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

Abstract

In geodesy and geophysics, spherical harmonic techniques are popular for modelling topography and potential fields with ever-increasing spatial resolution. For ultra-high-degree spherical harmonic modelling, i.e. degree 10,000 or more, classical algorithms need to be extended to avoid under- or overflow problems associated with the computation of associated Legendre functions (ALFs). In this work, two quadrature algorithms—the Gauss–Legendre (GL) quadrature and the quadrature following Driscoll/Healy (DH)—and their implementation for the purpose of ultra-high (surface) spherical harmonic analysis of spheroid functions are reviewed and modified for application to ultra-high degree. We extend the implementation of the algorithms in the SHTOOLS software package (v2.8) by (1) the X-number (or Extended Range Arithmetic) method for accurate computation of ALFs and (2) OpenMP directives enabling parallel processing within the analysis. Our modifications are shown to achieve feasible computation times and a very high precision: a degree-21,600 band-limited (=frequency limited) spheroid topographic function may be harmonically analysed with a maximum space-domain error of $3 \times 10^{-5}$ and $5 \times 10^{-5}$ m in 6 and 17 h using 14 CPUs for the GL and for the DH quadrature, respectively. While not being inferior in terms of precision, the GL quadrature outperforms the DH algorithm in terms of computation time. In the second part of the paper, we apply the modified quadrature algorithm to represent for—the first time—gridded topography models for Earth, Moon and Mars as ultra-high-degree series expansions comprising more than 2 billion coefficients. For the Earth’s topography, we achieve a resolution of harmonic degree 43,200 (equivalent to ~500 m in the space domain), for the Moon of degree 46,080 (equivalent to ~120 m) and Mars to degree 23,040 (equivalent to ~460 m). For the quality of the representation of the topographic functions in spherical harmonics, we use the residual space-domain error as an indicator, reaching a standard deviation of 3.1 m for Earth, 1.9 m for Mars and 0.9 m for Moon. Analysing the precision of the quadrature for the chosen expansion degrees, we demonstrate limitations in the implementation of the algorithms related to the determination of the zonal coefficients, which, however, do not exceed 3, 0.03 and 1 mm in case of Earth, Mars and Moon, respectively. We investigate and interpret the planetary topography spectra in a comparative manner. Our analysis reveals a disparity between the topographic power of Earth’s bathymetry and continental topography, shows the limited resolution of altimetry-derived depth (Earth) and topography (Moon, Mars) data and detects artefacts in the SRTM15 PLUS data set. As such, ultra-high-degree spherical harmonic modelling is directly beneficial for global inspection of topography and other functions given on a sphere. As a general conclusion, our study shows that ultra-high-degree spherical harmonic modelling to degree ~46,000 has become possible with adequate accuracy and acceptable computation time. Our software modifications will be freely distributed to fill a current availability gap in ultra-high-degree analysis software.