# Critical Configurations of Fundamental Range Networks

- Soren W. Henriksen,
- Armando Mancini and
- Bernard H. Chovitz

Published Online: 15 MAR 2013

DOI: 10.1029/GM015p0001

Copyright © 1972 the American Geophysical Union.

Book Title

## The Use of Artificial Satellites for Geodesy

Additional Information

#### How to Cite

Blaha, G. (1972) Critical Configurations of Fundamental Range Networks, in The Use of Artificial Satellites for Geodesy (eds S. W. Henriksen, A. Mancini and B. H. Chovitz), American Geophysical Union, Washington, D. C.. doi: 10.1029/GM015p0001

#### Publication History

- Published Online: 15 MAR 2013
- Published Print: 1 JAN 1972

#### Book Series:

#### ISBN Information

Print ISBN: 9780875900155

Online ISBN: 9781118663646

- Summary
- Chapter
- References

### Keywords:

- Geometric geodesy;
- Ground stations;
- Nonsingular network;
- Quads and notations;
- Range networks

### Summary

A range network is defined to consist of ground stations and targets where only distances between the two sets of points are observed. Such a network is said to be fundamental when only those six constraints are used that are needed to define the coordinate system for an adjustment. When ground stations or targets have certain configurations, a unique adjustment in terms of coordinates may be impossible, even when the number of observations is sufficient and the coordinate system is uniquely defined. Such configurations, resulting in the singular solutions, are said to be critical. This paper, being strictly theoretical, focuses on singular solutions rather than on numerical problems in some ill-conditioned systems of fundamental range networks. Because it is a summary, the paper is not intended to pursue the cases of linear dependence of the columns in the coefficient matrix of observation equations (a straightforward but extremely lengthy procedure). The critical configurations are presented in two separate groups. The first deals with ground stations lying all in a plane and the second deals with ground stations generally distributed. The two kinds of problems require different mathematical treatments and lead to quite different conclusions. A typical critical configuration when all ground stations are in a plane arises when they all lie on one second-order curve. When ground stations are generally distributed, a typical critical configuration can be represented by all points of a network (ground stations and targets) lying on one second-order surface. Because the singular solutions are inherent to them regardless of the number of co-observing ground stations, these two cases constitute the main contribution of the fundamental range network analysis. If they and some other more complex distributions of points summarized in this paper are avoided, an adjustment of range networks yields a unique solution.