Generalized Bremmer series with rational approximation for the scattering of waves in inhomogeneous media

The Bremmer series solution of the wave equation in generally inhomogeneous media, requires the introduction of pseudodifferential operators. In this paper, sparse matrix representations of these pseudodifferential operators are derived. The authors focus on designing sparse matrices, keeping the accuracy high at the cost of ignoring any critical scattering-angle phenomena. Such matrix representations follow from rational approximations of the vertical slowness and the transverse Laplace operator symbols, and of the vertical derivative, as they appear in the parabolic equation method. Sparse matrix representations lead to a fast algorithm. An optimization procedure is followed to minimize the errors, in the high-frequency limit, for a given discretization rate. The Bremmer series solver consists of three steps: directional decomposition into up- and downgoing waves, one-way propagation, and interaction of the counterpropagating constituents. Each of these steps is represented by a sparse matrix equation. The resulting algorithm provides an improvement of the parabolic equation method, in particular for transient wave phenomena, and extends the latter method, systematically, for backscattered waves.