Estimation of Accuracy for Solving Incorrect Geodetic Problems

Abstract

SUMMARY The most reliable way for calculating linear equations by the method of least squares, which can be used for solving incorrect geodetic problems, is based on matrix factorization. This method is called singular expansion. Other methods require less machine time and memory. But they are less effective in accounting for input errors, round-off errors, and linear dependence. The methodology of such scientific research is that for any matrix A and any two orthogonal matrices U and V there is a matrix Σ connected by the formula. The idea of the singular decomposition is that properly chosen matrices U and V turn most of the matrix Σ entries into zeros and make this matrix diagonal with non-negative entries. The novelty and relevance of scientific solutions lie in the expediency of applying the singular decomposition of the matrix when obtaining linear equations of the least squares method, which can be used to solve incorrect geodetic problems. The goal of scientific research is to obtain a stable solution of parametric equations of corrections to measurement results in incorrect geodetic problems and its accuracy estimation. The method of solving normal equations by successive elimination of unknown variables (Gaussian elimination) is quite common in geodesy but does not provide stable solutions for ill-conditioned or incorrect geodetic problems. Therefore, in the case of unstable systems of equations, it is proposed to use the method of singular decomposition of the matrix. In computational mathematics this method is called SVD. The SVD singular decomposition method makes it possible to obtain stable solutions for both stable and inherently unstable problems. Such an opportunity to solve precisely incorrect geodetic problems is connected with the use of some limit τ, which can be selected based on the relative errors of the matrix of coefficients of the parametric correction equations and the vector of geodetic measurement results. Moreover, the solution of the system of normal equations obtained by the SVD method will have the smallest magnitude. Thus, applying the singular decomposition of the matrix of coefficients of parametric equations of corrections to the results of geodetic measurements, we obtained new formulas for estimating the accuracy of the least squares method when solving incorrect geodetic problems. The derived formulas have a compact form and make it possible to calculate elements and accuracy estimates quite easily, practically neglecting the complex procedure of finding the inverse of the matrix of coefficients of normal equations.