The application of the geometric method of local geoid model determination which requires the fitting of geometric surfaces to known geoid heights to enable geoid heights of new points to be interpolated involves the use of least squares technique for computation of the models' parameters. The selection of polynomial geometric surfaces depends on the size of the study area, the variation of the geoid heights and the number of measurement points. The accuracy of the geometric geoid model increases as the number of observation points approximates the number of geometric surface terms. But in most cases, the number of observation points is not considered. To this effect, this paper presents the relationship between geometric surfaces terms and observation points numbers and effect in the accuracy of geometric geoid models. A total of 23 points of known local gravimetric geoid heights were used. Two polynomial geometric (third and fifth degrees) surfaces were fitted to the geoid heights at various observation point numbers and compared to determine the relationship between the number of model terms and that of observation points and effect in the accuracy of the models. Least squares adjustment technique was applied to obtain the model parameters. The differences between the models and the known geoid heights of the points were computed and used to obtain the RMSEs as well as the accuracy of the models. The obtained results showed that the accuracy of the polynomial geometric geoid models tends to the highest as the number of measurement points approximates the number of the model terms and in a unique solution where the number of observation points is equal to the number of the polynomial geometric model terms, the model accuracy is highest. The paper recommends that the geometric method of local geoid model determination should be strictly applied in small areas. Where the method will be applied in considerable large areas, higher degrees polynomial geometric surfaces with a larger number of terms approximating the number of observation points should be applied. This will enable a proper fit of the polynomial surface to the known geoid heights, as well as high accuracy to be obtained.

As the application of gravity data in applied sciences such as geodesy, geodynamics, astronomy, physics and geophysics for earth shape determination, geoid model determination, computation of terrestrial mass displacement, orbit computation of natural and artificial celestial bodies, realization of force standards and derived quantities and density distribution in the different layers in the upper crust and having considered the cost of direct gravity survey, the study presents modelling local gravity anomalies from processed observed gravity measurements for geodetic application in Benin City. A total of 22 points were used. The points were respectively observed with CHC900 dual frequency GNSS receivers and SCINTREX CG-5 Autograv to obtain their coordinates and absolute gravity values. The theoretical gravity values of the points were computed on the Clarke 1880 ellipsoid to obtain their local gravity anomalies. The free air and the Bouguer corrections were applied to the computed gravity anomalies to obtain the free air and the Bouguer gravity anomalies of the points. Least squares adjustment technique was applied to obtain the model variables coefficient/parameters, as well as to fit the fifth-degree polynomial interpolation surface to the computed free air and the Bouguer gravity anomalies. Kriging method was applied using Surfer 12 software to plot the computed and the models' free air and Bouguer gravity anomalies. Microsoft Excel programs were developed for the application of the models in the study area. The Root Mean Square Errors (RMSEs) and the standard errors of the two models were computed to obtain the dependability, as well as reliability of the models. It is recommended that whenever either free air or Bouguer gravity anomalies of points within Benin City are to be obtained for application in applied sciences, the determined models should be applied.

The conversion of geometric as well as ellipsoidal heights from GNSS observations to practical heights for engineering constructions has necessitated the determination of the local geoid model of areas. Benin City is a developing area which requires a local geoid model for conversion of geometric heights to orthometric heights for physical developments in the area. This paper is on the best local geoid model of Benin City, Nigeria by comparing three gravimetric-geometric geoid models of the study area. GNSS and gravimetric observations were carried out on 49 points to respectively obtain their coordinates and absolute gravity values. The theoretical gravity values of the points were computed on the Clarke 1880 ellipsoid, subtracted from the absolute gravity values and corrected for the air (free air) to obtain the free air gravity anomalies of the points. The computed free air gravity anomalies were applied to compute the geoid heights of the points using the integration of the modified Stokes integral. Three geometric geoid surfaces (plane, second degree and third degree surfaces) were fitted to the computed gravimetric geoid heights using the least squares technique to obtain the gravimetric-geometric geoid models of the study area. The RMSE of the three gravimetric-geometric geoid models were computed to determine their (the models) accuracy. The three gravimetric-geometric geoid models were compared using their accuracy to obtain the most suitable geoid model of the study area. The results of the comparison showed that the third degree gravimetric-geometric geoid model is most suitable for application in the study area. It is recommended that ellipsoidal heights obtained from GNSS observation within Benin City, Nigeria should be converted to orthometric heights using the third degree geoid model.

The geometric heights obtained from GNSS observations cannot be used for engineering works as they are not reduced to the geoid. This study presents practical local geoid modelling from gravimetric observations using the modified Stokes integral for engineering applications in Benin City. A total of 52 points were observed with GNSS receivers and a gravimeter to respectively obtain their positions and absolute gravity values. The theoretical gravity values of the points were computed on the Clarke 1880 ellipsoid to obtain their local gravity anomalies. The modified Stokes integral was applied to compute the geoid heights of the points. The combined topographic effect was applied to the computed geoid heights of the points to obtain their precise geoid heights. The mean of the precise geoid heights of the points was computed to obtain the local gravimetric geoid model of the study area. The determined geoid model was validated for its reliability as well as the accuracy using the RMSE index. It is recommended that the use of assumed, as well as handheld GPS receiver heights for engineering works should be totally abolished as this study has established the local geoid model of Benin City.

The conversion of geometric as well as ellipsoidal heights from GNSS observations to practical heights for engineering constructions has necessitated the determination of the local geoid model of areas. Benin City is a developing area which requires a local geoid model for conversion of geometric heights to orthometric heights for physical developments in the area. This paper is on the best local geoid model of Benin City, Nigeria by comparing three gravimetric-geometric geoid models of the study area. GNSS and gravimetric observations were carried out on 49 points to respectively obtain their coordinates and absolute gravity values. The theoretical gravity values of the points were computed on the Clarke 1880 ellipsoid, subtracted from the absolute gravity values and corrected for the air (free air) to obtain the free air gravity anomalies of the points. The computed free air gravity anomalies were applied to compute the geoid heights of the points using the integration of the modified Stokes integral. Three geometric geoid surfaces (plane, second degree and third degree surfaces) were fitted to the computed gravimetric geoid heights using the least squares technique to obtain the gravimetric-geometric geoid models of the study area. The RMSE of the three gravimetric-geometric geoid models were computed to determine their (the models) accuracy. The three gravimetric-geometric geoid models were compared using their accuracy to obtain the most suitable geoid model of the study area. The results of the comparison showed that the third degree gravimetric-geometric geoid model is most suitable for application in the study area. It is recommended that ellipsoidal heights obtained from GNSS observation within Benin City, Nigeria should be converted to orthometric heights using the third degree geoid model.