Inversion for a sphere’s geometrical, elastic, position ... · but du pr´esent rapport est...

Defence R&D Canada – Atlantic

DEFENCE DÉFENSE&

Inversion for a sphere’s geometrical, elastic,

position, and radial-motion parameters in a

waveguide

John A. Fawcett

Technical Memorandum

DRDC Atlantic TM 2010-029

May 2010

Copy No. _____

Defence Research andDevelopment Canada

Recherche et développementpour la défense Canada

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Inversion for a sphere’s geometrical, elastic,position, and radial-motion parameters in awaveguideJohn A. Fawcett

Defence R&D Canada – AtlanticTechnical MemorandumDRDC Atlantic TM 2010-029May 2010

Principal Author

John A. Fawcett

Approved by

David HopkinHead/Signatures

Approved for release by

Calvin HyattHead/Document Review Panel

c© Her Majesty the Queen in Right of Canada as represented by the Minister ofNational Defence, 2010

c© Sa Majeste la Reine (en droit du Canada), telle que representee par le ministrede la Defense nationale, 2010

Original signed by John A. Fawcett

Original signed by David Hopkin

Original signed by Ron Kuwahara for

Abstract

The broadband scattering characteristics of a target may be used to distinguish itsechos from those of clutter. This type of classification is of interest in many sonarapplications: ASW, torpedo, mines, and diver detection. In a shallow water situa-tion, the original sonar pulse and the echo from the target will consist of a sequenceof pulses corresponding to the various combinations of incident and backscatteredmultipath arrivals. Thus, the interference effects of the waveguide propagation canhave a significant effect upon the received echo and thus also affect the classificationof the target from the echo. In addition, the target may be moving, in which casethe received echo is Doppler-shifted. The purpose of this report is to investigate thesimultaneous determination of a spherical target’s position (range and depth) withina waveguide, its radial speed, its radius, its shell thickness, and the elastic parame-ters of the shell. It will be shown that many of these parameters can be accuratelyestimated from a single wideband echo. The expected uncertainties of the parameterestimates will also be investigated.

Resume

Il est possible de faire la distinction entre les echos d’une cible et ceux du clutter,au moyen des caracteristiques de diffusion a large bande de cette cible. Ce type declassification sert a un grand nombre d’applications sonar : guerre anti-sous-marine,defense contre les torpilles, chasse aux mines et detection des plongeurs. En eau peuprofonde, l’impulsion sonar initiale et l’echo de la cible consistent en une sequenced’impulsions correspondant aux diverses combinaisons de signaux incidents et designaux retrodiffuses recus par trajets multiples. Les effets d’interference de la propa-gation dans le guide d’ondes peuvent donc avoir une importante incidence sur l’echorecu, et par consequent sur la classification de la cible a partir de cet echo. De plus, lacible peut etre en mouvement, auquel cas l’echo recu subit un decalage Doppler. Lebut du present rapport est d’examiner la determination simultanee des parametresde position d’une cible spherique (distance et profondeur) a l’interieur d’un guided’ondes, de sa vitesse radiale, de son rayon ainsi que de l’epaisseur et de l’elasticitede sa coquille. Il sera montre que bon nombre de ces parametres peuvent etre evaluesde facon precise a partir d’un seul echo a large bande. Les incertitudes prevues desevaluations de parametres seront aussi examinees.

DRDC Atlantic TM 2010-029 i


ii DRDC Atlantic TM 2010-029

Executive summary

Inversion for a sphere’s geometrical, elastic, position,and radial-motion parameters in a waveguide

John A. Fawcett; DRDC Atlantic TM 2010-029; Defence R&D Canada – Atlantic;May 2010.

Background: The detection and classification of a sonar echo is a problem of muchinterest for many sonar applications: ASW, torpedo defence, minehunting and diverdetection. A wideband echo from an object contains much useful information aboutthe object itself. However, this echo is also affected by the object’s unknown positionwithin the acoustic waveguide due to the propagation-induced interference effects. Inthis report, an inversion method is described which allows for the simultaneous esti-mation of a spherical target’s geometrical, elastic, positional, and motion parameters.

Principal results: It was found that it was possible to accurately invert a broadbandecho from a spherical target to estimate its geometrical, speed, and elastic parameters,even when the position of the target within the waveguide is not accurately known.

Significance of results: It is shown in this report that a wideband echo contains alarge amount of information about the scatterer. With the assumption of a sphericalshape, the size, the shell thickness, and the sphere’s elastic parameters as well as itsposition and radial speed can be estimated. The values of these parameters couldthen be used to make a decision about whether the detected object is a threat (forexample, moving diver, underwater vehicle, mine, etc.) or can be ignored.

Future work: In future work, we would like to investigate the relaxation of theassumption of the target shape being spherical. It may be that even when an objectis non-spherical that the inversion method for an effective spherical scatterer stillyields useful classification information or, it may be, that addition shape parametersare required in the inversion. In addition, we would like to validate this approachwith experimental data.

DRDC Atlantic TM 2010-029 iii

Sommaire

Inversion for a sphere’s geometrical, elastic, position,and radial-motion parameters in a waveguide

John A. Fawcett ; DRDC Atlantic TM 2010-029 ; R & D pour la defense Canada –Atlantique ; mai 2010.

Introduction : La detection et la classification d’un echo sonar presentent un interetconsiderable pour de nombreuses applications sonar : guerre anti-sous-marine, defensecontre les torpilles, deminage et detection des plongeurs. L’echo a large bande d’unecible constitue une importante source d’information utile sur cet objet. Cependant,en raison des effets d’interference dus a la propagation, la position inconnue de l’objetdans le guide d’ondes a aussi une incidence sur l’echo. Le present rapport decrit unemethode d’inversion permettant l’evaluation simultanee des parametres de geometrie,d’elasticite, de position et de mouvement d’une cible spherique.

Resultats : Il a ete constate qu’il est possible d’inverser avec precision un echo a largebande produit par une cible spherique, afin d’evaluer les parametres de geometrie, devitesse et d’elasticite de cette cible, meme si sa position exacte dans le guide d’ondesn’est pas connue.

