速度-应力声波方程混合交错网格有限差分数值模拟及逆时偏移  

Mixed staggered grid finite difference numerical simulation and reverse time migration of velocity⁃stress acoustic equation

作　　者：胡自多[1,2] 刘威 宋家文[3] 曾庆才 田彦灿 韩令贺[1,2] HU Ziduo;LIU Wei;SONG Jiawen;ZENG Qingcai;TIAN Yancan;HAN Linghe(PetroChina Research Institute of Petroleum Exploration&Development-Northwest,Lanzhou,Gansu 730020,China;PetroChina Key Laboratory of Reservoir Description,Lanzhou,Gansu 730020,China;GRI,BGP Inc.,CNPC,Zhuozhou,Hebei 072750,China;PetroChina Research Institute of Petroleum Exploration and Development,Beijing 100083,China)

机构地区：[1]中国石油勘探开发研究院西北分院,甘肃兰州730020 [2]中国石油集团油藏描述重点实验室,甘肃兰州730020 [3]东方地球物理公司研究院,河北涿州072750 [4]中国石油勘探开发研究院,北京100083

出　　处：《石油地球物理勘探》2023年第5期1084-1100,共17页Oil Geophysical Prospecting

基　　金：中国石油集团前瞻性基础研究项目“物探岩石物理与前沿储备技术研究”(2021DJ3501)资助。

摘　　要：用常规交错网格有限差分法(C‐SFD)进行速度—应力声波方程数值模拟,虽然空间差分算子能达到2M阶差分精度(M表示空间差分算子中与差分中心点等距的坐标轴网格点的组数),但差分离散声波方程仅具有2阶差分精度,所以其模拟精度低、稳定性差。为此,联合利用坐标轴网格点和非坐标轴网格点构建空间差分算子近似一阶空间偏导数,建立了一种适用于速度—应力声波方程的混合交错网格有限差分法(M‐SFD),并基于时空域频散关系和泰勒级数展开建立差分系数求解方程组,导出了差分系数通解。M‐SFD给出的差分离散声波方程可达到4、6、8阶、甚至任意偶数阶差分精度。频散分析表明:在特定的Courant条件数取值(如Courant条件数r=0.3)条件下,C‐SFD通过调整M的取值,无法将数值频散误差控制在1%以内;M‐SFD通过调整M和N的取值(N表示空间差分算子中与差分中心点等距的非坐标轴网格点的组数),基本可以将频散误差控制在1‰以内,甚至可以控制在0.1‰以内。稳定性分析显示:M取值相同时,M‐SFD的稳定性强于C‐SFD。数值模拟实例表明:计算效率基本相同时,M‐SFD比C‐SFD能更有效地压制数值频散,模拟精度更高;M‐SFD还能采用比C‐SFD更大的时间采样间隔以获得更高的计算效率,且模拟精度更高。进一步将M‐SFD推广应用于逆时偏移,M‐SFD作为逆时偏移中的波场传播算子,能够有效消除由于数值频散造成的成像假象,从而提高深层的构造成像精度和分辨率。The conventional staggered grid finite difference method(C‐SFD)is widely used to simulate the velocitystress acoustic equation.Although the spatial difference operator can reach the order of 2M difference accuracy(M represents the number of sets of axial grid points equidistant from the center point of the spatial difference operator),the discretized difference acoustic equation only has the second‐order difference accuracy,resulting in low simulation accuracy and poor stability.In this paper,the axial grid points and off‐axial grid points are used to construct the spatial difference operator to approximate the first‐order spatial partial derivative,and a mixed staggered grid finite difference method(M‐SFD)suitable for the velocity‐stress acoustic equation simulation is constructed.Based on the time‐space dispersion relationship and Taylor series expansion,the difference coefficient solution equations are established,and the analytical solution of the difference coefficient is derived.The discretized difference acoustic equation given by M‐SFD can reach 4th‐order,6th‐order,8th‐order,or even any even order difference accuracy.The dispersion analysis shows that under the condition of specific Courant condition number value(such as Courant Condition number r=0.3),C‐SFD cannot control the numerical dispersion error within 1%by adjusting the value of M.By adjusting the values of M and N(N represents the number of sets of off‐axial grid points equidistant from the center point of the spatial difference operator),M‐SFD can basically control the dispersion error within 1‰or even within 0.1‰.Stability analysis shows that when M values are the same,the stability of M‐SFD is stronger than that of C‐SFD.Numerical simulation examples show that when the computational efficiency is almost the same,M‐SFD can more effectively suppress numerical dispersion than C‐SFD,resulting in higher simulation accuracy.M‐SFD can also use a larger time sampling interval than C‐SFD to achieve higher co

关 键 词：混合交错网格 数值频散 差分系数计算 数值模拟 逆时偏移 

分 类 号：P631[天文地球—地质矿产勘探]