Positioning is considered as one of the most important problems in civil engineering and surveying works. Today, satellite techniques are widely used for positioning. Global navigation satellite systems provide three-dimensional coordinates (latitude, longitude, and height), related to the reference ellipsoid surface. In most engineering projects, however, orthometric height related to the geoid is required. Therefore, separation between ellipsoid and geoid need to be known to convert ellipsoidal height into geoidal height which nearly approximates mean sea level. The ellipsoid is a mathematical surface, while the geoid is a physical one. This leads to problem of modelling the geoid. In this paper, multiple regression method was used to model the geoid in a defined area that resembles a big city. Riyadh city, capital of Saudi Arabia has been used as a case study. It has been found that fifth degree polynomials provide a model with accuracy of 1.8 centimeters.
Nowadays fully three-dimensional positions - latitude, longitude and height are commonly provided by Global Positioning Systems (GPS) with WGS84 ellipsoid as a reference surface. The knowledge of terrain heights is very essential for several surveying and civil engineering works. The type of heights needed for such applications is a height above mean sea level (MSL), usually called orthometric height. A precise geoid model is required to transform GPS-based geodetic heights to orthometric height to allow such data to be directly used in surveying and civil engineering applications . The process of geoid modelling can only be performed if dense networks of precise levelling or gravity and astronomical deflections of vertical are made available.
Several regional or local geoid models have been developed, such as: Kamguia, et al  (Cameroon); Merry  (South Africa); Aleem KF  (Port Harcourt, Nigeria); Dawod & Hosny  (Egypt); Manandhar & Shanker  (Nepal); Younis  &  (Palestine). Developed and tested geoid models include: Milbert & Smith , developed a GEOID96 model for the United States
and when testing 2729 GPS points on leveled benchmarks found a RMS error of 5.8cm with no offsets or trends removed. Zhan-Ji  showed that a geometric geoid model developed for Hong Kong would provide an accuracy of 3 cm and a relative geoid height difference less than 1 ppm for a base line longer than 30 km. Zan-Ji  constructed the Hong Kong gravimetric geoid with an accuracy of about 2 cm when compared with geoid heights of 31 check points derived from GPS/Leveling data. Lin 11], generated two geometric geoid models and one gravimetric-geometric geoid model for Tainan City using the GPS and leveling data. He used fifth- and seventh-degree polynomial, and achieved orthometric height accuracy of 2-4 cm.
Eteje  formed a geoid model using eight benchmarks. To interpolate geoid heights of new points, they used bi-cubic geoid model and obtained an accuracy of ± 1.7cm. Younis (7) constructed a geoid model for Palestine using finite element method. Using 2nd and 3rd order polynomial, the accuracy of 1-3 cm have been achieved. Kamguia, et al , formed a geometric geoid model for a small area in Cameroon and obtained an accuracy of 14 cm. After a four-parameter fitting to the GPS/levelled data, the accuracy reduced to 11 cm.
There are two approaches for developing a geoid model: geometric and gravimetric approaches. Geometric is the one most applicable for surveying works of small extent, i. e. city extent or
less. In this method, ellipsoidal heights h is measured by GNSS at
points with known orthometric heights H. Then, the difference
N (geoidal height) between h and H, known as geoidal height,
is derived at all known points (N = h-H). Conversion between
h and H at a point requires that N must be known at that point.
Therefore, geoid modelling became an important problem in
geodesy (Figure 1).
The simplest case of geometric geoid modelling is to consider
that, the geoid and the ellipsoid are parallel. In this case we need
a single control point to provide a constant height shift for all
measured points in the vicinity. For more accuracy, we consider
the geoid as a tilt plane. In this case, three control points are
needed. But this consideration is still convenient only for a small
area. The geoid is much more complex in shape. Therefore, more
control points would be required to generate a reliable model by
using one of the known numerical methods such as polynomials.
In such approach, obtained geometric geoid model is a function of
x and y (N= f (x, y)) is only valid over the area covered by the known
control points. Hence, for small areas where sufficient control
points are available the geometric approach is recommended
to form a precise local geoid model . Gravimetric approach,
however, provides a uniform grid of geoidal heights over a large
area, in contrast to the scattered point values over a limited area
provided by the geometric method. However, it requires a complex
numerical integration of gravity anomalies to determine a geoidal
Generally, in big cities, it is possible to find sufficient
benchmarks with known orthometric heights. In this paper
geometric approach will be adopted to form a geoid model for a
limited region: Riyadh city, capital of Saudi Arabia, which extends
approximately 35 km x 35 km. 110 control points and 9 check
points were available for the geoid modelling process. Figure 2
shows the distribution of these points among Riyadh City. After
testing polynomials of different degrees, considering the density
of control points to be about one per one square kilometer (110
control points distributed on about 1000 km2) and using multiple
regression method, it was found that, the following equation gives
the best result:
Using the above equation, the geoidal height N was calculated
for all check points (9 points). The difference between calculated
and existing value of N for all check point does not exceed 1.8
cm, while RMS error is less than 1 cm. All polynomials of higher
degree will improve the accuracy with a very small rate. This
result is highly superior to values obtained from EGM 2008 Model
for the same points, where the difference exceeds 70 cm. It also
competes very well with results of previous studies outlined in
the literature review of this paper where best results are between
1 cm and 4 cm.
The accuracy obtained for a geoid model using geometric
approach is much better to rely on than the global model EGM
2008. Therefore, it is recommended to apply the geometric method
for big cities and for regions that contain enough benchmarks
by which the model is generated. The degree of polynomial
representing the model, and its parameters, may change from
one region to another depending on extend and shape of the
region being studied, and, on the number of existing benchmarks.
Once an appropriate model for an area is obtained, it can be used
directly through configuration process of the receivers before
starting the survey works.
Engineers, Khalid Aldossary, Omar Alrasheed, Abdullah
Almogbel and Faisal Aldos carried out the field observations both
for levelling and GNSS observations for Bench Mark and Test points
during their final year graduation project at the Civil Engineering
Department, King Saud University under the guidance of Engineer
Yasser Sayed Ahmad. They are all acknowledged for the effort they
Younis G (2019) Finite Element Method for Precise Geoid Modeling for GNSS Positioning in Palestine. 2nd International Conference on Civil Engineering, 25th to 26th November 2019, Ramallah, Palestinian Territories.
Milbert DG, Dru Smith A (1996) Converting GPS Height into NAVD88 Elevation with the GEOID96 Geoid Height Model. National Geodetic Survey, NOAA. GIS/LIS'96 Denver, Colorado.