Study On The Extended Theories Of Generalized Total Least Squares And Their Applications In Surveying Data Processing

Total least squares(TLS)can effectively solve the issue of parameter estimation in the errors-in-variables(EIV)model.It has attracted attention by researchers in many fields and various kinds of algorithms were proposed.In addition,related scholars investigated some extended TLS algorithms(such as gross error processing and variance component estimation)to meet the application needs.However,some of the existing studies are premature and most of them are based on standard EIV model,limitations exist when dealing with some nonlinear problems,such as 3D similarity transformation with large rotation angles and multivariate nonlinear regression.Currently there are few studies on the generalized TLS along with its extended theories.To solve the above problems,related theories on TLS and generalized TLS are systematically investigated from five aspects,including basic solution,prediction,parameter estimation with equality constraints,gross error processing,and variance component estimation(VCE).Main contents of this thesis include:(1)Various kinds of weighted total least squares(WTLS)algorithm are systematically investigated.The WTLS algorithms for the EIV model,Partial EIV model,generalized EIV model,and nonlinear Gauss-Helmert model are derived using different ways.Moreover,all WTLS solutions obtained by Lagrange multiplier method are reformulated in the form of standard least squares,which lays the foundation for investigating some extended theories on TLS,such as gross error processing and variance component estimation.(2)To make up the limited applicability of the existing TLS prediction models in datum transformation problem,a generalized total least squares prediction(GTLSP)model is proposed.The transformation equation of common points is abstracted as a generalized EIV model and the equation for new point transformation is incorporated into the functional model as well.Then the iterative solution is derived based on the Gauss-Newton approach of nonlinear least squares and Lagrange multiplier method.Finally,the performance of GTLSP algorithm is verified by two experiments involving 3D similarity transformation with large rotation angles and map rectification.The results show that GTLSP algorithm can improve the statistical accuracy of the transformed coordinates compared with LS and WTLS algorithm.(3)The generalized constrained TLS(GCTLS)methods based on generalized EIV model and nonlinear Gauss-Helmert model are investigated,respectively.The GCTLS algorithms are derived in the form of constrained LS(CLS).Finally,some experiments are employed to verify the validity and compare the computational efficiency of different GCTLS algorithms.(4)A robust WTLS(RWTLS)algorithm for the partial EIV model with fully correlated covariance matrix is proposed.The standardized residuals are employed to construct the weight factor function.Therefore,the algorithm possesses good robustness in both the observation and structure spaces.The experimental results of line fitting and 2D coordinate transformation show that RWTLS algorithm possesses better robustness than the general robust estimation and the robust total least squares algorithm directly constructed with original residuals.(5)Based on the WTLS solution with standard LS form,the data snooping algorithm is extended to generalized EIV model.Two experiments including datum unification of 3D laser scanning and hydrological curve fitting are adopted to test the performance of resisting gross errors of the algorithm.The results indicate that the proposed algorithm can consider all kinds of random errors and effectively reduce the influence of gross errors in nonlinear problems.(6)Local test statistics of GCTLS are constructed based on the sensitivity analysis method,and then the data snooping algorithm is further extended to constrained TLS problem.The performance of the algorithm is verified with simulated and real data sets of 3D datum transformation.(7)The least square variance component estimation(LS-VCE)is extended to nonlinear Gauss-Helmert model.The results show that the LS-VCE algorithm based on nonlinear Gauss-Helmert model can consider all kinds of random errors and adjust their relative weight ratios in nonlinear problems,thereby the accuracy of estimated parameters is improved.(8)The variance component estimation method for GTLSP(GTLSP-VCE)is investigated,and the iterative VCE formula is derived.A simulated experiment of 3D similarity transformation with large rotation angles is employed to test the performance of GTLSP-VCE algorithm.The results show that when the priori variance matrix is not unreasonable,GTLSP-VCE not only can improve the accuracy of the transformed coordinates compared with the GTLSP algorithm,but also can provide proper precision information.