W.E. Featherstone

Curtin University of Technology, Western Australia



This report gives the Chair’s perspective of work undertaken by IAG SSG 3.177 between August 1999 and April 2001. It is important to acknowledge that, due to time and space constraints, not all SSG members’ activities have been included; for this I apologise.

The primary objective of this SSG is to construct synthetic models of the Earth’s gravity field for use in geodesy. Such models were not widely available to the geodetic research community, which is at odds with some other areas of the Earth sciences. Instead, geodetic gravity field researchers tended to rely on empirical methods to validate their results. The availability of a synthetic gravity field model avoids this undesirable scenario and can give a more independent and objective validation of the procedures used.

The ultimate outcome of this SSG will be theories, methodologies/software and synthetic models, which will probably be distributed via the IAG or IGGC. It is anticipated that these will allow for objective testing of the theories and methodologies used in gravity field determination and modelling. Indeed, they may even contribute to the resolution of some of the procedural differences currently encountered between gravity field researchers around the world.

A significant progression since the last report to the IAG is that new synthetic gravity fields (including grids of self-consistent gravity anomalies, geoid heights and vertical deflections at the geoid) and software packages for constructing synthetic gravity fields have been made available via the SSG’s web-page (



Full Members  
Will Featherstone (Australia, Chair) Yoichi Fukuda (Japan)
Erik Grafarend (Germany) Roger Haagmans (Norway)
Roger Hipkin (United Kingdom) Simon Holmes (Australia)
Christopher Jekeli (USA) Rüdiger Lehman (Germany)
Zdenek Martinec (The Czech Republic) Yuri Neyman (Russia)
Roland Pail (Austria) Gabor Papp (Hungary)
Doug Robertson (USA) Walter Schuh (Austria)
Gabriel Strykowski (Denmark) Gyula Toth (Hungary)
Ilias Tziavos (Greece) Petr Vanicek (Canada)
Peter Vajda (The Slovak Republic) Martin Vermeer (Finland)


Corresponding Members  
Giampietro Allasia (Italy)  Sten Claessens (The Netherlands)
Hans Engels (Germany) Jonathan Evans (United Kingdom)
Rene Forsberg (Denmark) Jonathan Kirby (Australia)
Michael Kuhn (Germany) Gabi Laske (USA)
Adam Dziewonski (USA)  Marcel Mojzes (The Slovak Republic)
Edward Osada (Poland) Mensur Omerbasie (Canada)
Spiros Pagiatakis (Canada) Dean Provins (Canada)
Michael Sideris (Canada) Dru Smith (USA)
Hans Sünkel (Austria) Christian Tscherning (Denmark)
Judit Benedek (Hungary)  



Some members have used ultra-high-degree spherical harmonics to construct synthetic gravity field models. One such model, computed at Curtin University of Technology, Australia, extends to degree 5400 (to 45 degrees latitude) and to degree 2700 (over the whole Earth). These limits are set by computer underflow and overflow errors in IEEE double precision. The ultra-high-degree coefficients have been created artificially by scaling and recycling EGM96 and GPM98 coefficients (ie. an 'effects' model). Fully normalised associated Legendre functions have been modified to avoid numerical instabilities (Holmes and Featherstone, 2001). The degree-variance of this synthetic model has been constrained to follow that of the Tscherning-Rapp model beyond degree-1800.

This synthetic gravity field has been used to construct self-consistent geoid heights and gravity anomalies at the geoid, which can then be used to test geoid-computation algorithms and software as follows. The synthetic spherical harmonics are used to compute the magnitude of gravity at the geoid (defined by the synthetic model), then normal gravity at the ellipsoid subtracted to yield gravity anomalies. These synthetic gravity anomalies are used as input into geoid computation algorithms and software, then the geoid output is compared with the self-consistent, synthetic geoid heights. Any differences between the computed and synthetic geoid can then be attributed to algorithmic and/or software errors.

Featherstone (1999) has used this approach and an earlier version of this synthetic model over Western Australia to test the theories and software used to produce AUSGeoid98. This study shows that, when using error-free gravity data (which, of course, is not the case in practice), the standard deviation of the error committed due to the algorithms and computer software is ±0.008m (max=0.035m, min=–0.035m, mean=0.000m).

