1. Introduction

Surveying with the advancement of modern science and technology has undergone an epoch-making transformation to break the spatial limitations of classical surveys, and to enter a new stage of development of modern surveying. One of the most important discovery that science and technology has provided the geo-scientific community due to its global coverage and free access is the Global Navigation Satellite Systems (GNSS) ^[1]. This GNSS technology such as Global Positioning System (GPS) furnishes the principal technology for geomatic and geodetic activities for both develop and developing countries. As an example, it is well known that in Ghana the GPS has been adopted as a viable tool for the majority of geodetic surveys due to its numerous advantages over classical methods of surveying. It is worth stating that the GPS ellipsoid of reference is the World Geodetic System 1984 (WGS84) and thus provides coordinates in latitude, longitude and ellipsoidal height. It is pertinent to note that it was indicated in ^[2] that assimilating GPS data into the mapping system of a country like Ghana cannot be done straightforwardly without appropriate mathematical conversions to determine transformation parameters. In an effort to have such integration into the mapping system of Ghana, the initial step is to convert the geodetic coordinates to the cartesian coordinate system. The method of directly converting geodetic coordinates to cartesian coordinates can be carried out using Bowring forward equation ^{[3, 4]}. Conversely, it was emphasized by ^[5] that the reverse transformation is more complex considering the latitude and the geodetic height. This problem has been attributed to the inability of the latitude to be separated from the radius of curvature in the prime vertical when carrying out the reverse approach. Thus, creating a lot of research interests among geodesist and mathematicians. It is worth mentioning that several methods have been developed in estimating the latitude and geodetic height respectively. These methods as stated in ^[6] could be divided into three categories namely; Exact or closed form approaches, Approximation methods, and Iterative methods.

A comparative study of the above mentioned techniques have been carried out by researchers from various countries. For instance, ^[7] tested two direct and four indirect methods on a region covering Australia and found to give acceptable results for the computation of and . It was concluded in their study that Lin and Wang’s method was appreciably faster than the other five methods and was recommended for use in Australia. Reference ^[8] reviewed exact transformation formulas and compared with the approximation methods by considering their computational complexity and sensitivity to computer round off error. The author concluded that the exact transformation formulas produced negligible errors in coordinate transformation and thus recommend to be used in practice. Reference ^[9] evaluated four non-iterative and four iterative methods in transforming cartesian coordinates to its corresponding geodetic coordinates. It was found that the iterative methods were faster than the non-iterative and converges to a sub millimetre accuracy exceeding the requirements of any practical application. In ^[10] a general overview of ten iterative methods and one direct method were presented. The author concluded that the direct procedure is fairly simple and straightforward compared with the iterative methods that requires an approximation to get at the desired solution. Reference ^[11] demonstrated that the Bowring’s inverse method in terms of computational speed was faster than Borkowski. However, it was shown that Lin and Wang method is faster than Bowring’s method ^[12].

Several authors have proposed new techniques for transforming geocentric coordinates to geodetic coordinates. For example, ^[13] introduced a closed-form algebraic technique that is applicable globally including the poles regardless of the eccentricity value of the ellipsoids. The proposed method produced refine values when compared with Bowring Inverse equation. In ^[14] an exact and relatively simple analytical transform of the rectangular coordinates to the geodetic coordinates was presented. The method was based on one solution of the quartic equation. Reference ^[15] developed a more robust algorithm that yields more accurate results than most of the published algorithms. A significant attribute of the procedure is that the processing time is nearly a constant. A vector-based algorithm to transform cartesian coordinate to geodetic coordinates was introduced by ^[16] and was further extended to triaxial ellipsoids ^[17]. Both algorithms showed identical result in accuracy, similar computer processing unit requirements, and also worked well for celestial bodies having significantly greater flattening than the Earth. Reference ^[18] compared computational intelligence algorithms with the conventional methods in transforming geocentric coordinates into geodetic coordinates. Among all the methods, the differential search algorithm applicable to numerical optimization problems yielded a very high level of accuracy.

It should be known that the situation in Ghana is different as researchers have only utilized the Bowring inverse equation in their datum transformation. For instance, in ^[19] the Bowring Inverse equation was applied in datum transformation between WGS84 and War Office datum. The same inverse method was implemented also in ^[2] and ^[20]. In view of the above development, it can be seen that such a comparative study on the applicability of the reverse methods within the Ghana geodetic reference network has not been fully investigated. This study thus compares seven of the reverse techniques developed by researchers namely; Simple Iteration, Bowring Inverse equation, Lin and Wang method, Paul’s method, method of Successive Substitution, Newton Raphson approach, and Borkowski’s method within the Ghana geodetic reference network. It is worth stating that that these are not the only techniques for computing geodetic latitude () from cartesian coordinate (X, Y, Z). Hence, this study is not definitive since only a small selection of the published techniques are chosen. However, the choice of the methods was influenced by its gaining popularity in usage within the last two decades by researchers.

