Statistics of Echoes From a Directional Sonar Beam Insonifying Finite Numbers of Single Scatterers and Patches of Scatterers

When a sonar beam sweeps across a field of scatterers and insonifies a different set of scatterers in each ping, the echoes can fluctuate significantly from ping to ping. For a homogeneous spatial distribution of scatterers in which there are a large number of scatterers in each sonar resolution cell, the echoes tend to be Rayleigh distributed. However, the echoes can be strongly nonRayleigh when there is a small number of scatterers in the sonar resolution cell-a condition that can occur with a narrow beam sonar. For scatterers randomly distributed in the sonar beam, the corresponding random weighting associated with the beampattern also significantly contributes toward a non-Rayleigh distribution when the number of scatterers is small. In this paper, a general formulation for echo statistics is developed by combining equations derived by Ehrenberg etal. [Proc. Conf. Eng. Ocean Environ., vol. 1, pp. 61-64,1972 and J. Acoust. Soc. Amer., vol. 69, pp. 955-962, 1981] and Barakat [Optica Acta, vol. 21, pp. 903-921,1974] to account for a directional sonar beam involving an arbitrary finite number of scatterers, each with an arbitrary echo probability density function (pdf), each randomly located in the beam, and each correspondingly randomly weighted by the beampattern. Theoretical predictions are made, along with numerical simulations for validation, for a range of cases including: 1) a range of number of scatterers each randomly located in the beam and 2) several different echo pdfs of the scatterers. Here, a "single scatterer" could be a patch of scatterers whose overall dimensions are much smaller than the resolution cell of the sonar beam and multiple scatterers could be multiple small patches. Although the application is intended for volumetric patches, the formulation could be applied to areal patches under appropriate conditions. The formulation applies directly to the geometry of the short-range direct-path sonar in which there are no reflections due to boundaries such as the seafloor and sea surface.