Volume scattering: Echo peak PDF

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The statistical properties of the backscattered acoustic echo from clouds of scatterers are examined. These clouds could be plankton, nekton, bubbles, etc., or a combination. The clouds of interest are sufficiently dense so that echoes from the individual scatterers overlap. The scatterers are assumed to be randomly distributed in space so that, at high enough acoustic frequencies, the individual echoes are not correlated (random phase). The probability density function (PDF) of the maximum echo value achieved in a depth gate has been analytically derived, computer simulated, and measured in the field. It is shown to be quite different than the Rayleigh PDF of the envelope, even for gate durations smaller than one transmission ping duration. An approximate analytical expression for the echo peak PDF is derived by first modeling the echo as a discrete ‘‘stepped’’ waveform where the ‘‘steps’’ are statistically independent of each other. The problem is then solved by straightforward extremal statistics. There is excellent agreement between the theoretical PDF, computer simulations, and field data. While we are concerned with gate durations several times one ping duration, the analytical expression for the PDF is shown to accurately describe the PDF over ranges of the gate duration less than the ping duration to much greater than the ping duration. From the echo peak PDF one can (1) estimate the abundance of occasional large scatterers that are within the clouds (for example large fish feeding on plankton) since their occurrence is differentiable from the rest of the cloud in the PDF, and (2) clearly differentiate between conditions where the echoes overlap and do not overlap. Once it is determined whether or not they overlap, lower or upper bounds in scatterer density (number per unit volume) may be calculated. Both of these applications can be performed in situ.