ler 
ng 
a 
an 
on 
ds, 
ion 
of 
lC 
)) 
GPS 
Yodel 
As 
ncept 
as an 
ining 
t> In 
ed in 
elling 
| as if 
jerous 
f this 
uthor. 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
question "What accuracy level can be reached using this 
approach?" is still an improvable issue as related to many 
conditions and one of them is used method of geoid modelling. 
In this paper, geoid modelling is evaluated as a surface fitting 
problem according to GPS and Levelling data and the focus is 
modelling the geoid of a local area as an analytical surface to 
serve practical geodetic applications. During implementation of 
the subject, two different interpolation methods are considered. 
There are several interpolation methods to transform point data, 
however, it is important to determine, which one is more 
appropriate and give better and clear parameter solution for the 
implemented data and the problem. Modern spatial 
interpolation methods are based on geometric and geostatistical 
aspects. 
Here in, two interpolation methods, Inverse Distance Weighting 
and Geostatistical Kriging were applied for generating a digital 
geoid heights model in a local area. The study area is 45x50 
km“ and covered by heterogeneous and appropriately 
distributed 301 reference points of which positions are known 
in ITRF datum and orthometric heights are in Turkish National 
Vertical Datum. 
Density of the reference points is appropriate for the geoid 
modelling in the region (1 point 3 km). There is not a specific 
reason for appreciating especially these two interpolation 
methods in the study. The main purpose of the study is to 
emphasize the necessity of a precise geoid model to serve 
practical purposes of geodesy. The presented interpolation 
methods were programmed as to be computation modules of a 
geoid modelling software and the results of the computation 
algorithms are going to be mentioned more than the 
mathematical bases of the methods because of that they can be 
reached in the literature easily (see Isaaks and Srivastava, 1989; 
Watson, 1992; Yanalak 1997). 
2. LOCAL PRECISE GEOID MODELS TO SERVE 
PRACTICAL GEODETIC APPLICATIONS 
A precise geoid model is an important part of the geodetic 
infrastructure. Geodetic infrastructure includes whole of the 
geodetic networks, which realize the coordinate systems and 
datum definitions as the bases of geodetic works (Aksoy et al 
1999). 
Restructuring and revising efforts of geodetic infrastructure 
continues in Turkey and also structuring the national geoid 
model of Turkey is a part of these efforts. So far, regional 
geoid model of Turkey revised several times. The latest geoid 
model is "Turkey Geoid 1999A" (TG99A). TG99A geoid 
model satisfies necessary accuracy for producing large scale 
maps and routine surveying applications. However, in some 
parts, this gravimetric based refined geoid model stays weak in 
accuracy for practical geodetic applications. 
In this case, determining local precise geoid models using more 
intensive data in areas where national geoid model does not 
have sufficient absolute accuracy is purposed as a solution. By 
this way, weaker parts of the national geoid model will be 
supported by local these local solutions and the component of 
geodetic infrastructure as well. 
From this view point, to serve the practical geodetic and 
surveying applications, determining the geoid model of a 
77 
limited area as an analytical surface using homogeneously 
distributed and GPS/Levelling reference points with appropriate 
density constitutes the subject of this study. 
3. SURFACE FITTING METHODS IN MODELLING 
GPS/ LEVELLING GEOID IN A LOCAL AREA 
Among the geoid modelling techniques as based on geometrical 
approach, fitting a surface, which depends on the reference 
points that are chosen in the critical and characteristic locations 
of the field to represent trend of geoid surface, is a common 
method in small areas for local studies. Representing geoid 
heights as mathematically formulated surface and calculating 
the geoid heights in new measured points according to GPS 
technique constitutes the idea in these kinds of studies (Erol and 
Celik, 2004). 
However, in a local area, determined geoid model with surface 
fitting technique works with in the area that surrounded and 
covered by reference points properly and these kinds of models 
does not provide reliable results for the extrapolation points. 
One another important handicap in local geoid determination is 
datum inconsistency problem but in this study, this problem is 
not going to be considered because of the focus is on testing 
surface fitting algorithms as a geometrical approach for 
modelling a local geoid according to GPS and Levelling data. 
There are several important factors that affect the accuracy of 
GPS/ Levelling geoid model (Erol and Celik, 2004). These are; 
— Distribution and number of reference stations. (GPS/ 
Levelling stations) (these points must be distributed 
homogeneously to the coverage area of the model and have 
to be chosen to figure out the changes of geoid surface). 
— The accuracy of GPS derived ellipsoidal heights (h) and the 
heights derived from levelling measurements (H). 
— Characteristic of the geoid surface in the area. 
— Used method while modelling the geoid (researches showed 
that there is not a unique model works properly for realizing 
the geoid surface of different areas) (Erol and Celik, 2004). 
Interpolation methods are most common approaches that are 
used while modelling the geoid heights (N) in a local arca. 
There are different interpolation algorithms and each of them 
can have different results when interpreting your data. Inverse 
distance weighting and geostatistical Kriging are two of them 
and very popular also in modelling spatial data as well as in 
evaluation of some other data sets of different disciplines 
(Anonym, 1999), 
In this study these two methods have been chosen to model the 
geoid undulations in a local area and to prove that they give 
satisfying results to serve practical applications in geodesy. By 
applying them, the provided results and the performances of 
both arc compared each other. These two interpolation methods 
constitutes solution algorithms of a geoid modelling software 
that is still in developing process as the product of a thesis 
study, and because of that in this study, these two methods were 
evaluated to investigate and compare the performance of 
interpolation algorithms in local geoid modelling. 
Inverse Distance Weighting (IDW) is a weighted average 
interpolator. The power parameter controls how the weighting 
factors drop off as distance from the reference point increases 
= 
MERE 
i 
SESS Ze ze Zt