On ill-conditioning in 3D Similarity Transformation

Year 2015, Volume: 2 Issue: 1, 47 - 53, 02.03.2015

Orhan Kurt

https://doi.org/10.30897/ijegeo.302761

Abstract

3D similarity transformations are often used for datum transformation in Geomatic Engineering. The
transformations are routinely performed between the point coordinates evaluated from GNSS (Global
Navigation Satellite Systems) observations and the point coordinates given in national datum. A mathematical
model established among two different 3D coordinate systems contains tree type datum parameters which are
translations, rotations and a scale. Coefficients of the parameters reflect large variations from each other’s. So,
the unknown coefficient matrix in normal equations is to be ill-conditioned. In this study, it is indicated how
the ill-condition structure is reduced to acceptable level by means of choosing at different unit of unknown
parameters. Finally, the transformation from Kocaeli Metropolitan Municipality GNSS network coordinates to
Turkish Geodetic Datum is used as a numerical example.

Keywords

3D similarity transformation, linear equation system, ill-conditioning

References

• Leick, A. (1995). GPS Satellite Surveying, Second Edition, John Willey and Sons, Inc., ISBN 0-471-30626-6, Printed in USA.
• Hofmann-Wellenhof, B., Lichtenegger, H. and Wasle, E (2008). Global Positioning System, Theory and Practice, Fourth Edition, ISBN 3-211-82839-7, Springer-Verlag Wien New York.
• Öztürk, E., (1991). Adjustment Computation, Volume I, (in Turkish), Karadeniz Technical University, Architecture and Engineering Faculty, General Press Number: 119, Faculty Press Number: 38, Trabzon.
• Öztürk, E. and Şerbetçi, M (1992). Adjustment Computation, Volume III, (in Turkish), Karadeniz Technical University, Architecture and Engineering Faculty, General Press Number: 144, Faculty Press Number: 40, Trabzon.
• Kurt, O. (2007). Fundamental Coordinate Systems, Lecture Notes (in Turkish), Kocaeli University, Engineering Faculty, Geomatic Engineering Department, Kocaeli, http://orhankurt.jimdo.com/.
• Kurt, O. (2010). 1st Technical Report for Geodetic Studies, Productions of 1/1000 scaled numerical photogrammetric base map and colored orthophoto for Kocaeli Metropolitan Municipality (in Turkish), Kocaeli, Turkey.
• Press, W.H., Teukolsky, S.A., Vetterling, W. T. and Flannery, B.P. (2002). Numerical Recipes in C, The Art of Scientific Computing, Second, Edition, Cambridge University Pres, ISBN 0-521-43108-5.