Two staggered-grid finite-difference (SGFD) methods with fourth- and sixth-order accuracy in time have been developed recently based on two new SGFD stencils. The SGFD coefficients are determined by Taylor-series expansion (TE), which is accurate only nearby zero wavenumber. We adopt the new two SGFD stencils and optimize the SGFD coefficients by minimizing the errors between the wavenumber responses of the SGFD operators and the first-order k (wavenumber)-space operator in a least-squares (LS) sense. We solve the LS problems by performing weighted pseudo-inverse of nonsquare matrices to obtain the SGFD coefficients, and to yield two LS based SGFD methods. Dispersion analysis and numerical examples demonstrate that our LS based SGFD methods can preserve the original fourth- and sixth-order temporal accuracy and achieve higher spatial accuracy than the existing TE based time-space domain SGFD methods.
The staggered-grid finite-difference (SGFD) (Virieux, 1984) method has been widely used in seismic wave propagation modeling. Most of the SGFD applications adopt the traditional (2M, 2) scheme, which uses 2M-order Taylorseries expansion (TE) based FD operator to discretize spatial derivatives, and 2nd-order TE based FD operator to discretize temporal derivative. Although high-order spatial accuracy can be achieved by using a long stencil length, the temporal accuracy is only second-order. When a coarse time step is used, the traditional scheme suffers from obvious temporal dispersion during long time wave propagation.
Recently, Tan and Huang (2014a) propose two new SGFD methods with fourth-order and sixth-order accuracy in time respectively by incorporating a few of off-axial grid points into the standard SGFD stencil. The two methods are denoted as (2M, 4) and (2M, 6). The FD coefficients are determined in the time-space domain using TE approach. Althouth high-order temporal accuracy has been achieved, the TE based (2M, 4) and (2M, 6) methods still suffer from obvious spatial disperion when a large grid size or a short stencil length is adopted. Tan and Huang (2014b) continue to improve the spatial accuracy by using a nonlinear optimization to seek the optimal FD coefficients. However, the optimization requires repeated iterations, and the procedure may be time-consuming.