一种利用Nystrom离散与FFT快速褶积的散射地震波并行计算方法  

A parallel computation method for scattered seismic waves using Nystr m discretization and FFT fast convolution

作　　者：徐杨杨[1] 孙建国[1] 商耀达 XU YangYang;SUN JianGuo;SHANG YaoDa(College of Geo-exploration Science and Technology,Jilin University,Changchun 130026,China)

机构地区：[1]吉林大学地球探测科学与技术学院,长春130026

出　　处：《地球物理学报》2021年第8期2877-2887,共11页Chinese Journal of Geophysics

基　　金：国家自然科学基金(41974135);国家重点研发计划(2019YFC0312004)联合资助.

摘　　要：利用数值方法解Lippermann-Schwinger(L-S)方程的主要困难在于系数矩阵存储和线性方程组求解.这主要是因为L-S方程的积分部分是一个空间褶积,在离散后将导致一个满秩矩阵,进而形成一个大型或超大型代数方程组.因此,在利用L-S解决地震波散射问题时,一般是利用散射级数法而非数值方法.然而,散射级数法的计算精度和收敛性强烈地依赖于速度扰动的强度,而克服这种依赖性的一个可能的途径就是对现有的数值方法进行改进或是建立新的数值求解方案.在这种思想指导下,首先对L-S方程进行改写,得到一个与原L-S方程等价的积分方程(等价L-S方程).然后,对等价L-S方程进行逐点归一化处理,并利用Nystrom法对经归一化处理的等价L-S方程(归一化等价L-S方程)进行离散,并用FFT计算空间褶积.之所以这样选择是由于归一化等价L-S方程经Nystrom法离散生成的系数阵为一个Toeplitz阵,可利用其Toeplitz性质降低存储空间;而FFT可以将矩矢空间褶积转化为乘积,且积分核部分只要计算一次即可.进一步,为节约正演计算时间,设计了进程级和线程级相结合的MPI+OpenMP并行模式.数值试验表明,与传统的积分方程数值算法相比,利用等价L-S方程、Nystrom离散和FFT快速褶积的计算方案可极大地降低存储需求,进而在保证精度的同时提高计算效率.The storage of coefficient matrix and solution of linear equations are the main factors of limiting the application of the Lippermann-Schwinger(L-S)integral equation to solving the seismic scattering problem.Because the integral part of the L-S equation is a spatial convolution,which can lead to a full rank matrix after discretization,thus forming a large or very large algebraic equation set.Therefore,when L-S equation is used to solve the seismic wave scattering problem,the scattering series method is generally used instead of the numerical method.However,the computational accuracy and convergence of the scattering series method strongly depend on the intensity of velocity perturbation,and one possible way to overcome such dependence is to improve the existing numerical methods or to establish new numerical solutions.Under the guidance of this idea,the L-S equation is first rewritten to obtain an integral equation equivalent to the original L-S equation(equivalent L-S equation).Then,the equivalent L-S equation is normalized point by point,and is discretized by the Nystrom method.The spatial convolution is calculated by Fast Fourier Transform(FFT).The coefficient matrix generated by the normalized equivalent L-S equation discretized by the Nystrom method is a Toeplitz matrix,and its Toeplitz property can be used to reduce the storage space.FFT is adopted to transform the convolution into a product,and the integral kernel only needs to be calculated once.Furthermore,in order to save computing time,MPI+OpenMP parallel mode combining process level and thread level are designed.Numerical experiments show that compared with the traditional integral equation numerical algorithm,the calculation scheme based on the equivalent L-S equation,Nystrom discretization and FFT fast convolution can greatly reduce the storage and CPU time required,and improve the calculation efficiency with sufficient accuracy.

关 键 词：L-S积分方程 快速傅里叶变换 Nystrom法 MPI+OpenMP并行 

分 类 号：P631[天文地球—地质矿产勘探]