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 ([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

,

.

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

,

,

.

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.

[1] A. D. Whalen,

*Detection of Signals in Noise*, New York: Academic Press, 1971.