UDC: 
DOI: 
10.22389/0016-7126-2019-952-10-2-9
1 Neiman Yu.M.
2 Sugaipova L.S.
3 Popadyev V.V.
Year: 
№: 
952
Pages: 
2-9

Moscow State University of Geodesy and Cartography (MIIGAiK)

1, 
2, 

Center of Geodesy, Cartography and SDI

3, 
Abstract:
As we know the spherical functions are traditionally used in geodesy for modeling the gravitational field of the Earth. But the gravitational field is not stationary either in space or in time (but the latter is beyond the scope of this article) and can change quite strongly in various directions. By its nature, the spherical functions do not fully display the local features of the field. With this in mind it is advisable to use spatially localized basis functions. So it is convenient to divide the region under consideration into segments with a nearly stationary field. The complexity of the field in each segment can be characterized by means of an anisotropic matrix resulting from the covariance analysis of the field. If we approach the modeling in this way there can arise a problem of poor coherence of local models on segments’ borders. To solve the above mentioned problem it is proposed in this article to use new basis functions with Mahalanobis metric instead of the usual Euclidean distance. The Mahalanobis metric and the quadratic form generalizing this metric enables us to take into account the structure of the field when determining the distance between the points and to make the modeling process continuous.
References: 
1.   Gofman-Vellengof B., Morits G. Fizicheskaya geodeziya. / Per. s angl. pod red. Yu. M. Neimana. - Moskva: Izd-vo MIIGAiK, - 2007. - 410 p.
2.   Neiman Yu. M., Sugaipova L. S. O kovariatsionnom analize neodnorodnogo gravitatsionnogo polya Zemli. Izv. vuzov. Geodeziya i aerofotos"emka, - No 5. - 2013. - pp. 15–22.
3.   Darbehesti N., Featherstone W. E. (2009) Nonstationary covariance function modelling in 2D least-squares collocation. Geodesy, 83, pp. 495-508,
4.   Mahalanobis P. C. (1936) On the generalized distance in statistics. Proceedings of the National Institute of Sciences of India, Volume 2, 1, pp. 49-55,
5.   Michel V. (2013) Lectures on Constructive Approximation. Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball. Springer, New York, 324 p.
6.   Paciorek C. J., Schervish M. J. (2003) Nonstationary Covariance Functions for Gaussian Process Regression, Department of Statistics. Carnegie Mellon University, Neural Information Processing Systems. URL: papers.nips.cc/paper/2350-nonstationary-covariance-functions-for-gaussian-process .
Citation:
Neiman Yu.M., 
Sugaipova L.S., 
Popadyev V.V., 
(2019) Sfericheskie radial'nye bazisnye funktsii s metrikoi Makhalanobisa dlya otobrazheniya lokal'nykh osobennostei polya [Spheric radial basis functions in Mahalanobis measuring for displaying local field features]. Geodesy and Cartography = Geodezija i kartografija, 80, 10, pp. 2-9. (In Russian). DOI: 10.22389/0016-7126-2019-952-10-2-9
ARTICLE
Received: 26.08.2019
Accepted: 08.10.2019
Published: 20.11.2019

Content

2019 October DOI:
10.22389/0016-7126-2019-952-10

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