A similar study was conducted by Novak et al. (2001) to validate the Stokes-Helmert and modified kernel theories used at the University of New Brunswick, Canada. This shows that the theory and software, used for some recent Canadian geoid models, is capable of producing a geoid with centimetre accuracy. The standard deviation of the error committed when using quadrature-based numerical integration of the modified Stokes's formula is ±0.010m (max=0.039m, min=–0.030m, mean=0.003m) and when using the fast Fourier transform is ±0.011m (max=0.045m, min=–0.036m, mean=0.003m). Recent work at the University of New Brunswick shows that extreme care must be used when dealing with terrain corrections during the construction of mean Helmert anomalies in mountainous regions.

Experiments, due to be reported at the 2001 IAG General Assembly, will use a refinement of the above synthetic gravity field to test the gravimetric computation of vertical deflections using modifications of Vening-Meinesz's formulae. Regular geographic grids of self-consistent geoid heights, gravity anomalies and vertical deflections (at the geoid) over Greece (18E-30E, 34N-42N) and Australia (108E-160E, 8S-45S) are available at the SSG’s web-site. The Greek data are given on a 5' by 5' grid (degree 2160), and the Australian synthetic data are given on a 2' by 2' grid (degree 5400).



Point-mass models continue to be a useful means of modelling the external gravity field. Through the numerical (or even analytic; see later) integration of Newton’s integrals, self-consistent values of the gravitational potential and acceleration can be generated (ie. a 'source' model).

Claessens et al. (2001) use 500 free-positioned point masses beneath the Perth region of Western Australia to construct a geoid model consistent with gravity observations. This study led to the identification of some quite serious errors in the Australian marine gravity database, as well as confirming the misfit between satellite-altimeter-derived gravity anomalies in near-coastal regions. As such, this synthetic gravity field has indirectly found an additional application in detecting errors in regional gravity data.

During the same study, an issue relevant to generating a synthetic gravity field from free-positioned point masses was identified. Specifically, if gravity observations are used to attempt to construct a synthetic field that is as realistic as possible, large masses may be positioned in areas devoid of observations. These subsequently cause very large synthetic gravity and geoid values to be produced in these areas, which are not necessarily realistic. This study implies that such synthetic fields should use fixed-positioned masses, which also reduces the computational burden.

Allasia (2001) has developed analytic solutions of Newton's integral for a continuous mass-density distribution. This paper has set a theoretical framework, but no practical application of this method has yet been made. It is thus recommended that SSG members begin to collaborate with Allasia to undertake practical computations to generate a synthetic gravity field. This method appears to have the potential to generate a completely error-free (ie. with no approximations) synthetic gravity field. It may also be possible to generate gravity field quantities inside the topographic masses, but further work is probably required.

Lehmann (2000) has produced a synthetic gravity field model using MATLAB (version 4.2 or later) script files, principally to test altimetry-gravimetry problems. However, this synthetic field can also be adapted, or used directly, to generate other gravity field quantities, such as spherical harmonics between degrees 11 and 2160. This synthetic field is based on an axisymmetric model of the Earth that is made as realistic as possible. Pseudo-random, un-correlated noise can be introduced into this model. A copy of the 'user manual' and the MATLAB script files are available via the SSG’s web-site.

Haagmans (2000) has constructed a global synthetic gravity field model that can generate gravity field quantities exterior to the Earth's surface at various spatial resolutions, and at aircraft and satellite altitudes. The model is based on a spherical harmonic expansion of an isostatically compensated topography and the EGM96 global geopotential model. The maximum degree is 2160, which corresponds to a spatial resolution of 5' by 5'. This synthetic field is available directly from the author.

Work on forward gravity field modelling of prism-based mass-density models continues. Papp and Benedek (2000) have used Newtonian integration of a three-dimensional topographic mass-density model to determine curvature of the plumblines. Nagy et al. (2000) have published a review-type paper on determining gravity field quantities from prisms. Papp (2000), Benedek (2000) and Kuhn (2000) have presented papers related to the use of mass-density data in synthetic modelling of the gravity field and to geoid computation.

Papp and his group continue to develop the three-dimensional model of the lithosphere in Central Europe. The depth-density model of the sediments has been modified according to the research results of the Eötvös Loránd Geophysical Institute by separating the sediments into two groups (Transdanube/West Hungary and Great Hungarian Plane/East Hungary). The lithospheric model was also extended towards the East (Romania), where the Transylvanian Basin and the Vrancea region (plate subduction) are dominant.

