贴体网格剖分的地震资料逆时偏移方法研究

发布时间：2018-08-13 20:05

【摘要】：有限差分方法(Finite Difference Method,FDM)作为一种快速而精确的数值方法,在波动方程正演数值模拟领域得到了最为广泛的应用,然而,当在不规则或者地表起伏不平的区域中模拟时,基于规则网格的有限差分法对波动方程离散时会产生阶梯状近似,影响模拟的精度,给有限差分方法的求解带来困难。通过求解椭圆型偏微分方程生成的贴体网格为这种复杂区域中的有限差分波场模拟问题提供了一种有效的手段。借助贴体网格以及链式法则,将空间导数的计算从不规则的物理区域转换到规则的计算域,在计算域中继续使用有限差分方法。相比其他的不规则网格剖分方法,如插值法、映射法以及非结构网格,贴体网格在普遍性、精度和稳定性方面都有优势。本文在复杂区域综合应用了贴体网格、声波方程正演模拟、逆时偏移(Reverse Time Migration,RTM)以及完全匹配层(Perfectly Matched Layer,PML)等技术。首先通过求解椭圆偏微分方程法生成贴体网格,根据链式法则代入顺序不同,将二阶位移形式的二维声波方程改写到曲线网格中,得到了两种形式的贴体坐标二维声波方程,第二种形式的波动方程相比第一种形式而言,形式对称、更紧凑。为了压制截断边界造成的人工边界反射,推导了两种形式的声波方程对应的PML方程。在复杂起伏地表区域中,分部求和(Summation-by-Parts,SBP)有限差分方法的使用能够确保曲线网格中非均匀介质数值模拟的稳定性。SBP有限差分法特别适合求解含变系数的导数项。因此,对两种形式的波动方程及PML针对性地采用了不同的差分方法:第一种形式中,导数项的系数可预先求得,采用中心差分方法进行数值离散;第二种形式中含变系数,因此采用显式的二阶精度SBP有限差分方法进行离散,并采用Fourier谱分析方法讨论了这种离散格式的稳定性。与中心有限差分方法对比,分部求和有限差分方法的稳定性更高。四阶精度的有限差分方法在降低存储需求和提高效率方面有很大的优势。在二阶精度中心差分方法和SBP有限差分方法的基础上,将两种形式的贴体网格声波方程和PML中空间导数项的离散均扩展到四阶精度。采用Fourier谱分析方法研究了该四阶SBP差分离散格式的稳定性,得到了离散方程的稳定性条件,并验证了四阶精度SBP有限差分方法比中心有限差分方法的稳定性更高。同时,在同样尺寸的模型中比较了二阶和四阶精度SBP有限差分方法的计算时间及存储需求,验证了四阶方法的高精度、低频散特性,显示了四阶方法在提高运算效率、降低存储方面的能力。本论文将贴体网格的方法分别应用于地面地震和VSP(Vertical Seismic Profiling,垂直地震剖面)资料的逆时偏移研究,得到了曲线坐标系中各自对应的成像数据体,为解决实际地震采集中观测系统不规则、地表起伏等情况下的数据处理难题提供一种有效的技术手段。
[Abstract]:As a fast and accurate numerical method, finite difference method (Finite Difference) has been widely used in the field of forward numerical simulation of wave equations. The finite difference method based on regular grid will produce ladder approximation when the wave equation is discretized, which will affect the accuracy of the simulation and make it difficult to solve the finite difference method. The body-fitted meshes generated by solving elliptic partial differential equations provide an effective means for the finite difference wave field simulation in this complex domain. With the help of body-fitted meshes and chain rules, the computation of spatial derivatives is transformed from irregular physical regions to regular computational domains, and the finite difference method is used in the computational domain. Compared with other irregular mesh generation methods, such as interpolation, mapping and unstructured meshes, body-fitted meshes have advantages in universality, accuracy and stability. In this paper, the techniques of body-fitted mesh, forward modeling of acoustic wave equation, inverse time migration (Reverse Time migration and perfectly matched layer (Perfectly Matched are applied in complex regions. Firstly, the body-fitted grid is generated by solving the elliptic partial differential equation. According to the different order of the chain rule, the two-dimensional acoustic wave equation in the form of second-order displacement is rewritten into the curvilinear grid, and two kinds of two-dimensional acoustic wave equations with body-fitted coordinates are obtained. The second form of wave equation is more symmetrical and compact than the first one. In order to suppress the artificial boundary reflection caused by truncated boundary, the PML equation corresponding to two kinds of acoustic wave equations is derived. In complex undulating surface regions, the use of the Summation-by-Partss-SBP finite difference method can ensure the stability of numerical simulation of non-uniform media in curved grids. The SBP finite difference method is particularly suitable for solving derivative terms with variable coefficients. In the first form, the coefficient of the derivative term can be obtained in advance, and the central difference method is used to discretize the two forms of wave equation and PML. Therefore, the explicit second-order precision SBP finite difference method is used for discretization, and the stability of the scheme is discussed by using the Fourier spectrum analysis method. Compared with the central finite difference method, the partial summation finite difference method is more stable. The fourth order finite-difference method has great advantages in reducing storage requirements and improving efficiency. Based on the second-order precision central difference method and the SBP finite difference method, the dispersion of the two kinds of body-fitted mesh acoustic equations and the spatial derivative term in PML are extended to the fourth order accuracy. The stability of the fourth order SBP difference discrete scheme is studied by using the Fourier spectrum analysis method. The stability conditions of the discrete equation are obtained, and the stability of the fourth order precision SBP finite difference method is proved to be higher than that of the central finite difference method. At the same time, the computation time and storage requirement of the second-order and fourth-order precision SBP finite difference method are compared in the model of the same size. The high precision and low frequency dispersion characteristics of the fourth-order method are verified, and it is shown that the fourth-order method is improving the computational efficiency. Reduce storage capabilities. In this paper, the method of body-fitted grid is applied to the inverse time migration of surface seismic and VSP (Vertical Seismic Profiling, vertical seismic profiles, and the corresponding imaging data volume in the curvilinear coordinate system is obtained. In order to solve the problem of data processing in the case of irregular observation system and surface undulation in actual seismic acquisition, this paper provides an effective technical means.
【学位授予单位】：中国石油大学(北京)
【学位级别】：博士
【学位授予年份】：2016
【分类号】：P631.4

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