Asymptotic Ray Method for the Double Square Root Equation

Abstract

The parabolic wave equation describes wave propagation in a preferable direction, which is usually horizontal in underwater acoustics and vertical in seismic applications. For dense receiver arrays (receiver spacing is less than signal wavelength), this equation can be used for propagating the recorded wavefield back into the medium for imaging sources and scattering objects. Similarly, for multiple source and receiver array acquisition, typical for seismic exploration and potentially beneficial for ocean acoustics, one can model data in one run using an extension of the parabolic equation—the pseudo-differential Double Square Root (DSR) equation. This extended equation allows for the modeling and imaging of multi-source data but operates in higher-dimensional space (source, receiver coordinates, and time), which makes its numerical computation time-consuming. In this paper, we apply a faster ray method for solving the DSR equation. We develop algorithms for both kinematic and dynamic ray tracing applicable to either data modeling or true-amplitude recovery. Our results can be used per se or as a basis for the future development of more elaborated asymptotic techniques that provide accurate and computationally feasible results.