Heft 17/1986

Schriftenreihe
des Instituts für Geodäsie


 
Heft 17/1986

EISSFELLER, Bernd /
HEIN, Günter W.

A Contribution to 3d-Operational Geodesy

Part 4
The Observation Equations of Satellite Geodesy in the Model of Integrated Geodesy

187 S.

Auflage:  800

ISSN:  0173-1009

Inhaltsverzeichnis

Introduction

 


Inhaltsverzeichnis

1.  Introduction 5
 
2.  Orbit Integration 6
     2.1  Equation of motion 6
     2.2  Principle of integration of the equation of motion 6
     2.3  The solution of the homogeneous problem 10
     2.4  The solution of the inhomogeneous or perturbed problem 26
 
3.  The Acceleration Model 41
     3.1  Coordinate reference frames 42
     3.2  Gravity acceleration of the soild earth 45
     3.3  Accelration due to air-drag 47
     3.4  Acceleration due to solar radiation pressure 52
     3.5  Acceleration due to the attraction of sun and moon 53
     3.6  Acceleration due to the tides of the solid earth 54
 
4.  The Observation Equations of Satellite Geodesy 56
     4.1  General form of observation equations 57
     4.2  The vector    of model parameters 59
     4.3  Determination of  δ (t)  by the linearization principle
            of integrated geodesy

60
     4.4  The determination of the necessary Jacobi matrices 63
     4.5  Numerical determination of the derivatives of the
            coordinate vector    with respect to the parameters
            , ... ,


125
     4.6  Numerical integration of the vector  δ 129
     4.7  The position vector of the ground station, its linearized
            form and corresponding time-derivative

130
     4.8  Further parametrization of the vector of earth rotation
            parameters and of its time-derivative

137
     4.9  Final expressions for the linear observation equations
            of satellite measurements

139
 
5.  References 172
 
Acknowledgements 173
 
Appendix A.  Rotation matrices 174
Appendix B.  Canonical transformations 175
Appendix C.  Jacobi matrices, gradients, differentiation rules 180
Appendix D.  Reference system, transformations 185
 

 
Introduction

This work is the fourth part of a series on the development of operational or integrated geodesy. After having published the observation equations for geodetic measurements of terrestrial type (HEIN, 1982a), the concept of a solution (HEIN, 1982b), the first operational software OPERA on the processing of terrestrial data in the integrated adjustment model (HEIN and LANDAU, 1983), we have tried to outline here in detail the observation equations of satellite geodesy including orbit determination in that unified model.
Thereby we refer to the main principle outlined already by
MORITZ (1980, p. 225 to 230) and other developments of the Stuttgart scholl on a slightly different approach to operational geodesy, e.g. (GRAFAREND, 1979; 1981; 1982), in particular (GRAFAREND and LIVIERATOS, 1978; GRAFAREND and HEINZ, 1978) - to mention only some of the publications. The present state-to-the-art in integrated geodesy is summarized in HEIN (1986).

With the detailed study of the satellite observations and the orbit determination in the integrated model we had two things in mind:

(i)  With respect to theory the classical textbook of KAULA (1966) was and is still the fundamental source for all theoretical developments in satellite geodesy. However, the style and approach of how it was presented, seem to separate space geodesy from the other terrestrial parts - at least at the very first sight. Thus, it was our intention to present satellite geodesy in the same context we have discussed the terrestrial measurements. In addition, we tried to fit it into the integrated geodesy adjustment model in order to end up with a consistent approach to geodesy. This, however, can only be the first trial on the way there.

(ii)  The present theory offers also a new computational possibility to orbit determination and the processing of satellite observations. There is no doubt that a realization of the theory into an operational software package still requires a lot of efforts. The interested reader will easily recognize that the determination of appropriate covariance in the general collocation algorithm is the crucial point due to a heavy load of time-consuming calculations for it. However, this does not mean that a numerical realization is impossible. Grid-structured and/or equally-spaced data and subsequent use of Toeplitz matrices can overcome these difficulties.

Some other comments related to the source of the above mentioned difficulties. Although we are publishing this report under the main head line: A Contribution to 3D-Operational Geodesy, it is no longer threedimensional! The consideration of satellite geodesy requires the parameter time, thus it is already a fourdimensional approach.

