###### H. van der Marel

**Publications on Geodesy 31, Delft, 1988. 135 pagina's. **** ISBN-13: 978 90 6132 237 5. ISBN-10: 90 6132 237 5.**

## Abstract

In this thesis several aspects of the scientific data reduction for the astronomical satellite Hipparcos are discussed. The Faculty of Geodesy of the -Delft University of Technology participates in the data reduction in the framework of the international FAST consortium. Hipparcos (an acronym for HIgh Precision PARallax Collecting Satellite) is scheduled for launch in the spring of 1989 under supervision of the European Space Agency (ESA). During its operational life time of 2.5 years the satellite will scan the celestial sky in a slowly precessing motion and measure the angles between stars which 0 are 60 apart. The observations will be done in the visible part of the electromagnetic spectrum. The Hipparcos data reduction aims at the construction of a precise star catalogue: The catalogue will contain the position, annual proper motion and annual parallax of about 110,000 stars, up to visual magnitude 12-13. The accuracy will be a few milliarcseconds and a few milliarcseconds per year respectively.

Besides a short introduction of the Hipparcos mission, the scientific objectives and the measurement principle, and a brief analysis of the data reduction as a whole, three topics are discussed in this thesis:

- model assumptions, estimability and accuracy of the great circle reduction,
- attitude smoothing, which improves the results of the great circle reduction,
- the numerical methods for the great circle reduction.

These subjects all concern one phase of the data reduction: the so-called great circle reduction. The great circle reduction comprises a half-daily least squares solution of some 80,000 observations with 2,000 unknown star abscissae and some 50 instrumental parameters. Depending on the solution method chosen, also some 18,000, or in case of attitude smoothing 600, attitude parameters have to be solved. The great circle reduction is a relatively modest adjustment problem in the complete data reduction, but one which must be solved several times per day over a period of several years.

The first four chapters are of an introductory nature. In chapter 2, which is more or less self contained, the scientific objectives and possible -geodetic- applications of the Hipparcos catalogue are sketched. In chapter 3 the Hipparcos measurement principle and raw data treatment are described and in chapter 4 a start is made with the description of the data reduction. It is in this chapter that the great circle reduction, the main subject of this thesis, is introduced and placed within the total data reduction.

The model assumptions, estimability and accuracy of the great circle reduction results are investigated in chapter 5. The great gircle reduction processes only observations of stars within a small band (2^{0}) on the celestial sphere. Therefore, only one coordinate can be improved, viz. the abscissa on a reference great circle chosen somewhere in the middle of the band. The ordinates are not improved, i.e. they are fixed on their approximate values, which results in errors in the estimated star abscissae. By iterating the complete data reduction several times, in order to obtain better approximate values for the ordinates, the modelling error finally becomes very small and can be neglected. In chapter 5 analytical formulae for the magnitude of this error are derived. Further we investigate, one by one, the estimability of the instrumental parameters. They appear generally to be

estimable. At the end of this chapter the covariance function of the star abscissae is computed for a regular star network using Fourier analysis. Throughout this chapter analytical results are compared with test computations on simulated data.

Chapter 6 is devoted to attitude smoothing. Smoothing of the attitude improves not only the quality of the attitude parameters, but also the quality of the star abscissae. We will consider in particular numerical smoothing with B-splines; the attitude is modelled by a series expansion using the above mentioned B-splines as base functions. The number of attitude parameters is reduced considerably; instead of the 18,000 geometric attitude parameters now only 600 are needed. But if the degree of smoothing is too high, systematic errors are introduced. The number of parameters have been chosen in such a way that the extra error introduced by smoothing is negligible.

Chapters 7 and 8 deal with the numerical methods for solving the sparse systems of quations which arise during the great circle reduction. Choleski factorization of the normal equations has been chosen as solution method. Optimization of the calculations is worthwhile, since such a system has to be solved several times per day. Computing time and memory requirements depend on the order in which the unknowns are eliminated. The best order appears to be: first the attitude unknowns, then the star unknowns and finally the instrumental unknowns. However, in the case of attitude smoothing it is better to eliminate the star unknowns first, and then the attitude and

instrumental unknowns. Also the order in which the star parameters are eliminated, or in the case of attitude smoothing the attitude parameters, is important. Therefore, in chapter 8 several reordering procedures are evaluated. It turns out that the so-called banker's algorithm, which operates on the graph of the system, gives the best results in both cases. But also a synthetic ordering, which orders the star abscissae modulo 60^{0}, gives good results. The same algorithm, but then modulo 360^{0}, can be applied to the attitude unknowns for smoothing.

Finally, in chapter 9, methods are given for handling certain ambiguities in the data. Although the Hipparcos instrument is able to measure phases very accurately, the integer number of periods must follow from approximate data. This results in a large number of so-called grid step errors of about 1" (100 times the precision of measurement). These errors must be detected and corrected during the great circle reduction. Some strategies are discussed in chapter 9. The most successful strategy is based on an approximate sequential adjustment, which can be applied before and after the least squares adjustment .

In the appendices descriptions are given of the FAST great circle reduction software (appendix A) and of the simulated data used in simulation experiments with the great circle reduction software (appendix B). The results of these simulation experiments are used throughout this thesis for

illustration. Finally, appendix C contains some background material on the numerical methods for solving large sparse systems of linear equations having a positive definite matrix.

## Contents

- Preface iii
- Abstract v
- Acknowledgements vii
- Abbreviations viii

- Introduction 1
- Scientific objectives of the Hipparcos mission 3
- Hipparcos measurement principle 17
- Geometric aspects of the Hipparcos data reduction 27
- Great circle reduction 49
- Attitude smoothing 85
- Numerical techniques for the great circle reduction 115
- Ordering of the unknowns during the great circle reduction 143
- Grid step ambiguity handling 171

Appendices:

- Delft great circle reduction software 185
- Simulated data for the great circle reduction 193
- Computer solution of least squares problems 199

- References 221