21. Elementary Mathematical Models for GNSS Positioning

In 1984, the author got the opportunity to visit the US National Geodetic Survey near Washington, D.C., and, guided by Dr. Benjamin W. Remondi, could contribute to the development of civilian software for processing data of the primarily military Global Positioning System (GPS), the US version of a Global Navigation Satellite System (GNSS). In parallel, the former Soviet Union developed its own military Global Navigation Satellite System (GLONASS). In these early days of development, the two systems were far from completion which occurred for GPS in late 1995 and for GLONASS in early 1996.

After his return, the author was not only impressed by the incredibly innovative potential of satellite-based navigation, but also decided to “simply write a book on GPS”. The reader might ask why GLONASS was not an issue; the answer is that in those days it was difficult to get official information on the Russian system. Thus, the intention was realized and “GPS – theory and practice”, coauthored by Herbert Lichtenegger and Jim Collins, was published by Springer, Wien New York in 1992. Nine years later, by 2001, the fifth edition was released.

In the early years of the new millenium, additional systems were either concepted or realized, e.g., the Chinese BeiDou system as a two-step approach with the first step as a regional and the second step as a global system and the European Galileo, a remarkable development of a system under civilian control. The Galileo definition phase was completed in 2003. These new developments made necessary a complete revision of the “GPS-book”; consequently, the author together with the coauthors Herbert Lichtenegger and Elmar Wasle published “GNSS – GPS, GLONASS, Galileo & more” again by Springer Wien New York in 2008.

Now the story approaches its end; when the author got the invitation to contribute to the Springer Handbook of Mathematical Geodesy the mathematical models for GNSS positioning, he argued that on the one hand that he had published this issue several years ago and on the other hand that essentially the same topics and much more is contained in the just recently released Springer Handbook of Global Navigation Satellite Systems. Nevertheless, the editors of this handbook claimed that “this is just what we want and need”. Therefore, the author felt honored and contacted the Publishing Company Springer and asked if it is possible to republish parts of his book on GNSS in this Springer Handbook of Mathematical Geodesy volume mainly unchanged but updated where requested and supplemented by measurement examples – and he got the permission. Thus, essentially, this contribution is extracted from “GNSS - GPS, GLONASS, Galileo & more”. Due to the fact that these chapters were originally written by the author only and the coauthors did not contribute to them, the names of the coauthors have been omitted for the current publication.

Since almost a decade has passed by with respect to the release of the GNSS book, much more experience has been gained by applying these mathematical models. This enables the author to spoil the readers with numerical examples for some of the models. These examples should justify the re-publication of the well-established models. The measurements, the processing of the data and the production of the respective figures were carried out by Mathias Duregger, a young and very talented student assistant at our institute at the Graz University of Technology. His very industrious support is gratefully acknowledged.