Proceedings, XXth congress: Proceedings, XXth congress

Altan, M. Orhan

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Multivolume work

Persistent identifier:: 1663674213

Title:: Proceedings, XXth congress

Sub title:: Istanbul, 12 - 23 July 2004

Year of publication:: 2004

Place of publication:: Istanbul

Publisher of the original:: [Verlag nicht ermittelbar]

Identifier (digital):: 1663674213

Language:: English

Additional Notes:: Erscheinungsdatum des Originals ist aus dem Copyrightjahr ermittelt.; Auch bezeichnet als XXth International Congress for Photogrammetry and Remote Sensing

Editor:: Altan, M. Orhan

Corporations:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing, Commission Primary Data Acquisition

Adapter:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing, Commission Primary Data Acquisition

Founder of work:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing, Commission Primary Data Acquisition

Other corporate:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing, Commission Primary Data Acquisition

Document type:: Multivolume work

Volume

Persistent identifier:: 166368779X

Title:: Proceedings, XXth congress

Scope:: IV, 226 Seiten

Year of publication:: 2004

Place of publication:: Istanbul

Publisher of the original:: [Verlag nicht ermittelbar]

Identifier (digital):: 166368779X

Illustration:: Illustrationen, Diagramme

Signature of the source:: ZS 312(35,B8)

Language:: English

Additional Notes:: Erscheinungsdatum des Originals ist aus dem Copyrightjahr ermittelt.

Usage licence:: Attribution 4.0 International (CC BY 4.0)

Editor:: Altan, M. Orhan

Corporations:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing

Adapter:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing

Founder of work:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing

Other corporate:: International Society for Photogrammetry and Remote Sensing, Congress, 20., 2004, Istanbul; International Society for Photogrammetry and Remote Sensing

Publisher of the digital copy:: Technische Informationsbibliothek Hannover

Place of publication of the digital copy:: Hannover

Year of publication of the original:: 2019

Document type:: Volume

Collection:: Earth sciences

Chapter

Title:: APPROACH OF THE HUNGARIAN GEOID SURFACE WITH SEQUENCE OF NEURAL NETWORKS P. Zaletnyik, L. Völgyesi, B. Paláncz

Document type:: Multivolume work

Structure type:: Chapter

ıbul 2004 degree, Ly of the yr mean of the ev. [m] 180 ^ order e a new > to the has been CURAL 1) neural Gaussian cause the he most Figure 2 ‘geodetic ‘he RBF tions, or it ped curve (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.

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