Portee : Le present rapport montre qu’un echo a large bande constitue une impor-tante source d’information sur le diffuseur. En presumant qu’une cible est spherique,il est possible d’evaluer sa grosseur, l’epaisseur de sa coquille, son elasticite, sa po-sition et sa vitesse radiale. Les valeurs de ces parametres pourraient ensuite servir adeterminer si l’objet detecte represente une menace (p. ex. un plongeur en mouve-ment, un vehicule sous-marin, une mine, etc.) ou non.

Recherches futures : Dans le cadre de recherches futures, nous aimerions etudierl’assouplissement de l’hypothese d’une cible spherique : soit la methode d’inversionpour un diffuseur spherique fournit de l’information utile sur la classification memedans le cas d’un objet non spherique, soit des parametres supplementaires sur la formesont necessaires a l’inversion. Aussi, nous aimerions valider cette approche avec desdonnees experimentales.

iv DRDC Atlantic TM 2010-029

Table of contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Executive summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Sommaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Modelling of echo from a moving sphere . . . . . . . . . . . . . . . . . . . 3

3 Inversion of observed echo for spherical, location and motion parameters . 6

4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.1 Some echo computations . . . . . . . . . . . . . . . . . . . . . . . . 8

4.2 Some Two-dimensional Cost Function Surfaces . . . . . . . . . . . . 10

4.3 Simulated annealing results . . . . . . . . . . . . . . . . . . . . . . . 14

5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

DRDC Atlantic TM 2010-029 v

List of figures

Figure 1: The computed echo (10 Hz frequency spacing) from the sphere inthe waveguide (source/receiver depth = 10 m, sphere depth=1.5 m) using (a) one multipath term (direct) (b) (9 incident, 9scattered) (c) (21 incident, 21 scattered) - only those which fallwithin the time window . . . . . . . . . . . . . . . . . . . . . . . 9

Figure 2: The computed echo (10 Hz frequency spacing) from the sphere inthe waveguide (source/receiver depth = 10 m, sphere depth=1.5 m) for (a) sphere moving at 10 m/s towards thesource/receiver (b) stationary sphere (c) sphere moving away at10 m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Figure 3: The cost function surfaces as a function of range and radial speed:(a) STTC (b) SD. All parameters except range and radial speedare set to their true values . . . . . . . . . . . . . . . . . . . . . . 11

Figure 4: The cost function surfaces as a function of range and depth: (a)STTC (b) SD. All parameters except range and depth are set totheir true values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Figure 5: The cost function surfaces as a function of shell compressionalspeed and sphere radius: (a) STTC (b) SD. All parameters exceptsphere radius and shell compressional speed are set to their truevalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Figure 6: The ZTTC cost function surface as a function of radial speed andrange. All parameters except for the radial speed and the initialrange are set to their true values. . . . . . . . . . . . . . . . . . . 13

Figure 7: The ZTTC cost function surface as a function of compressionalsound speed and the radius. All parameters except for the radialspeed and the initial range are set to their true values. . . . . . . 13

Figure 8: The 16-kHz [2 18] Chirp pulse: (a) incident pulse - for direct pathpropagation from source/sphere and sphere/receiver with nosphere scattering (b) received echo . . . . . . . . . . . . . . . . . . 14

Figure 9: The STTC cost function surface as a function of radial speed andrange (Chirp pulse). All parameters except for the radial speedand the initial range are set to their true values. . . . . . . . . . . 15

vi DRDC Atlantic TM 2010-029

Figure 10: The STTC cost function surface as a function of compressionalsound speed and the radius (Chirp pulse). All parameters exceptfor the radial speed and the initial range are set to their true values. 15

Figure 11: The noisy echo from the incident Ricker pulse used for the inversion 16

Figure 12: The noisy Chirp echo used (a) for the inversion and (b) thematch-filtered time series . . . . . . . . . . . . . . . . . . . . . . 17

Figure 13: The distribution of four of the parameter values from thesimulated annealing method for cost function values τ < 0.35. . . 18

Figure 14: The distribution of the other four parameter values from thesimulated annealing method for cost function values τ < 0.35. . . 18

Figure 15: The distribution of four of the parameter values from thesimulated annealing method for cost function values τ < 0.80. . . 19

Figure 16: The distribution of the other four parameter values from thesimulated annealing method for cost function values τ < 0.80. . . 19

List of tables

Table 1: The search intervals for the sphere’s parameters - Ricker Pulse . . 20

Table 2: The search intervals for the sphere’s parameters - Chirp Pulse . . 20

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viii DRDC Atlantic TM 2010-029

1 Introduction

Over the last few years, there has been much interest in the discrimination of ob-jects of interest from clutter on the basis of the spectral features of a wideband sonarecho. This has been investigated for buried and proud mine and general target/clutterclassification [1-5]. Unfortunately, a target’s location within the ocean waveguide willalso affect its echo characteristics. For example, in Ref. 6 it was shown how theecho from an aluminum sphere on a seabed, was significantly different in amplitudeand phase, from that expected in free-space (i.e., no surrounding boundary effects).Thus, an understanding of the target scattering process in a waveguide is impor-tant in understanding target detection and classification in a shallow water scenario.A shelled-sphere serves as a simple model for a target. In the case of a stationarysphere, it can be used to model a sea mine on the seabed, near the surface, or floatingsomewhere in the water column. A moving sphere can be used to represent a movingvehicle or a swimming diver. In previous reports and papers [7-9] we described a mul-tipath expansion method for modelling the scattering from general, elastic spheres ina waveguide. The spherical target was considered for the modelling and simulation ofechos because the solution to the scattering problem in free space can be determinedanalytically and is thus computationally faster than having to use, for example, afinite element code. In [8,9] the scattering/propagation model was used to generatethe echos for three classes of spheres. Each class of sphere had fixed material prop-erties (these properties varied across classes) but had varying radii and relative shellthicknesses. The classification of the broadband echos into one of 6 possible classes(the three classes were additionally subdivided into thick and thin shell subclasses)was investigated. A portion of the echos were used to define a training set and theremainder were used as a test set. It was possible to obtain very good classificationresults when all the echos corresponded to the target in free space( i.e., there are noupper and lower boundaries). However, it was found that if the echos in the testingset were those from the target in the waveguide at an unknown depth, then it wasoften necessary to either know the depth of the target or to generate sets of echoscorresponding to a range of hypothesized depths in order to obtain good classificationresults. Other authors have also considered classification within a waveguide utilizingvarious approaches to the problem such as time-frequency features [10], waveguide-invariant features[11], and time-reversal[12].

