GJI Seismology An optimized implicit finite-difference scheme for the two-dimensional Helmholtz equation

We have developed a generic expression of implicit finite-difference (FD) operators for second derivatives that is suitable for optimizing parts of FD coefficients. Then, the implicit FD operators are applied to derive the implicit FD scheme for the 2-D Helmholtz equation. The approximation accuracy of the implicit FD scheme can be increased by optimizing parts of the coefficients. Thus, parts of implicit FD coefficients are optimized based on the minimum error of FD dispersion relations. Within a certain range of error, the optimized coefficients can extend the scope of application of the implicit FD schemes to a large range of wavenumber for the selected grid spacing, K. Different integral upper limits, Kmax, of K during the optimization procedure can determine different sets of optimized coefficients. The optimized coefficients obtained by a large value of Kmax are more suitable for a large range of K of forward modeling. Hence, under a certain range of error, we can use a large spatial sampling interval with optimized coefficients instead of a small spatial sampling interval with Taylor series expansion method coefficients, which can reduce the memory consumption and computational workload for forward modeling