The finite-difference method has been widely used in seismic forward modeling, RTM (Reverse Time Migration) and FWI (Full Waveform Inversion) because of its easy implementation, small memory and low computational cost. Numerical dispersion, due to discretization of time and space derivatives, seriously affects the forward modeling accuracy of the finite-difference method. So suppressing the numerical dispersion to improve the forward modeling accuracy is the key problem for the finite-difference method. In the frequency-space domain, the mixed-grid finite-difference method is often used, which can effectively improve the forward modeling accuracy. In the time-space domain, the traditional 2Mth-order finite-difference method is commonly used, which essentially has only 2nd-order accuracy. The time-space domain 2Mth-order finite-difference method, in which the difference coefficients are determined by satisfying time-space dispersion relation, has relatively high modeling accuracy, but its dispersion curves are still divergent. Although the rhombus-grid finite-difference method has indeed high modeling accuracy, it requires high computational cost. In this article, by introducing the mixed-grid strategy from the frequency-space domain to the time-space domain, we propose a new kind of mixed-grid 2M+ N style finite-difference method, and derive the approach for calculating the finite-difference coefficients by satisfying time-space domain dispersion relation. In addition, we conduct dispersion analysis, stability analysis, and numerical simulation. The results demonstrate that (1) with almost the same computational cost, the traditional 2Mth-order finite-difference method has seriously numerical dispersion mainly in the time dispersion, and has the lowest modeling accuracy. The time-space domain 2Mth-order finite-difference method has some time dispersion and space dispersion, and has relatively high modeling accuracy. The mixed-grid 2M+N style finite-difference method has no obvious numerical dispersion, and so has the highest modeling accuracy. (2) When M is not too big, a bigger N value can hardly decrease the numerical dispersion and increase the forward modeling accuracy, but will increase the computational cost. It suggests that we should take use of the mixed-grid 2M+ N (N = 1) style finite-difference method for general conditions. The rhombus-grid is a special shape of the mixed-grid 2M+N style finite-difference method, in which the N value is usually too big, so the rhombus-grid finite-difference method is not the optimal choice. (3) The mixed-grid 2M+N (N=1) style finite-difference method has stronger stability than the traditional 2Mth-order finite-difference method, and has almost the same stability as the time-space domain 2Mth-order finite-difference method. A bigger N value will slightly improve the stability, but also increases the computational amount. (4) Numerical modeling on homogeneous model, layer model and salt model further demonstrates the mixed-grid 2M+ N style finite-difference method can effectively suppress the numerical dispersion, and improve the modeling accuracy. In the end, we effectively eliminate most of the energy reflected from the artificial boundary by adopting an NPML (Nearly Perfectly Matched Layer) absorbing boundary, and carry out forward modeling and RTM on the salt model. There is no obvious numerical dispersion and boundary reflection on the shot gathers of the forward modeling, and RTM results have really good quality. All of these prove the validity and applicability of the mixed-grid 2M+ N style finite-difference method suggested in this study.
