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Figure 4. Cutting out the unreliable data of the south east region 
Cutting out these data the number of points of the teaching set 
was reduced to 7438 and that of the testing set was reduced to 
184910. Executing again the teaching and testing procedures 
the results of these networks were significantly better. 
  
Min Max Mean | St. dev. 
[m] [m] [m] [m] 
  
Teaching set (7438 
-().2 
points, 4“ order) 0.234 | 0.284 | 0.000 | 0.049 
  
Testing set (184910 
points, 4" order) «0.351 | 0.291 0.000 | 0.050 
  
  
  
  
  
  
  
Table 4. Quality of the 4% order network cutting out the south 
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east region 
The standard deviation is reduced to 5 cm, and the maximum 
errors are lower with 10-15 cm than the maximum errors of the 
first sequence of neural networks. Figure 5 shows the 
differences between the estimated and the original geoid heights 
leaving out the south east region. The accuracy of the original 
geoid heights was about €3-4 cm. In Figure 5 the errors smaller 
than 4 cm are indicated with white color. According to Figure 5 
in the greatest area of Hungary the errors of the estimation are 
of the same order as the errors of the original data. The 
estimation via sequence of neural networks provides a good 
approximation of the geoid heights in Hungary. 
  
  
  
16.00 17.00 18.00 19.00 20.00 21.00 22.00 2300 B 20 
:024 
Figure 5. Differences between the estimated and the original 
geoid heights cutting out the south east region 
4. SUMMARY 
In this research a sequence of neural networks was applied to 
approximate the geoid surface in the area of Hungary. To 
analyze the result, the errors of the estimation were compared 
with the errors of other approximation methods, with 
polynomial fitting and with a single RBF neural network. In 
both cases the sequence of neural networks proved to be better. 
On the basis of our research can be statred that using this 
method the error of the estimation can be reduced efficiently, 
even in the case of a morphologically so sophisticated data 
structure as a geoid. 
For the approximation of the geoid surface a gravimetric geoid 
solution was used with 211680 known geoid heights in a 
regular grid. 8484 points were selected for the teaching set from 
the whole database, and the approximation method was tested 
in every known point. In accordance with the results the 
teaching set can represent quite well the whole database of the 
known geoid heights. 
Cutting out an area with unreliable data outside of Hungary the 
estimation was improved significantly. The standard deviation 
of the errors of estimation was reduced to 5 cm and this 
accuracy is of the same order as the accuracy of the original 
data. 
5. REFERENCES 
Kenyeres A. 1999: Phys. Cem. Earth (A), 24, pp. 85-90. 
Palancz B., Vólgyesi L. 2003: High accuracy data 
representation via sequence of neural networks, Acta Geod. 
Geoph. Hung., Vol. 38 (3), pp. 337-343 
Papp G. Kalmár J. 1996: In: Proceedings of the 7 
International Meeting on Alphine Gravimetry, Osterreichische 
Beiträge zu Meteorologie und Geophysik, pp. 95-96. 
Tóth Gy., Rózsa Sz. 2000: New Datasets and Techniques — an 
Improvement in the Hungarian Geoid Solution, Paper presented 
at Gravity, Geoid and Geodynamics Conference, Banf, Alberta, 
Canada 
Zaletnyik P. 2003: Neurális hálózatok alkalmazasa a 
geodéziában, MSc. Thesis, Budapest University of Technology 
and Economics 
6. ACKNOWLEDGEMENTS 
| Our investigations were supported by the Hungarian National 
Research Fund (OTKA), contract No. T-046718. 
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