Abstract
In order to accelerate the spherical harmonic synthesis and/or analysis of arbitrary function on the unit sphere, we developed a pair of procedures to transform between a truncated spherical harmonic expansion and the corresponding two-dimensional Fourier series. First, we obtained an analytic expression of the sine/cosine series coefficient of the \(4 \pi \) fully normalized associated Legendre function in terms of the rectangle values of the Wigner d function. Then, we elaborated the existing method to transform the coefficients of the surface spherical harmonic expansion to those of the double Fourier series so as to be capable with arbitrary high degree and order. Next, we created a new method to transform inversely a given double Fourier series to the corresponding surface spherical harmonic expansion. The key of the new method is a couple of new recurrence formulas to compute the inverse transformation coefficients: a decreasing-order, fixed-degree, and fixed-wavenumber three-term formula for general terms, and an increasing-degree-and-order and fixed-wavenumber two-term formula for diagonal terms. Meanwhile, the two seed values are analytically prepared. Both of the forward and inverse transformation procedures are confirmed to be sufficiently accurate and applicable to an extremely high degree/order/wavenumber as \(2^{30}\,{\approx }\,10^9\). The developed procedures will be useful not only in the synthesis and analysis of the spherical harmonic expansion of arbitrary high degree and order, but also in the evaluation of the derivatives and integrals of the spherical harmonic expansion.
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References
Alpert B, Rokhlin V (1989) A fast algorithm for the evaluation of Legendre expansions. Yale U/Department of Computer Science Report 671
Cheong HB (2000) Double Fourier series on a sphere: applications to elliptic and vorticity equations. J Comput Phys 157:327–349
Cheong HB, Park JR, Kang HG (2012) Fourier-series representation and projection of spherical harmonic functions. J Geod 86:975–990
Clenshaw CW (1955) A note on the summation of Chebyshev series. Math Tables Other Aids Comput 9:110–118
Colombo OL (1981) Numerical methods for harmonic analysis on the sphere. OSU/Department Geodesy Science Survey Report 310
Cooley JW, Tukey JW (1965) An algorithm for the machine calculation of complex Fourier series. Math Comput 19:297–301
Dilts GA (1985) Computation of spherical harmonic expansion coefficients via FFT’s. J Comput Phys 57:439–453
Driscoll JR, Healy D (1989) Computing Fourier transforms and convolutions on the 2-sphere. In: Proceedings of the 34th IEEE FOCS, pp 344–349
Driscoll JR, Healy D (1994) Computing Fourier transforms and convolutions on the 2-sphere. Adv Appl Math 15:202–250
Driscoll JR, Healy D, Rockmore D (1997) Fast discrete polynomial transforms with applications to data analysis for distance transitive graphs. SIAM J Comput 26:1066–1099
Egersdorfer R, Egersdorfer L (1936) Formeln und Tabellen der zugeordneten Kugelfunktionen 1. Art von \(n=1\) bis \(n=20\). Reichs fur Wett Wiss I:18–47
Edmonds AR (1957) Angular momentum in quantum mechanics. Princeton Univ Press, Princeton
Elovitz M (1989) A test of a modified algorithm for computing spherical harmonic coefficients using an FFT. J Comput Phys 80:506–511
Foldvary L (2015) Sine series expansion of associated Legendre functions. Acta Geod Geophys 50:243–259
Fukushima T (2011) Efficient parallel computation of all-pairs N-body acceleration by do loop folding. Astron J 142:18–22
Fukushima T (2017) Rectangular rotation of spherical harmonic expansion of arbitrary high degree and order. J Geod. doi:10.1007/s00190-017-1004-3
Ghobadi-Far K, Sharifi MA, Sneeuw N (2016) 2D Fourier series representation of gravitational functionals in spherical coordinates. J Geod 90:871–881
Gooding RH (1971) A recurrence relation for inclination functions and their derivatives. Celest Mech 4:91–98
Gooding RH, Wagner CA (2008) On the inclination function and a rapid stable procedure for their evaluation together with derivatives. Celest Mech Dyn Astron 101:247–272
Gooding RH, Wagner CA (2010) On a Fortran procedure for rotating spherical harmonic coefficients. Celest Mech Dyn Astron 108:95–106
Gruber C (2011) A study on the Fourier composition of the associated Legendre functions; suitable for applications in ultra-high resolution. Scientific technical report STR11/04, GFZ. Potsdam
Gruber C, Abrykosov O (2014) High resolution spherical and ellipsoidal harmonic expansions by fast Fourier transform. Stud Geophys Geod 58:595–608
Gruber C, Abrykosov O (2016) On computation and use of Fourier coefficients for associated Legendre functions. J Geod 90:525–535