In this report, instead of considering a classification approach, an inversion approachis considered. This has the advantage that instead of considering a fixed set of targetsand their echos for classification purposes, we can allow a large number of parameters,including the position of the sphere within the waveguide, to be unknown. However,we do make the assumption that the target is spherical in shape so that we can quicklygenerate the scattered echo for the many realisations of the parameters. In principle, ifwe used a more general numerical target scattering model or an approximate method,

DRDC Atlantic TM 2010-029 1

then the methods of this report could also be used.

We will discuss in Section 2 the propagation model used in the pulse generation; inparticular, we discuss how the Doppler effect is incorporated as this is a new additionto the method of [7]. In Section 3, a simulated annealing method to estimate thesphere’s elastic and positional parameters is described. In Section 4, some numericalexamples are presented. First, the computed echos for some moving spheres are pre-sented. Then, using the echo for the a known target as the received echo, we comparethe echos generated by varying two of the sphere’s parameters with the others heldfixed. The resulting cost function surfaces give an indication which parameters can beestimated unambiguously and which may be difficult to uniquely estimate. Finally,we show the performance of a simulated annealing optimization method for the pa-rameter estimation. The echos from two different spheres, for different incident pulsesand amounts of noise, are considered. The various parameters and the associated costfunction values are retained during the computations and this allows us to displaythe range of parameters which yield cost function values below a specified threshold.

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2 Modelling of echo from a moving sphere

In order to model the scattering from a sphere in a waveguide, we utilize the modeldescribed in [7]. In the propagation model,the sphere is characterized by its scatteringcoefficients an - the coefficient of the outgoing Spherical Harmonic hn(kR)Pn(cos φ)which results from an incident spherical harmonic function jn(kR)Pn(cos φ). Herek = ω/c for c the waveguide soundspeed , ω ≡ 2πf for f the frequency and φ isthe angle measured off the vertical (z) axis through the sphere. The coefficients {an}are a function of frequency. The farfield (i.e., kR large) expression for the free-fieldscattered pressure field from the sphere is given by

pscff (φ) =

exp(ikR)

kRS(φ, ω) (1)

where

S(φ, ω) =N∑

n=0

−i(2n + 1)an(ω)Pn(cos(φ)) (2)

and the waveguide field scattered back to a receiver at depth zr for a source at zs isgiven by

psc(zr, zs) =R∑

n1=1

R∑

n2=1

V (n1)V (n2)exp(i2πω/c(Dn1 + Dn2))

Dn1Dn2

S(φn2 − φn1, ω). (3)

Here, we are considering R raypaths incident upon the sphere from the source locationand then R rays from the sphere centre to the receiver. We are considering the sourceand receiver to both be at the same horizontal range (r = 0) but possibly at differentdepths. The terms Vn represent the effects of the multiple reflections off the seabed andthe upper surface and Dn represents the distances along these paths. The scatteringat the sphere is contained within the scattering function S(Δφ, ω) where Δφ is theangular difference between the incoming and outgoing rays. This model was validatedin [7] by comparing it with an exact wavenumber integral approach.

The exact solution to the scattering problem from a moving object or boundary isdifficult [13]. A moving object has hydrodynamic boundary conditions (particle veloc-ity boundary condition) and the elastic boundary conditions for a stationary sphereare not exactly the correct boundary conditions for a moving sphere. The method wedescribe below basically considers the scattering-strength from the sphere as if it wasstationary. However, the motion is accounted for in terms of the frequency Doppler-shifts along the incident and scattered ray paths and the time delay with respect topulse-propagation along the incident path. Similar approaches for scattering from amoving target are described using a modal approach [14] and a ray approach [15].

We consider each of the R2 ray path pairs in Eq.(3). For the incident path, wesolve a quadratic equation to determine the position of the moving sphere when


the incident waveform arrives at the sphere. This defines the position used for theray path pair computations and it is slightly different for the various incident raypaths. We wish to compute the received echo at a set of equi-spaced frequencies�f = {fi}, i = 1, ..., Nf . However, we consider the scattering function of the spherein its own frame of reference. Thus each receive frequency, fi, is compensated forits Doppler shift along the ray path of the scattered signal. This is done with theformula,

f spi = fi(1 + vr/c| sin(φrec)|).) (4)

Here vr denotes the radial velocity of the sphere, φrec denotes the angle the ray fromthe sphere to the receiver makes with the vertical, and f sp

i denotes the ith frequencyin the sphere’s frame-of-reference. For each ray path pair which defines the angles φn1

and φn2 in Eq.(3), we first evaluate the scattering function at the set of frequencies{fi}; in this case, however, we compute it also for i > Nf to accommodate possibleDoppler-stretching in the spectrum. It would be preferable to evaluate the scatteringfunction S(Δφ; ω) at the exact values of f sp

i ; however, the scattering coefficients{an(f)} are computed at a fixed set of frequencies. We use spline interpolation ofS(Δφ, ωi) to compute the values of S at {f sp

i }, i = 1, ..., Nf . For each frequency, theecho spectrum must be multiplied by the appropriate incident pulse weighting. Forthis we wish to transform the receive frequency back to the source frequency,

f srci = fi

1 + vr/c| sin(φrec)|1 − vr/c| sin(φsrc)| (5)

and these frequencies are then used in the incident source spectral function Ω(f) tocompute the source weighting. In the case of a simple analytic waveform such as aRicker (second time derivative of a Gaussian pulse) pulse we simply evaluate Ω(f src

i )analytically. For more complicated waveforms, where an analytic specification is moredifficult, the source spectrum is precomputed at a fine frequency spacing (using zero-padding in the time domain) and then spline interpolation used to compute thesource weighting at the desired frequencies. Finally, after summing over all the pathcombinations, we obtain the received spectrum, as a function of frequency. The FourierTransform of this spectrum yields the received pulse in the time domain.

