Regularization Algorithms For Solving Ill-posed Matrix Equations Arising From Geophysical Inversions

Geophysical inversion is a key technique in geophysical exploration. Geophysical inversion often relates to solving ill-posed, large matrix equations. Theoretically, regularization is an effective method in dealing with ill-posed problems. Its application in practical geophysical inversion problems, however, still has many difficulties in selecting regularization parameters. Based on systematic study of regularization, this dissertation presents a few newly developed regularization methods that apply Active-Set algorithm, Differential Evolution algorithm, and a few others. These regularization methods focus on solving ill-posed matrix equations arising from practical problems. Main results of the dissertation include:Tikhonov regularization and Active-Set algorithm are applied together to the geophysical inversion problems so that the problems with non-negative parameters are converted into problems of non-negative damped least square algorithm, which can be further solved by the Active-Set algorithm. The improved recursive algorithm is further verified by numerical simulation. Satisfactory results are obtained by applying this algorithm to electrical conductivity imagery inversion.Furthermore, differential evolution algorithms are also studied. To improve the rate of convergence of Differential Evolution algorithms, two new Tikhonov regularization algorithms are proposed that respectively employ Adaptive Recursive Differential Evolution (ARDE) algorithm based on population entropy and Particle Swarm Optimization and Differential Evolution (PSODE) algorithm. Without any compromise in effectiveness, these two algorithms both improve the convergence speed and thus reduce computation cost. Still further, a new DE algorithm based on LSQS, which inherits the advantages of both LSQR and DE, is designed. This new algorithm avoids the common difficulty of regularization parameter selection in Tikhonov and TSVD algorithms. It also displays superior stability, independence on initial values, unlikelihood of local extrema, and faster convergence. This algorithm is particularly suitable to solve the problem of selection of regularization parameters in the study of geophysical inversion.Finally, the selection method of regularization term is also studied. A regularization term with a second order regularization operator is introduced to propose a mixed regularization method with two parameters. The L-curve criterion, discrepancy principle, and generalized cross-validation are applied to determine the optimal value of the regularization parameter. The validity and superiority of the proposed method is verified by numerical simulation of the theoretical model.Based numeric simulations and results from actual data processing, it is found that the newly developed A-TR regularization algorithm, HDE regularization algorithm are effective in certain conditions. The A-TR regularization algorithm is applicable to inversion problems that require non-negative parameters, whereas HDE regularization algorithm is applicable to those inversion problems whose selection of regularization parameters is difficult. Because of non-negative conductivity, A-TR algorithm will get more detailed and reliable imagery for dual-frequency conductivity inversion problems investigated by this paper.