REPORT OF IAG SPECIAL STUDY GROUP 3.177 SYNTHETIC MODELLING OF THE EARTH’S GRAVITY FIELD (1996-1999)
W. E. Featherstone
Curtin University of Technology, Western Australia
http://www.cage.curtin.edu.au/~will/iagssg3177.html
 
INTRODUCTION
 In April 1996, the Executive Committee of IAG Section III (Determination of the Gravity Field) endorsed the out-of-cycle creation of SSG 3.177 "Synthetic Modelling of the Earth’s Gravity Field". This can be interpreted as a logical continuation and extension of the work undertaken by Sub-Commission 2 (Numerical Approximation and Methods) of IAG Section IV (General Theory and Methodology), which is summarised by Klees (1995) and Vermeer (1995). The primary objective of IAG SSG 3.177 is to develop the theoretical basis and computational methods necessary to construct a synthetic model of the Earth’s gravity field for use in geodesy.

At present, gravity field researchers often rely on empirical error estimates to validate their results, which are generally based on observed data. A classical example can be seen in gravimetric geoid determination, where the computed models are often compared with GPS and geodetic levelling data that have their own error budgets. In addition, these data types have different physical and geometrical interpretations. The availability of a synthetic gravity field model would avoid this somewhat undesirable scenario and give a more independent validation of the procedures presently used. However, a synthetic gravity field model is not widely available to the geodetic research community, which is at odds with other areas of the geosciences. The most notable example of this is the Preliminary Reference Earth Model or PREM (Dziewonski & Anderson 1981), which has made tremendous contributions to seismology. An additional rationale for the activities of SSG 3.177 can be found in the preface of Moritz (1990), who argues that the emphasis of physical geodesy should change from the treatment of only the external gravity field to that of both the internal and external gravity field. Since the Earth’s internal structure generates the (interior and exterior) gravity field, this should be considered in gravity field modelling.

This report gives the Chair’s perspective of the work undertaken by SSG 3.177 since its creation and the future problems to be addressed, assuming that the tenure of the group will be extended for a further four years. It is important to acknowledge that, due to time constraints, not all members’ activities have been included in this report; for this I apologise. It is expected that the final outcome of SSG 3.177 will be the theories, methodologies/software and a synthetic model, which will probably be distributed via the International Geoid Service (IGeS) and the Bureau Gravimetrique International (BGI). It is anticipated that the synthetic model will allow for an objective test of the various theories and methodologies in use and may contribute to the resolution of some of the procedural differences currently encountered between gravity field researchers around the world.
SOME PREVIOUS WORK (pre-1996)An elementary synthetic model of the Earth’s gravity field is generated by the normal ellipsoid (Moritz 1968a,b) which is often (incorrectly) assumed to be error-free. However, as can be concluded from the magnitude of gravity anomalies and geoid undulations, this model is far too simplistic. Moreover, it does not allow for the testing of modern gravity field determination techniques. Refined models of the global gravity field, usually based on spherical harmonics, have been developed with increasing degree and now reach degree 1800 (eg. Wenzel 1998). However, neither are these spherical harmonic models error-free, because of the quality and treatment of the observed data used in their construction. Accordingly, a synthetic gravity field model is also desired to calibrate and test the construction of these models. Nevertheless, spherical harmonic models can be assumed to generate an error-free synthetic gravity field and therefore used to test, for example, residual geoid determination techniques (Tziavos, 1996), which will be discussed later.

In addition to the normal ellipsoid and spherical harmonic models, the external gravity fields generated by point-mass models have been used for a number of years. Vermeer (1995) gives a useful review of these approaches. Essentially, point masses of various magnitudes are placed at points inside the Earth, such that the superposition of their gravitational acceleration and potential (computed using Newton’s integrals) replicate the external gravity field (eg. Vermeer 1982, Barthelmes et al. 1991, Vermeer 1992, Lehmann 1993a, Sünkel 1982). Point-mass models have also been used to generate gravity gradients (Vermeer 1989a,b, Balmino et al. 1991, Vermeer 1994) and gravity disturbances (Zhang & Blais 1993). Logically, the use of point-mass models has been extended to the use of digital density models (eg Heikkinen 1981, Martinec & Pec 1987, Marussi 1980, Moritz 1989, Tscherning & Sünkel 1981) to better replicate the continuous nature of Earth’s density distribution.