There is a limit to the length of the time series we can compute based upon thefrequency increment used in the frequency computations. We multiply the frequencyspectra by a phase factor exp(iωΔt) corresponding to a time shift of Δt so that thecomputed time series will start only a little before the expected arrival of the echo.However, for a frequency spacing of Δf Hz then we can only compute the time seriesaccurately until Δt + 1/Δf . There is also a trade-off with respect to the number ofmultipath terms included in the spectral sum. For example, the higher-order termsmay arrive (in terms of travel time) outside of the computational time window andhence do not contribute to the time series within the desired time window. However,they can enter computationally into the time series through periodic aliasing effects.


In order to mitigate these effects, we will include only the multipath terms withpredicted echo times which fall into the time window corresponding to the frequencyincrement used.


3 Inversion of observed echo for spherical,location and motion parameters

In this paper, we estimate the parameters of a spherical target in a waveguide. Theset of unknown parameters, �p, for the sphere are: (1) shell compressional velocity(2) shell shear velocity (3) radius of sphere (4) relative thickness of the shell and (5)the density of the shell material. There are also parameters for: (6) the depth in thewaveguide (7) the range in the waveguide and (8) the radial velocity of the sphere.For simplicity, the interior (the region contained within the shell) of the sphere istaken to be evacuated. Otherwise, we could also estimate the elastic parameters ofthe material contained in the interior region. This would result in a larger inversionproblem. We will suppose that we have an observed echo e(t) which we will normalizeto have unity norm over its time window. The optimization problem is to minimizewith respect to the parameter vector �p the cost function

Ct(�p) = (1 − max(|e(t) ⊗ s(t; �p)|)2) (6)

where ⊗ denotes the time cross-correlation operation and we have normalized s(t; �p)to have unity norm. We have shown absolute values, but in fact we will consider themaximum amplitude of the envelope. This is a sliding cross-correlation; that is, thenormalized replica echo s(t; �p) is moved across the time window and the maximumcorrelation selected. This means the value is insensitive to the actual time of thereturn. This is analogous to traditional match-filtering of a received echo, exceptthat we are utilizing many different replicas corresponding to different values of theparameter vector �p. We refer to Ct(�p) as the Sliding Time Cross Correlation (STCC)cost function. We can also consider Eq.(6) for a time lag of zero

C0(�p) = (1 − |e(t) ⊗ s(t; �p)|2τ=0). (7)

In this case, the absolute value of the time scale is important; however, the range ofthe target is an unknown parameter so that this value can effectively be varied untils(t) is aligned with e(t). This expression is denoted as the Zero-offset Time CrossCorrelation (ZTCC) cost function.

Numerically, we use the computed spectrum for the replica echo to form its timeseries. The sliding cross-correlation of the data time series e(t) and s(t) is computedusing an IMSL Fortran routine, DRCORL. The envelope of this cross-correlationtime series is then computed (with FFT zero-padding we double the sampling ofthis function) and quadratic interpolation about the discrete peak is used to furtherrefine the maximum value, thus mitigating time-discretization effects. In the case ofEq.(7) we just take an inner product of the two normalized time series to compute|e(t) ⊗ s(t; �p)|2τ=0.


Another approach is to consider the minimization of a cost function based uponthe differences of the absolute values of the spectra (observed and modelled), CS(�p).In this case, we take the observed times series and compute its FFT spectrum andnormalize the vector of spectral amplitudes to have unity norm. This is then comparedto the corresponding vector of Nf modelled values,

CS(�p) =

Nf∑

i=1

(|SR(fi)| − |SD(fi)|)2/2. (8)

where SR is the normalized spectrum for the modelled replica and SD is the normal-ized spectrum for the received (data) echo. This is denoted as the Spectral Difference(SD) cost function.

Tesei et al [16] used a search over compressional and shear speeds and attenuationsto match a broad band echo (in that paper, the position and speed of the sphere isnot an issue). For the optimization in this paper, we used a straightforward simulatedannealing approach which has been widely-used in geoacoustical inversion problems(for example, [17-18]). In this approach, a large number of sweeps through the set ofparameters is carried out. For each sweep, each parameter is considered one at a timeand a value is drawn from an uniform random distribution within the parameter’sallowed range of values. The new parameter value is accepted if it decreases thecomputed cost function; if the cost function increases, the new parameter value isalso accepted for a random number rε[0 1] if

r < exp(−ΔC/T ) (9)

where T is called the temperature and ΔC is the difference between the cost-functionvalue for the new parameter values and the present cost function value. The tem-perature is initially set at a relatively high value so that most changes are acceptedand is gradually lowered to very small values at the end of the sweeps. As the sphereparameters are varied, the sphere’s scattering coefficients for all the frequencies, needto be recomputed each time. However, when the location and radial speed parametersare varied, only the effect of the waveguide propagation and the re-interpolation ofthe scattering function for Doppler-shifted frequencies need to be recomputed. Foreach sweep of the parameter vector’s values, we will, in fact, vary the range and radialspeed parameters 11 times and the depth parameter 21 times. This is done because:(a) the re-computation of the cost function is relatively cheap for these parametervariations and (b) as we shall see, the cost function is very sensitive to these param-eters. All the various sphere parameter values and the computed cost function valuesare saved as the algorithm progresses. This allowed us to compute histograms of theparameter values corresponding to cost function values below a certain threshold.


