3D Fourth-Order Staggered-Grid Finite-Difference Schemes: Stability and Grid Dispersion

We investigated stability and grid dispersion in the 3D fourth-order in space, second-order in time, displacement-stress staggered-grid finite-difference scheme. Though only displacement-stress scheme is explicitly treated, all results also apply to the velocity-stress and displacement-velocity-stress finite-difference schemes. We derived independent stability conditions for the P and S waves by exact separation of equations for the two types of waves. Since the S -wave group velocity can differ from the actual velocity as much as 5% for the sampling ratio 1/5 (that is usually used in modeling), we recommend to sample a minimum S wavelength by six grid spacings. Grid dispersion is strongest for a wave propagating in the direction of a coordinate axis and weakest for a wave propagating along a body diagonal. Grid dispersion in the fourth-order scheme for the sampling ratios s = 1/5 and s = 1/6 is smaller than grid dispersion in the second-order scheme for s = 1/10 and s = 1/12, respectively.