Exponential scattering and K-distributed reverberation

Active sonar systems have been developed using larger arrays and broadband sources to counter the detrimental effects of reverberation in shallow water operational areas. Increasing array size and transmit waveform bandwidth improve the signal-to-noise power ratio (SNR) after matched filtering and beamforming by reducing the size of the range-bearing resolution cell and thus decreasing reverberation power levels. This can also have the adverse effect of increasing the tails of the probability density function (PDF) of the reverberation envelope, resulting in an increase in the probability of false alarm. A model is proposed that links the number of scatterers in a resolution cell to the resulting PDF of the matched filter envelope. In particular, it is shown that when the amplitude of the echo from a single scatterer is exponentially distributed, the envelope PDF follows the K-distribution with the shape parameter related to the number of scatterers within the resolution cell. The probability of detection for an additive nonfluctuating target in K-distributed reverberation is then considered and seen to be well approximated by a location shifted gamma distribution. The SNR required to obtain a probability of detection and false alarm performance specification is then used to determine that, as long as the target is not over-resolved, decreasing the size of the resolution cell always results in an improvement in performance. The amount of improvement is then quantified by the signal excess, which was seen to less than one decibel per doubling of bandwidth when the reverberation is extremely non-Rayleigh as opposed to the expected 3 dB gain when the reverberation is Rayleigh distributed