On the computation and approximation of ultra-high-degree spherical harmonic series |
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Authors: | Christopher Jekeli Jong Ki Lee Jay H Kwon |
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Institution: | (1) Division of Geodesy and Geospatial Science, School of Earth Sciences, Ohio State University, 125 South Oval Mall, Columbus, OH 43210, USA;(2) Department of Geoinformatics, University of Seoul, 90 Jennong-dong Dongdaemun-gu, Seoul, 130-743, South Korea |
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Abstract: | Spherical harmonic series, commonly used to represent the Earth’s gravitational field, are now routinely expanded to ultra-high
degree (> 2,000), where the computations of the associated Legendre functions exhibit extremely large ranges (thousands of
orders) of magnitudes with varying latitude. We show that in the degree-and-order domain, (ℓ,m), of these functions (with full ortho-normalization), their rather stable oscillatory behavior is distinctly separated from
a region of very strong attenuation by a simple linear relationship: , where θ is the polar angle. Derivatives and integrals of associated Legendre functions have these same characteristics.
This leads to an operational approach to the computation of spherical harmonic series, including derivatives and integrals
of such series, that neglects the numerically insignificant functions on the basis of the above empirical relationship and
obviates any concern about their broad range of magnitudes in the recursion formulas that are used to compute them. Tests
with a simulated gravitational field show that the errors in so doing can be made less than the data noise at all latitudes
and up to expansion degree of at least 10,800. Neglecting numerically insignificant terms in the spherical harmonic series
also offers a computational savings of at least one third. |
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Keywords: | Legendre functions Spherical harmonic series Recursion formulas |
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