Wavelet And Homotopy Methods For The Inverse Problems Of Wave Equation In Seismic Prospecting

Wavelet analysis is an international forward research field in recent years. Itnot only contains abundant mathematical theories, but also is a powerful methodand tool in engineering—it brings new ideas to many fields. The homotopymethod is a widely convergent method for solving nonlinear system of equations,and also has important applications in many fields. The theory and practice ofinverse problem of wave equation in seismic prospecting indicate that local ex-tremum and ill-posedness are the bottlenecks restricting the development of theinversion theory. Therefore, this dissertation draws the wavelet analysis and thehomotopy method into the inversion process, and carries out a series of studiesof numerical inversion methods, which have the abilities of saving computationalcosts, noise suppression, global convergence and easy realization of the procedure.Consequently, we acquires some concrete wavelet, homotopy inversion methods.Firstly, the principle of wavelet multiscale inversion is introduced to the gen-eral parameter estimation problem. Basing on this principle and combining themature and simple Galerkin method, we design a wavelet multiscale Galerkininversion method. Numerical simulations are carried out for the inverse problemof one-dimensional wave equation in seismic prospecting. The numerical resultsand tests of noise suppression indicate the method's e?ectiveness.Secondly, for the insu?ciencies of traditional nonlinear minimization meth-ods, the wavelet multiscale inversion method and the stable and quickly conver-gent regularized Gauss Newton method are combined, and which are successfullyapplied to solve the velocity inverse problem of 2–D wave equation in seismicprospecting. Compared with simple iterative methods, the risk of trapped inlocal extremum is greatly reduced. Compared with the multigrid multiscale in-version method, it owns a quick decomposition and reconstruction algorithm.The operation is more simple, and the procedure is more easily to realize.Thirdly, the widely convergent homotopy method is introduced to the para-meter estimation problem. The basic principle of homotopy inversion method isdescribed. The widely convergent and stable method—the regularized homo-topy inversion method is constructed by combining the Tikhonov regularizationfor solving ill-posed problems with the homotopy method, which mixed togetherthe advantages of the homotopy method and Tikhonov regularization. It theo-...