###### W. Baarda

**Publications on Geodesy 41, Delft, 1995. 148 pagina's. **** ISBN-13: 978 90 6132 253 5. ISBN-10: 90 6132 253 7.**

## Abstract

1. Introduction.

2. A preliminary consideration of the three-dimensional S-coordinate frame preferred by the author.

3.1 An estimation of the position of the centre of mass of the earth in an S-coordinate frame according to Section 2, with possible consequences for the linking up of mathematical models in geodesy.

3.2 An intermezzo treating by means of quaternions the mathematical formulation of the S-coordinate frame chosen, and some consequences.

3.2.0 Introduction.

3.2.1 The similarity transformation including the gravity potential. A form element for this potential.

3.2.2.1 The formula for the three-dimensional S-coordinate. The S-transformation as a connection onto assumed fixed coordinates.

3.2.2.2 The relation between two- and three-dimensional S-coordinates. An elegant formula for the three-dimensional S-coordinate as a function of three intrinsic quantities.

3.2.3 Application of the law of propagation of variances by means of isomorphic matrices.

3.2.4 The construction of Criterion Matrices. The important theorem stating that sphericity of three-dimensional point- and relative standard ellipsoids is not conserved in an S-transformation, contrary to the corresponding property of circularity in the two-dimensional situation.

4.1 Again the connection between gravimetric and geometric theory, treated sketchily but with concentration on the fundamentals. The linking up of a mathematical model by dimensionless quantities. An objectionable interpretation of compound quantities as "free-air reduction to the geoid". A more appropriate definition of "relative sea-topography".

4.2 An analysis of the modified integral formulas of Stokes and Hotine, based on ideas of Rummel and Teunissen. A choice for the present satellite era. The lasting influence of the transition from sea to land.

4.3 The influence of P_{M}P_{C} ≠ 0, P_{C} being the centre of mass of the earth and P_{M} the origin of a quasi-centric S-coordinate frame for terrestrial data.

4.4 Once more the integral formula of Hotine. Effects of first degree spherical harmonics. A suggestion for application. An afterthought.

5. Supplementary remarks on the linking up of the gravimetric-geometric model. Correction terms in the modified integral formulas of Stokes, Hotine and Vening Meinesz, deviating from the terms found in the literature.

6. Possible consequences of the gravimetric-geometric S-system for (terrestrial) mechanics. The corresponding dimensionless time quantity.

7.1 A sketch of problems in point positioning on the earth by means of satellite observations.

7.2 The effect of P_{M}P_{C} ≠ 0 on launch data of a satellite.

7.3 Correction of DΠ-quantities (orbit data) for earth rotation.

8.1 An alternative way of writing the formulas of the Kepler ellipse for the computation of a satellite orbit. Difference formulas for dimensionless quantities, such as the dimensionless time interval.

8.2.1 The linking up of the mathematical model from section 8.1. The influence of P_{M}P_{C} ≠ 0.

8.2.2 Comparison of the S-system in satellite orbit computation with the S-system in physical geodesy; possible small differences in scale in mass, potential and time.

8.2.3 Questions arising when rewriting the higher-order terms of orbit computations by means of the dimensionless quantities introduced.

9.1 Establishment of control by satellite measurements. The bird's-tail construction. Difference formulas with an appraisal of the influence of P_{M}P_{C} ≠ 0.

9.2 An investigation into possibilities for the estimation of P_{M}P_{C} ≠ 0.

9.3.1 A more realistic process of measurement by means of series of pseudo-distances (distance measurements from one station with the same but unknown length scale).

9.3.2 Synchronous measurements in several stations, with an estimation of the maximum distance between stations if a (practically acceptable) elimination of the influence of orbit errors is to be attained. Measurement of the distance differences.

10. A short after-consideration. The application of dimensionless quantities in satellite gradiometry.

11. Concluding word with remarks concerning relativity theory, a possible influence of the choice of terrestrial datum points on the precision of the determination of points of satellite orbits, doubts about the alleged precision of computed quantities in physical geodesy obtained by satellite gradiometry.

Notes and References.