4 Numerical Examples4.1 Some echo computationsWe start the numerical examples with some sample pulse computations. We considera waveguide 20 m deep. The water has a sound speed of 1500 m/s and a density of1 g/cm3. The seabed has a compressional sound speed of 1700 m/s, an attenuation of0.25 dB/λ and a density of 1.5 g/cm3. In the first example, we consider the sphere at100 m horizontal range and a depth of 1.5 m. The source/receiver are at a depth of10 m (i.e., midwater column). The sphere is steel-shelled with an evacuated interiorand is initially stationary. It has a radius of 0.253 m and a shell thickness of 5 mm.The compressional and shear speeds for steel are taken to be 5950 m/s and 3240 m/sand the density is taken to be 7.7 g/cm3. The source waveform is a 5 kHz Rickerwavelet. In Fig. 1, we investigate the effect of the number of multipath terms inthe echo computations. In Fig.1a, a portion of the echo computed using only thedirect incident/scattered path contribution is shown (normalized to have a maximumamplitude of unity); this corresponds to the free space echo from the sphere. In theinversion examples, we use 9 incident and scattered terms to compute the replicaechos. The echo resulting from using these number of terms is shown in Fig. 1b.The accuracy of this representation can be compared to the echo computed using21 incident and scattered terms as shown in Fig. 1c. As can be seen, the agreementbetween Figs.1b and 1c is very good - there are some small differences but they arevery difficult to see. This gives us confidence that using 9 incident and scattered termswill be sufficiently accurate for the inversion. These computations were done for thesphere moving with a horizontal radial speed of 10 m/s. In Figs. 2a-2c the effectsof radial motion on the echo are shown. In the first plot, the sphere is moving witha radial speed of -10 m/s (towards the source/receiver), in the second the sphere isstationary and in the third, it is moving with a speed of +10 m/s. As is expectedthe arrival time of the echo changes due to the motion of the sphere during thetransit time of incident pulse. Somewhat surprising is that, aside from the overallshift in time, the time series for the speeds ±10m/s agree more with each other inappearance than with the stationary sphere. In the case of a stationary target, the raypairs for incident/scattered (rayj,rayk) yield exactly the same echo as the raypath-pair (rayk,rayj). For a moving target, the distance moved by the object during thetransit time of the incident pulse depends only on the incident ray path length. Inthis case, there is a small splitting of the arrival times of the echos from the 2 setsof rays discussed above. The echo from these 2 sets of rays results from the coherentaddition of the echos from slightly different arrival times. Although the radial speedhas different signs between Fig.2a and 2c, the splitting of the various multipathsresults in these 2 timeseries appearing more similar than the one for the stationarysphere.


−1

0

1

−1

0

1

130 140 150 160 170 180 190−1

0

1

Time(msec)

Nor

mal

ized

Rec

eive

d S

igna

l

(a) Direct/Direct term

(b) 9 incident/scattered terms

(c) 21 incident/scattered terms

Figure 1: The computed echo (10 Hz frequency spacing) from the sphere in the waveg-uide (source/receiver depth = 10 m, sphere depth =1.5 m) using (a) one multipathterm (direct) (b) (9 incident, 9 scattered) (c) (21 incident, 21 scattered) - only thosewhich fall within the time window

−1

0

1

−1

0

1

130 135 140 145 150 155 160 165−1

0

1

Time(msec)

Nor

mal

ized

Rec

eive

d S

igna

l

(a) −10 m/s

(b) 0 m/s

(c) 10 m/s

Figure 2: The computed echo (10 Hz frequency spacing) from the sphere in thewaveguide (source/receiver depth = 10 m, sphere depth =1.5 m) for (a) sphere movingat 10 m/s towards the source/receiver (b) stationary sphere (c) sphere moving awayat 10 m/s


4.2 Some Two-dimensional Cost Function SurfacesWe now consider some two-dimensional cost function surfaces for the moving sphere.We fix 6 of the sphere’s 8 parameters at their correct values. Then we vary thetwo selected parameters and compute the time domain (STTC) and spectral cost-functions (SD) as given by Eqs.(6) and (8). In the first example, we consider therange of the target unknown in the interval [98 102]m and the radial speed of thetarget unknown in the interval [-20 20]m/s. This interval of unknown speeds islarge corresponding to ±40 knots. The somewhat restricted interval of the unknowninitial ranges of the sphere is due to the fact that one would have fair estimate ofthis parameter from the arrival time of the echo. We first generate the echo for anincident 5-kHz Ricker pulse using 21 incident and scattered paths with the correctrange (100 m) and radial velocity (10 m/s) values. In addition, this echo is generatedusing a frequency spacing of 5 Hz in the spectral computations. The echo is computedsuch that t = 0 corresponds to the 2-way travel time for a sphere 95 m away andthe first 100 msec of the echo is used. The STTC and SD cost functions are thencomputed as the range and radial velocity values are varied. We use 9 incident andscattered paths and 10-Hz frequency spacing to generate the replicas. In Fig. 3a weshow the two-dimensional STTC cost function surface and in Fig.3b the SpectralDifference surface. In both cases, the range has been unambiguously determined butthe sign of the radial speed is ambiguous. (Recall that the absolute arrival time ofthe signal does not affect either of these 2 cost functions). The results of Fig. 3 areconsistent with our previous discussion regarding the echos of Fig. 2. It is difficult toquantitatively compare the STTC and SD cost function surfaces. We would expect thecross-correlation method to have a sharper minimum with respect to the parametersthan the spectral cost function which does not account for the relative phases of thespectra. In Fig.4, we repeat the computations as for Fig. 3 but now vary the rangeand depth of the target with the other parameters held constant. In Fig.4a it canbe seen that the global minimum is precisely determined as a function of the rangeand is very precisely determined as a function of the depth of the sphere. There isanother secondary local minimum observable at about 1.4 m off the seabed instead of1.5 m below the sea surface and at a range of 98.4 m; however, this minimal value issignificantly higher than the global minimum. The SD surface of Fig. 4b is similar tothat of Fig.4a but there is more ambiguity. The sensitivity of the STTC cost functionto the depth and range parameters indicate that a simulated annealing method willhave to sample these parameters quite finely in order to locate the global minimum.

We now consider the case of varying the outer radii of the sphere and its compressionalspeed. The ratio of the shear velocity to the compression velocity is fixed so that as thecompressional speed is increased (decreased) the shear speed is also changed. It wasexpected that these parameters would be somewhat correlated. As the diameter ofthe sphere is increased, the time for any waves to circumnavigate the sphere becomeslarger; however, by increasing the sound speed the increasing travel time is reduced.


