ellipsoidal approximation

The problem of datum transformation; determination of parameters for transferring curvilinear coordinate from one ellipsoid to another, is one of the main problems in geometrical geodesy. The problem draws the attentions of many researchers due to its role in the integration of all types of data in the geospatial database framework. Although the problem is one of the oldest geometrical problems by its nature, it is still challenging because of the newly introduced Earth gravitational models and precise global coordinate measurements using the global positioning systems. Different methods have been introduced by many famous geodesists like Molodensky (1962), Vanicek (1986) and others.In this paper, we developed a full mathematical model for determination of datum transformation parameters based on ellipsoidal approximation. It is theoretically and numerically compared with the previously developed model with spherical approximation. For small area, both models lead to the same accuracy while we expect to achieve higher with the ellipsoidal approximation in wider area.     Moreover, lack of ellipsoidal height in the old data sets is one of the main obstacles for the implementation of the classical transformation schemes. Herein, we introduced two methods for solving this problem. The Earth Gravitational Models (EGMs) which were wieldy available in the new century, thanks to the Earth gravity field’s dedicated missions, were employed to get an estimate of the geoidal heights of the data point with enough accuracy. Alternatively, the idea of the widely used polynomial approximating correcting surface was considered to model the geoid height at the area of computation. The numerical results showed that the second alternative was most helpful. Higher accuracy and better fitness in terms of statistical goodness of fit criteria were the outcomes of the implementation of the polynomial approximating correcting surface.     In order to show the performance of the ellipsoidal approximation as well as the idea of polynomial correcting surface, 150 points were selected in the Nigeria. The curvilinear coordinates of the data points were given both in the CLARCK-1880 (local old coordinates) and the World Geodetic System 1984 (WGS84) as the global new coordinates. The old coordinates of the data points were geodetic latitudes, geodetic longitudes and orthometric heights where the new coordinate set is fully geodetic components. A quadratic polynomial mathematical model was employed to approximate the geoid surface in the country. The achieved results showed its reasonable accuracy.