Exact extrapolation and immersive modelling with finite-difference injection

Abstract

In numerical modelling of wave propagation, the finite-difference (FD) injection method enables the re-introduction of simulated wavefields in model subdomains with machine precision, enabling the efficient calculation of waveforms after localized model alterations. By rewriting the FD-injection method in terms of sets of equivalent sources, we show how the same principles can be applied to achieve on-the-fly wavefield extrapolation using Kirchhoff-Helmholtz (KH)-like integrals. The resulting extrapolation methods are numerically exact when used in conjunction with FD-computed Green's functions. Since FD injection only relies on the linearity of the wave equation and compactness of FD stencils in space, the methods can be applied to both staggered and non-staggered discretizations with arbitrary-order spatial operators. Examples for both types of discretizations show how these extrapolators can be used to truncate models with exact absorbing or immersive boundary conditions. Such immersive modelling involves the evaluation of KH-type extrapolation and representation integrals in the same simulation, which include the long-range interactions missing from conventional FD injection.