Research On Several Key Problems Of Ill-posed And Robust Total Least Squares Method

In the classical adjustment,there is a basic assumption that only the observation vector is affected by random errors.And the optimal estimation of parameters are calculated under the criteria which ensuring the norm of observation residuals to be minimal.In fact,because the coefficient matrix is available through measurements or it is an idealized approximation of the true operator,the observation vector and the coefficient matrix are contaminated by some noise at the same time,which is widely existent in the measurement data processing.If the least squares method is used,the result will be biased.In order to improve the accuracy of parameters,studying new theory and method of measurement data processing that is necessary.In recent years,total least squares method which can deal with the errors of observation and coefficient matrix has been widely attended and studied by survey researchers.On the basis of the deficiencies of ill-posed total least squares and robust total least squares method,their algorithms are studied and theories are expanded in data processing of surveying and mapping.The principal work and conclusion are generalized as follows:(1)Three methods of total least squares are summarized comprehensively.The conceptions and algorithms of mixed LS-TLS and weighted total least squares are introduced.At last,through two examples of linear fitting and geoid refine,least squares method,total least squares method,mixed LS-TLS method and weighted total least squares method are discussed.A linear fitting example shows that weighted total least squares method is the best.But in the geoid's experiment,the results of the instance demonstrate the accuracy of weighted total least squares method is not higher than other methods.(2)Ill-posed problems of total least squares are investigated systematically.The cause and the methods of ill-posed problems are analyzed.Total least squares model is more affected by ill-posed problems,and the solution of total least squares is more unstable than least squares method.The algorithms of ill-posed total least squares problem are summarized as four types.Ridge estimation,generalized regularization algorithm,virtual observation method and truncated SVD method are concluded.In view of the randomness of regularization parameter,an improved regularization algorithm is proposed.The formula is deduced in detail and the corresponding iterative method is given.The results of numerical simulation and spatial trilateration network adjustment demonstrate the effectiveness and correctness of the improved regularization algorithm.Single Tikhonov regularization can tend to the unstable and unaccurate solution,a hybrid regularization algorithm incorporating both Tikhonov and total variation regularization is put forward.The solution formula is deduced and specific iterative steps are given.The numerical simulation results show that the correctness of the hybrid regularization algorithm.The regularization algorithms of weighted total least squares and Partial EIV model are deduced respectively to ill-posed problems.(3)The influence of gross error on the parameter estimation is analyzed,and data detection and robust estimation method of least squares model are discussed.When the observation vector and the coefficient matrix are contaminated by gross error at the same time,robust estimation algorithms are discussed,including gross error detection method under the condition of equal rights,the iteration method under the condition of unequal rights,and robust total least squares algorithm based on median.To avoid choosing the improper equivalent weight function and its threshold,the t robust estimation and EM algorithm of linear EIV model are proposed.The results of plane fitting show that t robust estimation can effectively deal with gross errors by selecting the appropriate degrees of freedom.And the iterative steps of Partial EIV model robust estimation algorithm are given.(4)Total least squares of determining parameters for probability integral method is presented.The model linearization of probability integral method is derived,and the iterative steps of nonlinear total least squares are provided.Xieqiao coal face 11316 in Huainan as an example,the ridge methods of least squares and total least squares are used respectively to calculate the vertical deformation parameters according to the actual observation of part points and considering the influence of ill-posed coefficient matrix of the observation equation.The comparison and analysis of the experiment results show that the the ridge algorithm of total least squares has higher precision in solving the prediction parameters.And the fitting parameters are affected by the initial values.