The Inverse Problem Of Mathematical Physics Logging In Some Applications

The field of inverse problem is a relatively new area of Mathematical research. It has bright future in application on radars , sonar and prospecting for petroleum, etc. The inverse problem in geophysical logging has broadly attracted our attention in recent years. The high performance solving process and corresponding numeric algorithm are needed in practice since the ill posedness of most inverse problems has been considered. The question of geophysical logging recover is come down to the solution of the first kind of Fredholm integral equations, but it is ill-posed problem. This problem should been solved by special method for getting approximate solution.1.This paper sums up of the solution on ill-posed problems in chapter I.2. As the bases of inverse problem, the direct scattering problem is solved by layer potential theory for Dirichlet and impedance boundary. The results of examples are given.3. Many results have get on the exterior Problem of 2-D Helmholtz equation. In this paper, a Helmoltz equation is transformed into the second integral equation containing a Cauchy singularity by applying potential theory. We obtain numerical solution by utilizing Nysto|¨m method. These numerical solutions show that this method is simple and effective.4. This paper deals with the actual data basing on Walsh recover for evaluating the territorial heterogeneity of reservoir.5. Beginning with the ill posedness this paper obtains the regularized solution by a priori choosing regularizing parameters based on Tikhonov regularization. It deals with the actual datas. The result shows that this method improves speed and precision of recover calculation.6. The Backus-Gilbert method is used in physical geography,in fact it is a regularization method. This paper deals with the actual data basing on Backus-Gilbert method for evaluating the territorial heterogeneity of reservoir.