Sine series expansion of associated Legendre functions
Tóm tắt
The most regularly used mathematical tools for representing the geopotential globally are the spherical harmonics, which consists of the longitude-dependent Fourier transform and of the latitude-dependent associated Legendre functions. While the former is by definition a Fourier series, the latter also can be formed to that. An alternative formulation for the sine series expansion of associated Legendre polynomials has been derived based on well-known recurrence formulae. The resulted formulae are subsequently empirically tested for errors to determine the limitations of its use, and strong dependence on the co-latitude has been found.
Tài liệu tham khảo
Abramowitz M, Stegun IA (1965) Handbook of mathematical functions. Dover Publications, New York
Cheong HB, Park JR, Kang HG (2012) Fourier-series representation and projection of spherical harmonic functions. J Geod 86(11):975–990
Dilts GA (1985) Computation of spherical harmonic expansion coefficients via FFT’s. J Comput Phys 57(3):439–453
Dunster TM (2010) Legendre and related functions. In: Olver Frank W J, Lozier Daniel M, Boisvert Ronald F, Clark Charles W (eds) NIST handbook of mathematical functions. Cambridge University Press, Cambridge, p 968
Elovitz M, Hill F, Duvall TL (1989) A test of a modified algorithm for computing spherical harmonic coefficients using an FFT. J Comput Phys 80(2):506–511
Gilbert EG, Otterman J, Riordan JF (1960) A tabulation of Fourier transforms of trigonometric functions and Legendre polynomials. Computation Dept., Willow Run Laboratories, University of Michigan, Ann Arbor, Michigan, p 24
Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman and Co., San Francisco, W. H
Hofsommer DJ, Potters ML (1960) Table of Fourier coefficients of associated Legendre functions. Proc KNAW Ser A 63(5):460–480
Mughal AM, Ye X, Iqbal K (2006) Computational algorithm for higher order Legendre polynomial and Gaussian quadrature method. In: Proceedings of the 2006 international conference on scientific computing, Las Vegas, NV, pp 73–77, 26–29 June 2006
Ricardi LJ, Burrows ML (1972) A recurrence technique for expanding a function in spherical harmonics. IEEE Trans Comput 21(6):583–585
Smylie DE (2013) Earth dynamics: deformations and oscillations of the rotating earth. Cambridge University Press, New York
Sneeuw N, Bun R (1996) Global spherical harmonic computation by two-dimensional Fourier methods. J Geod 70:224–232
Swarztrauber PN (1993) The vector harmonic transform method for solving partial differential equations in spherical geometry. Mon Weather Rev 121:3415–3437