旋转网格和常规网格混合的时空域声波有限差分正演

压制数值频散,提高正演模拟精度,一直是有限差分正演模拟研究的重要内容.基于时空域频散关系的有限差分法,比基于空间域频散关系的传统有限差分法,模拟精度更高.时空域声波方程数值模拟,普遍采用常规十字交叉型高阶有限差分格式.而在频率-空间域,普遍采用旋转网格和常规网格混合的有限差分格式,有效提高了模拟精度和计算效率.本文将频率-空间域混合网格有限差分的思想引入到时空域,提出了时空域混合网格2 M+N型声波方程有限差分方法.首先推导出基于时空域频散关系的混合网格差分系数计算方法,然后进行频散分析、稳定性分析,并和传统高阶、时空域高阶有限差分法对比,结果表明:计算量相同时,新方法能有效压制数值频散,显著提高模拟精度;新方法相比传统2 M阶有限差分法,稳定性增强,与时空域2 M阶有限差分法稳定性基本相当.最后利用新方法进行均匀介质、层状介质、盐丘模型的数值模拟和盐丘模型的逆时偏移,模拟效果和成像质量进一步证实了该方法的有效性和普遍适用性.

Scalar wave equation modeling using the mixed-grid finite-difference method in the time-space domain

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.#br#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.#br#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.#br#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.