Research On Homotopy Algorithms Of GPR Problem

As the basis of data processing of GPR (Ground Penetrating Radar) and a key link of the imaging process, the numerical simulation of GPR provokes more and more concern and interest. The forward problem and the inverse problem of the electromagnetic propagation in anisotropy media underground based on Maxwell equation have increasingly been an advanced task in the research of the fundamental theory of GPR. That the idea of homotopy, which is a basic conception of algebraic topology, is used in the solution of nonlinear operator equation will be able to eliminate the weakness that the traditional numerical iteration falls easily into local convergence, broaden the rigorous restrictions on selecting initial guess, and come to a large-scale convergent method which is efficient and pragmatic. This thesis, in the process of the solution to the inverse problems of GPR, draws into homotopy method and makes a series of study of numerical inversion algorithm that has simultaneously less computation load, powerful antinoise capacity, wide-ranging convergence and easy accomplishment of the program; consequently actualizes the fully nonlinear inversion of GPR problem in real sense.This thesis mainly includes that: (1) inverse problem models used for data interpretation of GPR based on Maxwell equation and absorbing boundary condition have been constructed. (2) wide-ranging convergent homotopy method has been drawn into inversion procedure of the identification to the underground parameter and in the process of it, wide-ranging convergent and steady homotopy regularization inversion algorithm, homotopy parameter differential algorithm and homotopy adaptive inversion algorithm have been constructed, which theoretically proves global convergence of algorithm and effectiveness of algorithm has also been verified by numerical test. (3) homotopy-wavelet hybrid algorithm and homotopy-multigrid hybrid algorithm, which have both wide-ranging convergent character and effective reduction of amount of inversion, have been constructed through combination of the idea of multi-scale inversion with homotopy inversion algorithm. (4) By means of connection of EM record with well-log data, a series of algorithms for homotopy- well-log constraint have been constructed, which improves longitudinal resolving power of inversion of GPR.Concrete synopses in each Chapter are as the follows:In Chapter 2, continuous models of inverse problems of GPR and corresponding finite difference discretization models have been constructed by means of ascertaining Maxwell equations and absorbing boundary condition. In Chapter 3, which is the focal one of this thesis, the principle of homotopy algorithm and the basic train of thought of homotopy inversion are stated at some length. Combining Tikhonov regularization with homotopy method, this thesis constructs homotopy regularization algorithm, which mixes together the advantages of both. The algorithm avoids inelasticity of the inverse problems, and extends initial selection and algorithm convergence. Furthermore, the Chapter constructs homotopy parameter differential inversion algorithm and, in terms of theory, strictly proves global convergence of the algorithm. At length, the Chapter completes the designment of homotopy adaptive algorithm, which can be adaptive to select regularization parameter through the delimitation of threshold, and which greatly improves the flexibility and practicality of the algorithm.In Chapter 4 and Chapter 5, homotopy-wavelet hybrid inversion algorithm and homotopy-multigrid hybrid inversion algorithm are constructed by utilizing respectively wavelet multi-scale and multigrid multi-scale decomposition, which makes original inverse problems transformed into the inverse problem alignment in a sequence of nested subspace, in the maximum scale adopts wide-ranging convergent homotopy inversion algorithm as a means of global search, makes the search zone bigger and guarantees in the maximum scale the solution can be done in the global limits. The algorithm makes use of the respective advantages of the wavelet analyses and multigrid, and possesses the global search capacity with rapid convergence speed and favorable inversion effect. Global convergence in the maximum scale still can be guaranteed even though large quantities of local minimum exists in multi-solution or target function.In Chapter 6, homotopy- well-log constraint hybrid inversion algorithm is constructed by making use of the well-log constraint condition. The algorithm combines the lateral data of the surface with the longitudinal data of the well-log, and improves the longitudinal resolving capacity of GPR inversion. In addition, two more multi-scale algorithms correspondent with the algorithm have been constructed through the usage of multigrid and wavelet analyses, which strengthens the stability and antinoise capacity of the inversion algorithm and reduces the computation load of the inverse problem solution.The inversion algorithm and its application contributed by this thesis solve, to some extent, some puzzles that exist in the process of solution of the numerical value of GPR. These methods are all in generality; they are of certain theoretical significance and widespread practical value thereby.