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In this paper, we investigate a bathymetry problem, the seabottom surface reconstruction, using side scan sonars. A sonar emits pulses of sounds and detects echoes. The sound waves bounce off the surface, and the sonar receives the backscattered signal. Gauging water depth, using acoustic technology, involves measuring the time taken for sound waves to travel between the vessel and the seabottom and back again. To describe sound propagation in a fluctuating ocean, a mathematical model, based on the radiative transfer equation, is used. The solution of the direct problem, consisting of determining the wave energy flux density, has been obtained in the single and double scattering approximations. Further, we consider inverse problem which involves the determination of the bathymetric function, which describes the seabottom level variation from the average horizontal plane. As a solution to the problem, a non-linear differential equation has been deduced under assumptions for the directivity pattern of the receiving antenna. An adaptive numerical method is developed for the analysis of the bathymetry problem. Computational experiments, showing the influence of a double scattering signal on the seabottom reconstruction, are conducted.

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