Resolution of a Rank-Deficient Adjustment Model Via an Isomorphic Geometrical Setup with Tensor Structure.

Abstract

Abstract : This study develops the rank deficient adjustment theory in a geometrical manner. In accordance with most least-squares (L.S.) applications, the adjustment model is considered linear or linearized. The fundamental building blocks consist of orthonormal vectors spanning the spaces and surfaces linked to the L.S. setup. From this setup to the desired results including the variance-covariance matrices, the standard adjustment quantities can be represented by first and second order tensors. It is thus possible to express them in terms of the components of the above vectors, allowing for an easy and clearcut geometrical interpretation of the L.S. process. By virtue of such a vectorization, the propagation of the contravariant and covariant metric tensors is shown to fit perfectly the variance covariance propagation law and even to establish a weight propagation law. The opportunity to obtain variance-covariances and weights as a coherent part of the geometrical development provides the motivation for using tensor structure in the analysis of various L.S. methods and their properties. An algorithm furnishing the pseudoinverse of a positive semi-definite matrix, which could be useful for its straightforward geometrical interpretation as well as for its computational efficiency, is developed as a by-product of this analysis. The Choleski algorithm for positive-definite as well as positive semi-definite matrices is interpreted in terms of orthonormal vector components.