Research On The Regularization Solutions Of Ill-Posed Problems In Geodesy

The ill-posed problems include the ill-conditioned and the rank-deficient ones in Geodesy, which exist widely in GPS data processing, deformation analysis, Geodesy inversion, gravity field continuation downward, and so on. Systematical research on the theory and methods of processing the ill-posed problems is an important task and grows to a significant subject direction in Geodesy. Based on TIKHONOV regularization method and considering sufficiently the practice of Geodesy, the main thread of the choice of the regularizer and the determination of the smoothing parameters is grasped and the in-depth research on the ill-posed problems in Geodesy is carried out. A framework of the theory and methods of systematically processing the ill-posed problems has been established and TIKHONOV regularization method has been evolved in this paper. The paper contains the following contents:1 Deriving the unified expression of solutions of the ill-posed problems in GeodesyThe common mathematical models of the ill-posed problems in Geodesy are analyzed, such as the collocation model, the semi-parametric model, the free net adjustment model and the ill-conditioned model, etc. It is discovered that their solutions can be expressed by a relation formula and all of them can be derived based on TIKHONOV regularization theorem. The unified formula helps to hold the commonness of the ill-posed problems and analyze their individuality. In practice, we should not only consider the basic theory, but also find the optimum algorithm, which is beneficial to deepening the investigation.2 Investigation on the improved algorithms of overcoming the ill-condition(1) For the case that the ridge parameter is difficult to determine, L-curve method and its Matlab program is investigated systematically. The comparisons are carried out among the L-curve method, the ridge mark method and the GCV method in order to illustrate the effect of L-curve method.(2) A new method of overcoming the ill-condition -Two-Step Method is proposed. The theorem, the characteristic of solutions and the applicability of Two-Step Method are discussed. The new method not only improves the results of LS greatly, but also has an advantage over ridge estimate and truncated singular value method.(3) A new singular value modification scheme is proposed. Based on SVD technique and considering the compromise between the distinguishing rate and the variance of the solution, a new singular value modification scheme has been proposed if the singular values decrease gradually, whose key is separating the singular values into two parts and modifying them separately. The examples show that the new scheme is very effective when the condition number of the normal matrix is less than 1010. Compared with other methods, the new scheme improves the precision and accuracy of the computation results obviously.3 Investigation on new approaches of mitigating the ill-condition in GPS rapid positioning using single frequency GPS receiversThe new approaches are investigated in GPS rapid positioning using several-epoch single frequency phase data. Firstly, the structure characteristic of the normal matrix in above case is analyzed. Then, in the light of the characteristic, based on the TIKHONOV regularization theorem, two new regularizers are designed to mitigate the ill-condition of the normal matrix in GPS rapidpositioning. The accurate float ambiguity solutions and their MSEM (Mean Squared Error Matrix) are obtained using several-epoch single frequency phase data. Combining with LAMBDA method, the new approaches can fix the integer ambiguities correctly and quickly using MSEM instead of the covariance matrix of the ambiguities. Compared with the traditional methods, the new approaches improve the efficiency in rapid positioning obviously. The comparisons are carried out between the new approaches and the traditional methods using several actual baselines and the results of the new approaches are verified. The two new approaches of mitigating the ill-condition of...