Determination of finite difference coefficients for the acoustic wave equation using regularized least-squares inversion

Finite difference (FD) solutions of wave equations have been proven useful in exploration seismology. To yield reliable and interpretable results, the numerically induced error should be minimized over a range of frequencies and angles of propagation. Grid dispersion is one of the key numerical problems and there exist some methods to solve this problem in the literature. Traditionally, the spatial FD operator coefficients are only determined in the spatial domain; however, the wave equation is solved in the temporal and spatial domain simultaneously. Recently, some methods based on the joint temporal-spatial domains have been proposed to address this problem. Variable length coefficients methods are proposed in the literature to improve efficiency while preserving accuracy by using longer operators in the low velocity regions and shorter operators in the high velocity regions. To cope with the ill-conditioning of the linear system induced by long stencil FD operators, we study in this paper a regularizing simplified least-squares model to minimize the phase velocity error in the joint temporal-spatial domain with a variable length of coefficients. Different from our previous study, we determine FD coefficients on the regular grid instead of on the staggered grid. Though the regular grid FD methods are less precise, however, with a little increase of the operator length, the precision can be improved. Stability of the numerical solutions is enhanced by the regularization. Numerical simulations made on one-dimensional to three-dimensional examples show that our scheme needs shorter operators and preserves accuracy compared with the previous methods.