Optimization of Geodetic Networks

Abstract

Optimization of geodetic networks is classified in different orders. (1) Zero-order design solves the geodetic datum problem of how to transform geodetic measurements into absolute Cartesian coordinates. (2) First-order and (3) second-order design are closely related to find an optimaI configuration and an optimal observational weight within a geodetic net. (4) As a third-order design, the problem of introducing new observations into an existing geodetic net referred to as geodetic Bayesian estimation is discussed. In the Introduction, the general Gauss-Markov model, BLUE, least squares and BLIMBE are reviewed. By experimental design, a geodetic experiment is defined. For geodetic loss functions of cost, time, material, on the one hand, and of the dispersion matrix of the net coordinates on the other, five different classes are introduced: invariants for point transformations, eigen values and condition numbers that have the property to produce a homogeneous and isotropic geodetic net, invariants of derived quantities, maximal distribution functions, and norms of the dispersion matrix. The interpretation of a geodetic net in terms of stochastic processes is reviewed, especially the Taylor-Karman decomposition (1) is solved by the Moore-Penrose inverse, the Tykhonov inverse and the bias problem. (2) gives a review of critical configuration designs, three examples for simulation and nonlinear programming of type A designs (scalar risk functions). Although the configuration design is strong nonlinear, many problems in observational weight design (3) are linear. For type B designs (given the explicit structure of the dispersion matrix of net coordinates) three-dimensional examples solved by simplex algorithms are presented. Literature on the Schreiber Theorem of optimal weight distribution in a geodetic net is collected. Finally in (4) the geodetic Bayesian approach is formulated.