2(Vt/c) cos 0
2(Vt/c) cos &E
f (cos Be)
TLi + TL2
where T is the duration of the transmitted pulse. The echo energy level (EEL) received from the target at a hydrophone on the receiver array is then
where TS is the target strength. The noise-limited signal excess (SEn) can be defined as
SEn = ESL — TL1 — TL2 + TS — No + AGn — A — L (10.7)
where No is the noise spectral level, AGn the array gain against noise, A is the threshold on the SNR required for detection, and L is a loss term to account for time spreading and system losses.
Cox (1989) noted that the problem of estimating reverberation-limited bistatic sonar performance is more complicated than the monostatic case since it involves summing the contributions of a large number of scatterers ensonified by numerous propagation paths that differ in angle of incidence and position on the beam patterns of the source and receiver. Let Ro represent the reverberation spectral level, then the signal excess can be written to anticipate both noise- and reverberation-limited cases as
SE = ESL — TL1 — TL2 — [(No — AGn) © Ro] + TS — A — L (10.8)
where © represents power summation (Chapter 6, see Equation (6.1)).
Equations (10.7) and (10.8) do not include the effects of the direct blast. Analogies to the direct blast in monostatic sonars are the fathometer returns, which are the multiple surface- and bottom-reflected arrivals following each transmission. Comparisons of selected monostatic and bistatic parameters are presented in Table 10.3.
The Navy interim surface ship model (NISSM) II, developed by Weinberg (1973), is a computer program designed to predict the performance of active monostatic sonar systems using ray-theoretical techniques. The measure of performance is expressed in terms of probability of detection versus target range for a given false-alarm rate. Selected intermediate results can also be displayed, including ray traces, TL, and boundary and volume reverberation. This model is applicable to range-independent ocean environments.
Navy interim surface ship model uses a cell-scattering model for reverberation. The most time-consuming phase in the execution of NISSM is the volume reverberation computation. The volume scattering strength is expressed in terms of a column (or integrated) scattering strength, and a careful integration of the volume reverberation integral is performed, which considers multipath structure.
Because the NISSM II model encompasses all the modeling categories addressed thus far, and because it has been widely used in the sonar-modeling community, it is considered to be representative of the more general class of active sonar models. Examples of selected NISSM outputs are also presented in this section.
The depth-dependent sound speed c(z) and the inclination angle (0) at a point on a ray in the x-z plane can be related by Snell's law (see Figure 10.2) as c(z)
where c(z) is the sound speed as a function of depth (z), 0 the inclination angle and cv the vertex sound speed. Snell's law implies that the range (x)
272 Sonar performance models and travel time (T) along the ray can be calculated as fz c(z)|dz|
Officer (1958, chapter 2) provided a rigorous development of these equations.
The range derivative (9x/9cv) is given by d x d cv d x.
(which equals zero at caustics) and is used in computing the geometrical spreading loss factor (nsp):
cv tan 0 tan O0 x— d cv where O0 is the inclination angle of the ray at a point source of unit magnitude situated at xa = 0, za = z0.
The relative geometrical acoustic pressure field at field point (x, y) at time (t) is p = nspe-1"(t-T) (10.13)
where m = 2nf and f is the frequency.
Since the range, travel time and range-derivative integrals are symmetric with respect to the initial and final depths, it follows that the pressure satisfies the law of reciprocity. This law states that the acoustic pressure will remain unchanged if the source and receiver positions are interchanged. The sound-speed profile is approximated by a continuous function of depth (Weinberg, 1971) using Leroy's (1969) sound-speed formula. Earth curvature corrections are also applied.
The relative acoustic pressure along a ray is found by multiplying the relative geometrical acoustic pressure (p) by the surface, bottom and absorption loss factors:
and the TL is given by
where Nsp = -20log10 nsp is the geometrical spreading loss (dB), Nsur = -20log10 nsur the surface loss per bounce (dB/bounce), nsur the number zz of surface bounces, N^ot = —20logj0 nbot the bottom loss per bounce (dB / bounce), n^ot the number of bottom bounces, A = — 20 log^(a)/loge (10) the absorption coefficient (dBkm ), a the absorption in nepers per unit distance, t the elapsed time (s), T the travel time (s), $ the accumulated phase shift (rad), S the arc length (km), rn = 2nf the radian frequency (rad s ) and f the acoustic frequency (Hz).
The surface loss may be input as a table of surface loss per bounce versus angle of incidence. The bottom loss may either be input as a table of bottom loss per bounce versus angle of incidence, or internally computed either by marine geophysical survey (MGS) data or by an empirical equation.
The absorption coefficient can be represented in one of three ways: (1) input as a table of absorption per unit distance versus frequency; (2) calculated by Thorp's (1967) equation; or (3) calculated by the equation attributed to H.R. Hall and W.H. Watson (1967, unpublished manuscript).
Eigenrays are found by tracing a preselected fan of test rays to the target depth. When two adjacent rays of the same type bracket a target range, a cubic spline interpolation is performed to determine the eigenray. The principal ray types were defined previously in Figure 4.2.
Shadow zone propagation is characterized according to Pekeris (1946) as
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