Modeling the Solution of the Acoustic Inverse Problem of Scattering for a Three-Dimensional Nonstationary Medium

The inverse problem of acoustic sounding of a three-dimensional nonstationary medium is considered, based on the Cauchy problem for the wave equation with a sound speed coefficient depending on the spatial coordinates and time. The data in the inverse problem are measurements of time-dependent acoustic pressure in some spatial domain. Using these data, it is necessary to determine the positions of local acoustic inhomogeneities (spatial sound speed distributions), which change over time. A special idealized sounding model is used, in which, in particular, it is assumed that the spatial sound speed distribution changes little in the interval between source time pulses. With such a model, the inverse problem is reduced to solving three-dimensional Fredholm linear integral equations for each sounding time interval. Using these solutions, the spatial sound speed distributions are calculated in each sounding time interval. When a special (plane-layer) geometric scheme for the location of the observation and sounding domains is included in the sounding scheme, the inverse problem can be reduced to solving systems of one-dimensional linear Fredholm integral equations, which are solved by well-known methods for regularizing ill-posed problems. This makes it possible to solve the three-dimensional inverse problem of determining the nonstationary sound speed distribution in the sounded medium on a personal computer of average performance for fairly detailed spatial grids in a few minutes. The efficiency of the corresponding algorithm for solving a three-dimensional nonstationary inverse sounding problem in the case of moving local acoustic inhomogeneities is illustrated by solving a number of model problems.