Gruber C, Novak P, Sebera J (2011) FFT-based high-performance spherical harmonic transformation. Stud Geophys Geod 55:489–500
Healy D, Kostelec P, Rockmore D (2004) Towards safe and effective high-order Legendre transforms with applications to FFTs for the 2-sphere. Adv Comput Math 21:59–105
Healy D, Rockmore D, Kostelec P, Moore S (2003) FFTs for the 2-sphere: improvements and variations. J Fourier Anal Appl 9:341–385
Healy D, Rockmore D, Moore S (1996) An FFT for the 2-sphere and applications. J Fourier Anal Appl 9:341–385
Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San Francisco
Hobson EW (1931) The theory of spherical and ellipsoidal harmonics. Cambridge Univ Press, Cambridge
Hofsommer DJ, Potters ML (1960) Table of Fourier coefficients of associated Legendre functions. Proc KNAW Ser A Math Sci 63:460–466
Ito T, Fukushima T (1997) Parallelized extrapolation method and its application to the orbital dynamics. Astron J 114:1260–1267
Kellogg OD (1929) Foundations of potential theory. Springer, Berlin
Kostelec PJ, Rockmore DN (2008) FFTs on the rotation group. J Fourier Anal Appl 14:145–179
Maslen D (1998) Efficient computation of Fourier transform on compact groups. J Fourier Anal Appl 4:19–52
Moazezi S, Zomorrodian H, Siahkoohi HR, Azmoudeh-Ardalan A, Gholami A (2016) Fast ultrahigh-degree global spherical harmonic synthesis on nonequispaced grid points at irregular surfaces. J Geod 90:853–870
Mohlenkamp MJ (1999) A fast transform for spherical harmonics. J Fourier Anal Appl 5:159–184
Moritz H (1980) Advanced physical geodesy. Herbert Wichmann, Karlsruhe
Moritz H (2000) Geodetic reference system 1980. J Geod 74:128–162
Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2012) The development and evaluation of the Earth gravitational model 2008 (EGM2008). J Geophys Res 117:B04406
Rexer M, Hirt C (2015a) Ultra-high degree surface spherical harmonic analysis using the Gauss–Legendre and the Driscoll/Healy quadrature theorem and application to planetary topography models of Earth, Mars and Moon. Surv Geophys 36:803–830
Rexer M, Hirt C (2015b) Spectral analysis of the Earth’s topographic potential via 2D-DFT: a new data-based degree variance model to degree 90,000. J Geod 89:887–909
Ricardi LJ, Burrows ML (1972) A recurrence technique for expanding a function in spherical harmonics. IEEE Trans Comput 21:583–585
Risbo T (1996) Fourier transform summation of Legendre series and D-functions. J Geod 70:383–396
Rokhlin V, Tygert M (2006) Fast algorithms for spherical harmonic expansions. SIAM J Comput 27:1903–1928
Schuster A (1902) On some definite integrals and a new method of reducing a function of spherical coordinates to a series of spherical harmonics. Proc R Soc Lond 71:97–101
Schuster A (1903) On some definite integrals and a new method of reducing a function of spherical coordinates to a series of spherical harmonics. Philos Trans R Soc Lond A 200:181–223
Smith B, Sandwell D (2003) Accuracy and resolution of shuttle radar topography mission data. Geophys Res Lett 32:L21S01
Sneeuw NJ, Bun R (1996) Global spherical harmonic computation by two-dimensional Fourier methods. J Geod 70:224–232
Stacey FD, Davis PM (2008) Physics of the Earth, 4th edn. Cambridge Univ Press, Cambridge
Swarztrauber PN (1979) On the spectral approximation of discrete scalar and vector functions on the sphere. SIAM J Numer Anal 16:934–949
Swarztrauber PN (1993) The vector harmonic transform method for solving partial differential equations in spherical geometry. Mon Weather Rev 121:3415–3427
Tachikawa T, Hato M, Kaku M, Iwasaki A (2011) Characteristics of ASTER GDEM version 2. In: Proceedings of the IEEE international geoscience and remote sensing symposium, pp 3657–3660
Tscherning CC, Poder K (1982) Some geodetic applications of Clenshaw summation. Boll Geod Sci Aff 41:349–375
Wandelt BD, Gorski KM (2001) Fast convolution on the sphere. Phys Rev D 63:123002
Wigner EP (1931) Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren. Vieweg Verlag, Braunschweig
Zucker PA (1991) Smoothing and desmoothing in the Fourier approach to spherical coefficient determination. Proc IAG Symp 107:533–542
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The author appreciates valuable suggestions and fruitful comments by anonymous referees to improve the readability of the article.
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Fukushima, T. Transformation between surface spherical harmonic expansion of arbitrary high degree and order and double Fourier series on sphere. J Geod 92, 123–130 (2018). https://doi.org/10.1007/s00190-017-1049-3
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DOI: https://doi.org/10.1007/s00190-017-1049-3