Spe

ed(m

/s)

−20

−10

0

10

200.2

0.4

0.6

0.8

Range(m)98 98.5 99 99.5 100 100.5 101 101.5 102

−20

0

20

0.1

0.2

0.3

(b)

(a)

Figure 3: The cost function surfaces as a function of range and radial speed: (a)STTC (b) SD. All parameters except range and radial speed are set to their truevalues

Dep

th(m

)

5

10

15 0.2

0.4

0.6

0.8

Range(m)97 98 99 100 101 102 103

5

10

15 0.1

0.2

0.3

0.4

(a)

(b)

Figure 4: The cost function surfaces as a function of range and depth: (a) STTC (b)SD. All parameters except range and depth are set to their true values


0.2

0.3

0.4

0.5

Compressional speed (m/s)

Rad

ius(

m)

3000 3500 4000 4500 5000 5500 6000 6500 7000

0.2

0.3

0.4

0.5

0.2

0.4

0.6

0.8

0.10.20.30.40.50.6

(a)

(b)

Figure 5: The cost function surfaces as a function of shell compressional speed andsphere radius: (a) STTC (b) SD. All parameters except sphere radius and shell com-pressional speed are set to their true values.

The two-dimensional cost function surfaces (STTC and SD) are shown in Figs.5aand 5b. As can be seen, the resulting surfaces are quite complicated with severallocal minimum. The optimal result in both Figs.5a and 5b are within a linear-likefeature indicating a simple correlation between the radius and compressional speedparameters.

In Figs.6 and 7 we repeat the cost-function surface computations for the range/radialspeed and the radius/compressional speed parameters but now for the correlationtime-offset set to zero, Eq.(7), (ZTTC). As shown, the ambiguities in the cost functionsurfaces are much reduced. Recall, for these surfaces the arrival time of the replicais important. Thus, for example in Fig. 6, there is no longer the ± ambiguity inthe radial speed. However, it can be seen that there is a linear feature in the costfunction surface. It is the range of the sphere when the incident pulse arrives which isimportant in terms of the echo’s arrival time. A linear combination of different initialranges and sphere radial speeds yields the same range for the sphere at the time ofpulse incidence resulting in this linear feature. The cost function surface of Fig. 7 issignificantly different than those of Fig. 5. However, there is still a somewhat linearappearance to the surface.

The range/radial speed surface of Fig.4 shows an ambiguity for the simple Rickerwavelet. In terms of match-filtering, a more complex time-frequency waveform is moreDoppler-sensitive than the compact Ricker wavelet. In Fig. 8 we show a 2-18 kHzincident Chirp (20 msec) pulse and the resulting echo time series. The steel sphere in


Rad

ial S

peed

(m/s

)

Range(m)98 98.5 99 99.5 100 100.5 101 101.5 102

−20

−15

−10

−5

0

5

10

15

20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 6: The ZTTC cost function surface as a function of radial speed and range.All parameters except for the radial speed and the initial range are set to their truevalues.

Compressional Speed (m/s)

Rad

ius(

m)

3000 3500 4000 4500 5000 5500 6000 6500 7000

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 7: The ZTTC cost function surface as a function of compressional sound speedand the radius. All parameters except for the radial speed and the initial range areset to their true values.


135 140 145 150 155 160−0.1

−0.05

0

0.05

0.1

Nor

mal

ized

Inci

dent

Pul

se

135 140 145 150 155 160 165 170 175 180−0.1

−0.05

0

0.05

0.1

Time(msec)

Nor

mal

ized

Rec

eive

d S

igna

l

(a)

(b)

Figure 8: The 16-kHz [2 18] Chirp pulse: (a) incident pulse - for direct path propa-gation from source/sphere and sphere/receiver with no sphere scattering (b) receivedecho

this example has a radius of 0.4 m and a shell thickness of 8 mm. In Figs. 9 and 10 weshow the corresponding STTC cost function surfaces for variations of the range/radialspeed and the compressional velocity/radius. As shown in Fig. 9, the ambiguity withrespect to the sign of the radial speed has disappeared. The two-dimensional surfacefor the radius/compressional sound speed in Fig. 10 is quite similar to that of Fig.5for the Ricker pulse.

4.3 Simulated annealing resultsIn the previous section, we varied two of the sphere’s parameters, with the other valuesfixed at their true values, in order to generate a cost function surface which expressedsome of the uncertainty and nonlinearity of the parameter estimation. In this sectionwe show the results of simulated annealing runs using the method described in Section3.

For the first inversion example, we consider the echo from an incident 5-kHz Rickerwavelet. Random white noise is filtered in the Fourier domain to have the same averagespectral amplitude as the Ricker wavelet and is added to the echo. In Fig .11 we showthe resulting noisy signal. It can be seen that for this case, that although the latterportions of the echo are obscured with noise, the initial strong multipath arrivals areclearly visible. The second inversion example is for the incident Chirp pulse and thebigger sphere. In this case, we add a very significant amount of random white noise.


Ranges(m)

Spe

ed(m

/s)

98 98.5 99 99.5 100 100.5 101 101.5 102

−20

−15

−10

−5

0

5

10

15

20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 9: The STTC cost function surface as a function of radial speed and range(Chirp pulse). All parameters except for the radial speed and the initial range are setto their true values.

Compressional Speed(m/s)

Rad

ius(

m)

3000 3500 4000 4500 5000 5500 6000 6500 7000

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 10: The STTC cost function surface as a function of compressional soundspeed and the radius (Chirp pulse). All parameters except for the radial speed andthe initial range are set to their true values.


0 20 40 60 80 100

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time(msec)

Nor

mal

ized

Rec

eive

d S

igna

l

Figure 11: The noisy echo from the incident Ricker pulse used for the inversion

As can be seen in Fig.12a the echo from the sphere is not visible. However, in Fig.12bwe show the results of match-filtering (the process one would use in normal practice)this noisy echo with the incident Chirp pulse. There, a set of arrivals are discernible.

