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Fourier expansion, geopotential, ultra-high degree Legendre functions, Spherical Harmonics
Abstract:
The associated Legendre functions were historically calculated as closed power series. With the growing need for higher degrees and associated orders recursive algorithms have been developed, highly efficient for numerical processing. As ground–based gravity measurements are available that can be combined with existing and upcoming datasets of satellite systems ultrahigh degree spherical harmonic representation and transformation of the fields becomes a necessity. Moreover, for applications in spectral domain it is in general desirable to process the associated Legendre function directly, without a recursive antecessor that predefines the order of the sequence. The closed power series mentioned can not serve beyond certain degrees due to alternating signs with extraordinarily large rational numbers, leading to a considerable loss in numerical precision. The Fourier transform and recursive relations between the Fourier coefficients themselves instead turn out to be stable and widely useful.
