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

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 T. Wittwer (B) · R. Klees Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands e-mail: [email protected] R. Klees e-mail: [email protected] K. Seitz · B. Heck Geodetic Institute, University of Karlsruhe, Englerstr. 7, 76128 Karlsruhe, Germany e-mail: [email protected] B. Heck e-mail: [email protected] 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.