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Elementary Mathematical Models for GNSS Positioning

  • Bernhard Hofmann-WellenhofEmail author
Living reference work entry
Part of the Springer Reference Naturwissenschaften book series (SRN)

Zusammenfassung

Der Autor bekam im Jahr 1984 die Gelegenheit, am US National Geodetic Survey nahe Washington (District of Columbia) einen wissenschaftlichen Aufenthalt zu verbringen. Unter der Betreuung von Dr. Benjamin Remondi konnte er an der Entwicklung einer Software für zivile Anwendungen für die Datenauswertung des primär militärischen Global Positioning System (GPS), der amerikanischen Version eines globalen Navigationssatellitensystems (GNSS) mitwirken. Parallel zu den Vereinigten Staaten von Amerika entwickelte die ehemalige Sowjetunion ihr eigenes militärisches globales Navigationssatellitensystem (GLONASS). In diesem frühen Entwicklungsstadium waren die beiden Systeme noch im Aufbau. Die sogenannte Full Operational Capability (FOC) wurde für GPS in der zweiten Jahreshälfte 1995 und für GLONASS wenig später in der ersten Jahreshälfte 1996 erreicht.

In den Anfangsjahren des neuen Milleniums wurden neue Systeme entweder konzipiert oder es wurde bereits mit deren Umsetzung begonnen. Beispiele sind das chinesische BeiDou System (in der ersten Ausbaustufe ein regionales und erst in der zweiten Ausbaustufe ein globales System) und das europäische Galileo, das als besonders bemerkenswert anzusehen ist, da es sich nicht um ein militärisches System, sondern um ein System unter ziviler Kontrolle handelt. Die Definitionsphase von Galileo wurde im Jahr 2003 abgeschlossen.

Zusammen mit den Koautoren Herbert Lichtenegger und Elmar Wasle hatte der Autor im Jahr 2001 im Springer Verlag Wien New York die fünfte Auflage von ,,GPS – Theory and Practice“ publiziert. Diese neuen Entwicklungen machten eine vollständige Überarbeitung notwendig, die im Jahr 2008 unter dem Titel ,,GNSS – GPS, GLONASS, Galileo & more“ wiederum im Springer Verlag Wien New York erschien. Nun beginnt sich der Kreis zu schließen. Als der Autor von den Herausgebern des Springer Handbook of Mathematical Geodesy die Einladung bekam, einen Beitrag ,,Mathematical Models for GNSS Positioning“ zu schreiben, gab er zu bedenken, dass er selbst einerseits dieses Thema bereits vor einigen Jahren publiziert hätte und dass andererseits die im wesentlichen gleichen Themen mit noch viel mehr zusätzlichen Themengebieten im gerade erst veröffentlichten Springer Handbook of Global Navigation Satellite Systems erschienen seien. Die Herausgeber des vorliegenden Handbuchs aber beharrten darauf, ,,dieses Thema ist genau das, was wir wollen und benötigen“. Der Autor fühlte sich sehr geehrt und kontaktierte den Springer Verlag mit der Frage, ob es möglich wäre und erlaubt sei, Teile des GNSS-Buchs im vorliegenden Springer Handbook of Mathematical Geodesy in aktualisierter Form und ergänzt durch Messbeispiele wieder zu veröffentlichen. Der Springer Verlag sprach umgehend seine Zustimmung und Erlaubnis aus. Folglich ist der vorliegende Beitrag ein Auszug aus den Kapiteln 5, 6 und 7 des Buchs ,,GNSS – GPS, GLONASS, Galileo & more“. Da diese Kapitel in der Originalversion nur vom Autor erstellt wurden und die Koautoren dazu nicht beigetragen hatten, scheinen die Namen der Koautoren des GNSS-Buchs in dieser Publikation nicht auf.

Da nahezu eine Dekade seit der Veröffentlichung des GNSS-Buchs verstrichen ist, liegt viel mehr Erfahrung für die Anwendung der mathematischen Modelle vor. Somit kann der Autor die Leserinnen und Leser mit numerischen Beispielen von im Feld gemessenen Daten verwöhnen. Diese Beispiele sollten die Wiederveröffentlichung der bereits etablierten mathematischen Modelle rechtfertigen. Die Messungen, das Auswerten der Daten und die Generierung der zugehörigen Figuren wurden von Mathias Duregger, einem jungen und sehr talentierten Studienassistenten an unserem Institut für Geodäsie der Technischen Universität Graz durchgeführt. Für seinen Fleiß, seine Gründlichkeit und seine kontinuierliche Unterstützung bedanke ich mich sehr herzlich.

Keywords

GNSS Satellite-based positioning GNSS positioning Observables Data combination Point positioning Differential positioning Relative positioning Ambiguity resolution Linearization 

Abstract

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”, chapters 5, 6, and 7. 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.

Notes

Acknowledgements

With kind permission of the Springer Publishing Company, this contribution to the Handbook of Mathematical Geodesy is basically a part taken over from the book “GNSS – GPS, GLONASS, Galileo & more” published in 2008 and written by Herbert Lichtenegger, Elmar Wasle, and myself. The republication in this Handbook offered the chance to include the experience obtained in the years having passed since then by using the mathematical models. Therefore, I asked Mathias Duregger, student assistant at the Graz University of Technology, Institute of Geodesy, to carry out measurement campaigns, to process the data, and to prepare representative figures. These measurement examples are illustrative supplements to the pure theory. The ambitious support by Mathias and his independent and extremely reliable work are gratefully acknowledged.

I would also like to thank Katrin Huber, James J. Spilker and Guenter Hein who gave the permission to publish the contents of figures they had used in one of their publications or presentations. These figures themselves have been prepared again by Mathias Duregger.

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Copyright information

© Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Working Group Navigation, Institute of GeodesyGraz University of TechnologyGrazAustria

Section editors and affiliations

  • Willi Freeden
    • 1
  1. 1.Geomathematics GroupUniversity of KaiserslauternKaiserslauternGermany

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