A note on numerical solutions of least squares adjustment in GNU project gama

Abstract

Project Gama for adjustment of geodetic networks was started at the department of mapping and cartography, Faculty of Civil Engineering, Czech TU Prague, in 1998. Formerly it was planned to be only a local project with main goal to demonstrate students the power of object programming and at the same time to be a free independent tool for comparison of adjustment results from other sources. The Gama project received the official status of GNU software in 2001 and now contains a C++ library (including small C++ matrix/vector template library gmatvec) and two programs gama-local and gama-g3, that correspond to two development branches of the project. Stable branch of the Gama project is represented by command line program gama-local for adjustment of three-dimensional geodetic networks in a local coordinates system. New development branch of the project (gama-g3) is aimed to the adjustment of geodetic networks in global geocentric system. The stable branch (gama-local) enables common adjustment of possibly correlated horizontal directions and distances, horizontal angles, slope distances and zenith angles, height differences, observed coordinates (used in sequential adjustment, etc.) and observed coordinate differences (vectors). Although such an adjustment model is now obsoleted by global positioning systems, it can still serve as an educational tool for demonstrating adjustment procedures to students and as a starting platform for the development of new branch of the project (gama-g3). Numerical solution of least squares adjustment in geodesy is most commonly based on the solution of normal equations. As the Gama project was also meant to be a comparison tool, it was desirable to use a different method and Singular Value Decomposition (SVD) was implemented as the main numerical algorithm. As an testing alternative Gama implements another algorithm from the family of orthogonal decompositions based on Gram-Schmidt orthogonalization (GSO). Practical experience with both algorithms are discussed. In the Gama project geodetic input data are described in Extensible Markup Language (XML). The primary motivation for usage of XML was to define structured input data for adjustment of local geodetic network. The most important feature of XML is probably the ease of defining a grammar for user data (a class of XML documents) that consequently can be validated even independently of our applications. One of the goals of the Gama project is to build a collection of model geodetic networks described in XML. The lack of reliable testing data was one of major obstacles when testing implementation of numerical solution of the geodetic network adjustment. © 2009 Springer Berlin Heidelberg.