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Proceedings, XXth congress

International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXX V, Part B-YF. Istanbul 2004

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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
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
Figure 5. Differences between the estimated and the original
geoid heights cutting out the south east region
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
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,
Zaletnyik P. 2003: Neurális hálózatok alkalmazasa a
geodéziában, MSc. Thesis, Budapest University of Technology
and Economics
| Our investigations were supported by the Hungarian National
Research Fund (OTKA), contract No. T-046718.
— — m juwed O9. 3.
Y. MM m Py
gent a.