Theoretical Background

The analysis presented in this paper is based on the acoustic scattering model for elastic cylinder presented in (Stanton, 1988). In order to estimate the scattered eld due to a cylinder of

nite length, the volume ow per unit length of the scattered eld of an in nitely long cylinder is integrated over a nite distance. Many scatterers posses elastic properties, so conversion of

compressional waves into shear waves has to be taken into account. The assumption made for the in nite cylinder is that there is no absorption, dispersion, or nonlinearity in the cylinder or the surrounding medium (Stanton, 1988). The scattering from ends of the cylinder is ignored, and the receiver-target separation must be great enough to be in the rst Fresnel zone (i.e,L <<2pr ;

whereL is the length of the cylinder or of the insoni ed "spot" of a longer cylinder, is the

acoustic wavelength andris the range from the axis of the cylinder to the receiver or eld point).

Stanton (Stanton, 1988) shows that taking into account arbitrary transmitter direction, receiver position and cylinder orientation, and assuming thatr >> L, the expression of the scattered

pressure is given by 8.1: Pscatter(k) = P0 eikr r L sin( ) X1 m=0 "msin( m)e i m_cos(m ) (8.1)

wherek is the acoustic wavenumber; Po is the amplitude of the incident plane wave; r is the

source-target separation; = 1₂kL(!ri !rr) !rc ,!rr the unit direction of receiver, !ri the unit

direction of incident plane wave,!rc the unit direction of cylinder axis; "m is Neumann's number:

"0 = 1; "m>0 = 2; m is scattering phase angle; and is the azimuth angle of the arbitrarily

oriented cylinder. In equation 8.1, sin( ) _{represents the beam pattern. If it is assumed to be} Gaussian, the effective length insoni ed is given byLb =p2 e s0=2 2 where_s

0 is the distance

from the maximum response on the bottom to the closest point of approach of the cylinder. The following considerations have to be incorporated in the model: spherical spreading by replacingPobyP1=r, and the bottom effects. Assuming a at bottom comprising a homogeneous lossy half-space with sound speed cs and density s, (cw is sound speed in water and w is

density in water) the pressure is reduced byTwsTswe 2 szs. Here,Tws = 2 scs=( wcw+ scs)

is the normal incidence plane wave transmission coef cient from water to sediment and

Tsw = 2 wcw=( wcw + scs)is the normal incidence plane wave transmission coef cient from

below surface. The scattered pressure becomes: Pscatter(k) = P1TwsTsw ei(kwrw+ksrs(1+i s)) (rw+rs)2 b L!p 2 e s0=2 P2 X1 m=0 "msin ( m)e i m_{cos (m )} (8.2) wherekwandksare the wavenumbers in the water and sediment, respectively,rwandrsde ne the

path lengths in water and in the sediment, and s = s=ks:The solution is for continuous wave

signals of in nite duration. For a band-limited, nite duration pulse, a time series can be created from Fourier synthesis of solutions over a discrete range of wavenumberskn,n= 0;1; :::; N. The

impulse response for the jth_{sample of the time series is given by:}

hscatter(tj) = 2 fs N2_c s n=Xnmax n=nmin Pscatter(kn)e2 i(j 1)(n 1) (8.3)

wherenminandnmaxare determined from the upper and lower frequency band.

8.2.2 Time-Frequency Analysis

Time-frequency methods are powerful tools for studying variations in spectral components. The spectrum's time dependency of the return signal could be a strong indicator of the target acoustic signature. The generalized time-frequency representation can be expressed in term of the kernel '( ; ), which determines the properties of the distribution (Mecklenbrauker &

F.Hlawatsch, 1997):

C(t; $) = 1

4 2

Z Z Z

f (u =2)f(u+ =2)'( ; )e j t j $+ju dud d (8.4)

The Wigner distribution can be derived from the generalized time-frequency representation for the unity value of the kernel:

W(t; $) = 1 2

Z

f (u =2)f(u+ =2)e j $d (8.5)

The Wigner distribution function is a time-frequency analysis tool that can be used to illustrate the time-frequency properties of a signal, and it can be interpreted as a function that indicates the distribution of the signal energy over the time-frequency space. One disadvantage of the Wigner distribution is that it sometimes indicates intensity in the regions where one would expect zero

values. This effect can be minimized by choosing a kernel that has the form'( ; ) = e 2 2= _,

which yields the Choi-Williams distribution:

C(t; $) = 1 4 3=2 Z Z ₁ p ₂ = f (u =2)f(u+ =2)e (u t)2₌ 2 _{j $} dud (8.6)

Choosing this kernel, the marginals are satis ed and the distribution is real. In addition, if the parameter has a large value, the Choi-Williams distribution approaches the Wigner distribution, since the kernel approaches one. For small values, it satis es the reduced interference criterion.