三维高阶深度偏移方程及其数值求解方法

THE 3-D HIGH ORDER DEPTH MIGRATION EQUATION AND ITS NUMERICAL SOLUTION METHOD

从三维非均匀介质中的波动方程出发，利用拟微分算子理论，Ｐａｄｅ逼近方法及因式分解技巧，获得了非均匀介质的三维高阶深度偏移方程，相应地提出了逐次低阶方程方法、低阶方程组方法及分裂方法等３种求解方法．与二维情形不同，以上每一种方法在数值求解时均存在由测线坐标ｙ的出现而带来的困难．为了克服这一困难，我们提出了差分算子分解方法，避免了近年来人们竞相研究的ｘ－ｙ方向微分算子分裂带来的分裂误差，保持了应有的相容性，解决了这一令人烦恼的问题．

In this paper a 3-D high order depth migration equation is derived by usingthe theory of pseudo-differential operator and Pade approximation method andthe factorization technique.This migration equation is based on the 3-D wawe equation of inhomogeneous media. Three numerical solution methods are proposed for this kind of migration equation. They are the method of low order equation time by time and the method of low order system of equations and the splitting method.In distinction from that of 2-D,there are difficulties about numerical solution in each method above. It Is owing to the appearence of the crossllne-y.In order to overcome those difficulties ,we proposed the factorization method of difference opelators.It avoided the splitting error brought by the splitting of the differential operator in the x and y direction and reserved the necessary consistency of the.So this troubled and difficult problem is solved. The method proposed in this paper is tested with synthetic section.Both the numerical results and the theoretical analyses show the effectiveness of the method.