Coordinate Transformations

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.

Most developing countries like Ghana are yet to adopt the geocentric datum for its surveying and mapping purposes. It is well known and documented that nongeocentric datums based on its establishment have more distortions in height compared with satellite datums. Most authors have argued that combining such height with horizontal positions (latitude and longitude) in the transformation process could
introduce unwanted distortions to the network. This is because the local geodetic height in most cases is assumed to be determined to a lower accuracy compared with the horizontal positions. In the light of this, a transformation model was proposed by Featherstone and Vaníček (1999) which avoids the use of height in both global and local datums in coordinate transformation. It was confirmed that adopting such a
method reduces the effect of distortions caused by geodetic height on the transformation parameters estimated. Therefore, this paper applied Featherstone and Vaníček (FV) model for the first time to a set of common points coordinates in Ghana geodetic reference network. The FV model was used to transform coordinates from global datum (WGS84) to local datum (Accra datum). The results obtained based on
the Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) in both Eastings and Northings were satisfactory. Thus, a RMSE value of 0.66 m and 0.96 m were obtained for the Eastings and Northings while 0.76 m and 0.73 m were the MAE values achieved. Also, the FV model attained a transformation accuracy of 0.49 m. Hence, this study will serve as a preliminary investigation in avoiding the use of height in
coordinate transformation within Ghana’s geodetic reference network.

Coordinate transformation is one of the commonest activities in geodesy and other geospatial disciplines. Transformation of coordinates facilitates the integration of geodetic coordinates of points obtained from different sources onto a common geodetic reference datum. To achieve this requires the determination of transformation parameters between the different geodetic datums. Generally, the method of ordinary least squares (OLS) approach has been mostly applied in Ghana for estimating such parameters. However, literature has shown that the ordinary least squares approach consider errors in only the observation matrix. In view of this, the present study applied the general least squares (GLS) technique which models out errors in both the observation and data matrices. Hence, this study investigated and compared the efficiency of GLS with OLS using a four parameter similarity transformation model. Coordinates were transformed between the two reference datum (Accra and Leigon) adopted in Ghana for its surveying and mapping purposes. In order to ascertain the precision of the derived transformation parameters (an angle of rotation, a scale and two translation values), the reference standard deviation and the standard deviations of the individual parameters were estimated. Statistical indicators used to evaluate the performance of the least squares methods were the Mean Squared Error (MSE), Mean Absolute Error (MAE), and the Mean Bias Error (MBE). The adequacy of the generated least squares models was further evaluated using t-test hypothesis on the derived parameters and Legates and McCabe (LM) index. The results based on test data revealed that both the OLS and GLS models attained identical coordinate transformation results. Hence, the GLS could be used as an alternative technique to transform plane coordinates within the Ghana geodetic reference network.

Ghana has been utilizing Global Navigation Satellite Systems (GNSS) technology for geodetic purposes. In order to apply the GNSS acquired data locally, there is the need to transform coordinates from the geocentric datum to the Ghana local geodetic datum (nongeocentric). This is usually carried out by applying least squares adjustment technique to a coordinate transformation model. The present study compared both the ordinary least squares and total least squares adjustment techniques in 3D coordinate transformation between the
Ghana local geodetic datum and the global geocentric datum. These least squares techniques were applied to the Bursa-Wolf and Molodensky-Badekas model. The statistical analytical results revealed that the ordinary least squares and total least squares gave closely related transformation results. The main motivation of this work is to ascertain the extent of applicability and performance of ordinary least squares and total least squares for the first time in Ghana geodetic reference network. The reason is that, although the aforementioned
least squares methods have been compared and reported in several studies, such kind of work is yet to be done in the Ghana geodetic reference network. Therefore, this study will help Land surveyors and other built environment practitioners in Ghana to know the efficiency of
the two least squares methods and apply them accordingly.

The Molodensky-Badekas model is one of the similarity transformation models used in Ghana for transferring Global Positioning System (GPS) coordinates from the geocentric World Geodetic System 1984 (WGS 84) ellipsoid to the local non-geocentric coordinate system, and vice versa. The objective of the Molodensky-Badekas model is to introduce a centroid to cater for the correlation that exists between the parameters when used over a small portion on the earth surface. However,
the Molodensky-Badekas model performance depends on a particular centroid method adopted and the adjustment technique used. By virtue of literature covered, it was realised that the arithmetic
mean centroid has been the most widely used. In view of this, the present study developed and tested two new hybrid centroid techniques known as the harmonic-quadratic mean and arithmetic-quadratic mean centroids. The proposed hybrid approaches were compared with the geometric mean, harmonic mean, median, quadratic mean and arithmetic mean. In addition, the Total Least Squares (TLS) technique was used to compute the transformation parameters with varying centroid techniques to investigate and assess their accuracies in precise GPS datum transformation parameters estimation within the Ghana Geodetic Reference Network. Statistical indicators such as Mean Error (ME), Mean Squared Error (MSE), Standard Deviation (SD), and Mean Horizontal Position Error (MHPE) were used to assess the centroid techniques performance. The results attained show that the Harmonic-Quadratic Mean produced reliable coordinate transformation
results within the Ghana geodetic reference network and thus could serve as practical alternative technique to the frequently used arithmetic mean.