The first inversion used 3600 sweeps through the set of 8 parameters. The cost func-tion we use in these examples is the Sliding Time Cross Correlation. The first 5parameters are the sphere’s compressional speed, ratio of shear speed/compressionalspeed, density, radius, and relative shell thickness (in logarithmic (base 10) spac-ing). The second parameter was used instead of the shear velocity in order to ensurethat the ratio of these 2 speeds lay within physical bounds (i.e., vp >=

√2vs). The

shell thickness was described logarithmically (base 10) so that there would be a fineenough sampling of thin shell-thicknesses. The last 3 parameters are the depth, therange and the radial speed of the sphere. As discussed previously, the sphere’s scat-tering coefficients depend only on the first 5 parameters. Once these coefficients havebeen computed, the last 3 parameters can be varied and the waveguide response effi-ciently computed. Thus, for each sweep we vary these parameters 21,11, and 11 timesrespectively. We used 2 different cooling schedules: for the first 5 parameters, a tem-perature is initialized at a value of 0.07 and decreased by a factor of 0.9997, for thelast 3 parameters, the temperature is initialized at a value of 0.02 and decreased ateach sweep by a factor of 0.9994. This defines the value of T which is used in Eq.(9).These values for the cooling schedule parameters were obtained through numericalexperimentation. They yielded good parameter estimates while at the same time lead-ing to a good sampling of parameter space. The reason for the 2 different schedulesis that the cost function is much more sensitive to the range and depth parameters


−0.1

−0.05

0

0.05

0.1

Nor

mal

ized

Rec

eive

d S

igna

l

0 20 40 60 80 100−0.1

−0.05

0

0.05

0.1

Time(msec)

Nor

mal

iaze

d M

atch−F

ilter

ed S

igna

l

(a)

(b)

Figure 12: The noisy Chirp echo used (a) for the inversion and (b) the match-filteredtime series

(and to a certain extent the radial speed) relative to the other parameters. With thistemperature scheme, the depth, range, and radial parameters converge fairly quicklywith the other parameters varying more widely. We also reinitialized the randomnumber generator for the simulated annealing realizations with 7 different seeds for7 different runs. For each parameter vector realization, the parameter vector and thecorresponding cost function value is saved. The simulated annealing approach of thispaper could likely be made more efficient. However, in our approach the parameterspace is well-sampled and we can display histograms of the values of the parametersfor which the cost function is below a specified value. The search intervals used forthe 8 parameters for this inversion are given in Table 1.

In Figs.13 and 14 we show the histograms of the parameters (a composite of the 7simulated annealing initializations) which yield a cost below a threshold τ = 0.35.The distributions are reasonable: the sphere radius, its relative thickness (logarithm),and the compressional and shear speed are well-determined. In Fig.14 we see that thedensity is relatively poorly-determined. The position of the sphere is very accuratelydetermined as is its absolute radial speed. The sign of the radial speed is not well-resolved which is consistent with the two-dimensional cost function surface of Fig.3.If we make the threshold τ smaller then the distributions of the parameter valuesbecome tighter.

For the second inversion, the larger (radius=0.4 m, thickness=8 mm) is consideredwith the noisy echo of Fig. 12. Because of the large increase in the amount of noise in


0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60

0.5

−2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.20

0.5

0 2000 4000 6000 8000 100000

0.2

0.4

0 1000 2000 3000 4000 5000 6000 7000 80000

0.5

Parameter Value

Rel

ativ

e O

ccur

renc

e

Radius(m)

Relative Thickness (Log)


Shear Speed (m/s)

Figure 13: The distribution of four of the parameter values from the simulated an-nealing method for cost function values τ < 0.35.

0 2 4 6 8 100

0.1

0.2

0 0.5 1 1.5 2 2.5 30

0.5

1

98.5 99 99.5 100 100.5 101 101.50

0.5

1

−20 −15 −10 −5 0 5 10 15 200

0.5

Parameter Value

Rel

ativ

e O

ccur

renc

e

density (g/cm3)

depth (m)

range (m)

radial speed (m/s)

Figure 14: The distribution of the other four parameter values from the simulatedannealing method for cost function values τ < 0.35.


0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60

0.5

−3 −2.5 −2 −1.5 −1 −0.50

0.5

0 2000 4000 6000 8000 100000

0.1

0.2

0 1000 2000 3000 4000 5000 6000 7000 80000

0.5

Parameter Value

Rel

ativ

e O

ccur

renc

e

Radius(m)

Relative Thickness (Log)


Shear Speed (m/s)

Figure 15: The distribution of four of the parameter values from the simulated an-nealing method for cost function values τ < 0.80.

0 2 4 6 8 100

0.1

0.2

0 0.5 1 1.5 2 2.5 30

0.5

1

98.5 99 99.5 100 100.5 101 101.50

0.5

1

−20 −15 −10 −5 0 5 10 15 200

0.5

1

Parameter Value

Rel

ativ

e O

ccur

renc

e

density (g/cm3)

depth(m)

range(m)

radial speed (m/s)

Figure 16: The distribution of the other four parameter values from the simulatedannealing method for cost function values τ < 0.80.


Parameter Min MaxVp(m/s) 300 9300Vs/Vp .05 .7

Radius(m) 0.15 0.5Rel.Thick(Log) -2. -.1Density(g/cm3) 1 9

Depth(m) 0.5 18.5Range(m) 98. 102.vr(m/s) -20 20.

Table 1: The search intervals for the sphere’s parameters - Ricker Pulse

Parameter Min MaxVp(m/s) 800 8800Vs/Vp .05 .7

Radius(m) 0.15 0.6Rel.Thick(Log) -3. -.1Density(g/cm3) 1 9

Depth(m) 0.5 18.5Range(m) 98. 102.vr(m/s) -20 20.

Table 2: The search intervals for the sphere’s parameters - Chirp Pulse

the echo, the simulated annealing parameters need to be modified somewhat. Here,we use 4800 sweeps per run. For the first 5 parameters, the initial temperature wasset to 0.035 and a cooling factor of 0.9998 was used. for the depth, range, and radialspeed parameters, the initial temperature was set to 0.01 and a cooling factor of0.9994 was used. The parameter search intervals are shown in Table 2.

