GENERALIZED INVERSES OF NONLINEAR MAPPINGS AND THE NONLINEAR GEODETICDATUM PROBLEM

Motivated by the existing theory of the geometric characteristics of l inear generalized inverses of linear mappings, an attempt is made to e stablish a corresponding mathematical theory for nonlinear generalized inverses of nonlinear mappings in finite-dimensional spaces. The theo ry relies on the concept of fiberings consisting of disjoint manifolds (fibers) in which the domain and range spaces of the mappings are par titioned. Fiberings replace the quotient spaces generated by some char acteristic subspaces in the linear case. In addition to the simple gen eralized inverse, the minimum-distance and the x(0)-nearest generalize d inverse are introduced and characterized, in analogy with the least- squares and the minimum-norm generalized inverses of the linear case. The theory is specialized to the geodetic mapping from network coordin ates to observables and the nonlinear transformations (Baarda's S-tran sformations) between different solutions are defined with the help of transformation parameters obtained from the solution of nonlinear equa tions. In particular, the transformations from any solution to an re-n earest solution (corresponding to Meissl's inner solution) are given f or two-and three-dimensional networks for both the similarity and the rigid transformation case. Finally the nonlinear theory is specialized to the linear case with the help of the singular-value decomposition and algebraic expressions with specific geometric meaning are given fo r all possible types of generalized inverses.