Proceedings, XXth congress: Proceedings, XXth congress

Altan, M. Orhan

Bibliographic data

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

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 120 Int W T T si m ea tre w| Fc G nt ef sk si Wi Te ha es es th re sig

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