抗频散有限差分波动方程数值模拟及逆时偏移

Anti-dispersion finite difference simulation and reverse-time migration for wave equations

数值频散是有限差分法求解波动方程时的最突出问题,严重降低了波场模拟的分辨率,通常使用更精细的计算网格或较长的差分算子来解决,但都会显著地增加计算成本。为此,本文构造了一种新的波动方程差分格式压制数值频散,通过在常规的差分方程中增加了频散校正项,能够有效地衰减高波数成分,抑制频散;根据相速度和群速度对频散的影响,推导了二阶和四阶差分格式频散校正项的最优参数值,当校正参数等于该值时,相应频散曲线图中相速度最接近群速度,频散误差最小。数值模拟和逆时偏移的实验结果表明:本文构造的抗频散算法对数值频散的抑制效果明显,新方法的二阶差分抗频散性能不低于常规四阶差分,而计算效率近似于常规二阶差分;抗频散差分格式与同阶常规差分格式相比,逆时偏移成像效果更好,精度更高。

Numerical dispersion is the most prominent problem in solving wave equations by the finite difference method, which seriously affects the wave field modeling and decreases the resolution of wave field in the simulation process.Usually refined computational grids or high order difference operators are used to solve this problem.But all these approaches increase the computational complexity.In this work we propose a new finitedifference model.Firstly by adding a dispersion correction term to the conventional finite-difference equations,the finite-difference based wave equation method is constructed to weaken the numerical dispersion.Secondly by analyzing the relation between phase velocity/group velocity and the dispersion,the optimal parameters of the correction terms in the second and fourth order finite-difference scheme are derived.When the optimal values are chosen for correction parameters,the corresponding dispersion curves show that the dispersion error is minimum while the phase velocity has the best match with group velocity.The numerical simulation and reverse-time migration results suggest that the proposed algorithm has an obvious suppression on numerical dispersion,and the performance of the second order finite-difference in the new method is equivalent to or even better than the conventional fourth order finite-difference,but the computational cost is about the same as that of conventional second order finite-difference.Additionally, the anti-dispersion finite-difference scheme has a better imaging result and higher accuracy than the conventional method with the same order in reverse time migration.