## Jr

where Dtra = 20logj0 dtra is the reference level of the transmitting array, p1 = —20logj0 |p1l the transmission loss of the incident ray,

Figure 10.3 Closed ray path used in reverberation calculations in the NISSM active sonar model (Weinberg, 1973).

Figure 10.3 Closed ray path used in reverberation calculations in the NISSM active sonar model (Weinberg, 1973).

Figure 10.4 Ensonified region used in reverberation calculations in the NISSM active sonar model (Weinberg, 1973).

Figure 10.4 Ensonified region used in reverberation calculations in the NISSM active sonar model (Weinberg, 1973).

P2 = —20log10 |p21 the transmission loss of the backscattered ray, Ntra = —20log10 rçtra the transmitter response (dB), Nrec = —20log1o nrec the receiver response (dB), R the region containing the scatterers corresponding to rays with travel time (t): T — t < t < T, t the pulse length (s), T = to + t1 + t2;0 < to < t , to the initial time, t1 the travel time of incident ray, t2 the travel time of backscattered ray and ¡i the backscattering strength (volume, surface or bottom).

Navy interim surface ship model usually computes eight paths to each representative scatterer [AR(i)]. Thus, there may be as many as 56 (i.e. 82 — 8) differently oriented closed paths per scatterer.

For volume reverberation, the expression for prev is replaced by the double summation (refer to Figure 10.4 for an illustration of the ensonified region)

Ipvoll2 = £ I dtra^tra (öJ pfVfVec (ö^) | ^Ol^ A*(i j Az« A0

where A0 is the horizontal beamwidth (rad), x(i) the range of the (i, j)-th scat-terer (km), z() the depth of the (i, j)-th scatterer (km) and = 10 log10 the volume scattering strength per unit volume of ocean.

The surface reverberation is calculated using an expression similar to that for prev (Equation (10.17)), except now Usur = 10log10 M-sur is the surface scattering constant per unit area of sea surface (dB) (Chapman and Harris, 1962).

Bottom reverberation is calculated in a manner analogous to that for surface reverberation, but where Ubot = 10log10 M-bot is the bottom scattering constant per unit area of sea floor (dB) (Mackenzie, 1961).

The total reverberation is assumed to be the random-phase addition of surface and bottom echoes from directly above and beneath the sonar (also referred to as fathometer returns), together with volume, surface and bottom reverberation.

10.3.3 Target echo

Target echo (or echo signal level) at a particular time is determined by i 2 Ipechol2 = £ |^trantra (0$) pfpffec (og) & (10.19)

where Utar = 10log10 M-tar is the target strength. The summation is taken over all closed paths with round-trip travel times between T - t and T.

10.3.4 Noise

Noise is separated into two components: self-noise and ambient noise. Assuming isotropic noise fields:

where DI = — 10logj0 nDl is the directivity index (dB), Psef = 20logj0 pseif the self-noise spectrum density and Pamb = 20logi0 pamb the ambient noise spectrum density.

276 Sonar performance models 10.3.5 Signal-to-noise ratio

Target echo to masking background, for a narrowband process, may be approximated by the signal-to-noise ratio (s/n):

A/noisepno;se + ^/revprev where Pecho = 20logj0 pecho is the target spectrum density at time T (dB), Prev = 20logj0 prev the reverberation spectrum density at time T (dB), Pnoise = 20logj0 pnoise the noise spectrum density at the beamformer output (dB), Afecho the equivalent bandwidth for the received echo (Hz), Afrev the equivalent bandwidth for the received reverberation (Hz) and Afnojse the equivalent bandwidth for the received noise (Hz).

Assuming that the target echo is Gaussian distributed and centered about the receiving (Doppler shifted) frequency with a standard deviation of t/2, then

where Af is the receiving bandwidth (Hz), t the pulse length (s) and where the function $ (x) is defined as

1 Cx

is the normal probability function (e.g. Hassab, 1989; Burdic, 1991; Ziomek, 1995).

Reverberation is also assumed to be Gaussian with a standard deviation of t/2, but is centered about the transmitting frequency. Therefore, reverberation energy falling outside the echo band is Afrevp^ev, where

Afrev = $(2TAfDop + T Af) — $(2TAfDop — t Af) (10.24)

and where

cs where Afuop is the Doppler shift (Hz), f the frequency (Hz), Vcl the closing speed (kms—1) and cs the sound speed (kms—1).

[Note that for passive systems, the factor of 2 in Equation (10.25) is removed.] If the noise-spectrum density can be considered constant over the receiving band, then

10.3.6 Probability of detection

Probability of detection is modeled using a narrowband, square-law envelope detector (Figure 10.5). The input x(t) is assumed to be either a stationary zero-mean Gaussian signal, s(t) and noise, n(t), or noise alone. It is assumed that the signal spectrum has the same bandwidth as the narrowband filter and is centered on it, and that the noise is flat over the frequency interval. Samples of the squared-envelope, which are taken every 1/B seconds, are accumulated for an observation time of T seconds. The threshold is fixed.

If Pf is a given probability of a false alarm, a quantity A is determined by inverting the following expression:

where M = BT + 1. Once A is known, the probability of detection (Pd) is given by

where S/N is the maximum value with respect to time of s/n at a particular target.

### 10.3.7 Model outputs

Navy interim surface ship model II can generate data appropriate for graphical presentation of ray diagrams, TL versus range reverberation (surface, bottom, volume and total) versus time, SNR versus range and probability of detection versus range. Sample outputs are illustrated for reverberation (Figure 10.6), SNR (Figure 10.7) and probability of detection

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