Benchmarking virtual backscatter in parabolic evolution

Abstract

A perturbative method has recently been presented for improving parabolic equations to account for intermediate backscattering induced by slow range dependence of the acoustic index of refraction. In realistic cases, it yields solutions to the full wave equation from a truly one-way algorithm, and generalizes a number of special results of this kind which have been known for many years. Numerically validating the method, though, involves certain unique difficulties, because the physical corrections it introduces to local evolution are accompanied by medium dependence of the projection operator that defines the input and output wave fields. A method for benchmarking is presented here, which permits numerical isolation of range-local effects from the endpoint projections, and also from particular details of the choice of one-way model used. To illustrate that the physical content of the new terms can be sensibly validated in this manner, an exact solution is considered, where they are shown to correct the ''idealized'' one way equation on which current parabolic algorithms are based. This solution is used to illustrate the relation of energy conservation to one-way evolution in the presence of slow range dependence. In particular, it demonstrates that energy conservation alone is not a sufficient requirement to produce correct parabolic evolution, even when it pertains.