Modelling Orthometric Heights from a Combination of Ellipsoidal Heights and Gravimetric Geoid Model in Rivers State, Nigeria

Kurotamuno P Jackson,Elochukwu C Moka

doi:10.4236/ijg.2020.114011

Abstract

Many applications in geodesy, hydrography and engineering require geoid-related heights. Spirit leveling which is the traditional means of obtaining geoid- or mean sea level-related heights is slow, time-consuming and costly. Global Navigation Satellite Systems (GNSS) offer faster and relatively cheaper way of obtaining geoid-related heights when geoidal undulation is applied to ellipsoidal heights. However, difficulties involved in determining acceptable geoid height have seriously hampered the application of GNSS for leveling in Rivers State, thus necessitating the need to develop an acceptable geoid model which will serve as a means of conversion of GNSS-delivered ellipsoidal heights to their orthometric heights equivalent. In pursuance of this objective, a detailed gravimetric geoid has been evaluated for Rivers State, Nigeria. The computation of the geoid was carried out by the traditional remove-restore procedure. The Earth Geopotential Model 2008 (EGM08) was applied as the reference field for both the remove and restore parts of the procedures; spherical Fast Fourier Transform (FFT) was employed for the evaluation of the Molodenskii’s integral formula for the height anomaly, (ζ) to yield the quasi-geoid; while the Residual Terrain Modelling (RTM) was done by prism integration. The classical gravimetric geoid over Rivers State was obtained from the rigorously evaluated quasi-geoid by adding the quasi-geoid to geoid (N - ζ) correction it. The minimum and maximum geoid height values are 18.599 m and 20.114 m respectively with standard deviation of 0.345 m across the study area. Comparison of the gravimetric geoidal heights with the GPS/Leveling-derived geoidal heights of 13 stations across Rivers State, Nigeria showed that the absolute agreement with respect to the GPS/leveling datum is generally better than 7 cm root mean squares (r.m.s) error. Results also showed that combining both GPS heights and the computed Rivers State geoid model can give orthometric heights accurate to 3 cm post-fit using a 4-parameter empirical model. The geoid model can thus serve as a good alternative to traditional leveling when used with GPS leveling, particularly for third order leveling in the study area.

Highlights

Measurements derived from Global Navigation Satellite Systems such as GlobalPositioning System (GPS) provide position of points which are commonly evaluated in a terrestrial three-dimensional Cartesian Coordinates
Difficulties involved in determining acceptable geoid height have seriously hampered the application of Global Navigation Satellite Systems (GNSS) for leveling in Rivers State, necessitating the need to develop an acceptable geoid model which will serve as a means of conversion of GNSS-delivered ellipsoidal heights to their orthometric heights equivalent
While ellipsoidal heights (h) are well known as heights reckoned from a defined reference ellipsoid, orthometric heights which are required in most engineering and hydrographic applications are reckoned from the geoid

Summary

Introduction

Measurements derived from Global Navigation Satellite Systems such as GlobalPositioning System (GPS) provide position of points which are commonly evaluated in a terrestrial three-dimensional Cartesian Coordinates. To obtain the equivalent geodetic coordinates in terms of latitude (φ), longitude (λ), and, ellipsoidal height (h), the resulting X, Y, and Z co-ordinates of the GPS points are transformed, employing the parameters of the reference ellipsoid. The separation between the two heights system hinges on the difference between the reference ellipsoid and the geoid. This difference is referred to as geoidal height (N). If the ellipsoidal height (h) derived from GPS observations and the geoid-ellipsoid separation (N) of a station is known, the orthometric height (H) of the station can be readily be computed directly from Equation (1) [1]: H= h − N (1)

Objectives

Methods

Conclusion