Research On The Total Least Squares And Its Applications In Surveying Data Processing

Least squares is the most fundamental method in measurement data processing, and is widely used. For the classical least squares, the observation errors are considered while the errors of coefficient matrix are not considered. In the measurement data processing, the errors of coefficient matrix are existent. From 1980, the total least squares was named by Gene H. Golub,it is widely studied in mathematics, and widely used in automatic control, signal processing, image processing, medicine, statistic, and so on. Recently, the total least squares is studied in surveying and mapping science, but is concerned by more and more scholars. It is becoming a hot point in data processing of surveying and mapping science.In the paper, combined with the feature of surveying data, the mathematical model, adjustment criterion, solution formula and precision evaluation of the total least squares adjustment are systematically investigated. Then the total least squares theory is fell under the theoretical system of surveying adjustment solution from the solution method of mathematical field. And the ill-posed total least squares adjustment is studied in the detail. At last, through some examples, the applications are shown and discussed.The main conclusions and contributions of this thesis involve the aspects as following:Matrix decomposition is the main solution method to the total least squares. Firstly, the relationship between matrix decomposition and surveying adjustment solution model are studied. The relative contents of QR decomposition and SVD decomposition are summarized. The relationship between decomposition and matrix generalized inverse is given. Then the relationship between matrix QR decomposition and surveying adjustment solution is discussed deeply. The solution formula of indirect adjustment using QR decomposition, the solution formula of least squares collocation using QR decomposition and SVD decomposition, the solution formula of rank defect free network using SVD decomposition are deduced in detail. Through some examples, the correctness of the adjustment model by matrix decomposition and the validity of the algorithms are verified.On the basis of already existing solutions of the total least squares problems, the solutions to the total least squares are further studied. Some solutions of the total least squares problems are systematically summarized. According to the method of prof. Schaffrin, one kind of simplified mathematical model, adjustment criterion, solution formula of the total least squares adjustment are presented; through the examples, correctness of the algorithm is verified. For the mixed total least squares problems(LS-TLS), the method of SVD decomposition is summarized, and one kind of he mixed total least squares (LS-TLS) adjustment model and criterion are presented, then the solution formula is deduced and through the examples, correctness of the algorithm is verified. The relationship between iterative method and SVD decomposition; the equivalence and consistency of these two methods are proved in theory.According to the feature of surveying data, the constrained and weighted total least squares adjustment are studied. The method of constrained total least squares deduced by prof. Schaffrin is summarized. One kind of simplified mathematical model and adjustment criterion of constrained total least squares adjustment are presented, and the formula is deduced. The method of weighted total least squares deduced by prof. Schaffrin is summarized. The relationship between iterative method and SVD decomposition of weighted total least squares is further demonstrated, and the conversion method of these methods are given. The iterative method of the scaled total least squares is deduced, and its equivalence to SVD decomposition is proved.According to the ill-posed total least squares adjustment problems, the methods to overcome or weaken ill-condition are studied. The formula of ridge estimation is deduced and the methods of chosing ridge parameter are discussed, then combined with the real example, the analysis is done. According to Tikhonov theory, the regularized solution formula of the ill-posed total least squares adjustment problems is deduced. For the ill-posed total least squares adjustment problems, the truncated SVD decomposition method is analyzed, through the example, some methods to choose the truncated value k are given.At last, combined with the surveying examples, the applications of the total least squares are analyzed. The applications of the total least squares in Auto Regressive model, inversion data processing, the datum transformation of quasi-geoid and sphere target fixing of point cloud dada are discussed. The differences and same between TLS and LS are compared.