Heft 17

Schriftenreihe des Studiengangs Geodäsie und Geoinformation
der Universität der Bundeswehr München



  • Contents
  • Introduction
  • Bestellung



Heft 17

A Contribution to 3D-Operational Geodesy
Part 4:
The Observation Equations of Satellite Geodesy in the Model of Integrated Geodesy

Autoren: B. Eissfeller und G. W. Hein

Universität der Bundeswehr München, Neubiberg, 1986
190 Seiten







Orbit Integration

Equation of Motion

Principle of Integration of the Equation of Motion

  • The Homogeneous Problem
  • The Inhomogeneous or Perturbed Problem
    (Method of Variation of Constants)

The Solution of the Homogeneous Problem

  • Kepler Orbital Elements
  • Poincaré Orbital Elements
  • Modified Poincaré Orbital Elements

The Solution of the Inhomogeneous or Perturbed Problem

  • Transformation of the Inhomogeneous Problem into Modified Poincaré Orbital Elements
  • Determination of the Jacobi Matrix of Kepler Elements and Modified Poincaré Orbital Elements
  • Determination of the Jacobi Matrix X
  • Determination of the Jacobi Matrix X'
  • Determination of the Inverse Y
  • The Solution by the Method of Successive Approximations



The Acceleration Model

Coordinate Reference Frames

  • The Inertial Reference System
  • The Earth-Fixed Reference System
  • Transformations

Gravity Acceleration of the Solid Earth

Acceleration Due to Air-Drag

  • Velocity of the Satellite Relative to the Atmosphere of the Earth
  • The Modified Harris-Priester Model Atmosphere

Acceleration Due to Solar Radiation Pressure

Acceleration Due to the Attraction of Sun and Moon

Acceleration Due to the Tides of the Solid Earth



The Observation Equations of Satellite Geodesy

General Form of Observation Equations

The Vector p of Model Parameters

Determination of dv(t) by the Linearization Principle of Integrated Geodesy

The Determination of the Necessary Jacobi Matrices

Numerical Determination of the Derivatives of the Coordinate Vector v with Respect to the Parameters p1, ... , p4

Numerical Integration of the Vector dg

The Position Vector of the Ground Station, its Linearized Form and Corresponding Time-Derivative

Further Parametrization of the Vector of Earth Rotation Parameters and of its Time-Derivative

Final Expressions for the Linear Observation Equations of Satellite Measurements

  • Type S 1:
    Direction Measurements
  • Type S 2:
    Distance Measurements from a Terrestrial Ground Station to a Satellite
  • Type S 3.1:
    Doppler Frequency Shift (Range-Rate)
  • Type S 3.2:
    Doppler Count
  • Type S 4.1:
    Satellite-to-Satellite Tracking (Intersatellite Laser Distances)
  • Type S 4.2:
    Satellite-to-Satellite Tracking (Range-Rate)
  • Type S 5.1:
    Interferometric Time Delays
  • Type S 5.2:
    Diefferenced Interferometric Time Delays (Doppler Differences)
  • Type S 6:
    Altimeter Measurements








Appendix A - Rotation Matrices

Appendix B - Canical Transformations

Appendix C - Jacobi Matrices, gradients, Differentiations Rules

Appendix D - Reference Systems, Transformations




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 school 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-of-the-art in integrated geodesy is summarized in Hein (1986).

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

  • 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.
  • 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 covariances in the general collocation algorithm is the crucial point due to a heavy load of time-consuming calculations for its. 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 considerartion 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 paragraph 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.