Full text: Proceedings, XXth congress (Part 8)

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P. Zaletnyik ^ * , L. Vólgyesi * , B. Paláncz " 
* Department of Geodesy and Surveying, Budapest University of Technology and Economics, H-1521 Budapest, Hungary, e-mail: 
volgyesi@eik.bme.hu, zaletnyikp@hotmail.com 
? Department of Photogrammetry and Geoinformatics, Budapest University of Technology and Economics, H-1521 Budapest, 
Hungary, e-mail: palanez@epito.bme.hu 
KEY WORDS: Neural, Artificial Intelligence, Surface, Data Mining, Geodesy 
For global data representation, like the approximation of a surface, algebraic or trigonometric polynomials may be used. However, 
polynomial approaches are limited concerning their accuracy. In the last decade neural networks were applied very successfully in 
many fields of data mining and representation. 
In this research sequence of neural networks has been employed to high accuracy regression in 3D as data representation in form 
z = f(x,y). The first term of this series of networks estimates the values of the dependent variable as it is usual, 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 
effectively. To illustrate this method the geoid of Hungary was estimated. To approach this surface, a RBF neural network has been 
employed with 35 neurons having Gaussian activation functions. We used this type of network, because the radial basis type 
activation function proved to be the most efficient in case of function approximation problems. According to our experience, the 
iteration process is converging rapidly, and after 3-4 iteration steps there were no further significant change in the values. 
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, it is about 60 percents. 
These computations were carried out with the symbolic-numeric integrated system Mathematica. 
1. INTRODUCTION N=h-H : (1) 
Nowadays the GPS measurements are one of the most 
frequently used technique in geodesy. With this technique where N = geoid height 
ellipsoidal height can be reckoned. However in the engineering h = ellipsoidal height 
practice orthometric heights (height above sea level) are used. H = orthometric height. 
The orthometric heights are determined by levelling. 
Transforming the GPS-derived ellipsoidal heights to The transformation of ellipsoidal heights to orthometric ones 
orthometric heights it is important to know the distance between requires that the geoid height must refer to the same reference 
the ellipsoidal and the geoid surface, called the geoid height or ellipsoid (to the WGS-84 ellipsoid). 
geoid undulation. 
With the spreading of the GPS measurements the need for a 
In geodesy two types of Earth’s figure are distinguished. One of good geoid model has been increased. The geoid can be 
them is the physical or topographic earth’ surface, that is the calculated from different types of input data. The simplest 
real surface of the Earth with the mountains, seas and plains. method is to use GPS and levelling points, where both the 
The other is the mathematical or the theoretical figure of the ellipsoidal and  orthometric heights are given. Another 
Earth. This is the shape of the free water surface be balanced by possibility and the most commonly used technique for precise 
the gravity field only, this surface is one of the equipotential determination of geoid is using the gravimetric solution, carried 
surfaces of the gravity. The equipotential surface at the mean out by the Stokes-integral. 
sea level (MSL) is called geoid. 
In Hungary considerable investigations are in progress for the 
The geoid is the reference surface of the orthometric heights, determination of the separation of the geoid: lithospheric geoid 
but the reference surface of the GPS measurements is an solution (Papp and Kalmar 1996), gravimetric solution HGR97 
ellipsoid, the WGS-84. The geoid height can be computed with (Kenyeres 1999), HGTUB98 and HGTUB2000 solution (Téth 
a simple subtraction: and Rózsa 2000). 
The HGTUB2000 geoid heights were used for our 
investigations. This gravimetric solution was based on 
terrestrial gravity data, height data and the EGM96 geopotential 
model, and was computed with the 1D Spherical FFT method 
* Corresponding author. 

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