Journal of Geosciences and Geomatics
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Journal of Geosciences and Geomatics. 2016, 4(1), 1-7
DOI: 10.12691/jgg-4-1-1
Open AccessArticle

Accuracy Assessment of Cartesian (X, Y, Z) to Geodetic Coordinates (φ, λ, h) Transformation Procedures in Precise 3D Coordinate Transformation – A Case Study of Ghana Geodetic Reference Network

Bernard Kumi-Boateng1, and Yao Yevenyo Ziggah1, 2

1Department of Geomatic Engineering, University of Mines and Technology, Tarkwa, Ghana

2Department of Surveying and Mapping, China University of Geosciences, Wuhan, P.R. China

Pub. Date: January 07, 2016

Cite this paper:
Bernard Kumi-Boateng and Yao Yevenyo Ziggah. Accuracy Assessment of Cartesian (X, Y, Z) to Geodetic Coordinates (φ, λ, h) Transformation Procedures in Precise 3D Coordinate Transformation – A Case Study of Ghana Geodetic Reference Network. Journal of Geosciences and Geomatics. 2016; 4(1):1-7. doi: 10.12691/jgg-4-1-1


Ghana a developing country still adopt the non-geocentric ellipsoid known as the War Office 1926 as its horizontal datum for all surveying and mapping activities. Currently, the Survey and Mapping Division of Lands Commission in Ghana has adopted the satellite positioning technology such as Global Positioning System based on a geocentric ellipsoid (World Geodetic System 1984 (WGS84)) for its geodetic surveys. It is therefore necessary to establish a functional relationship between these two different reference frames. To accomplish this task, the Bursa-Wolf transformation model was applied in this study to obtain seven transformation parameters namely; three translations, three rotations and a scale factor. These parameters were then used to transform the WGS84 data into the War office system. However, Ghana’s national coordinate system is a projected grid coordinate and thus the new War Office coordinates (X, Y, Z) obtained are not applicable. There is therefore the need to project these coordinates onto the transverse Mercator of Ghana. To do this, the new war office data (X, Y, Z) attained must first be transformed into geodetic coordinates. The reverse conversion from cartesian (X, Y, Z) to its corresponding geodetic coordinate (φ, λ, h) is computation intensive with respect to the estimation of geodetic latitude and height. This study aimed at evaluating the performance of seven methods in transforming from cartesian coordinates to geodetic coordinates within the Ghana Geodetic Reference Network. The seven reverse techniques considered are Simple Iteration, Bowring Inverse equation, method of successive substitution, Paul’s method, Lin and Wang, Newton Raphson and Borkowski’s method. The obtained results were then projected onto the transverse Mercator projection to get the new projected grid coordinates in the Ghana national coordinate system. These results were compared with the existing coordinates to assess their performance. The authors proposed the Paul’s method to be a better fit for the Ghana geodetic reference network based on statistical indicators used to evaluate the reverse methods performance.

bursa-wolf model coordinate transformation geodetic coordinates geocentric coordinates

