Abstract
Most boundary value problems of the geopotential field have integral and series solutions in terms of Green’s convolution kernels. These solutions are advantageously evaluated using fast Spherical Harmonic Transforms (SHTs) for regular arrays of simulated or observed global data. However, the computational comlexity and numerical conditioning of SHTs for relatively dense data are quite challenging and recent algorithmic developments warrant further investigations for geodetic and geophysical applications.
Global multiresolution application for scalar, vector and tensor fields on the Earth and its neighborhood require spherical harmonic analysis and synthesis using convolution filters with data decimationand dilation. For global spherical grid applications, efficient and reliable SHTs are needed just as Fast Fourier Transforms (FFTs) are used in regional planar applications.
With the availability of enormous quantities of space, surface and subsurface data, extensive data structuring and management are unavoidable for most array computations. Different methodologies imply very different strategies and conflicting claims often appear in the literature. Discussions of the implicit and other assumptions with simulated results would undoubtedly help to clarify the situation and help decide on appropriate data structuring strategies for different computational applications.
Keywords
- Discrete Fourier Transform
- Cosmic Microwave Background
- Geopotential Model
- Associate Legendre Function
- Hierarchical Data Format
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Adams, J.C. and P.N. Swarztrauber. SPHEREPACK 2.0: A Model Development Facility, 1997. http://www.scd.ucar.edu/softlib/SPHERE.html
Augenbaum, J.M. and C.S. Peskin. On the Construction of the Voronoi Mesh on a Sphere. Journal of Computational Physics, 59, pp. 177–192, 1985.
Blais, J.A.R. and D.A. Provins. Spherical Harmonic Analysis and Synthesis for Global Multiresolution Applications. Journal of Geodesy, vol. 76, no. 1, pp. 29–35, 2002.
Bond, J.R., R.G. Crittenden, A.H. Jaffe and L. Knox. Computing Challenges of the Cosmic Microwave Background. Computing in Science and Engineering, March–April, pp.21–35, 1999.
Colombo, O. Numerical Methods for Harmonic Analysis on the Sphere. Ohio State University, Report no. 310, 1981.
Crittenden, R.G. and N.G. Turok. Exactly Azimuthal Pixelizations of the Sky, 1998. http://xxx.lanl.gov/list/astro-ph/9806[374] (I.e. article 374 in the 9806 directory).
Driscoll, J.R. and D.M. Healy, Jr. Comuting Fourier Transforms and Convolutions on the 2-Sphere. Advances in Applied Mathematics, 15, pp. 202–250, 1994.
Gold, C.M. Problems with Handling Spatial Data — The Voronoi’s Approach. CISM Journal ACSGC, vol. 45, no. 1, pp. 65–80, 1991.
Gorski, K.M., E. Hivon and B.C. Wandlt. Analysis Issues for Large CMB Data Sets. Proceedings: Evolution of Large Scale Structure, Garching, 1998.
Lemoine, F.G., D.E. Smith, L. Kunz, R. Smith, E.C. Pavlis, N.K. Pavlis, S.M. Klosko, D.S. Chinn, M.H. Torrence, R.G. Williamson, C.M. Fox, K.E. Rachlin, Y.M Wang, S.C. Kenyon, R. Salman, R. Trimmer, R.H. Rapp and R.S. Nerem. The Development of the NASA GSFC and NIMA Joint Geopotential Model. International Symposium Gravity, Geoid and Marine Geodesy, Tokyo, International Association of Geodesy Symposia, Vol. 117, pp. 461–469, Springer-Verlag, 1996.
Lukatela, H. Hepparchus Geopositioning Model: An Overview. Proceedings of AUTOCARTO 8, 1987. http://www.geodyssey.com.
Mohlenkamp, M.J. A Fast Transform for Spherical Harmonics. The Journal of Fourier Analysis and Applications, vol. 5, nos. 2/3, pp. 159–184, 1999.
Muciaccia P.F., P. Natoli and N. Vittorio. Fast Spherical Harmonic Analysis: A Quick Algorithm for Generating and/or Inverting Full-Sky, High Resolution cosmic Microwave Background Anisotropy Maps. The Astrophysical Journal, 488, pp. L63–L66, 1997.
NCSA Introduction to HDF5 Release 1.0, National Center for Supercoming Applications, 1999.
Paul, R.H. Recurrence Relations for the Integrals of Associated Legendre Functions. Bulletin Géodésique, Vol. 52, pp. 177–190, 1978.
Ricardi, L.J. and M.L. Burrows. A Recurrence Technique for Expanding a Function in Spherical Harmonics. IEEE Transactions on Computers, June, pp. 583–585, 1972.
Short, N.M., Jr., R.F. Cromp, W.J. Campbell, J.C. Tilton, J.L. LeMoigne, G. Fekete, N.S. Netanyahu and G. Gylvain. AI Challenges within NASA’s Mission to Planet Earth. Workshop Notes for the 1994 National Conf. on Artificial Intelligence, C.L. Mason (Ed.).
Sneeuw, N. Global Spherical Harmonic Analysis by Least-Squares and Numerical Quadrature Methods in Historical Perspective. Geophys. J. Int. (1994) 118, 707–716.
Swarztrauber, P.N. On the Spectral Approximation of Discre Scalar and Vector Functions on the Sphere. SIAM Journal of Num. Analysis, Vol. 16, No. 6, pp. 934–949, 1979.
Tan, C.J., J.A.R. Blais and D.A. Provins. Large Imagery Data Structuring Using Hierarchical Data Format for Parallel Computing and Visualization. High Performance Computing Systems and Applications, edited by A. Pollard, D. J.K. Mewhort and D.F. Weaver, Kluwer Academic Publisher, Chapter 39, pp. 371–386, 2000.
Varshalovich, D.A., A.N. Moskalev and V.K. Khersonskij. Quantum Theory of Angular Momentum. World Scientific Publishing, Singapore, 1988.
Wandelt, B.D., E. Hivon and K.M. Gorski. Topological Analysis of High-Resolution CMB Maps. Theoretical Astrophysics Centre, Copenhagen, Denmark, 1998, 4 pages.
Wenzel, G. Ultra High Degree Geopotential Model GPM3E97A to Degree and Order 1800 Tailored to Europe. Proceedings of the Second Continental Workshop on the Geoid in Europe, Budapest, 1998.
Wenzel, G. Ultra High Degree Geopotential Models GPM98A, B and C to Degree 1800. Preprint, Bulletin of International Geoid Service, Milan, 1998.
Wenzel, G. Hochaufloesende Kugelfunktionsmodelle fuer das Gravitationspotential der Erde. Wissenschaftliche Arbeiten der Fachrichtung Vermessungswesen der Universitat Hannover, Nr. 135, Hannover, 1985.
Zheng, C., J. Nie and J.A.R. Blais. Applicability of the Hepparchus Software in Geoscience Information Systems. Proceedings of the Canadian 1994 GIS Conference in Ottawa, pp. 434–442.
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Blais, J.A.R., Provins, D.A. (2003). Optimization of Computations in Global Geopotential Field Applications. In: Sloot, P.M.A., Abramson, D., Bogdanov, A.V., Gorbachev, Y.E., Dongarra, J.J., Zomaya, A.Y. (eds) Computational Science — ICCS 2003. ICCS 2003. Lecture Notes in Computer Science, vol 2658. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44862-4_65
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