Research On Parameter Estimation Method For Ill-posed Mixed Additive And Multiplicative Random Error Model

In the geodetic measurement data processing field,traditional surveying adjustment theories and methods are developed on the basis of the measurement random errors as additive random error.However,with the rapid development of modern surveying and mapping technology,the measurement random errors with electromagnetic waves as the carrier show as multiplicative or mixed additive and multiplicative random error,and traditional surveying adjustment theories and methods based on additive random error can no longer meet the data processing needs of new-type random error.At present,there are relatively few research results on the multiplicative and mixed additive and multiplicative random error model in the field of geodesy at home and abroad,and no literatures research the ill-posed problems in them.Thus,how to estimate the parameters of ill-posed multiplicative and mixed additive and multiplicative random model is a worthy topic to research.Based on common solutions of ill-posed problems,this paper proposes some new methods to process the ill-posed multiplicative and mixed additive and multiplicative random error model for obtaining more reasonable parameter estimation and precision information,and the specific research contents are as follows:1)The weighted least squares regularization iterative solution of ill-posed multiplicative random error model is researched.In order to make the parameter estimation and precision information of ill-posed multiplicative random error model more accurate,this paper first derives the weighted least squares regularization iterative solution and covariance matrix of parameter estimation based on the Tikhonov regularization principle.Then,considering that the gradual iterative process will affect the final parameter estimation and precision information when using the weighted least squares regularization iterative solution to calculate,the scaled unscented transformation(SUT)method that can calculate the mean and covariance matrix of nonlinear function by weighting without derivation operation is used for precision estimation of ill-posed multiplicative random error model.Finally,the experiment research and analysis results show that the weighted least squares regularization iterative solution can effectively reduce the ill-posed model and obtain more reasonable parameter estimation and precision information than the existing solutions,while the SUT method can obtain accurate parameter estimation and precision information than the weighted least squares regularization iterative solution,which verifies the validity and reliability of the proposed methods in this paper.2)The virtual observation iterative solution of ill-posed mixed additive and multiplicative random error model is researched.In order to make the parameter estimation and precision information of ill-posed mixed additive and multiplicative random error model more accurate,this paper derives the virtual observation iterative solution of ill-posed mixed additive and multiplicative random error model by combining the observation equation of mixed additive and multiplicative random error model with the virtual observation equation based on the virtual observation principle,and further proposes the A-optimal design method to determine the regularization parameter of virtual observation iterative solution based on the A-optimal design principle.Finally,the experiment research and analysis results show that the A-optimal design method can determine reasonable regularization parameter,and the virtual observation iterative solution can effectively reduce the ill-posed model and obtain more accurate parameter estimation and precision information than the existing solutions,which verifies the validity and reliability of the proposed methods in this paper.3)The ridge estimation iterative solution of ill-posed mixed additive and multiplicative random error model with equality constraints is researched.Aiming at the existing solutions of mixed additive and multiplicative random error model solution without considering the prior equality constraints information when adjusting,this paper first introduces the equality constraints condition on the basis of mixed additive and multiplicative random error model,establishes the mixed additive and multiplicative random error model with equality constraints and further derives the weighted least squares iterative solution and its covariance matrix based on the Lagrangian extremum principle.Then,considering the ill-posed condition of coefficient matrix,the ridge estimation iterative solution of ill-posed mixed additive and multiplicative random error model with equality constraints is proposed based on the ridge estimation principle,and the U curve method to determine the ridge parameter is derived.Finally,the experiment research and analysis results show that the weighted least squares iterative solution can obtain more reasonable parameter estimation and precision information than the existing solutions,which has strong applicability,while the ridge estimation iterative solution that considers the ill-posed coefficient matrix can obtain more accurate parameter estimation and precision information,which verifies the validity and reliability of the proposed methods in this paper.4)The exhaustive searching method for ill-posed mixed additive and multiplicative random error model with inequality constraints is researched.Considering that the parameter estimation problem for mixed additive and multiplicative random error model with inequality constraints has not been researched,this paper first establishes the mixed additive and multiplicative random error model with inequality constraints based on the inequality constraints equation and mixed additive and multiplicative random error model,and the specific parameter estimation algorithm steps is given by applying the ideas of active constraint,inactive constraint and exhaustive searching method.Then,aiming at the ill-posed condition of coefficient matrix,the weighted least squares regularization iterative solution for ill-posed mixed additive and multiplicative random error model is first derived,and further based on the idea of exhaustive searching method,the specific parameter estimation algorithm steps for ill-posed mixed additive and multiplicative random error model with inequality constraints is given.Finally,the experiment research and analysis results show that when the coefficient matrix is non-ill-posed,the mixed additive and multiplicative random error model with inequality constraints algorithm can obtain more accurate parameter estimation than the existing solutions.And when the coefficient matrix is ill-posed,the weighted least squares regularization iterative solution can obtain more reasonable parameter estimation than the existing solutions,while using the ill-posed mixed additive and multiplicative random error model with inequality constraints algorithm can obtain more accurate parameter estimation than the weighted least squares regularization iterative solution,which verifies the validity and reliability of the proposed methods in this paper.