Relationship between Polynomial Geometric Surfaces Terms and Observation Points Numbers and Effect in the Accuracy of Geometric Geoid Models

Abstract

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.

Improving the Local Geometric Geoid Model of FCT Abuja Accuracy by Fitting a Higher Order/Degree Polynomial Surface

“…Polynomial geoid models, as well as surfaces, are fitted to the known geoid heights of points to enable the geoid heights of new points, be computed by interpolation using the coordinates of the points (Eteje et al, 2019). The Multiquadratic and Bicubic models had earlier been applied in the Federal Capital Territory, Abuja to determine its local geometric geoid model (Oluyori, 2019).…”

“…The Multiquadratic and Bicubic models had earlier been applied in the Federal Capital Territory, Abuja to determine its local geometric geoid model (Oluyori, 2019). In the attempts to improve the accuracy of the determined geoid model, having considered Schmidt et al (2003) and Eteje et al (2019), there is a need to apply a fifth-degree (Quintic) polynomial surface in the study area. The Multiquadratic and the Bicubic models as given by Oluyori (2019) and the fifth degree polynomial surface given by Awange et al (2010) and Eteje et al (2019) 4 20 3 2 19 2 3 18 4 17 5 16 5 15 3 14 2 2 13 3 12 4 11 4 10 3 9 2 8 2 7 3 6 2 5 4 2 3 2 1 0 xy a y x a y x a y x a y a x a xy a y x a y x a y a x a y a xy a y x a x a y a xy a x a y a x a a…”

“…Although the quadratic-based programs according to Schmidt et al (2003) gives a more stable result, the high-order local surfaces can fit more complex landform features and reliable only for very accurate data from GNSS/GPS space methods. The accuracy of the local geometric geoid model increases as the number of observations, as well as points used in the study, approximates that of the model parameters/terms (Eteje et al, 2019). According to Eteje et al, (2019), the accuracy of the local geometric geoid model is highest at a unique solution that is, when the number of the observation is equal that of the model parameters, as well as terms.…”

“…The accuracy of the local geometric geoid model increases as the number of observations, as well as points used in the study, approximates that of the model parameters/terms (Eteje et al, 2019). According to Eteje et al, (2019), the accuracy of the local geometric geoid model is highest at a unique solution that is, when the number of the observation is equal that of the model parameters, as well as terms. Here, the consideration is that the number of model terms increases as the degree increases.…”

“…Here, the consideration is that the number of model terms increases as the degree increases. Considering Schmidt et al (2003) and Eteje et al (2019), the improvement of the local geometric geoid model of the Federal Capital Territory (FCT), Abuja requires a higher degree polynomial geoid surface whose number of model terms, as well as parameters approximates the number of observations/points used in the study. As a result, this study presents improving the local geometric geoid model of FCT, Abuja accuracy by fitting a higher order polynomial surface.…”