基于分数阶拉普拉斯算子解耦的黏声介质地震正演模拟与逆时偏移

摘要

时间域常Q黏声波方程,由于含分数阶时间导数项,数值求解需要大量内存,计算效率低,不利于地震偏移的实施.通过一系列近似,可将该方程简化为介质频散效应和衰减效应解耦的分数阶拉普拉斯算子黏声波方程,数值求解内存需求少,计算效率高.本文采用交错网格有限差分逼近时间导数,改进的伪谱法计算空间导数,PML吸收边界去除边界反射,对该方程进行数值离散和地震正演模拟.开展地震数据的黏声介质逆时偏移,实现波场逆时延拓过程中同时完成频散校正和衰减补偿.改善深层构造的成像精度,数值结果表明,基于分数阶拉普拉斯算子解耦的黏声介质地震正演模拟与逆时偏移可大幅度提高地震模拟计算效率,偏移剖面明显优于常规声波偏移剖面,极大改善深层构造的成像品质.%Modeling seismic wave propagation in attenuating media accounts for the effective anelastic characteristics of the real earth.Numerical solution of the constant Q wave equation in the time domain requires a lot of memory and has low computational efficiency because of the fractional time derivative,which limits the wide use in solving inverse problems,e.g.,seismic migration.We evaluate a time-domain wave equation expressed by a second-order temporal derivative and two fractional Laplacian operators for modeling acoustic wave propagation in attenuating media.The wave equation introduces separated amplitude loss and phase velocity dispersion operators.The separated forms are more useful in compensating for attenuation loss in inverse problems,such as reverse time imaging by only reversing sign of the attenuation operator and leaving the sign of the dispersion operator unchanged.The other advantage of using our formulation over the traditional fractional time derivative approach is the avoidance of time history memory variables and thus it offers more economic computations.In numerical simulations,the temporal derivative is calculated with a staggered-grid finitedifference approach.The fractional Laplacians are calculated in the spatial frequency domain using extended Fourier pseudospectral implementation.We formulate the first-order constitutive equations with the perfectly matched layer absorbing boundaries.In order to verify the effectiveness of the method we conduct one-dimensional and two-dimensional wave field forward modeling and reverse time migration.Numerical results show that the method can conduct constant Q media wave field simulation accurately,and the viscoacoustic reverse time migration can correct dispersion and compensate loss at the same time,thus significantly improve the image quality.The image profile is better than that by acoustic reverse time migration.