An affine generalized optimal scheme with improved free-surface expression using adaptive strategy for frequency-domain elastic wave equation

Abstract

Effective frequency-domain numerical schemes are central for forward modeling and inversion of the elastic wave equation. The rotated optimal nine-point scheme is a highly utilized finite-difference (FD) numerical scheme. This scheme makes a weighted average of the derivative terms of the elastic wave equations in both the original and the rotated coordinate systems. In comparison with the classical nine-point scheme, it can simulate S-waves better and has higher accuracy at nearly the same computational cost. Nevertheless, this scheme limits the rotation angle to 45°; thus, the grid sampling intervals in the x- and z-directions need to be equal. Otherwise, the grid points will not lie on the axes, which dramatically complicates the scheme. Affine coordinate systems do not constrain axes to be perpendicular to each other, providing enhanced flexibility. Based on the affine coordinate transformations, we developed a new affine generalized optimal nine-point scheme. At the free surface, we applied the improved free-surface expression with an adaptive parameter-modified strategy. The new optimal scheme had no restriction that the rotation angle must be 45°. Dispersion analysis shows that our scheme can effectively reduce the required number of grid points per shear wavelength for both equal and unequal sampling intervals compared to the classical nine-point scheme. Moreover, this reduction improves with the increase of Poisson’s ratio. Three numerical examples demonstrate that our scheme can provide more accurate results than the classical nine-point scheme in terms of both the internal and free-surface grid points.