Research On The Ill-posed Total Least Squares Algorithm And Its Application

The ill-conditioned problem exists widely in geodetic data processing, it is so harmful to the estimation in data processing that some measurements or methods must be taken to eliminate or weaken the ill-condition of the system. By now, with the development of total least squares(TLS) and the expansion of its applications, the ill-posed total least squares has become one of the hot topic in geodetic data processing. In this paper, the study history and research status of the ill-posed total least squares problem both at home and abroad are introduced systematically. According to the deficiencies in the current methods on ill-conditioned problem, some new methods are proposed and applied in volcano deformation analysis. The main research and innovations in this paper are explained as follows:1. A new virtual observation method to ill-posed total least squares problem is proposed, the priori information of the parameters in indirect adjustment that the parameters should be independent each other is taken as the second observations which is so called virtual independent observations, then the virtual observations is regard as constraints with ill-posed observation equation are united together to solve the unknown parameters. The formulas are derived and the physical meaning of sub-criterion-parameter is given. Through two examples, the TLS-virtual observation method, TLS-L curve method, common TLS method and least squares(LS) method are compared and analyzed.2. According to the deficiencies existed in the biased estimation, on the base of the iteration method by correcting characteristic values to LS, the iteration method by correcting characteristic values to TLS and its improved algorithm are proposed. The new method proposed in this paper maintains the merits in the iteration method by correcting characteristic values to LS. Through the examples, the characteristics and defects of the new algorithm are analyzed in normal and ill-conditioned situation respectively.3. The ill-posed total least squares problem can be regard as unconstrained optimization problem in the field of numerical analysis. The Armijo inexact line search technique restart FR conjugate gradient method is taken to estimate the unknown parameters. Through the examples, the connection between TLS conjugate gradient method and regularized TLS method is explained.4. At the last, the three methods proposed in this paper are applied to solve the ill-conditioned problem in volcano inversion of Mogi model. The applicability and effectiveness of them are discussed and analyzed.