Taylor series derivation of nonstationary wavefield extrapolators

Abstract

Summary A Taylor series approach is used to derive two elemental and complementary wavefield extrapolators directly from the Helmholtz equation (i.e. the wave equation after a temporal Fourier transform). These extrapolators are for vertical propagation when velocity varies arbitrarily in the horizontal direction. The Helmholtz equation provides two alternative and exact pseudodifferential operator forms for the second derivative with respect to the vertical coordinate. These lead to alternative but approximate pseudodifferential operators for the n th vertical derivative as required in the Taylor series. When these approximate operators are substituted into the Taylor series, the series can be summed to give the two alternative extrapolators: PSPI (phase shift plus interpolation) and NSPS (nonstationary phase shift). The resulting extrapolators are Fourier integral operators and are approximate unless velocity is constant when they both simplify to ordinary phase shift. An investigation of the accuracy of these approximations is done using the composition theorem of pseudodifferential operators. The result suggests that NSPS and PSPI produce errors that are complementary in that they tend to cancel when the extrapolations are averaged together. A numerical example supports this conclusion.