Wave Equation Numerical Simulation and RTM With Mixed Staggered-Grid Finite-Difference Schemes

Abstract

For the conventional staggered-grid finite-difference scheme (C-SFD), although the spatial finite-difference (FD) operator can reach 2Mth-order accuracy, the FD discrete wave equation is the only second-order accuracy, leading to low modeling accuracy and poor stability. We proposed a new mixed staggered-grid finite-difference scheme (M-SFD) by constructing the spatial FD operator using axial and off-axial grid points jointly to approximate the first-order spatial partial derivative. This scheme is suitable for modeling the stress–velocity acoustic and elastic wave equation. Then, based on the time–space domain dispersion relation and the Taylor series expansion, we derived the analytical expression of the FD coefficients. Theoretically, the FD discrete acoustic wave equation and P- or S-wave in the FD discrete elastic wave equation given by M-SFD can reach the arbitrary even-order accuracy. For acoustic wave modeling, with almost identical computational costs, M-SFD can achieve higher modeling accuracy than C-SFD. Moreover, with a larger time step used in M-SFD than that used in C-SFD, M-SFD can achieve higher computational efficiency and reach higher modeling accuracy. For elastic wave simulation, compared to C-SFD, M-SFD can obtain higher modeling accuracy with almost the same computational efficiency when the FD coefficients are calculated based on the S-wave time–space domain dispersion relation. Solving the split elastic wave equation with M-SFD can further improve the modeling accuracy but will decrease the efficiency and increase the memory usage as well. Stability analysis shows that M-SFD has better stability than C-SFD for both acoustic and elastic wave simulations. Applying M-SFD to reverse time migration (RTM), the imaging artifacts caused by the numerical dispersion are effectively eliminated, which improves the imaging accuracy and resolution of deep formation.