Frequency-domain elastic wave modeling for vertical transversely isotropic media with an average-derivative optimal method

Abstract

Based on an average-derivative method (ADM), it is feasible to develop an optimal elastic wave finite-difference nine-point scheme for vertical transversely isotropic (VTI) media in the frequency domain. The optimization coefficients of ADM can be determined by minimizing phase-velocity dispersion error. Nevertheless, for the complex VTI media, the point-by-point optimization, which only considers a single anisotropic parameter pair, will require high computational cost. To address this issue, we take into account the other two low-cost schemes: the overall simultaneous optimization in terms of various anisotropy parameter pairs and the interpolation based on the completely computed optimization coefficients. Applying the 58 measured anisotropy parameter pairs of sedimentary rocks, we test the performances of point-by-point optimization, the overall optimization, interpolation, and classical nine-point method (CM). The results indicate that the number of grid points per qS wavelength of the interpolation scheme is closest to that of the point-by-point optimization scheme and much lower than that of the overall optimization scheme and CM. In addition, the surface of solid earth is a strong impedance-discontinuity surface (free surface), resulting in the more complicated wavefields. To realize the full-waveform simulation for elastic VTI media, it is necessary to implement the free-surface boundary condition in detail. Based on the equivalence between the stress-image and adaptive parameter-modified schemes for implementing the free-surface boundary condition in the time domain, we give a new frequency-domain free-surface expression for the VTI media. We still call this expression the stress-image scheme. Numerical experiments demonstrate that (1) the optimal ADM can produce more accurate results than CM at almost the same computational cost, (2) the interpolation scheme has almost identical accuracy with the point-by-point optimization scheme, and (3) the stress-image scheme is feasible for the implementation of free-surface boundary condition in a VTI medium.