时空域波动方程混合网格有限差分数值模拟

摘要

传统2M阶有限差分格式(Tradtional 2 Mth-order Finite-Difference Schemes,T2 M-FD)和时空域2 M阶有限差分格式(Time-Space Domain 2 Mth-order Finite-difference,TS2 M-FD)均是目前应用较普遍且具代表性的高精度有限差分方法.T2 M-FD仅基于空间域频散关系求解差分系数,模拟精度较低.TS2 M-FD基于时空域频散关系和平面波理论求解差分系数,模拟精度较高.T2 M-FD和TS2 M-FD的差分格式相同,都是只利用常规直角坐标系中坐标轴上的网格点差分近似波动方程中的Laplace算子,而没能充分利用旋转直角坐标系中距离差分中心点更近的网格点来进一步提高模拟精度.本次研究提出利用常规直角坐标系和旋转直角坐标系中的网格点一起差分近似波动方程中的Laplace算子,并将Laplace算子表示为常规直角坐标系中M个Laplace算子和旋转直角坐标系中N个Laplace算子的加权平均,构建出一种新的混合2 M+N型有限差分格式(M2 M+N-FD).推导出M2 M+N-FD基于时空域频散关系和平面波理论的差分系数计算方法,进行频散及稳定性分析.频散分析表明:与T2 M-FD和TS2 M-FD相比,M2 M+N-FD能更有效地压制数值频散,模拟精度更高.稳定性分析表明:M2 M+N-FD和TS2 M-FD的稳定性基本相当,比T2 M-FD的稳定性强.最后,利用M2 M+N-FD进行均匀介质和层状介质模型的数值模拟试验,并将其推广应用于Marmousi模型的逆时偏移,高精度的数值模拟结果和偏移成像质量证明了M2 M+N-FD的优越性和普遍适用性.%Traditional high-order finite-difference(FD)scheme(T2 M-FD)and time-space-domain high-order finite-difference scheme(TS2 M-FD)are the most widely used higher-accuracy numerical modeling methods for seismic wave equation.T2 M-FD,with its FD coefficients calculated only based on space-domain dispersion rela-tionship,has relatively low accuracy.TS2 M-FD,with its FD coefficients calculated based on time-space-domain dispersion relationship and plane wave theory,has relatively higher accuracy.However,T2 M-FD and TS2 M-FD have the same FD scheme only using the grid points in the general coordinate system to approximate the Laplace operator in the wave equation,having not taking full use of the grid points in the rotated coordinate system to fur-ther improve the modeling accuracy.We proposed to use the grid points in the general and rotated coordinate sys-tem together to conduct difference approximation for the Laplace operator,and constructed a new kind of mixed 2 M+N style FD schemes,M2 M+N-FD for short,and derived the approach for calculating the FD coefficients based on the time-space domain dispersion relationship and plane wave theory.And then we carried out disper-sion analysis and stability analysis. Dispersion analysis shows that,comparing to T2 M-FD and TS2 M-FD, M2 M+N-FD can more effectively suppress the numerical dispersion and have higher modeling accuracy.Stability analysis shows that,M2 M+N-FD has better stability than T2 M-FD,and has almost the same stability with TS2 M-FD.In the end,we conduct numerical modeling test on homogeneous and layer model with M2 M+N-FD, and implement RTM on Marmousi model with M2 M+N-FD.The high accuracy modeling and migration results demonstrate the superiority and universal applicability of M2 M+N-FD.