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[1]  Poku-Gyamfi, Y. and Gunter W. H. (2006). Framework for the Establishment of a Nationwide Network of Global Navigation Satellite System (GNSS): A Cost Effective Tool for Land Development in Ghana. 5th FIG Conference, 1-13.
[2]  Ayer, J. and Fosu, C. (2008). Map Coordinate Referencing and the Use of GPS Datasets in Ghana. Journal of Science and Technology, Vol. 28, No. 1, 1-18.
[3]  Leick, A. (2004). GPS Satellite Surveying. John Wiley and Sons, Inc., Hoboken, New Jersey, United States of America.
[4]  Sickle, J. V. (2010). Basic GIS Coordinates. CRC Press, Taylor and Francis Group, 2nd Ed., 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 USA, 7.
[5]  Jekeli, C. (2012). Geometric Reference Systems in Geodesy. Division of Geodetic Science, School of Earth Sciences Ohio State University, USA, 15.
[6]  Shu, C. and Li, F. (2010). An Iterative Algorithm to Compute Geodetic Coordinates. Computer and Geosciences, Vol. 36, 1145-1149.
[7]  Gerdan, G. P. and Deakin, R. E. (1999). Transforming Cartesian Coordinates (X, Y, Z) to Geographical Coordinates (φ, λ, h). The Australian Surveyor, Vol. 44, No. 1, 55-63.
[8]  Zhu, J. (1994). Conversion of Earth-Centered Earth-Fixed Coordinates to Geodetic Coordinates. IEEE Transactions on Aerospace and Electronic Systems, Vol. 30, No. 3, 957-961.
[9]  Fok, S. H. and Iz, H. B. (2003). A Comparative Analysis of the Performance of Iterative and Non-iterative Solutions to the Cartesian to Geodetic Coordinate Transformation. Journal of Geospatial Engineering, Vol. 5, No.2, 61-74.
[10]  Burtch, R. (2006). A Comparison of Methods Used in Rectangular to Geodetic Coordinate Transformations, ACSM Annual Conference and Technology Exhibition, Orlando, FL, 1-25.
[11]  Laskowski, P. (1991). Is Newton’s iteration faster than simple iteration for transformation between geocentric and geodetic coordinates? Bull Geod, 65:14–17.
[12]  Lin K.C. and Wang, J. (1995). Transformation from geocentric to geodetic coordinates using Newton’s iteration. Bull Geod, 69:14-17.
[13]  Vermeille, H. (2002). Direct transformation from geocentric coordinates to geodetic coordinates, Journal of Geodesy 76: 451-454.
[14]  Borkowski, K. M. (1989). Accurate Algorithms to Transform Geocentric to Geodetic Coordinates. Turon Radio Astronomy Observatory; Bull Geoid, Vol. 63, pp. 50-56.
[15]  Toms, R. (1996). An Improved Algorithm for Geocentric to Geodetic Coordinate Conversion. Fourteenth Workshop on Standards for Distributed Interactive Simulations Orlando, Lawrence Livermore National Laboratory, University of California, W-7405-Eng-48.
[16]  Feltens, J. (2007). Vector methods to compute azimuth, elevation, ellipsoidal normal, and the Cartesian (X, Y, Z) to geodetic (φ, λ, h) transformation. Journal of Geodesy, Vol. 82, Issue 8, 493-504.
[17]  Feltens, J. (2009). Vector method to compute the Cartesian (X, Y, Z) to geodetic (φ, λ, h) transformation on a triaxial ellipsoid. Journal of Geodesy, Vol. 83, Issue 2, 129-137.
[18]  Civicioglu, P. (2012). Transforming Geocentric Cartesian Coordinates to Geodetic Coordinates by using Differential Search Algorithm. Computers & Geosciences, Vol. 46, 229-247.
[19]  Ziggah, Y.Y., Youjian, H., Odutola, A.C. and Nguyen, T.T. (2013). Accuracy Assessment of Centroid Computation Methods in Precise GPS Coordinates Transformation Parameters Determination: A Case Study, Ghana, European Scientific Journal, Vol. 9, No. 15, 1-12.
[20]  Dzidefo, A. (2011). Determination of Transformation Parameters between the World Geodetic System 1984 and the Ghana Geodetic Network. Master’s Thesis, Department of Civil and Geomatic Engineering, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana.
[21]  Baabereyir, A. (2009). Urban environmental problems in Ghana: case study of social and environmental injustice in solid waste management in Accra and Sekondi-Takoradi. Thesis submitted to the Department of Geography, University of Nottingham for the Degree of Doctor of Philosophy, UK, 97.
[22]  Deakin, R.E. (2006). A Note on the Bursa-Wolf and Molodensky-Badekas Transformations. School of Mathematical and Geospatial Sciences, RMIT University, 1-21.
[23]  Deakin, R. E. and Hunter, M. N. (2010). Geometric Geodesy - Part A. School of Mathematical and Geospatial Sciences, RMIT University, Melbourne, Australia, 146.