Research On M Estimation Of The EIV Models And Its Application In Geodetic Data Processing

Gauss-Markov(Gauss-Markov)model is a classical model used in surveying field.However,errors can be found in the design matrix of the Gauss-Markov model in geodetic data processing.The Gauss-Markov model with random design matrix is called an EIV(Errors-in-Variable)model.The EIV model not only has the value of theoretical research,but also receives more and more attention in geodesy,data processing and related fields.In the framework of least squares,the adjustment theory of EIV model has been studied deeply,but there is very little research on theory and application of data processing under the M estimation criteria.Aiming at the theoretical and practical requirements of M estimation for EIV model,the M estimation theory of the Gauss-Markov model is extended up to the EIV model,which is the supplementary and improvement from the theory of data processing and analysis based on M-method.In this paper,the major contributions can be summarized as follows:(1)The data processing theory of Gauss-Markov model and the TLS(Total least square)theory is described systematically.On the basis of existing TLS techniques,new TLS solutions for the general EIV model,the mixed EIV model,and the partial EIV model are derived respectively.(2)For the general EIV model,the mixed EIV model,and the partial EIV model,The expressions and algorithms based on M estimation criteria are derived.It is derived theoretically that,the equivalent weight principle of Gauss-Markov model can be applied to the TLS solution expressions of the EIV model for robust estimation.In order to ensure the success of iteration process,we need to introduce a regulation factor in the equivalent weight matrix to avoid the ill-condition of the relevant matrix.(3)The Bahadur-type linear representation of basic vectors and their variance-covariance matrix are obtained by applying Bahadur linearization techniques into the implicit equation(s),in which the basic vectors are composed of observation vectors,estimators of the design matrix,adjustment vectors,residual vectors and unknown parameters.Further,the asymptotic normality of the parameter estimators under a large sample is obtained.(4)Based on the TLS residuals of the EIV model,the first-order and second-order unbiased estimators for the unit weight variance are derived,and the accuracy of the first-order estimator is discussed.The formulas for robustly estimating the unit weight variance is derived from the M-residuals of the EIV model,and the accuracy of the estimator is discussed when all the errors obey the same distribution.(5)In the case that the error vectors obey P-norm distribution,The redundant parameters and variance-covariance matrixes are discussed under the various M estimation criteria,including Lp estimation criterion,estimation criterion depending on Gauss density,and M estimation criterion independent of probability density.In the case that the error vectors obey the normal distribution,the redundant parameters are discussed.(6)The M estimation theory of the EIV model is applied to 2D and 3D geodetic coordinate transformations and plane fitting of LIDAR point cloud data.The beneficial results are obtained,and the relevant problems in the application are discussed.