Difference between revisions of "JSG T.34"

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(Created page with "<big>'''JSG 0.21: High resolution harmonic analysis and synthesis of potential fields'''</big> Chair: ''Sten Claessens (Australia)''<br> Affiliation:''Comm. 2 and GGOS'' __T...")
 
 
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<big>'''JSG 0.21: High resolution harmonic analysis and synthesis of potential fields'''</big>
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<big>'''JSG T.34: High-resolution harmonic series  of gravitational and topographic potential fields'''</big>
  
 
Chair: ''Sten Claessens (Australia)''<br>
 
Chair: ''Sten Claessens (Australia)''<br>
Affiliation:''Comm. 2 and GGOS''
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Affiliation:''Commission 2 and GGOS''
  
 
__TOC__
 
__TOC__
  
===Terms of Reference===
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===Introduction===
  
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).
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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.  
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.  
+
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.  
  
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.  
+
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.
  
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.  
+
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.
  
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.
+
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.  
  
 
===Objectives===
 
===Objectives===
  
The objectives of this study group are to:
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* 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.
* 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.
+
* 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 and spheroidal harmonic series inside the Brillouin sphere/spheroid.
+
* Study the divergence effect of ultra-high degree spherical, spheroidal and ellipsoidal harmonic series inside the Brillouin sphere, spheroid and/or ellipsoid.
* Verify the numerical performance of transformations between spherical and spheroidal harmonic coefficients to ultra-high degree and order.
+
* 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.
* 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===
 
===Program of activities===
  
* Providing a platform for increased cooperation between group members, facilitating and encouraging exchange of ideas and research results.
+
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.
* 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.
 
  
 
===Membership===
 
===Membership===
  
'' '''Sten Claessens (Australia), chair''' <br /> Hussein Abd-Elmotaal (Egypt) <br /> Oleh Abrykosov (Germany) <br /> Blažej Bucha (Slovakia) <br /> Toshio Fukushima (Japan) <br /> Thomas Grombein (Germany) <br /> Christian Gruber (Germany) <br /> Eliška Hamáčková (Czech Republic) <br /> Christian Hirt (Germany) <br /> Christopher Jekeli (USA) <br /> Otakar Nesvadba (Czech Republic) <br /> Moritz Rexer (Germany) <br /> Josef Sebera (Czech Republic) <br /> Kurt Seitz (Germany) <br />''
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'' '''Sten Claessens (Australia), chair ''' <br /> Hussein Abd-Elmotaal (Egypt) <br /> Blažej Bucha (Slovakia) <br /> Christoph Förste (Germany) <br /> Toshio Fukushima (Japan) <br /> Ropesh Goyal (India) <br /> Christian Hirt (Germany) <br /> Norbert Kühtreiber (Austria) <br /> Kurt Seitz (Germany) <br /> Elmas Sinem Ince (Germany) <br /> Michal Šprlák (Czech Republic) <br /> Philipp Zingerle (Germany) <br />''

Latest revision as of 11:52, 10 June 2020

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

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

Introduction

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.

Objectives

  • 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.

Membership

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)