Citation : Rexer M, Hirt C (2015) Ultra-high Degree Surface Spherical Harmonic Analysis using the Gauss-Legendre and the Driscoll/Healy Quadrature Theorem and Application to Planetary Topography Models of Earth, Mars and Moon, Surveys

In geodesy and geophysics, spherical-harmonic techniques are popular for modelling to7 pography and potential fields with ever-increasing spatial resolution. For ultra-high degree spherical 8 harmonic modelling, i.e. degree 10000 or more, classical algorithms need to be extended to avoid 9 underor overflow problems associated with the computation of Associated Legendre Functions 10 (ALFs). In this work two quadrature algorithms the Gauss-Legendre (GL) quadrature and the 11 quadrature following Driscoll/Healy (DH) and their implementation for the purpose of ultra-high 12 (surface) spherical harmonic analysis of spheroid functions are reviewed and modified for appli13 cation to ultra-high degree. We extend the implementation of the algorithms in the SHTOOLS 14 software package (v2.8) by 1) the X-number (or Extended Range Arithmetic) method for accurate 15 computation of ALFs and 2) OpenMP directives enabling parallel processing within the analysis. 16 Our modifications are shown to achieve feasible computation times and a very high precision: a 17 degree-21600 band-limited (=frequency limited) spheroid topographic function may be harmoni18 cally analyzed with a maximum space-domain error of 3 x 10−5 m and 5 x 10−5 m in 6 h and 17 19 h time using 14 CPUs for the GL and for the DH quadrature, respectively. While not being inferior 20 in terms of precision, the GL quadrature outperforms the DH algorithm in terms of computation 21 time. In the second part of the paper, we apply the modified quadrature algorithm to represent 22 for the first time gridded topography models for Earth, Moon and Mars as ultra-high degree series 23 expansions comprising more than 2 billion coefficients. For the Earth’s topography, we achieve a 24 resolution of harmonic degree 43,200 (equivalent to ∼ 500 m in the space domain), for the Moon 25 of degree 46,080 (equivalent to ∼ 120 m) and Mars to degree 23,040 (equivalent to ∼ 460 m). 26 For the quality of the representation of the topographic functions in spherical harmonics we use the 27 residual space domain error as an indicator, reaching a standard deviation of 3.1 m for Earth, 1.9 m 28 for Mars and 0.9 m for Moon. Analysing the precision of the quadrature for the chosen expansion 29 degrees, we demonstrate limitations in the implementation of the algorithms related to the deter30 mination of the zonal coefficients, which, however, do not exceed 3 mm, 0.03 mm and 1 mm in 31 case of Earth, Mars and Moon, respectively. We investigate and interpret the planetary topography 32 1 Institute for Astronomical and Physical Geodesy · Institute for Advanced Study, Technische Universität München Arcisstrasse 21, D-80333 München Tel.: +49-(0)89-289-23190 Fax: +49-(0)89-289-23178 E-mail: [email protected] 2 The Institute for Geophysical Research · Western Australian Geodesy Group · Department of Spatial Sciences, Curtin University of Technology GPO Box U1987, Perth, WA 6845 Tel.: +49 (0)89-289-23198 Fax: +49-(0)89-289-23178 E-mail: [email protected] 2 Moritz Rexer1, Christian Hirt2,1 spectra in a comparative manner. Our analysis reveals a disparity between the topographic power 33 of Earth’s bathymetry and continental topography, shows the limited resolution of altimetry-derived 34 depth (Earth) and topography (Moon, Mars) data and detects artifacts in the SRTM15 PLUS 35 data set. As such, ultra-high degree spherical harmonic modeling is directly beneficial for global 36 inspection of topography and other functions given on a sphere. As a general conclusion, our study 37 shows that ultra-high degree spherical harmonic modeling to degree ∼ 46, 000 has become possible 38 with adequate accuracy and acceptable computation time. Our software modifications will be freely 39 distributed to fill a current availability gap in ultra-high degree analysis software. 40