The reader who is interested in a quick-look how satellite geodesy fits into the integrated model is recommended to start with paragraphs 4.1 to 4.3 with a short look on the structure of the appropriate observation equations in 4.9.

This report can only be a first step on the inclusion of satellite geodesy in an unified approach to geodesy - although we needed much more time to develop it than earlier anticipated. There is also no doubt that in spite of careful typing and proof-reading still some (or more?) errors might be in it. Looking to the many formulas we hope that the reader has some understanding and tries to assist us in a better version.
 



References:

GRAFAREND, Erik W. (1979): Space-time geodesy. In: Bollettino di geodesia e scienze affini, Vol. 38, No. 2, pp. 305-340  -  ISSN:  0006-6710.

GRAFAREND, Erik W. (1981): Die Beobachtungsgleichungen der dreidimensionalen Geodäsie im Geometrie- und Schwereraum. Ein Beitrag zur operationellen Geodäsie. In: Zeitschrift für Vermessungswesen (ZfV), Vol. 106, pp. 411-429  -  ISSN:  0340-4560.

GRAFAREND, Erik W. (1982): Six lectures on geodesy and global geodynamics. In: Mitteilungen der geodätischen Institute der Technischen Universität Graz, Folge 41, pp. 531-672.

GRAFAREND, Erik W. / HEINZ, Karl (1982): Rank defect analysis of satellite geodetic network (II) - Dynamic mode. In: Manuscripta Geodaetica, Vol. 3, pp. 135-156  -  ISSN:  0340-8825.

GRAFAREND, Erik W. / LIVIERATOS, Evangelos (1978): Rank defect analysis of satellite geodetic network (I) - Geometric and semi-dynamic mode. In: Manuscripta Geodaetica, Vol. 3, pp. 107-134  -  ISSN:  0340-8825.

HEIN, Günter W. (1982a): A contribution to 3d-operational geodesy. Part 1: Principle and observational equations of terrestrial type. In: SIGL, Rudolf (Ed.): Proceedings of the International Symposium on Geodetic Networks and Computations of the International Association of Geodesy, Munich, August 31 to Sept. 5, 1981. Vol. 7: Combination of horizontal, vertical and gravity networks. Deutsche Geodätische Kommission, Reihe B, Nr. 258/VII. Verlag der Bayerischen Akademie der Wissenschaften, München, pp. 31-64  -  ISBN:  3-7696-8554-7.

HEIN, Günter W. (1982b): A contribution to 3d-operational geodesy. Part 2: Concepts of solutions. In: SIGL, Rudolf (Ed.): Proceedings of the International Symposium on Geodetic Networks and Computations of the International Association of Geodesy, Munich, August 31 to Sept. 5, 1981. Vol. 7: Combination of horizontal, vertical and gravity networks. Deutsche Geodätische Kommission, Reihe B, Nr. 258/VII. Verlag der Bayerischen Akademie der Wissenschaften, München, pp. 65-85  -  ISBN:  3-7696-8554-7.

HEIN, Günter W. (1986): Integrated Geodesy - State-of-the-Art 1986 Reference Text. In: SÜNKEL, Hans (Ed.): Mathematical and Numerical Techniques in Physical Geodesy. Lectures delivered at the Fourth International Summer School in the Mountains on Mathematical and Numerical Techniques in Physical Geodesy, Admont, Austria, August 25 to September 5, 1986. Lecture Notes in Earth Sciences, Vol. 7. Springer Verlag, Berlin / Heidelberg, pp. 506-548  -  ISSN:  0930-0317.

HEIN, Günter W. / LANDAU, Herbert (1983): A contribution to 3d-operational geodesy. Part 3: OPERA - A multi-purpose program for operational adjustment of geodetic observations of terrestrial type. Deutsche Geodätische Kommission, Reihe B, Nr. 264. Verlag der Bayerischen Akademie der Wissenschaften / Beck in Komm., München, 82 p.  -  ISBN:  3-7696-8557-1.

KAULA, William M. (1966): Theory of Satellite Geodesy. Applications of Satellites to Geodesy. Blaisdell Publ. Co., Waltham / Toronto / London, XIV, 124 p.

MORITZ, Helmut (1980): Advanced Physical Geodesy. Sammlung Wichmann, Neue Folge, Buchreihe 13. Herbert Wichmann Verlag, Karlsruhe, XIII, 500 p.  -  ISBN:  3-87907-106-3.
 


 
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