IGN Method and Matrices Equivalence of Regularization Jacobian Matrix in Solving NLS Survey Adjustment Problems

Abstract

NLS methods may have ill-posed problems in many actual survey adjustment applications. With the increasing applications of nonlinear least squares (NLS) problems of surveying adjustment, it is very important to understand the ill-posed attribute of NLS problems and find ways to solve them. Based on regularization theory, this paper presents a new stable function to help obtain stable results of the NLS problems. An improved Gauss-Newton (IGN) method incorporating the developed stable functions is implemented with the capability of solving the ill-posed problem of survey adjustment applications. Matrices equivalence between regularization Jacobian matrices in solving NLS ill-posed problems was proposed to this paper. Experiments using simulated data and actual survey data on an old bridge which need to be repaired proved the good performances of the developed method, which can also be used in many other NLS applications such as bundle adjustment problems with photogrammetry.