An implicit staggered-grid finite-difference method for seismic modelling

Abstract

We derive explicit and new implicit staggered-grid finite-difference (FD) formulas for derivatives of first order with any order of accuracy by a plane wave theory and Taylor's series expansion. Furthermore, we arrive at a practical algorithm such that the tridiagonal matrix equations are formed by the implicit FD formulas derived from the fractional expansion of derivatives. Our results demonstrate that the accuracy of a (2N+ 2)th-order implicit formula is nearly equivalent to or greater than that of a (4N)th-order explicit formula. The new implicit method only involves solving tridiagonal matrix equations. We also demonstrate that a (2N+ 2)th-order implicit formulation requires nearly the same amount of memory and computation as those of a (2N+ 4)th-order explicit formulation but attains the accuracy achieved by a (4N)th-order explicit formulation when additional cost of visiting arrays is not considered. Our analysis of efficiency and numerical modelling results for elastic wave propagation demonstrates that a high-order explicit staggered-grid method can be replaced by an implicit staggered-grid method of some order, which will increase the accuracy but not the computational cost.