The geophysical community is conducting work relevant to the construction of a synthetic gravity field for use in geodesy. These studies are based on forward modelling to generate gravity and magnetic fields due to reasonably sophisticated geological structures. One of these software packages, Noddy, uses Hjelt's dipping prism equations and frequency domain methods to calculate the potential field response from a three-dimensional geological model. Other geological forward modelling software, some of which is in the public domain, is given at

Work on the Reference Earth Model (REM), the follow-on from Dziewonski and Anderson's PREM, also continues, but that group seems to be focussing more on the seismic properties of the Earth.



In order not be too prescriptive over the future activities of SSG 3.177, the following are suggested directions to the Group. Firstly, it is important to recognise that different authors are investigating different, yet complementary, approaches to the construction of the synthetic gravity field. This in itself is essential so that there is cross fertilisation of ideas and, moreover, tests on the synthetic field(s) that may eventually be used as control.

It is realistic to expect that preliminary synthetic fields will continue to be customised to accommodate a specific area of study. For example, comparisons of approaches to Stokesian integration on a regional scale (eg. Novak et al., 2000) probably require a synthetic field of different functionality to that used for, say, detailed investigation of plumbline geometry (eg. Papp and Benedek, 2000). Of course, a 'complete' synthetic field could be used for both, but separate synthetic fields are more convenient initially. With this qualification, the following are offered as a list of characteristics and features that a complete synthetic gravity field model could contain.

For a complete synthetic gravity field model, which is constructed from geophysical forward modelling, the following should be considered:

·         Realistic models of the Earth’s topography by the densest available digital elevation models, which can be extended artificially to higher resolutions.

·         Realistic models of the mass-density distribution within the deep Earth, crust and topography, probably using a priori geophysical models from other disciplines.

·         Realistic models of the modes and depths of isostatic compensation and other boundaries that are characterised by large mass-density contrasts.

·         Realistic models of noise and systematic errors (correlated and un-correlated), which can be varied by the user for sensitivity analyses.

Most importantly, the model should rely on as few assumptions as possible so that it can be used to test the assumptions currently in use. In addition, the use of realistic and accepted models of the Earth should guarantee that the results from the synthetic field can be applied to the real Earth.

It is envisaged that complete synthetic gravity field models should at least offer the following features:

·         Generation of the synthetic field in different formats; these being point, grid or mean values of geoid, gravity anomalies, gravity gradients and vertical deflections.

·         A spherical harmonic series expansion with various spectral error characteristics.

·         Generation of point, grid and mean gravity data with various error characteristics.

·         Generation of vector gravity data above and within the Earth’s physical surfaces.

Ideally, the model will generate gravity to <1microGal and the geoid to <1mm at all frequencies, though this aim may prove to be over-optimistic; but let us try!



Allasia G (2001) Approximating potential integrals by cardinal basis interpolants on multivariate scattered data , Computers and Mathematics with Applications (in press)

Benedek (2000) The effect of point density of gravity data on the accuracy of geoid undulations investigated by 3D forward modelling. Presented to Gravity Geoid and Geodynamics 2000, Banff, Canada.

Claessens S, Featherstone WE, Barthelmes F (2001) Experiments with free-positioned point-mass geoid modelling in the Perth region of Western Australia, Geomatics Research Australasia, (submitted in 2000)

Featherstone WE (1999) Tests of two forms of Stokes's integral using a synthetic gravity field based on spherical harmonics, in: Festschrift Erik Grafarend, University of Stuttgart, Germany.

Haagmans R (2000) A synthetic Earth model for use in geodesy, Journal of Geodesy 74: 503-511.

Holmes SA, Featherstone WE (2001) A generalised approach to the Clenshaw summation and the recursive computation of very-high degree and order normalised associated Legendre functions, Journal of Geodesy (submitted in 1999).

Kuhn M (2000) Density modelling for geoid determination. Presented to Gravity Geoid and Geodynamics 2000, Banff, Canada.

Lehmann, R. (2000) Technical description of the axisymmetric experiment for the study of the altimetry-gravimetry problems of geodesy, (publications details unknown)

Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism, Journal of Geodesy, 74 (7/8): 552-560.

Novak P, Vanicek P, Veronneau M, Holmes S, Featherstone WE (2001) On the accuracy of modified Stokes’s integration in high-frequency gravimetric geoid determination, Journal of Geodesy, 74(11): 644-654.

Papp G (2000) On some error sources of geoid determination investigated by forward gravitational modelling. Presented to Gravity Geoid and Geodynamics 2000, Banff, Canada.

Papp G, Benedek J (2000) Numerical modelling of gravitational field lines – the effect of mass attraction on horizontal coordinates, Journal of Geodesy, 73(12): 648-659.


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