ıbul 2004 
degree, 
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(3) 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B-YF. Istanbul 2004 
  
In case of the geoid approximation the function has two input 
data, two variables. The RBF network output is formed by a 
weighted sum of the outputs of neurons: 
A 
n > ^" 
zÁs dO xe 5t -e:.] 
f x))7 Y we ! (4 
i=] 
where X, y = input data 
A, Cı, C2 = parameters of the activation function 
n = number of applied neurons 
w = weights of the neuron’s outputs 
The parameters (À, c,, c», w) are determined during a supervised 
learning algorithm, using a teaching set to minimize the 
deviation between the known geoid heights and the outputs of 
the network. 
The geoid heights are known in 211680 points, from these 
database 8484 points were selected for training the RBF 
network at a grid of Aq-2'30" x AX- 4'10" resolution. (The 
original grid’s resolution was A@=0’30" x AA= 0°50”). The 
training procedure was executed with different numbers of 
neurons. The best configuration was using 35 neurons 
(Zaletnyik 2003). After the training procedure the network was 
tested in the whole database with the 211680 points. The 
summarized statistical data of the training set (teaching points) 
and the testing set are in Table 2. 
According to our experience, the iteration process is converging 
rapidly, and after 3-4 iteration steps there was no further 
significant change in the values. Therefore in this study 4 
networks were used. The first was a RBF neural network, and 
then the later used neural networks had saturated line activation 
function. The network learned fairly well. The results of the 4" 
order network are summarized in Table 3. 
  
Min Max Mean St. dev. 
[m] [m] [m] [m] 
  
Teaching set (8484 
«0.3 362 
points, 4° order) 0.367 | 0.362 | 0.000 | 0.066 
  
  
Testing set (211680 
points, 4* order) -0.506 | 0.433 0.000 | 0.068 
  
  
  
  
  
  
Min Max Mean St. dev. 
[m] [m] [m] [m] 
  
Teaching set (8484 
points, RBF network) -0.367 | 0.585 0.000 | 0.098 
  
estins set (? 
Testing set (211680 1g 416 | 0.600 | 0.000 | 0.099 
  
  
  
  
  
points, RBF network) 
  
  
Table 2. Quality of the estimation with RBF neural network 
The results of the testing set and the teaching set are very 
similar, between the two standard deviations the difference is 1 
mm and the maximum, minimum values are also very close to 
each other. All things considered can be declared that the 
training set with the 8484 points can represent quite well the 
whole database of the known geoid heights. 
For our purposes the accuracy of the results was not enough. 
Generally the accuracy can be improved with increasing the 
number of the neurons, but in this case with more neurons the 
efficiency of the network decreased, the training procedure was 
slower and the improvement of the accuracy was not 
significant. Therefore to improve the estimation of the network 
we had to look for a new method. 
3.2 Sequence of neural networks 
To improve the approximation a sequence of neural networks 
has been applied. The first term of this series of networks 
estimates the values of the geoid heights, while the second term 
estimates the error of the first network, the third term estimates 
the error of the second network and so on. Assuming that the 
relative error of every network in this sequence is less than 
100%, the sum of the estimated error can be reduced very 
significantly and efficiently (Paláncz, Vólgyesi 2003). 
Table 3. Quality of the 4™ order network 
Comparing the results of the first network with the fourth 
network the value of standard deviations was reduced with 
about 30 percents. And comparing these results with the 
polynomial approach the improvement is more significant, 
about 60 percents. However the maximum errors are still too 
big. Figure 3 shows differences between the estimated and the 
original geoid heights. 
  
Figure 3. Differences between the estimated and the original 
geoid heights 
Examining the distribution of the errors it was noticed that the 
greatest errors are outside of Hungary, in the south-east region, 
in Romania. In that region the quality of the input data of the 
geoid solution was not reliable. This could be the reason of 
these big errors. For our purposes these data are not necessary, 
because we only try to find a good geoid approximation in the 
region of Hungary, so they can be left out cutting them along a 
line. The equation of this line is very simple: p=A+25. Figure 4 
shows this cutting line.