Research On Precision Estimation Method For Total Least Squares

How to improve the theory on nonlinear adjustment based on the development of geodesy is a valuable issue which is worthy of study.The total least squares(TLS)method is a kind of nonlinear adjustment method with considering both errors existing in observation vector and coefficient matrix.Compared with the various parameter estimation algorithms for TLS,the precision estimation of TLS is not given enough attention and waits for further research.On basis of error propagation law of nonlinear function,taking ideas of approximating functional expression and approximating probability distribution of function into consideration,this thesis studies improved methods or new methods for precision estimation of TLS and contrives to obtain more reasonable and accurate information of precision estimation.The detailed researches are as follows:The approximation function method for precision estimation based on the second order derivative is researched.With the help of the Gauss-Helmert(GH)model,the first-order approximate cofactors matrices and cross cofactors matrices for parameters estimates,residuals,observations and adjustments of observations are given for TLS adjustment.Based on the Gauss-Markov(GM)model,the second order Taylor expansions between parameters estimates and residuals,and errors are formulated.According to the error propagation formulae of nonlinear function,the bias calculation formulae for parameters estimates and residuals with more wide application are derived,the second-order covariance matrix and mean squared error matrix for parameters estimates are also given.The sigma point methods for precision estimation are researched,which consist of scaled unscented transformation(SUT)method and the Sterling interpolation method.In order to avoid the complex derivative calculation and deal with the situation where derivatives are not available,the SUT and the Sterling interpolation based on the determined sigma points(sampling points)are integrated into precision estimation of TLS.The whole process of precision estimation is divided into two parts: bias calculation and approximate covariance matrix or mean squared error matrix calculation.Two projects are designed to calculate the corresponding values of sigma points after nonlinear transformation,the project 1 is calculating the values by converting TLS iterative process into nonlinear function,the project 2 is directly calculating the values by TLS iterative process.Examples show that both the SUT and Sterling interpolation methods can yield second-order approximate results of precision estimation.And the two methods adoptingproject 2 have more wide applicability.The SUT method can obtain more slightly accurate results than the Sterling interpolation method while the Sterling interpolation is easier to implement than the SUT method.The adaptive Monte Carlo method for precision estimation is researched.The simulation number of Monte Carlo method generally is chosen subjectively and the result of that also can't be controlled directly.Besides those,the biases of parameters estimates,residuals and the estimate of variance of unit weight are not taken into consideration simultaneously.The adaptive Monte Carlo method is combined with precision estimation of TLS and the tolerance is analyzed and determined.By calculating biases of estimates and approximate covariance matrix of parameters estimates,the process of precision estimation based on adaptive Monte Carlo is given.With the help of the term of antithetic variates,the antithetic and adaptive Monte Carlo algorithm is proposed for biases of parameters estimates.Examples show that the adaptive Monte Carlo can determine the simulation number automatically and obtain stable and reasonable results for precision estimation with keeping the balance between accuracy and computation cost.The antithetic and adaptive Monte Carlo in this paper is better at convergence and computational efficiency for calculating biases of parameters estimates.The approximation function method,sigma point methods,and antithetic and adaptive Monte Carlo are applied for analyses on effects of focal parameters estimates on the Green function matrix in distributed slip inversion.Considering the elements of Green function matrix are the nonlinear functions of focal parameters estimates,the randomness of focal parameters estimates causes that distributed slip inversion becomes the parameter estimation problem of TLS.Effects of the length,width,depth and dip of fault with different variances on the corresponding displacements of unit strike slip dislocation,unit dip slip dislocation and unit tensile dislocation of fault are analyzed by biases of displacements,which can be expected to provide a certain basis for the use of TLS and the weight matrix of Green function matrix.Results of simulated fault show that compared with the other methods the sigma point methods perform better in computational efficiency.The second order term has dominant effect on nonlinear relationship between displacements and fault parameter in rectangular dislocation model.Biases of displacements appear about the area centering on the fault center within 5 kilometers,the main biases of displacements are near to fault.The corresponding displacements of unit tensile dislocation are most susceptible to fault parameters estimates,followed by unit dip slip dislocation and unit strike slip dislocation.In addition,the vertical displacement is more sensitive to fault parameters estimates than the plane displacements.TLS can be considered to treat distributed slip inversion when the bias of displacement approaches millimeter.