The geoid is used as a transformation linkage between ellipsoidal heights (h) determined from DGPS observations and orthometric heights (H). Widespread acceptability and adoption of GPS in local geospatial data acquisitions require the development of a local geoid model (N) for use to obtain orthometric heights in the absence of a national geoid model. Geoid model can be developed by gravimetric approach; global geopotential model (GGM); geometric technique among others. The conventional approach to GPS measurements is the use of one base reference station for field measurements. It has several drawbacks e.g. in signal range/coverage, accuracy degradation of results, etc. Based on Grashof's law of stability of triangles, this study was therefore based on dual reference base stations to improve on DGPS signal range and stability of results. Pro-online matrix solver was applied to the least squares observation equations of the two modelled FCT surfaces (multiquadratic and bicubic) to determine polynomial coefficients. The geoid undulation was computed and orthometric height generated for production of a topographical plan at 1m contour interval for elevation data in surveying, engineering and environmental applications. Skill =1 and bias = 0 were computed to confirm the predictive capability of the models and that no bias/errors were introduced into the respective modelling exercise. Diagnostic test also confirmed the viability and feasibility of providing vertical datum surface for FCT by this approach. Standard deviation (σ) as accuracy indicator was computed and the multi-quadratic model with σ =11cm was the better geoid surface for modelling of orthometric height in the FCT by the geometric method.

Consistency is an important characteristic in height systems which the mean sea level (msl) surface cannot guarantee. Only a geoid surface can provide height consistency. The quality of geoid undulation (N) will obviously affect the resulting orthometric height (H) determined from GNSS. The geoid undulation may be global, regional/national and local. Online software CSRS-PPP was used for post processing rinex data. 2008 was computed from AllTrans EGM2008 geoid calculator while h was used to compute from the relationship N= h-H. H is the existing orthometric height. Twenty-four controls with FCT 260 P as base reference station were used for this study. The computed standard deviation of differences in − 2008 (σ) is used as accuracy indicator and σ =0.419m .The root mean square error (RMSE) is 0.934m. This indicates the quality and reliability of the geoid undulation from the EGM2008 model. Comparing the observed and 2008 , the use of global models may not satisfy the accuracy level of orthometric height desired for local applications in the FCT, Abuja. GNSS (GPS) may be used along with local geoid model as a way to acquire acceptable orthometric height. The smaller the-2008 makes it better model. The range of 1.585m from (-2008) in this study is a strong indication that global models should be avoided as much as possible in local applications.

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 application of gravity anomalies for gravimetric geoid model determination has necessitated the computation of normal gravity on the WGS 84 ellipsoid using the International Gravity Formula. For local gravimetric geoid model determination, the gravity anomalies are computed on the local ellipsoid adopted for geodetic computation in the area/region of study. To determine precise local gravimetric geoid model in countries where Clarke 1880 ellipsoid is adopted for geodetic computation, the theoretical gravity must be computed on the adopted geodetic computation surface. As a result, this paper derives a local theoretical gravity model on the Clarke 1880 ellipsoid for practical local geoid model determination. The model was derived using the Clarke 1880 ellipsoid parameters, angular velocity, gravitational constant and the Somiglinana's closed formula for normal gravity. The derived model was presented in two forms, model A and model B. The two forms of the derived model were compared to determine their suitability as well as agreement. The comparison results show that the two forms of the model agree. Thus, any of the model forms can be applied for accurate normal gravity computation where Clarke 1880 ellipsoid is adopted as a computation surface.

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