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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 is

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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 ([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.

Reference

[1] A. D. Whalen, Detection of Signals in Noise, New York: Academic Press, 1971.