geometry and 2 m as the initia horizonta distance, the time series in Fig. 2a appear to be dispersed far ess than the time series in Fig. 7a. The unshifted and shifted frequency spectra of Fig. 2d aso exhibit this phenomenon. The same measurement geometry and scatterer is used in Fig. 3 but with a imestone bottom. Figures 3a and 3b show the scattered fied from a stationary disk and a disk moving toward the source at m/s. These time series are aso much ess dispersive than the scattered fied from a spherica scatterer. The unshifted and shifted frequency spectra in Fig. 3d are both ess distorted than the spectra with a spherica scatterer in Fig. 8d. This is because scattering from the disk is much stronger in the specuar direction than the other directions. Figure 3b shows the magnitude of the free-space pane-wave scattering function of the circuar disk. Comparing Figs. 3a and b, we can see that the scattering function of the sphere does not vary too much near the specuar refection direction. This eads to reativey uniform couping between different modes of the incident and scattered fied. On the other hand, the disk is highy directiona near the specuar refection direction and gives strongest couping between a given mode of the incident fied and the same mode of the scattered fied, i.e., diagona terms of a couping matrix. Since higher order modes attenuate more than ower order modes and the couping term between a ower order mode and a higher order mode is weaker, the received signa is dominated by the ower order modes of both the incident and scattered fied from the disk. Time-frequency spreading is aso significanty weaker than for a spherica scatterer. It is not aways true that the scattered fied is stronger when the target is moving toward the source than at rest in a waveguide. For a moving source in free space, the sound fied in the forward direction is aways more intense than that in the back direction because of the factor M cos in pressure, which accounts for free-space dynamics, where M is the Mach number and is the ange between the direction of motion and the direction of the fied point. 2 In a waveguide, athough there are simiar dynamica factors v v G cos, v v G m m, cos in the moda expression of Eq. 59, they are so cose to unity for ow Mach number motions of the source and target, respectivey, and are not the dominant factors for the changes of signa ampitudes. In a waveguide the fied magnitude can fuctuate rapidy as a function of position, frequency, and waveguide environment due to moda interference. The observed fuctuations in fied magnitude of the various exampes given are dominated by such changes in moda interference as a function of frequency due to Dopper shifting. For exampe, Fig. 6d shows that with a sit bottom and a target moving toward the source, the scattered fied is actuay weaker than the scattered fied from a stationary target because the moda interference with Dopper shifting is more destructive than without. IV. CONCLUSION Anaytica expressions for the three-dimensiona fied scattered by a moving target from a moving source to a moving receiver in a genera horizontay stratified ocean waveguide are derived from first principes using the time-domain J. Acoust. Soc. Am., Vo. 3, No., January 23 Y.-s. Lai and N. C. Makris: Dopper fied scattered by an object 237

16 formuation of Green s theorem. Spectra and moda representations of the Dopper-shifted scattered fied for a simpe harmonic source and a source with arbitrary time dependence are obtained. The expressions are vaid when the source and receiver are sufficienty far from the target that mutipe scattering between the target and waveguide boundaries can be negected and the scattered fied can be expressed as a inear function of the target s pane wave scattering function. The source, target, and receiver are assumed to move horizontay with ow Mach numbers, as is typica in many active sonar scenarios. The moda representation has a compeing physica interpretation exhibited by the fact that a simpe harmonic source that excites N modes in the waveguide, for exampe, wi excite roughy N 2 distinct harmonic components in the scattered fied due to couping between the incident modes and the scattered modes. The spectra representation, however, is more genera and can be used at coser ranges to the target. Simuations show that Dopper shifts induced in the scattered fied by target motion are highy dependent on the waveguide environment, target shape, and measurement geometry. For a highy dispersive waveguide that supports many trapped modes, the frequency spectrum of the fied scattered by a moving target typicay exhibits significant distortion compared to that of a stationary target or the same target moving in free space. Rounded scatterers with reativey omnidirectiona scattering functions, such as spherica scatterers, have greater couping between incident modes and scattered modes than fat objects that scatter strongest in the specuar direction. The scattered fied from an object in a mutimoda waveguide tends to suffer greater dispersion as the target becomes more rounded and the scattering becomes more omnidirectiona. It is noteworthy that when the target, source, or receiver are moving, the scattered fied no onger obeys reciprocity, as is evident in our present formuation. The concept of a timereversa mirror 9 therefore is not directy appicabe under motion of the target, source, or receiver. This is true in both free space and in a stratified medium. A new derivation for the Dopper shifted fied radiated to a moving receiver from a moving source in a stratified medium that proved advantageous in the present work is aso presented. The new moda formuation is more accurate than previous formuations, since for exampe, it accounts for variation in mode shape due to Dopper shift. APPENDIX A: SPECTRAL REPRESENTATION OF THE DOPPLER-SHIFTED FIELD RADIATED BY A MOVING SOURCE TO A MOVING RECEIVER IN A STRATIFIED WAVEGUIDE A spectra representation for the wave fied induced by a moving source and measured at a moving receiver has been presented in Ref. 4. An aternative derivation utiizing Eq. 5 is presented here to represent incident fieds in the scattering probem. The resut is consistent with prior research but is better suited to the probem at hand. The ocation of a moving source is denoted by r r v t, A where r is the initia ocation of the source at t and v is its horizonta veocity. For simpe-harmonic radiation at frequency, the source function q(r s,t )is qr s,t e it rs r v t. The ocation of a moving receiver is denoted by rr vt, A2 A3 where r is the initia ocation of the receiver at t and v is its horizonta veocity. After changing the variabes of integration in Eq. 5 from r to r s and V to V s, and appying Eq. A2, Eq. 5 becomes i r,t t dt Gr vt,tr v t,t e it. A4 Green s function for the time-domain scaar wave equation of the waveguide can be obtained by appying an inverse Fourier transform to Green s function for the Hemhotz equation at frequency, Gr,tr,t Grr ;e itt d, 2 A5 where the spectra representation of Green s function for the Hemhotz equation of a stratified waveguide is given by Grr ; d 2 2 i gz,z ;e i i ". A6 The depth dependent Green function g(z,z ;) in Eq. A6 is defined as gz,z ; d 2 Grr 2 ;e i i ", A7 where. With this Eq. A5 can be expressed as Gr,tr,t d e itt 2 d 2 2 i gz,z ;e i i ". A8 After inserting Eq. A8 into Eq. A4, the incident fied becomes i r,t t dt 2 de it e it 2 d 2 i gz,z ;e i i " vt v t. A9 For sufficienty ong duration t, integration over t eads to 238 J. Acoust. Soc. Am., Vo. 3, No., January 23 Y.-s. Lai and N. C. Makris: Dopper fied scattered by an object