Initial Data for the Parabolic Equation

Abstract

When the Helmholtz equation is replaced by the related parabolic equation, we obtain both a profit and a puzzle. The profit is the relative numerical ease of solving the parabolic equation. In this paper then consider the puzzle: What do we use for the required initial data for the parabolic equation? We adopt the position that the solution of the parabolic equation should correspond at long ranges to the solution of the Helmholtz equation at least in the case when the environment does not change with range. By considering the solution of the parabolic equation that reduces to the modal decomposition of arbitrary data at the initial range, we recommend that ΣΦ_n (z_o)Φ_n (z) be used for initial data. Here the Φ_n are the mode functions, z is the variable depth at the initial range, and z_o is the source depth. The fewer terms in above sum, the less variation in the solution of the parabolic equation, and therefore the greater the profit from numerical ease. This would normally suggest including only those modes whose effects can be accurately computed. If, on the other hand, phase errors are to be corrected by a technique such as Polyanskii’s it does no harm to include all modes with the weights given in the above sum.