2. Study Area

Ghana is a country located at the Western part of Africa sharing boarders with Cote D’Ivoire to the West, Togo to the East, Burkina Faso to the North and the Gulf of Guinea to the South. The country has a 239,460 km² generally consisting of low plains ^[21] with 2,093 km of international land borders. Figure 1 shows the study area.

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Figure 1. Map of Ghana showing the study area

This study covers the first phase of the ongoing project by the Ghana Survey and Mapping Division of Lands Commission in establishing a new geodetic reference network base on the International Terrestrial Reference System (ITRS) ^[1]. The choice of the five out of ten administrative regions for the first phase of the project was due to the following reasons; almost all the natural resources such as gold, bauxite, manganese, oil, timber, cocoa, diamond and many others found in the country are situated in these regions. Hence, contributing significantly to the economic growth of Ghana.

3. Methods

3.1. Transformation from Geodetic Coordinates to Cartesian Coordinates

Secondary data of common points in both WGS84 and War Office 1926 geodetic coordinates for the new and old Ghana geodetic reference network were obtained from the Survey and Mapping Division of Lands Commission in Ghana. The Bowring forward equation was first used to transform the geodetic coordinates into cartesian coordinates because it is a prerequisite in most 3D datum transformation parameter determination. The direct transformation of the geodetic coordinates above a reference ellipsoid to the cartesian coordinates (X, Y, Z) could be carried out using the Bowring forward equation. This equation is expressed mathematically as:

Where; φ = Latitude, λ = longitude, f = flattening , e = eccentricity. The radius of curvature in the prime vertical plane N, and the first eccentricity are given by

Figure 2 shows the geometry of the Bursa–Wolf transformation model. The X, Y, Z axes of system 1 are rotated by very small angles from the X, Y, Z axes of system 2, and the origins of the two systems are displaced by translations in the directions of the X, Y, Z axes of system 2 ^[22]. and are vectors of coordinates in both systems and t is a vector of translations.

The mathematical relationship between coordinates in both systems can be written in the form of a vector equation ^[22]); .

Alternatively, the Bursa–Wolf transformation may be written as

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Figure 2. Geometry of Bursa-Wolf model (adopted from [22])

The new War office rectangular coordinates obtained after applying the Bursa-Wolf parameters determined need to be transformed back into geodetic coordinates (latitude, longitude, ellipsoidal heights) to enable projection of the coordinate onto the transverse Mercator that is utilized in Ghana. This will also enable the coordinates to be expressed in the Ghana national projected grid coordinate system (Easting, Northing). The reverse transformation was carried out using seven inverse equation models. The various models applied are described in the subsequent section.

An approximate latitude value was computed from the equation

where, second eccentricity is given us with a and b being the semi-major and minor axis of the reference ellipsoid respectively. The approximate value, was then used in the right hand side (RHS) of the equation

to evaluate, on the left hand side (LHS). The new value, was then applied in the right hand side (RHS) to give the next value. This procedure was repeated until the difference between successive LHS values,, reaches an acceptable limit thus, the iteration converges to a solution of .

The starting value, was obtained from the relationship between the geocentric and parametric latitude using

The required latitude was then computed from

Where is the parametric latitude; a is the semi-major axis of the reference ellipsoid; is the perpendicular distance from the rotational axis; is the first eccentricity and is the second eccentricity expressed as .

This iterative method uses Newton Raphson Iteration to evaluate a scalar multiplier q of the normal vector to the ellipsoid ^[23]. Once q was calculated, simple relationships between cartesian coordinates of P and its normal projection Q onto the ellipsoid at P were used to evaluate the cartesian coordinates of Q. The initial approximation of the latitude was computed using;

The latitude, and height, h were finally estimated from the equations below;

The and were obtained from

It should be noted that h is negative if is less than.

Paul’s method is direct in so far as is obtained from a simple closed equation but only after several intermediate variables have been evaluated. Thus, having X, Y, Z for a point related to an ellipsoid, the latitude, was obtained by computing the following variables () in order from the following equations;

It should be noted that all square roots in have the same sign as Z .The geodetic latitude, was estimated from

Detailed description of the Paul’s method can be found in ^[7].

The method of Successive Substitution is popular because of its programming simplicity. This method is comparable to that of simple iteration. A starting value was first calculated from the relationship

This was used on the RHS of the equation below

to evaluate (and hence) on the LHS. This new value, was then applied on the RHS to give the next value, (and hence). The procedure was repeated until the difference between successive LHS values, reached an acceptable limit. Thus, the iteration converges to a solution for and hence φ.

The Newton Raphson iteration can improve the rate of convergence for in successive substitution solution for real roots of equation given in the form of an iterative equation as

where n represents iteration and the function is expressed as;

The derivative of the function is given as;

where, .

A starting latitude value, was obtained from

The iteration continued until the correction factor in reached an acceptable small magnitude.

