Research On Adjustment Models And Algorithms With Uncertainty

In the measurement data processing,uncertainty often exists in the coefficient matrix and observation vector of the adjustment model.The improved adjustment method for weakening the impact of uncertainty factors is one of the focused research projects in the field of surveying and mapping.If the uncertainty data is estimated by least squares(LS)or total least squares(TLS)when it is a fuzzy value or an interval values,the reliability of parameter estimation may be decreasing.An adjustment model is established for parameter estimation in which uncertainty is incorporated into the function model as a parameter.And an adjustment criterion(i.e.min-max adjustment criterion with uncertainty)is applied to the adjustment model in which the maximum possible uncertainty is minimized.It is an efficient solution for uncertainty data processing.Therefore,this contribution has further study the adjustment models and criteria and algorithms and applications based on 2-norm and Frobenius-norm(F-norm)with(partial)uncertainty by making using of the theories and methods for matrix analysis and uncertainty and adjustment criterion,which is based on the surveying data processing and the summarizing the research situation at home and abroad.And the main works and outcomes are summarized as follows:(1)According to the fact that the existing algorithms for adjustment model based on 2-norm with uncertainty is complex and less-efficient,this paper presents a directly iterative algorithm without applying singular value decomposition to calculate the model.The iterative procedures are also designed.And the algorithm is simple in the concept and easy in the implementation.The results of the experiments(random errors and systematic errors)of the unary linear fitting illustrate that the proposed algorithm could be practiced.And it has faster convergence,higher calculation efficiency,and better stability.(2)The remaining calculation formulas of the algorithm of QR decomposition-singular value decomposition(SVD)-solving equations also are derived for adjustment model based on 2-norm with part uncertainty.A directly iterative algorithm is researched.Its calculation formulas are derived.And its iterative processes are also presented.The algorithm is simple in the solution process and easy to implement without applying QR decomposition and SVD to calculate the model.The results of a two-dimensional coordinate transformation(random errors and systematic errors)and a settlement observation in foundation pit verify that the algorithm is correct,and has the advantages of faster convergence and higher calculation efficiency and better stability.(3)When the iterative algorithm is divergent,the calculation formulas of SVD-solving equations algorithm are derived for adjustment model based on F-norm with uncertainty.Another directly iterative algorithm is also researched when the iteration is convergence.The unknown parameter estimation is derived from the iterative algorithm.And the iterative procedures are also designed.The correctness and equivalence of two algorithms are demonstrated by the experiment(random errors and systematic errors and outlier)of the binary linear fitting.While the calculation efficiency of the SVD-solving equations algorithm is lower,the directly iterative algorithm is higher.Furthermore,the adjustment model is applied to the surface subsidence prediction.And the validity is proved by the results of the prediction.(4)An adjustment model and its criterion and its three algorithms are given based on F-norm with partial uncertainty when some columns of coefficient matrix are constant elements.And the calculation formulas of three algorithms are derived respectively.The equivalence and validity of three algorithms are proved through an example(random errors)of two-dimensional coordinate transformation.Their appropriateness and calculation efficiency are also compared.First,the QR decomposition-SVD-solving equations algorithm can be applied to the case of the iteration divergence,but the efficiency is lowest.Second,the SVD-iterative algorithm can be applied to the condition that coefficient matrix is nonsingular and iteration is convergent,and the efficiency take second place.Third,the calculation efficiency of directly iterative algorithm is highest in case of iteration convergence.The outcome is better when the adjustment model is applied to the plane fitting of point clouds from 3D laser scanning.(5)Taking into consideration independent and unequally weighted coefficient matrix and observation vector,the weighted adjustment models and its criterion based on 2-norm and F-norm with uncertainty are presented by adopting a weighted method from the theories for the relation between the vectorization of matrix product and Kronecker product,propagation of cofactor and Cholesky decomposition.Two directly iterative weighted algorithms are respectively designed for the two weighted adjustment models.Their calculation formulas are derived.And their calculated steps are also given.The validity of two adjustment methods is proved by the example of the unary linear fitting.