Similar to point-mass and digital density modelling is forward modelling, which involves the generation of a synthetic gravity field from geophysical structures that are chosen to be as realistic as possible. Analytic solutions for simple figures, such as cylinders or prisms, are available in many of the geophysical or potential theory textbooks (eg. Blakely 1995). However, in order to achieve these analytic solutions, the models used are very restrictive because they almost always rely on simple geometric figures of constant density (cf. Tscherning 1981). Arguably more sophisticated alternatives use topographic-isostatic models of the Earth (eg. Sünkel 1985, 1986, Grafarend & Engels 1993, Martinec 1993, 1994a,b, Grafarend et al. 1995, 1996).
 WORK OF THE SSG (1996-1999)It is important to state that this section summarises only some of the work undertaken by a few of the members of SSG 3.177; apologies are extended to those whose results have been omitted.MembersGeorges Balmino (BGI, France), Will Featherstone (Australia), Yoichi Fukada (Japan), Erik Grafarend (Germany), Chris Jekeli (USA), Ruediger Lehman (Germany), Zdenek Martinec (Czech Republic), Yuri Neyman (Russia), Edward Osada (Poland), Gabor Papp (Hungary), Doug Robertson (USA), Walter Schuh (Austria), Michael Sideris (Canada), Gabriel Strykowski (Denmark), Gyula Toth (Hungary), Ilias Tziavos (Greece), Petr Vanicek (Canada), Peter Vajda (Slovak Republic), Giovanna Venuti (IGeS, Italy), Martin Vermeer (Finland)Corresponding MembersFabio Boschetti (Australia), Hans Engels (Germany), Heiner Denker (Germany), Rene Forsberg (Denmark), Simon Holmes (Australia), Jonathan Evans (UK), Adam Dziewonski (USA), Marcel Mojzes (Slovak Republic), Spiros Pagiatakis (Canada), Roland Pail (Austria), Dru Smith (USA), Hans Sünkel (Austria) Synthetic Gravity Fields Based on Spherical Harmonics

As an extension of the work of Tziavos (1996), a high-degree spherical harmonic model can be used to construct a synthetic gravity field. Assuming harmonicity, this can be used to investigate some of the errors associated with gravity field modelling. Members, and some non-members, of SSG 3.177 have constructed two such models as follows.

Curtin/UNB models

An Australian Research Council-funded project between Curtin University of Technology (Australia) and The University of New Brunswick (Canada) has led to a synthetic spherical harmonic model, which extends to degree 5400. This high degree has been achieved by artificially scaling the coefficients and fully normalised associated Legendre functions to avoid numerical instabilities in the latter. The higher degree terms have been constructed by scaling and recycling the EGM96 coefficients. The degree-variance of this synthetic model has been constrained to follow that of the degree 1800 model of Wenzel (1998) and the Tscherning-Rapp model beyond 1800.

This synthetic field has been used to construct self-consistent and assumed error-free geoid heights and gravity anomalies at the geoid. These have been used to investigate the performance of modified kernels in regional gravimetric geoid determination. Table 1 summarises the results of preliminary studies in western Canada (Novak et al., 1999) and Western Australia (Featherstone, 1999) and indicates that modified kernels offer a slightly more preferable approach.
 

area kernel max. min. mean std.
western 

Canada

 unmodified 0.058 -0.041 0.008 0.011
modified 0.035 -0.035 0.000 0.008
Western  

Australia

 unmodified 0.033 -0.026 0.003 0.009
modified 0.026 -0.017 0.003 0.008
Table 1. Statistics of the differences between the synthetic geoid heights and geoid heights computed using unmodified and modified kernels (units in metres). European Space Agency/GOCE modelsAs part of a collaborative project "Refinement of Observation requirements for GOCE" for the European Space Agency (ESA), a consortium of investigators has also generated a synthetic gravity field model based on spherical harmonics. This model extends to degree 1800 through the use of the coefficients and numerical algorithms of Wenzel (1998). A FORTRAN program harmexg.f is available at http://www.gfy.ku.dk/~cct/goce-study.htm. and can be used to generate point and grid values of the gravity field.

This synthetic gravity field is consistent with a-priori statistical information using a simple random-number generator. The values of the spherical harmonic coefficients, or perturbations of these, are calculated corresponding to a normal distribution with a given variance and zero mean. Global gravity fields can be generated from the existing GPM98 coefficients (Wenzel, 1998) using the degree-variances as variances in the distribution, or for the GOCE study, the error-degree-variances are used to generate perturbed fields. Local gravity field models are also generated using the global coefficients, where the variances are scaled by the ratio between the local and the global variance. Some problems were encountered when using the error estimates of existing spherical harmonic models, because the coefficient errors do not depend on the local gravity field variation. Therefore, the synthetic model is now being designed to use error models that take into account the variability of the gravity field, the terrain and the spatial availability of these data.
 Point Mass ModellingPoint-mass models (eg. Lehmann 1993, Vermeer 1995) continue to be used as a useful means of determining the external gravity field. These have been used in the determination of the geoid (eg. Ihde et al., 1998). Through the numerical or analytic integration of Newton’s integrals, self-consistent values of the gravitational potential and acceleration can be generated. However, it can be argued that these approaches become limited when one attempts to generate gravity field quantities inside the Earth, because the gravity field is not harmonic in this region. Nevertheless, these models will probably continue to attract attention because of their ease of use and conceptual simplicity.

