International Archives of the Photogrammetry, Remote Sensing 
(Töth and Rözsa 2000). The accuracy of HGTUB2000 geoid 
heights is about +3-4 cm. The used geoid heights cover the area 
of 45?30'« q < 49°, 16°< A < 23°; the resolution of the grid is 
A@=0’30" x AX- 0750". So the actual geoid heights are known 
in 211680 points. The geoid heights in the area vary between 
37.0 and 47.1 m. Figure 1 shows the geoid surface in Hungary. 
  
Figure 1. The HGTUB2000 geoid surface in Hungary 
Instead of the application of this huge geoid database for 
practical purposes we tried to find a simple mathematical 
formula (an equation of surface of geoid forms in Hungary). 
Using this mathematical formula to compute geoid heights in 
arbitrary points in Hungary would be simpler than interpolating 
the geoid heights between known points, especially if it should 
be implemented in a computational procedure. 
2. POLYNOMIAL FITTING 
For global data representation, like the approximation of a 
surface, algebraic or trigonometric polynomials, least squares 
collocation or weighted linear interpolation may be applied. 
The interpolation or regression methods can be considered not 
only for computing unmeasured values but for compressing the 
data, too. In this case, with 211680 points, the data compression 
is a very important viewpoint. 
As a classical approximation model polynomial fitting was used 
to approximate the geoid heights as a function of geographic 
coordinates ©, À. 
The formula of the used 6" order fitting polynomial is the 
following: 
N =a, ta, Pay Atay 9 +a, @ Ata A a, gh 
a,:Q A+, 9: A a: A +a, @' +a, Ada pA (2) 
ta, 0A +a, A as 9° ta. Atay, 9 A 
ta, 02 Ara, pA +a, A +a, 5 a4: A+ 
ha T A +05 (p A ss o AT Ty (A t, Ae 
where a; 7 coefficients of the polynomial 
N = geoid height 
Q, À = geodetic latitude, longitude. 
Differences between known geoid heights and approximated 
values are characteristic of accuracy of geoid heights computed 
by polynomials. Increasing the degree of polynomials, first 
and Spatial Information Sciences, Vol XXXV, Part B-YF. Istanbul 2004 
accuracy was increased, then decreased above the sixth degree, 
because of the deterioration of conditions of equations. 
The most important statistical data describing the quality of the 
estimation are the followings: maximum, minimum error, mean 
value, standard deviation. These statistical data of the 
polynomial fitting are summarized in Table |. 
  
Min [m] Max [m] Mean [m] St. dev. [m] 
  
  
-0.812 0.722 0.000 0.180 
  
  
  
  
  
Table 1. Quality of the polynomial fitting 
The maximum accuracy resulted by applying 6" order 
polynomial was not enough for our purposes therefore a new 
method was needed to look for. As an alternative to the 
classical polynomial fitting a series of neural networks has been 
applied to approximate geoid heights. 
3. APPROXIMATION WITH SEQUENCE OF NEURAL 
NETWORKS 
3.1 Approximation with RBF neural network 
To estimate the geoid, a RBF (Radial Basis Function) 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. Figure 2 
illustrates the applied RBF network with input ©, A (geodetic 
latitude, longitude) and output N (geoid height). The RBF 
network consists of one hidden layer of activation functions, or 
neurons. 
  
  
  
Figure 2. Applied RBF network with one output 
The basis or activation function is a Gaussian bell-shaped curve 
with two parameters: 
fase 1 (3) 
where A = parameter of the function's width 
c = centre of the function 
x = input data 
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