Reference ^[7] indicated that Borkowski’s method is an indirect method that uses Newton’s iterative technique to solve for the parametric latitude,. The is expressed as

where the function and its derivative are expressed as;

and . was obtained by iteration after calculating the constants , q and c from

An initial approximation, was estimated using the relation;

After solving for, was computed from;

4. Results and Discussion

The derived parameters for transforming data from WGS 84 to War Office datum with their associated standard deviations using Bursa-Wolf transformation model are presented in Table 1 below. It was observed from the results of the translation parameters in Table 1 that, from the origin of the War Office to the origin of WGS 84, the X-axes that intersects the Greenwich Meridian and the Equator has a negative displacement. On the other hand, the Y-axes created to have a right angle to the X, Z-axes and the Z-axes passing through the Earth instantaneous pole has a positive displacement between the two reference systems (War Office and WGS 84 datum). The individual standard deviations for the obtained transformation parameters are also shown in Table 1. The calculated reference standard deviation and reference adjustment variance were also estimated to check the precision for the overall observations (Table 1).

Table 1. Summary of derived parameters results

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The translation parameter, revealing the existence of a negative displacement from the geocenter is because both X-axes in the two reference ellipsoids are moving in opposite directions. Conversely, and translation parameters from the geocenter evident from Table 1 above shows that, the axes of both reference ellipsoids move in the same direction. In Table 1, the standard deviations indicate the spread of the data about the mean measured in the same units as the data. They also indicate the limits of the error bound within which the most probable value (MPV) of the mean lies. Since the individual standard deviations computed for each translation parameter are small, it implies that the data points are closer to the MPV value of the mean which indicates that the parameters response will be fairly uniform when applied to the observation data within the study area. In addition, the smaller standard deviation obtained signifies a steep bell-shaped on the normal distribution curve.

The reverse transformation methods were evaluated based on the residuals generated from new projected grid coordinates and the existing projected coordinates within the Ghana geodetic reference network. The following statistical indicators were used as a performance criteria index (PCI) namely: Mean Square error (MSE), Root Mean Square error (RMSE), Nash-Sutcliffe Efficiency Index (N) and Modified Index of Agreement (D). Their mathematical expressions are given below:

Where n is the total number of observations used in the Bursa-Wolf model, O and P are the existing projected coordinates and predicted projected coordinates from the reverse techniques used. is the mean of the existing projected coordinates. To assess the quality of the transformation results the MSE, RMSE, N and MID were computed as shown in Table 2 and Table 3.

Table 2. Statistical Indicators results for Easting Coordinate

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Table 3. Statistical Indicators results for Northing Coordinate

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The MSE values obtained signifies good performance from the various reverse method. This is because it is well known that for 3D datum transformation between non-geocentric and geocentric datums, the acceptable accuracy is within ±1 m due to the distortions in the local network. Also, the MSE further showed the degree of fitness of the reverse methods to the existing data.

The Nash-Sutcliffe Efficiency Index (N) was used in this study as a measure of the reverse methods efficiency by comparing the new projected grid coordinates to existing grid coordinates. From Table 2 and Table 3, both methods attained the optimum value of 1. The MID varies from 0 to 1 with higher values indicating a better fit of the model. With this in mind, it can be stated that the reverse methods modelled values (new projected grid coordinates) have better agreement with the existing datasets.

A careful observation of Table 2 and Table 3 reveals that; Borkowski’s method, Bowring Inverse method and Lin and Wang have the same range of errors in Eastings and Northings respectively. Thus, both methods achieved identical RMSE value of 0.467604 m and 0.838824 m in both Easting and Northing coordinates. Simple Iteration and method of Successive Substitution were the other two methods having the same RMSE values in both coordinates system. Having compared all the seven methods it was observed from Table 2 that Newton Raphson’s method gave the least RMSE in the Eastings compared to the other methods. However, its error in the Northing coordinates was higher than all the other reverse methods used in the assessment. Conversely, Paul’s method had the least error in the RMSE in the Northing coordinates. With the exception of the Newton Raphson’s method the same RMSE was achieved in the Easting coordinates when compared with the other reverse techniques. Upon critical examination of Tables 2 and 3, the authors proposed the Paul’s method as the most viable approach to be used within the Ghana geodetic reference network when precise 3D coordinate transformation values are needed. This is because the RMSE deviation estimated showed that the Pauls method deviate from Newton Raphson by 0.000005 m in the Eastings. In the case of the Northing coordinates, Newton Raphson performed poorly by deviating 0.038753 m from the Paul’s method.

5. Conclusion

Seven reverse methods namely Simple Iteration, Bowring Inverse equation, Lin and Wang, Pauls method, Successive Substitution, Newton Raphson and Borkowski has been tested within the Ghana geodetic reference network using Bursa-Wolf transformation model. The results obtained showed that the seven methods produced acceptable results in the coordinate transformation. This further confirms the assertion made by ^[8] that the reverse methods are not only imperative in theory, but can also be used in practice. To conclude, the authors’ based on the results obtained recommend Pauls method as the most practicable technique for precise 3D coordinate transformation work within the Ghana geodetic reference network.

Acknowledgement

The authors wish to express their appreciation to the Ghana Survey and Mapping Division of Lands Commission for making available the data for this research.

Statement of Competing Interests

The authors have no competing interests.

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