基于Chebyshev走时逼近的三维多次反射射线计算  

Computation of Three Dimensional Multi-Reflection Rays Based on Traveltimes Numerical Approximation Using Chebyshev Polynomials

作　　者：孙建国[1] 苗贺 Sun Jianguo;Miao He(College of GeoExploration 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：《吉林大学学报（地球科学版）》2018年第3期890-899,共10页Journal of Jilin University：Earth Science Edition

基　　金：国家自然科学基金项目(41274120)~~

摘　　要：在利用有限差分等基于网格的数值分析方法解地震波走时所满足的程函方程时,由于速度模型的网格化离散等原因,会使走时在各网格节点之间不具有计算射线路径所要求的光滑性,即走时在邻近网格节点之间不具有连续的一阶导数。因此,直接利用网格节点走时计算射线路径会使最终的射线路径不光滑。为解决这个问题,已有研究者提出了基于B样条插值的逆向梯度方案(法)。然而,在速度发生突变时,B样条逆向梯度法所计算出的射线路径会具有较大的误差。针对这个问题,首先采用适合于解最小零偏差逼近及最佳平方逼近问题的Chebyshev多项式取代B样条对来自于分区多级计算方案的网格节点走时进行最佳逼近,得到在最小平方意义下的最优走时公式;然后采用与B样条逆向梯度法类似的计算过程得到光滑的射线路径。数值实验表明,利用Chebyshev多项式逼近走时可以得到具有很高精度的多次反射射线路径,在多次波偏移成像研究中具有潜在的价值。When using the grid methods,such as the finite difference method for solving the eikonal equation satisfied by seismic traveltimes,the traveltimes obtained at the adjacent grid points may not have the smoothness needed for computing ray trajectories.This is because of the discretization of the velocity model by grids.To solve this problem,some researchers proposed a backward gradient schemebased on a B-spline interpolation.However,the ray trajectories calculated by the B-spline interpolation scheme will have large errors when the velocity changes abruptly in some model region.Aiming at this problem,instead of the B-splines,we first utilized the Chebyshev polynomials,which are optimal for solving the zero deviation and the least square problem,to obtain the formulas that can optimally approximate the grid point traveltimes resulted from the finite different solution of the eikonal equation.Then,we used a computation procedure which is similar to that used in the B-spline scheme for obtaining smooth ray trajectories.The numerical test showed that the use of Chebyshev polynomials for approximating traveltimes led to smooth ray trajectories of multiply reflected waves with high accuracy.It should be investigated further in the future work.

关 键 词：多次反射 射线走时 射线路径 走时逼近 CHEBYSHEV多项式 

分 类 号：P631.4[天文地球—地质矿产勘探]