Gauss–Jacobi combinatorial adjustment and its modification

Abstract

Many determined nonlinear systems in geodesy are analytical. C. F. Gauss and C. G. J. Jacobi ever developed an adjustment technique, later called as Gauss–Jacobi combinatorial method, to obtain an optimal solution of an over-determined nonlinear system. Nowadays, facing new development in abstract algebra, the potential of these contributions becomes clearer that this method can analytically solve an over-determined nonlinear system only if the determined sub systems are analytical. An important subject, that determines whether this method is rigorous, is to discuss the relation between the combinatorial estimator and the nonlinear least squares estimator. In this paper, we first introduce the Gauss–Jacobi combinatorial method in the context of the Gauss–Newton iterative equation, and illustrate what manifests the accuracy of the Gauss–Jacobi combinatorial estimator. It shows that high nonlinearity, large residuals or ill conditioning can cause the Gauss–Jacobi combinatorial estimator to be inaccurate. Then, employing the high order Taylor approximation we propose a modified combinatorial method to improve the Gauss–Jacobi combinatorial estimator. Finally, in order to show when and why one should employ the high order combinatorial technique proposed, the first order combinatorial method and the second order combinatorial method are both applied to solve over-determined distance equations. It shows that the second order combinatorial estimator is more accurate than the first order combinatorial estimator.