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JSG 0.18: High resolution harmonic analysis and synthesis of potential fields

Chair: Sten Claessens (Australia)
Affiliation:Comm. 2 and GGOS


Terms of Reference

The gravitational fields of the Earth and other celestial bodies in the Solar System are customarily represented by a series of spherical harmonic coefficients. The models made up of these harmonic coefficients are used widely in a large range of applications within geodesy. In addition, spherical harmonics are now used in many other areas of science such as geomagnetism, particle physics, planetary geophysics, biochemistry and computer graphics, but one of the first applications of spherical harmonics was related to the gravitational potential, and geodesists are still at the forefront of research into spherical harmonics. This holds true especially when it comes to the extension of spherical harmonic series to ever higher degree and order (d/o). The maximum d/o of spherical harmonic series of the Earth’s gravitational potential has risen steadily over the past decades. The highest d/o models currently listed by the International Centre for Global Earth Models (ICGEM) have a maximum d/o of 2190. In recent years, spherical harmonic models of the topography and topographic potential to d/o 10,800 have been computed, and with ever-increasing computational prowess, expansions to even higher d/o are feasible. For comparison, the current highest-resolution global gravity model has a resolution of 7.2” in the space domain, which is roughly equivalent to d/o 90,000 in the frequency domain, while the highest-resolution global Digital Elevation Model has a resolution of 5 m, equivalent to d/o ~4,000,000.

The increasing maximum d/o of harmonic models has posed and continues to pose both theoretical and practical challenges for the geodetic community. For example, the computation of associated Legendre functions of the first kind, which are required for spherical harmonic analysis and synthesis, is traditionally subject to numerical instabilities and underflow/overflow problems. Much progress has been made on this issue by selection of suitable recurrence relations, summation strategies, and use of extended range arithmetic, but further improvements to efficiency may still be achieved.

There are further separate challenges in ultra-high d/o harmonic analysis (the forward harmonic transform) and synthesis (the inverse harmonic transform). Many methods for the forward harmonic transform exist, typically separated into least-squares and quadrature methods, and further comparison between the two at high d/o, including studying the influence of aliasing, is of interest. The inverse harmonic transform, including synthesis of a large variety of quantities, has received much interest in recent years. In moving towards higher d/o series, highly efficient algorithms for synthesis on irregular surfaces and/or in scattered point locations, are of utmost importance.

Another question that has occupied geodesists for many decades is whether there is a substantial benefit to the use of oblate ellipsoidal (or spheroidal) harmonics instead of spherical harmonics. The limitations of the spherical harmonic series for use on or near the Earth’s surface are becoming more and more apparent as the maximum d/o of the harmonic series increase. There are still open questions about the divergence effect and the amplification of the omission error in spherical and spheroidal harmonic series inside the Brillouin surface.

The Hotine-Jekeli transformation between spherical and spheroidal harmonic coefficients has proven very useful, in particular for spherical harmonic analysis of data on a reference ellipsoid. It has recently been improved upon and extended, while alternatives using surface spherical harmonics have also been proposed, but the performance of the transformations at very high d/o may be improved further. Direct use of spheroidal harmonic series requires (ratios of) associated Legendre functions of the second kind, and their stable and efficient computation is also of ongoing interest.


The objectives of this study group are to:

  • Create and compare stable and efficient methods for computation of ultra-high degree and order associated Legendre functions of the first and second kind (or ratios thereof), plus its derivatives and integrals.
  • Study the divergence effect of ultra-high degree spherical and spheroidal harmonic series inside the Brillouin sphere/spheroid.
  • Verify the numerical performance of transformations between spherical and spheroidal harmonic coefficients to ultra-high degree and order.
  • Compare least-squares and quadrature approaches to very high-degree and order spherical and spheroidal harmonic analysis.
  • Study efficient methods for ultra-high degree and order harmonic analysis (the forward harmonic transform) for a variety of data types and boundary surfaces.
  • Study efficient methods for ultra-high degree and order harmonic synthesis (the inverse harmonic transform) of point values and area means of all potential quantities of interest on regular and irregular surfaces.

Program of activities

  • Providing a platform for increased cooperation between group members, facilitating and encouraging exchange of ideas and research results.
  • Creating and updating a bibliographic list of relevant publications from both the geodetic community as well as other disciplines for the perusal of group members.
  • Organizing working meetings at international symposia and presenting research results in the appropriate sessions.


Sten Claessens (Australia), chair
Hussein Abd-Elmotaal (Egypt)
Oleh Abrykosov (Germany)
Blažej Bucha (Slovakia)
Toshio Fukushima (Japan)
Thomas Grombein (Germany)
Christian Gruber (Germany)
Eliška Hamáčková (Czech Republic)
Christian Hirt (Germany)
Christopher Jekeli (USA)
Otakar Nesvadba (Czech Republic)
Moritz Rexer (Germany)
Josef Sebera (Czech Republic)
Kurt Seitz (Germany)

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