Vajda and Vanicek (1997, 1998a, 1999a) seek point masses that are neither fixed or free, which is an alternative to the two existing approaches. Their strategy is as follows:

  1. Compute the sequence of surfaces of the truncated geoid with systematically decreasing values of the truncation parameter (the TG sequence) and the sequence of surfaces of the first derivative of the TG sequence, with respect to the truncation parameter, with systematically increasing values of the truncation parameter (the DTG sequence) from a geopotential model on or above the boundary.
  2. Construct a set of mass points: At the smallest numerically achievable value of the TG sequence, the positions of the maxima and minima of the truncated geoid are assigned as the horizontal positions of the sought mass points.
  3. The sought set of mass points is completed by finding their depths. In the DTG sequence, dimple events are observed and the depths of the sought point masses are determined from the instants of the dimple onsets.
  4. From this, a preliminary set of point masses is formed, with the number of point masses being equal to the number of horizontal positions determined in step 2. If the number of clearly observed dimple onsets is less than the number of highs and lows from step 2, the depth of the remaining point masses (those in horizontal positions from step 2 that do not display a clear dimple onset in step 3) may be estimated or assumed. Another option is to reject the remaining point masses and exclude them from the preliminary set of point masses.
The Reference Earth Model (REM)Following the success of PREM (Dziewonski & Anderson 1981), a new project is being led by the seismological community (http://www.mahi.ucsd.edu/Gabi/rem.html) to generate an extension called REM. The primary aim of this project is to construct a reference model of the Earth that fits a greater variety of geophysical constraints than PREM, which only considered the seismic properties of the Earth. Of importance to SSG 3.177, the instigators of REM seek the input of other earth scientists, and one of the stated objectives of REM is a model that will generate gravity and geoid values. Accordingly, the two groups’ activities are complementary and formal links were established in 1997.

At this stage of SSG 3.177, possibly the most useful aspect of the REM project is through the links to other geoscientists and recent global datasets that can be used in the construction of a synthetic gravity field. These are given at the REM web-site, with the most notable being a global topographic/bathymetric model and a global model of the crust (Mooney et al. 1998). Complementary data that are not linked to the REM web-site include the compensation depth of topographic masses (eg. Sünkel 1985, 1986, Grafarend & Engels 1993, Martinec 1993, 1994a,b, Grafarend et al. 1995, 1996) and the position of the Mohorovicic discontinuity (Martinec 1994b). However, it should be bourn in mind that the boundaries of different physical properties inside the Earth do not necessarily coincide, so the construction of the synthetic field does not have to be rigidly constrained to all these boundaries.Forward Modelling Using a-priori Geophysical DataAs part of the aim to make the synthetic gravity model as realistic as possible, complementary geophysical data can be used to place constraints on the density distribution in a forward modelling process (eg. Strykowski 1996, 1997a,b 1998a,b, 1999a,b, Toth 1996, 1998, Papp 1996b, Tziavos et al. 1996; Kakkuri & Wang, 1998; Wang, 1998). A study by Papp et al. (http://www.ggki.hu/a/gravity/geoid1.html) concentrates on the determination and evaluation of the lithospheric geoid in the Pannonian Basin, Hungary. This used geological and geophysical data concerning the structure of the lithosphere in the region, which were used to construct a volume element model of the crustal structure and density distribution. This was supplemented by a simple model of the lower crust derived from deep seismic sounding data and gravity inversion. The resulting geoid undulations were compared to an existing gravimetric quasi-geoid solution. Preliminary results showed agreements of ±0.22m. By fine-tuning the method and completing the geophysical model with the surface topography, the agreement was improved to ±0.10m. The latest version of the model of the lithosphere consists of 180,000 rectangular prisms of differing dimensions. It extends from the eastern Carpathians to the eastern Alps and from the western Carpathians to the Dinarides.

Pail (1999) has constructed a global synthetic gravity field from a spherical body with a realistic three-dimensional density structure. This synthetic body consists of PREM as a radial background model, superposed by a mantle density distribution based on seismic tomography data and an isostatically compensated (local Airy/Heiskanen and Vening-Meinesz smoothing) crustal layer. A topography function of arbitrary roughness is generated by means of a fractal approach. This structure has been to generate synthetic test fields predominantly for global applications. As an example, a satellite gravity gradiometry mission is simulated in order to compare 'inner' (statistical) and 'outer' (absolute) error estimates and the influence of a variety of orbit configurations, noisy and band-limited observations and inhomogeneous data coverage on harmonic coefficient recovery. An abstract and overview are given at: http://www.cis.tu-graz.ac.at/mggi/pail_diss.html.SUGGESTED FUTURE WORK (1999-onwards)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. With all this in mind, the following are offered as a list of functions that a complete synthetic gravity field model could have to make it as realistic as possible and, more importantly, useable to a wide range of ‘customers’.InputAssuming that the synthetic gravity field model will be 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 artificially extended to higher resolutions, perhaps by fractals.
Realistic models of the bulk density distribution within the deep earth, crust and mantle, possibly from a-priori geophysical models from other disciplines.
Realistic models of the modes and depths of isostatic compensation and other boundaries that are characterised by bulk density changes.
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.OuputIt is envisaged that the synthetic gravity field should at least offer the following features:
Generation of the synthetic gravity 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 gravity data above and within the Earth’s physical surfaces.
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