Spheroidal spline interpolation and its application in geodesy
Abstract
The aim of this paper is to study the spline interpolation problem in spheroidal geometry. We follow the minimization of the norm of the iterated Beltrami-Laplace and consecutive iterated Helmholtz operators for all functions belonging to an appropriate Hilbert space defined on the spheroid. By exploiting surface Green’s functions, reproducing kernels for discrete Dirichlet and Neumann conditions are constructed in the spheroidal geometry. According to a complete system of surface spheroidal harmonics, generalized Green’s functions are also defined. Based on the minimization problem and corresponding reproducing kernel, spline interpolant which minimizes the desired norm and satisfies the given discrete conditions is defined on the spheroidal surface. The application of the results in Geodesy is explained in the gravity data interpolation over the globe.
Keyword : spheroid, discrete Dirichlet and Neumann conditions, norm minization, spline interpolation, Green’s function, gravity data interpolation
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
Amiri-Simkooei, A. R., Hosseini-Asl, M., & Safari, A. (2018). Least squares 2D bi-Cubic spline approximation: Theory and applications. Measurement, 127, 366–378. https://doi.org/10.1016/j.measurement.2018.06.005
Baramidze, V., Lai, M. J., & Shum, C. K. (2006). Spherical splines for data interpolation and fitting. Society for Industrial and Applied Mathematics Journal on Scientific Computing, 28(1), 241–259. https://doi.org/10.1137/040620722
Castilho, C., & Machado, H. (2008). The -Vortex problem on a symmetric ellipsoid: a perturbation approach. Journal of Mathematical Physics, 49(2). https://doi.org/10.1063/1.2863515
Felsen, L. B., & Marcuritz, N. (1994). Radiation and Scattering of Waves. IEEE Press Series on Electromagnetic Waves. John Wiley Sons, Inc. https://doi.org/10.1109/9780470546307
Freeden, W., Nashed, M. Z., & Schreiner, M. (2018). Spherical sampling. Springer, Switzerland. https://doi.org/10.1007/978-3-319-71458-5
Freeden, W., & Gutting, M. (2013). Applied and numerical harmonic analysis. Springer, Basel.
Freeden, W., & Gutting, M. (2018). Integration and cubature methods: A geomathematically oriented course. Taylor Francis Group. https://doi.org/10.1201/9781315195674
Freeden, W., Gervens, T., & Schreiner, M. (1998). Constructive approximation on the sphere. Clarendon press, Oxford University Press, UK.
Freeden, W. (1984). Spherical spline interpolation: Basic theory and computational aspects. Journal of Computational and Applied Mathematics, 11(3), 367–375. https://doi.org/10.1016/0377-0427(84)90011-6
Freeden, W., & Schreiner, M. (2009). Spherical functions of mathematical geosciences. Springer-Verlag, Berlin Heidelberg. https://doi.org/10.1007/978-3-540-85112-7
Freeden, W. (1981). On spherical spline interpolation and approximation. Mathematical Methods in the Applied Sciences, 3(1), 551–575. https://doi.org/10.1002/mma.1670030139
Greenberg, M. D. (2015). Application of Green’s functions in science and engineering. Dover Books on Engineering, Library of cataloging in publication data. Courier Dover Publications.
Jekeli, C. (1988). The exact transformation between ellipsoidal and spherical harmonic expansions. Manuscripta geodaetica, 13, 106–113.
Keller, W., & Borkowski, A. (2019). Thin plate spline interpolation. Journal of Geodesy, 93, 1251–1269. https://doi.org/10.1007/s00190-019-01240-2
Klees, R., Tenzer, R., Prutkin, I., & Wittwer, T. (2008). A datadriven approach to local gravity field modelling using spherical radial basis functions. Journal of Geodesy, 82, 457–471. https://doi.org/10.1007/s00190-007-0196-3
Sloan, I. H., & Womersley, R. S. (2002). Good approximation on the sphere; with application to geodesy and the scattering of sound. Journal of Computational and Applied Mathematics, 149, 227–237. https://doi.org/10.1016/S0377-0427(02)00532-0
Szmytkowski, R. (2006). Closed form of the generalized Green’s function for the Helmholtz operator on the two dimensional unit sphere. Journal of Mathematical Physics, 47(6). https://doi.org/10.1063/1.2203430
Wahba, G. (1981). Spline interpolation and smoothing on the sphere. Society for Industrial and Applied Mathematics Journal on Scientific and Statistical Computing, 2(1), 5–16. https://doi.org/10.1137/0902002
Wahba, G. (1990). Spline models for observational data. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania. https://doi.org/10.1137/1.9781611970128