The resulting histogram distribution of values are shown in Figs. 15 and 16 (a com-posite of 7 simulated annealing initializations). Here the threshhold value used was0.8 instead of 0.35 because there is significantly more noise in this example and hencethe minimum cost function value is higher. Even for this case of very noisy data,the inversion results are good. The radii and the relative thickness of the sphere arewell-estimated. The compressional sound speed of the shell is somewhat ambiguousbut the shear speed seems better-estimated. The biggest difference with the inversionof the Ricker pulse is that now the radial speed is very accurately and unambiguouslyestimated.


5 Summary

We have shown in this paper that a single receiver echo, even with substantial noise,can be used to determine a sphere’s dimensional, elastic, position, and radial speedparameters. The sphere can be used as a very simple model for a target, possiblymoving. It allows for much more efficient computations than for generally-shapedtarget, yet still includes elastic-structural scattering effects. There is some uncertaintyin the estimates of the sphere’s elastic parameters as well as its radius and thickness.The sphere’s location and radial motion are particularly well determined. For thesimple Ricker incident pulse, there was an ambiguity with respect to the sign of theradial motion. This ambiguity was not present in the case of the Chirp incident pulse.This work was done for a single source and receiver. One would certainly expect thatthe estimates would be even more accurate if an array of sources and receivers wasused. One could also envision incorporating the inversion into a tracking algorithm.


References

[1] D.D. Sternlicht, A.W. Thompson, D.W. Lemonds, R.D. Dikeman, and M.T.Korporaal, “Image and signal classifcation for a buried object scanning sonar”,in Proceedings of IEEE/MTS Oceans 2002, Vol.1, pp.485-490, (2002).

[2] A. Tesei, J. Fawcett, and R.Lim, “Physics-based detection of objects buried inhigh-density-clutter saturated sediments”, Journal of Applied Acoustics,Vol.69, pp.422-437 (2008).

[3] F.B. Shin, D.H. Kil, and R.F. Wayland, “Active impulsive echo discriminationin shallow water by mapping target physics-based features to classifiers”, IEEEJournal of Oceanic Engineering, Vol.22, pp.66-80, 1997.

[4] P. Runkle, L. Carin, L. Couchman, J. Bucaro, and T. Yoder, “Multi-aspectidentification of submerged elastic targets via wave-based matched pursuits andhidden Markov models”, J. Acoust Soc., Vol. 106, pp. 605-616.

[5] J. Sildam and J. Fawcett, “Baseline classification of acoustical signatures ofmine-like objects”, DRDC Atlantic TM 2005-058, July 2005.

[6] J.A. Fawcett, W.L.J. Fox, and A. Maguer, “Modeling of scattering by objectson the seabed”, J. Acoust. Soc. Am., Vol. 104, pp.3296-3304 (1998).

[7] J.A. Fawcett, “Modelling broadband scattering from shelled spheres in awaveguide”, DRDC Atlantic TM 2007-270, October 2007.

[8] J.A. Fawcett and J. Sildam, “Broadband classification of spherical shells in awaveguide”, DRDC Atlantic TM 2008-217, November 2008.

[9] J.A Fawcett, “Broadband scattering from spherical shells in a waveguide:modeling and classification”, in proceedings of European Acoustics AssociationAcoustics’08, Paris, 29 June - 4 July, 2008.

[10] P. Chevret, N. Gache, and V. Zimpfer, ”Time-frequency filter for targetclassification”, J. Acoust. Soc. Am., Vol. 106, pp. 1829-1837, 1999.

[11] G.Okopal, P.J. Loughlin, and L. Cohen, ”Dispersion-invariant features forclassification”, J. Acoust. Soc. Am., Vol. 123, pp.832-841, 2008.

[12] N.Dasgupta and L. Carin, ”Time-reversal imaging for classification ofsubmerged elastic targets via Gibbs sampling and the Relevance VectorMachine”, J. Acoust Soc. Am, Vol. 117, pp.1999-2011 (2005).

[13] M. Gennaretti and C. Testa, “A boundary integral formulation for soundscattered by elastic moving bodies”, Journal of Sound and Vibration, 314,pp.712-737 (2008).

[14] Y. Lai and N.C. Makris, “Spectral and modal formulations for theDoppler-shifted field scattered by an object moving in a stratified medium”, J.Acoust. Soc. Am., Vol. 113, pp.223-244, 2003.


[15] K.V. Rao, N.M. Kee, P.W. Goalwin , and H. Schmidt, “Element levelsimulation of sonar echo from a moving target in an ocean waveguide”, inProceedings of IEEE/MTS Oceans 94, Vol.3, pp. 195-199, 1994.

[16] A. Tesei, P. Guerrini, and M. Zampolli, “Tank measurements of scattering froma resin-filled fiberglass spherical shell with internal flaws”, J. Acoust. Soc. Am.,Vol.124, pp. 827–840, 2008.

[17] S.E. Dosso, “Quantifying uncertainty in geoacoustic inversion. I. A fast Gibbssample approach”, J. Acoust. Soc. Am., Vol.111, pp. 129–142, 2002.

[18] S.E. Dosso and M.J. Wilmut, “Effects of incoherent and coherent sourcespectral information in geoacoustic inversion”, J. Acoust. Soc. Am., Vol.112,pp. 1390–1398, 2002


Distribution list


Internal distribution1 David Hopkin, SH/MAP

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1 Vincent Myers

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Inversion for a sphere’s geometrical, elastic, position, and radial-motion parameters in awaveguide

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Fawcett, J.A.

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The broadband scattering characteristics of a target may be used to distinguish its echos fromthose of clutter. This type of classification is of interest in many sonar applications: ASW, torpedo,mines, and diver detection. In a shallow water situation, the original sonar pulse and the echofrom the target will consist of a sequence of pulses corresponding to the various combinationsof incident and backscattered multipath arrivals. Thus, the interference effects of the waveguidepropagation can have a significant effect upon the received echo and thus also affect the clas-sification of the target from the echo. In addition, the target may be moving, in which case thereceived echo is Doppler-shifted. The purpose of this report is to investigate the simultaneousdetermination of a spherical target’s position (range and depth) within a waveguide, its radialspeed, its radius, its shell thickness, and the elastic parameters of the shell. It will be shownthat many of these parameters can be accurately estimated from a single wideband echo. Theexpected uncertainties of the parameter estimates will also be investigated.

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