A COMPARISON OF TECHNIQUES FOR THE TRANSFORMATION FROM CARTESIAN TO GEODETIC COORDINATES

1. A COMPARISON OF TECHNIQUESFOR THE TRANSFORMATION FROMCARTESIAN TO GEODETIC COORDINATES Sten Claessens The Western Australian Centre for Geodesy & The Institute for Geoscience Research Curtin University of Technology

2. Introduction (1/5) • Transformation from Cartesian to geodetic coordinates is performed very frequently • Transformation between local and global systems • Computation of geodetic coordinates from GNSS pseudo-range observations • Transformation from Cartesian to geodetic coordinates is not straightforward • An exact solution was long thought impossible • Many different methods exist • Iterative / Approximate • Exact

3. Introduction (2/5)

4. Introduction (3/5)

5. Introduction (4/5) • A [small] selection of transformation methods: • Iterative / Approximate • (e.g. Barbee 1982, Bartelme & Meissl 1975, Bowring 1976, Crocetto 1993, Feltens 2007, Fukushima 1999, 2006, Guo 2001, Heiskanen & Moritz 1967, Hekimoglu 1995, Jones 2002, Keeler & Nievergelt 1999, Laskowski 1991, Lin & Wang 1995, Pollard 2002, 2005, Sjöberg 1999, Toms 1995,1998, You 2000, Zhu 1993,1994) • Exact • (e.g. Borkowski 1987, 1989, Ecker 1967, Fotiou 1998, Frölich & Hansen 1976, Grafarend 2001, Heikkinen 1982, Hsu 1992, Lapaine 1991, Ozone 1985, Paul 1973, Pick 1967, 1985, Sugai 1967, Sünkel 1999, Vermeille 2002, 2004, Zhang et al 2005)

6. Introduction (5/5) • All methods perform differently in terms of: • Accuracy • Stability • Efficiency • Several studies to compare different methods have been performed • Comparison of the efficiency of various methods have vastly different outcomes!

7. Comparison (1/1) • Four methods are compared: • - Bowring (1976) • Iterative, Newton iteration • - Lin and Wang (1995) • Iterative, Newton iteration • - Fukushima (2006) • Iterative, Halley iteration • - Vermeille (2004) • Exact • Special attention is given to the comparison of the efficiency of the methods

8. Accuracy (1/5) Accuracy of the method of Bowring (1976) for points at the Earth’s surface

9. Accuracy (2/5) Accuracy of the method of Lin and Wang (1995) for points at the Earth’s surface

10. Accuracy (3/5) Accuracy of the method of Fukushima (2006) for points at the Earth’s surface

11. Accuracy (4/5) Accuracy of the method of Vermeille (2004) for points at the Earth’s surface

12. Accuracy (5/5) Maximum error of various transformation methods for a large range of heights

13. Efficiency (1/8)

14. Efficiency (2/8) • Factors that influence numerical efficiency • System specifications • Programming language • Floating point precision • Test setup • Convergence criteria • Latitude and height or latitude only • Batch computation or single point • Implementation

15. Efficiency (3/8) Relative CPU time for basic operators (right two columns from Fukushima 1999)

16. Efficiency (4/8) • Test setup • Convergence criteria: • All methods investigated here are not iterated, i.e. the solution of the first iteration is selected • Latitude and height or latitude only: • Comparisons should include the computation of both latitude and height, because both are generally required by the user • Batch computation or single point: • Tests of computation time should be conducted on a large number of points, and the computation of constants from the reference ellipsoid parameters should not be included in the computation time

17. Efficiency (5/8) • Implementation example: Bowring’s (1976) method • Common formulation: • Bowring’s implementation: • Fukushima’s implementation: • The choice of implementation has a large influence on the computation time!

18. Efficiency (6/8) • Implementation example: Bowring’s (1976) method • Further optimisation of the implementation: • or: • Which of these implementations is fastest depends on the system specifications

19. Efficiency (7/8) Operation count for computation of latitude and height (lowest numbers per operation in bold)

20. Efficiency (8/8) Operation count for computation of latitude and height (lowest numbers per operation in bold)

21. Conclusions (1/2) • Conclusions: • Many methods for the transformation from Cartesian to geodetic coordinates exist • Exact methods provide the highest accuracy while iterative methods require less computation time • Computation time depends heavily on system specifications, programming language, floating point precision and method of implementation • (many previous comparison studies don’t supply full test details and don’t optimise the method of implementation) • Computation time can best be assessed by an operation count

22. Conclusions (2/2) • Conclusions: • Bowring’s method has the lowest overall computation count • Lin and Wang’s method is faster on systems where square roots are relatively expensive in terms of computation time • Fukushima’s method is the fastest on systems where multiplication is performed much faster than division, and provides higher accuracy • Vermeille’s method provides the highest accuracy and is faster than most other exact methods