Wolfram Demonstrations Project

Sonar performance prediction can be viewed as a two-step process. In the first step, the statistical properties of the signal and noise fields at the sonar are estimated. The characteristic parameter is the signal to noise ratio (

). In the second stage,

is converted to probability of detection

using a mapping appropriate to the sonar type and method of operation. If the

is known exactly then, the probability of detection

is a number on the interval

. As a practical matter

is never known with total certainty. This Demonstration illustrates the effect of uncertainty in

on the estimation of probability of detection

Contributed by: Marshall Bradley

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In underwater acoustics, sonar performance is predicted with the aid of the sonar equation. A simple example of a sonar equation expressed on a decibel scale applicable to passive sonar is the following:

, where

is the signal to noise ratio (dB),

is the target source level (dB re 1 μPa at 1m),

is the transmission loss (dB re 1 m), and

is the ambient noise level (dB re 1

). In practical situations the individual terms in the sonar equation are not precisely known. Thus there is uncertainty associated with an

prediction.

If we let

denote the

, then a plausible initial assumption is that

is a normally distributed random variable with probability density function

where

is the mean decibel

level predicted by the modeling process and

is the measure of the decibel uncertainty associated with the prediction. For example, in a situation where acoustic performance prediction models predict favorable sonar performance with high certainty, we might have

and

. If we let the random variable

denote the signal to noise ratio measured on an intensity scale, then the relationship between the random variables

and

The random variable

is said to have a lognormal distribution and its probability density function is

The probability density functions

and

convey the same information about the uncertainty in

, only the scales are different. When measured on an intensity scale,

is defined on the interval

. If we measure

on a decibel scale, then it is defined on the interval

. The condition of absolutely no signal corresponds to

. Infinite signal corresponds to infinity on both scales.

If we assume that the sonar is optimized for a signal with random phase and Rayleigh amplitude, then the appropriate mapping from

to probability of detection

is ([1])

where

is the sonar false alarm rate and

is the

measured on an intensity scale. The function

is a monotonic increasing map from intensity

to probability of detection. It has the inverse function

with derivative

Because of the simple properties of the inverse function

, the probability density function

of probability of detection

can be found in closed form. The result is

or more completely

Due to the uncertainty in

, the probability of detection

is no longer a single number but rather a statistical distribution. The expected value and variance of the probability of detection are useful characterizations of this distribution. They are defined to be

Use the two sliders to control the selection of

the mean

(dB) and

the standard deviation of

(dB). When

is small, the distribution of probability of detection is peaked with a relatively small standard deviation (square root of variance). This is indicative of a useful prediction of probability of detection. However, when

is larger, the distribution of probability of detection is much more spread out and has a much larger standard deviation. In this latter circumstance, the predicted probability of detection may not be useful due to the large associated uncertainty.