Applications of optimized rational approximations to parabolic equation modeling

Abstract

Rational approximations designed using least-squares constraints are easy to obtain and have useful applications. The coefficients are defined in terms of accuracy constraints on the propagating part of the spectrum and stability constraints on the rest of the spectrum. Among the applications are a filtering operator as well as improved approximations for the split-step Padé solution [J. Acoust. Soc. Am. 93, 1736–1742 (1993)] and radiation boundary conditions for outgoing wave equations [Clayton and Engquist, Geophysics 45, 895–904 (1980)]. An efficient operator filter is obtained by designing a rational approximation that is the identity function (or a weighted function) over the propagating (or desired) spectrum and decays to zero elsewhere. Operator approximations have previously been applied to replace the operator in the interior of the domain (the parabolic wave equation), generate radiation boundary conditions, solve scattering problems, and generate initial conditions. With the improved approximations, it is possible to use range steps of more than a hundred wavelengths with the split-step Padé solution. By enforcing both accuracy and stability constraints, it should be possible to overcome stability problems [Howell and Trefethen, Geophysics 53, 593–603 (1988)] and obtain useful higher-order radiation boundary conditions for outgoing wave equations.