JSG T.34

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JSG T.34: High-resolution harmonic series of gravitational and topographic potential fields

Chair: Sten Claessens (Australia)
Affiliation:Commission 2 and GGOS


The resolution of models of the gravitational and topographic potential fields of the Earth and other celestial bodies in the Solar System has increased steadily over the last few decades. These models are most commonly represented as a spherical, spheroidal or ellipsoidal harmonic series. Harmonic series are used in many other areas of science such as geomagnetism, particle physics, planetary geophysics, biochemistry and computer graphics, but geodesists are at the forefront of research into high-resolution harmonic series.

In recent years, there has been increased interest and activity in high-resolution harmonic modelling (to spherical harmonic degree and order (d/o) 2190 and beyond). In 2019, the first model of the Earth’s gravitational potential in excess of d/o 2190 was listed by the International Centre for Global Earth Models (ICGEM). All high-resolution models of gravitational potential fields rely on forward modelling of topography to augment other sources of information. Harmonic models of solely the topographic potential are also becoming more common. Models of the Earth’s topographic potential up to spherical harmonic d/o 21,600 have been developed, and ICGEM has listed topographic gravity field models since 2014.

The development of high-resolution harmonic models has posed and continues to pose both theoretical and practical challenges for the geodetic community. One challenge is the combination of methods for ultra-high d/o harmonic analysis (the forward harmonic transform). Least-squares-type solutions with full normal equations are popular, but computationally prohibitive at ultra-high d/o. Alternatives are the use of block-diagonal techniques or numerical quadrature techniques. Optimal combination and comparison of the different techniques, including studying the influence of aliasing, requires further study.

A related issue is the development of methods for the optimal combination of data sources in the computation of high-degree harmonic models of the gravitational potential. Methods used for low-degree models cannot always suitably be applied at higher resolution.

Another challenge is dealing with ellipsoidal instead of spherical geometry. Much theory has been developed and applied in terms of spherical harmonics, but the limitations of the spherical harmonic series for use on or near the Earth’s surface have become apparent as the maximum d/o of the harmonic series has increased. The application of spheroidal or ellipsoidal harmonic series has become more widespread, but needs further theoretical development.

A specific example is spectral forward modelling of the topographic potential field in the ellipsoidal domain. Various methods have been proposed, but these are yet to be compared from both a theoretical and numerical standpoint. There are also still open questions about the divergence effect and the amplification of the omission error in spherical and spheroidal harmonic series inside the Brillouin surface.

A final challenge are numerical instabilities, underflow/overflow and computational efficiency problems in the forward and reverse harmonic transforms. Much progress has been made on this issue in recent years, but further improvements may still be achieved.


  • Develop and compare combined full least-squares, block-diagonal least-squares and quadrature approaches to very high-degree and order spherical, spheroidal and ellipsoidal harmonic analysis.
  • Develop and compare methods to compute high-resolution harmonic potential models using ellipsoidal geometry, either in terms of spherical, spheroidal or ellipsoidal harmonic series.
  • Study the divergence effect of ultra-high degree spherical, spheroidal and ellipsoidal harmonic series inside the Brillouin sphere, spheroid and/or ellipsoid.
  • Study efficient methods for ultra-high degree and order harmonic analysis (the forward harmonic transform) for a variety of data types and boundary surfaces, as well as harmonic synthesis (the reverse harmonic transform) of various quantities.

Program of activities

To facilitate achievement of these objectives, the group will provide a platform for increased collaboration between group members, encouraging exchange of ideas and research results. Working meetings of group members will be organized at major international conferences.


Sten Claessens (Australia), chair
Hussein Abd-Elmotaal (Egypt)
Blažej Bucha (Slovakia)
Christoph Förste (Germany)
Toshio Fukushima (Japan)
Ropesh Goyal (India)
Christian Hirt (Germany)
Norbert Kühtreiber (Austria)
Kurt Seitz (Germany)
Elmas Sinem Ince (Germany)
Michal Šprlák (Czech Republic)
Philipp Zingerle (Germany)