Elimination of temporal dispersion from the finite-difference solutions of wave equations in elastic and anelastic models

Abstract

Time integration of wave equations can be carried out with explicit time stepping using a finite-difference (FD) approximation. The wave equation is the partial differential equation that governs the wavefield that is solved for. The FD approximation gives another partial differential equation — the one solved in the computer for the FD wavefield. This approximation to time integration in numerical modeling produces a wavefield contaminated with temporal dispersion, particularly at high frequencies. We find how the Fourier transform can be used to relate the two partial differential equations and their solutions. Each of the two wavefields is then a time-frequency transformation of the other. First, this transformation allows temporal dispersion to be eliminated from the FD wavefield, and second, it allows temporal dispersion to be added to the exact wavefield. The two transforms are band-limited inverse operations. The transforms can be implemented by using time-step independent, noncausal time-varying digital filters that can be precomputed exactly from sums over Bessel functions. Their product becomes the symmetric Toeplitz matrix with the elements defined through the cardinal sine (sinc) function. For anelastic materials, the effect of numerical time dispersion in a wavefield propagating in a medium needs special treatment. Dispersion can be removed by using the time-frequency transform when the FD wavefield is modeled in a medium with the frequency-modified modulus relative to the physical modulus of interest. In the rheological model of the generalized Maxwell body, the frequency-modified modulus is written as a power series, which allows a term-by-term Fourier transform to the time domain. In a low-frequency approximation, the modified modulus obtains the same form as the physical modulus, and it can be implemented as changes in the unrelaxed modulus and shifts of the relaxation frequencies and their strengths of the physical modulus.