Combination of geodetic networks
The rigorous combination of terrestrial satellite geodetic networks is not easily accomplished. There are many factors to be considered. The more important are how to deal with terrestrial networks that are separated into horizontal and vertical components which are not usually coincident; the relation of each component to a different datum; and the existence of unmodeled systematic errors in terrestrial observables. Satellite networks are inherently three-dimensional and are relatively free of systematic errors. In view of these facts, and with present practical considerations in mind, fourteen alternate mathematical models for the combination of terrestrial and satellite geodetic networks are investigated, catalogued and categorized in this report. To understand the reasoning behind the formulation of the models presented and the interpretation of the results obtained, some basic definitions and properties of datums, and satellite and terrestrial networks are presented. Based on previous investigation and the authors interpretation of the problem of combining geodetic networks, the models under study are split into two major groups. The first group treats datum transformation parameters as known. While the second includes them as unknowns to be estimated in the combination procedure. Each model is investigated in terms of its dimensionality, unknown parameters to be estimated, observables, and the estimation procedure utilized. The group of three-dimensional models that treat the datum transformation parameters as unknowns to be estimated are themselves separated into two parts. The Bursa, Molodensky, and Veis models contain only one set of the rotation parameters each, while the Hotline, Krakisky-Thomson, and Vanicek-Wells models each contain two sets of unknown rotations. For the combination of terrestrial and satellite networks, the latter three models represent physical reality. The models that are not three-dimensional do not take advantage of the inherent tri-dimensionality of satellite networks. Thus, when the satellite network data is split into horizontal and vertical components for combination with terrestrial data, the covariance between the components is omitted. Even though the use of two and one-dimensional combination models are required at present due to the sparseness of adequate terrestrial data and the need for the solution of practical problems, it is not recommended for the future. The Bursa model is recommended for the combination of two or more satellite networks. However, when combining terrestrial and satellite networks, when datum transformation parameters are unknown, none of the Bursa, Molodensky, or Vies models are adequate. In this case, the Hotline or Krakiwsky-Thomson model which parametrize the lower order systematic errors in the terrestrial network, should be used. The combination of the Krakiwsky-Thomson and Vanicek –Wells models is seen to be the best, from a theoretical point of view, for the combination of a satellite and several terrestrial networks. Such a solution will yield the datum transformation parameters between each of the datums involved, the orientation of each datum with respect to the Average Terrestrial coordinate system, and parameters representing the overall systematic orientation and scale errors of each terrestrial network. No substantiative conclusions could be given based on the numerical testing carried out. A sparseness of adequate data prevented this. The numerical testing has not been wasted, however. The type and quality of data required for several models has been demonstrated. Further, the available data was utilized to substantiate the fact that the proposed solution of the Krakiwsky-Thomson model is possible.