Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic

• Main Authors: Wittwer, T. (author), Klees, R. (author), Seitz, K. (author), Heck, B. (author)
• Format: Article in Journal/Newspaper
• Language: English
• Published: Springer 2007
• Subjects: Spherical harmonic analysis; Spherical harmonic synthesis; Extended-range arithmetic; Numerical stability; North Pole; Seitz
• Online Access: http://resolver.tudelft.nl/uuid:99fb9dcf-e90e-4fa9-8deb-92c6d4e55593

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. Delft Institute of Earth Observation and Space Systems Aerospace Engineering