Uncertainty in Sonar Performance Prediction
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 .
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 is
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 or . 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 ()
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
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.
 A. D. Whalen, Detection of Signals in Noise, New York: Academic Press, 1971.