International Archives of the Photogrammetry, Remote Sensing 
(Anonym, 1999). This is fast and has relatively simple 
mathematical algorithm. 
In equation 2 and 3, the relations are given to calculate the 
geoid undulation point k as related to the surrounding reference 
points i=1,2,3 .. n. The point k and neighbor reference points 
contributed to the computation of geoid undulation value of 
point k according to certain weight values are illustrated in 
Figure 2 (in the figure i=7). 
n 
YN 77 
N'= À (2) 
= v. 
izl 
Il 
N’ = geoid undulation value of point k 
: . at x : 
N; = geoid undulation value of i" reference point 
. nfl > 
P; = weight of i" reference point 
where 
den (3) 
1 n 
1 
og 
where di = distance between interpolation point k and i" 
reference point in kilometer 
n = power of the distance, it can be 1, 2, 3 or 4. In the 
case study, n has chosen 3 empirically. 
  
Figure 2. Interpolation point k of which geoid undulation value 
is going to be computed and neighbor reference points that 
contributes the computation of geoid undulation at point k. 
In general, it is not usual to contribute all the reference points to 
the computation of geoid undulation value according to IDW 
interpolation method. Because of that a boundary is described 
to surround an area to include the reference points which are 
going to be contribute to the computation. There are several 
ways to describe the boundary. One of them is to determine a 
circle as being the interpolation point in the center of it. The 
radius of the circle is determined according to conditions of the 
cover area of the study and in this example a 3 km radius was 
described by considering the topographical properties of the 
local area. 
Kriging is a geostatistical approach to interpolate data based 
upon spatial variance and has proven useful and popular in 
many fields in geodesy as well. This is a considerably flexible 
method and similar to IDW whereby proximity and influence 
are assumed to be related, Kriging recognizes that spatial 
variance is a function of distance (Wilson, 1996). It can be 
custom fit to a data set by specifying the appropriate variogram 
and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
model. The variogram model mathematically specifies the 
spatial variability of the data set. The weights of reference 
points, which are applied to data points during calculations, are 
direct functions of the variogram model (Anonym, 1999). 
Computing an experimental variogram from your data is the 
only certain way to determine which variogram model you 
should use. A detailed variogram analysis can offer insights 
into the data that would not otherwise be available, and it 
allows for an objective assessment of the variogram scale and 
anisotropy (Anonym, 1999). In an other word structural effects 
have been accounted for the surfacc may be modeled on the 
basis of variations as a function of distance. A mathematical 
model is fit to this data in order to interpolate unknown values 
at other locations. Spatial variation may also be more 
predominate in certain directions. Anisotropy implies the 
preferred direction of higher or lower continuity between data 
points. Anisotropy is applied by specifying an anisotropy ratio 
which states: “Give more weighting to points located along one 
axis versus points located along another axis”. The relative 
weighting is defined by an anisotropy ratio (Wilson, 1996). 
The method based on the recognition that the spatial variation 
of any property, known as a ‘regionalized variable’, is too 
irregular to be modeled by a smooth mathematical function but 
can be described better by a stochastic surface. The 
interpolation proceeds by first exploring and then modelling the 
stochastic aspects of the regionalized variable. The resulting 
information is then used to estimate weights for interpolation 
points and in the Kriging method, the logic is similar. 
Geostatistics, similar to any form of statistics, has two main 
criteria which must be met. All statistical models have 
assumptions which need to be recognized and expectantly meet 
before the model is used. The second criterion, a statistical 
analysis, to perform on the data set, should be explored for 
normality and spatial variance with data transformations 
applied as necessarily. Geostatistical assumptions are reviewed 
by Issaaks and Srivastava, 1989. Regionalized variable theory 
assumes that spatial variation is the sum of three components 
(Wilson, 1996); 
— A first order effect, in another word, known as a *structural 
component’, which is defined by a constant trend. This part 
gives trend and is depicted graphically in a variogram as the 
“sill”. It implies that at these values of the ‘lag’ (distance) 
there is no spatial dependence between the data points 
based on variance. 
_ A second order effect is defined as a random spatially 
correlated component. Referred lo as “variance of 
differences” and this is a function of distance between sites 
(the distance is called as “lag”). Variance increascs from 
random noise (the nugget) to the sill as distance increases. 
The distance is important as it specifies the distance site 
differences are spatially dependent. 
— Random noise or residual error, known as the “nugget”. 
  
For a detailed explanation about Kriging Method Isaaks and 
Srivastava, 1989; Watson, 1992 and Deutsch and Journel, 1992 
can be seen. 
[n the case study, universal Kriging method was applied with 
linear variogram model. The results of the case study will be 
given under following title. 
78 
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