Improvement Of Collocation Methods And Its Applications

Least squares collocation is a mathematical method to study the shape of the earth and the gravitational field.Moritz made a systematic study on the least squares collocation,proposed the least squares collocation with system parameters.Then the application of least square collocation method in geodesy is also summarized by him.The collocation methods is widely used in the analysis of crust deformation and multi-beam bathymetry data processing,its results are good and correctly express actual situation.In this paper,the traditional methods of crust deformation analysis and multi-beam bathymetric data processing are introduced firstly,focus on the analysis of their advantages and disadvantages,considering existing problems in the application of collocation methods,some relevant improvement methods were proposed in this paper.The major contributions of this paper can be summarized as follows:(1)Because of the influences of precision of the instruments,observation environment,measurement workers and so on,the data we obtained contain random errors of different degrees.The elements of the coefficient matrix of trend term and observation vector were consisted of observed data or its functions in analysis of crust deformation problems,they are both contain random errors.Aiming at the issue of observation vector and coefficient matrix both contain errors in the crust deformation problem,total least squares collocation(TLSC)was proposed,errors of observation vector and coefficient matrix were taken as a set of error vector in this method.And then the specific formula for determining parameter and iterative algorithm were given under the generalized total least squares principle.Simulated horizontal crust deformation and the actual data from 2009 L'Aquila Italy earthquake was analyzed by using total least squares collocation and least squares collocation.The experiment result shows that the total least squares collocation and least squares collocation are same in the analysis of crust deformation,and giving theoretical basis in the paper.(2)The rationality of the results obtained by least squares collocation method to analyze crust deformation lies on fitting of the empirical covariance function.The robust least square collocation method based on observation initial median value(RLSC-OIMV)and the robust least square collocation method based on median parameter(RLSC-MP)are proposed when the observation data contains outliers.Firstly,the observation median value is used to obtain the initial weight matrix of observation data and median parameters method is used to obtain the initial solution of least squares collocation in RLSC-OIMV and RLSC-MP respectively.With the initial parameters of covariance function,the equivalent weight scheme is then applied to obtain the robust parameters estimation of covariance function and least squares collocation solutions through iteration.We apply the two proposed methods and the traditional methods to analyze GPS vertical displacement data of LuShan earthquake and the InSAR coseismic displacement data of L'Aquila earthquake in Italy.And the results show that the RLSC-OIMV is better and more robust than other methods.(3)In the process of dealing with multi-beam bathymetry data by least squares collocation,there are these problems that the quadric cruved mathematical model of trend term does not express accurately the whole variation trend of seafloor topography in general and the covariance function estimated by general method is incapable of accurately expressing statistical characteristics with the multi-beam bathymetry data contains gross errors or outliers.So the iterative algorithm of robust least squares collocation is proposed in this paper.Firstly,the initial weight matrix of observations and the initial parameters of covariance function are both given in this method,then the trend term is fitted by polyhedral function and equivalent weights scheme is applied into robust estimation in this method.Finally,the robust parameters of covariance function and solutions of least squares collocation are iteratively calculated.The experimental results show that the method proposed in this paper can express well the whole variation trend of seafloor topography and overcome the effect of gross error or outlier in multi-beam bathymetry data to a certain extent.Compared with the conventional robust method,the proposed method in this paper more effectively probe the outliers in bathymetry data with the robust and better predicted results.