A new method for numerical dispersion analysis of Laplace-domain 2-D elastic wave equation

Abstract

Elastic wave modelling in the Laplace domain is the foundation of Laplace-domain elastic wave full waveform inversion. In order to use Laplace-domain numerical modelling schemes efficiently, appropriate grid intervals must be chosen correctly. The determination of grid intervals is based on numerical dispersion analysis of Laplace-domain elastic wave equation. A new method of numerical dispersion analysis is developed for Laplace-domain 2-D elastic wave equation. The novelty of the method lies in two aspects: (1) Based on the concept of the pseudo-wavelength for both P- and S-waves, I introduce a general quantity which is a combination of the Laplace constant, velocity and grid interval. Therefore, the dispersion relations no longer directly depend on the concrete Laplace constant, velocity and grid interval. Just like the frequency-domain dispersion analysis, a general conclusion with regard to the general quantity can be made; (2) The ratio of numerical eigenvalue to analytical eigenvalue can be interpreted as the normalised attenuation propagation velocity. The number of grid points per S-wave pseudo-wavelength is determined by the accuracy of normalised P- and S-wave attenuation propagation velocities. Based on a commonly used finite-element scheme, the numbers of grid points per S-wave pseudo-wavelength are determined for different Poisson ratios and for different ratios of directional grid intervals. These results are important in applying the finite-element scheme to Laplace-domain elastic wave modelling and full waveform inversion. Comparisons with the analytical solution validate the criterion on the numbers of grid points per S-wave pseudo-wavelength.