改进的TTI介质纯P波方程正演模拟与逆时偏移

Numerical simulation and reverse time migration using an improved pure P-wave equation in tilted transversely isotropic media

逆时偏移作为一种高精度偏移方法已成为复杂构造成像的重要技术,描述纵波独立传播的延拓方程是各向异性介质逆时偏移的一个关键问题.在对VTI介质几个经典相速度近似公式回顾的基础上,针对常用于描述纯P波的Harlan近似公式在各向异性参数ε较大情况下近似精度较低的问题,本文对Harlan公式中的非椭圆项进行了修正,在非椭圆项前添加了一个与各向异性参数ε有关的修正系数,得到了三种改进型Harlan公式,并以近似精度最高的改进式为基础,推导了TTI介质纯P波方程.针对该伪微分方程,本文利用伪谱法和有限差分法联合实现波场延拓,对于常密度二阶方程,基于中心网格实现;对于一阶应力-速度方程则基于旋转交错网格实现.通过数值试验分析了TTI介质纯P波一阶应力-速度方程的近似精度,并以一阶纯P波方程为基础进行了TTI介质逆时偏移数值模拟试验.结果表明,本文给出的方法能够较准确地描述TTI介质纯P波波场特征,可以应用至各向异性介质逆时偏移.

Seismic anisotropy has been observed in many exploration areas, especially in sedimentary rocks. The most commonly used anisotropic model is the transverse isotropy （TI） with a vertical （VTI） or a titled （TTI） symmetric axis. For practical application, transverse isotropy is commonly considered under the acoustic approximation. In general, such acoustic approximation can be divided into two categories, coupled acoustic anisotropic wave equations and decoupled pure P-wave equations. One of the problems of the coupled acoustic equations is that they are not really free of shear waves. The artificial SV component is usually considered as numerical artifacts and may cause numerical instabilities in TTI media. However, decoupled pure P-wave equations are completely free from shear wave. In this paper, we firstly evaluate some commonly used phase velocity approximations and conclude that the Harlanrs approximation, widely used to derive pure P-wave equations, has low accuracy in a medium with large s. Then we present a method to improve its accuracy by modifying the anelliptic （fractional） term and obtain three modifications. Based on the most accurate one, we derive an improved TTI pure P-wave equation. The resulting wave equation can be formulated either as a second-order equation or as a system of first-order equations. The proposed TTI pure P-wave equation involves pseudo-differential operators which are difficult to be calculated by the finite- difference method. We use a hybrid pseudo-spectral/finite-difference （PS/FD） scheme to solve the TTI pure P-wave equation. The pseudo-differential operators are calculated by the PS method and the differential operators are calculated by the high-order rotated-staggered-grid FD method （RSGFD）. We test the hybrid PS/RSGFD scheme with various kinds of TTI models. In the first numerical example, we compare the first-order TTI pure P-wave equation with the counterpart elastic equation. We conclude that pure P-wave equation can provide good kinematic and dynamic approximations to the elastic wave equation for homogeneous media with weak-to-moderate anisotropy. The second test model is a heterogeneous wedge TTI model which contains sharp contrasts. This numerical example demonstrates such a hybrid PS/RSGFD scheme can provide a stable P-wave propagator. To further verify the proposed algorithm, we implement an anisotropic reverse-time migration （RTM） example on a small region of BP2007 TTI model. These numerical examples demonstrate the potential of the proposed method.