Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic |
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| Authors: | Tobias Wittwer Roland Klees Kurt Seitz Bernhard Heck |
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| Institution: | (1) Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands;(2) Geodetic Institute, University of Karlsruhe, Englerstr. 7, 76128 Karlsruhe, Germany |
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| Abstract: | We present software for spherical harmonic analysis (SHA) and spherical harmonic synthesis (SHS), which can be used for essentially
arbitrary degrees and all co-latitudes in the interval (0°, 180°). The routines use extended-range floating-point arithmetic,
in particular for the computation of the associated Legendre functions. The price to be paid is an increased computation time;
for degree 3,000, the extended-range arithmetic SHS program takes 49 times longer than its standard arithmetic counterpart.
The extended-range SHS and SHA routines allow us to test existing routines for SHA and SHS. A comparison with the publicly
available SHS routine GEOGFG18 by Wenzel and HARMONIC SYNTH by Holmes and Pavlis confirms what is known about the stability of these programs. GEOGFG18 gives errors <1 mm for latitudes -89°57.5′, 89°57.5′] and maximum degree 1,800. Higher degrees significantly limit the range
of acceptable latitudes for a given accuracy. HARMONIC SYNTH gives good results up to degree 2,700 for almost the whole latitude range. The errors increase towards the North pole and
exceed 1 mm at latitude 82° for degree 2,700. For a maximum degree 3,000, HARMONIC SYNTH produces errors exceeding 1 mm at latitudes of about 60°, whereas GEOGFG18 is limited to latitudes below 45°. Further extending the latitudinal band towards the poles may produce errors of several
metres for both programs. A SHA of a uniform random signal on the sphere shows significant errors beyond degree 1,700 for
the SHA program SHA by Heck and Seitz. |
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| Keywords: | Spherical harmonic analysis Spherical harmonic synthesis Extended-range arithmetic Numerical stability |
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