Material Interface in the Finite-Difference Modeling: A Fundamental View

Abstract

ABSTRACT By analyzing the equations of motion and constitutive relations in the wavenumber domain, we gain important insight into attributes determining the accuracy of finite-difference (FD) schemes. We present heterogeneous formulations of the equations of motion and constitutive relations for four configurations of a wavefield in an elastic isotropic medium. We Fourier-transform the entire equations to the wavenumber domain. Subsequently, we apply the band-limited inverse Fourier transform back to the space domain. We analyze consequences of spatial discretization and wavenumber band limitation. The heterogeneity of the medium and the Nyquist-wavenumber band limitation of the entire equations has important implications for an FD modeling: The grid representation of the heterogeneous medium must be limited by the Nyquist wavenumber. The wavenumber band limitation replaces spatial derivatives both in the homogeneous medium and across a material interface by continuous spatial convolutions. The latter means that the wavenumber band limitation removes discontinuities of the spatial derivatives of the particle velocity and stress at the material interface. This allows to apply proper FD operators across material interfaces. A wavenumber band-limited heterogeneous formulation of the equations of motion and constitutive relations is the general condition for a heterogeneous FD scheme.