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 |

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 Modelling**Point-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:

- 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.
- 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.
- 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.
- 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.

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 Data**As 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

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’.**Input**Assuming 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. |

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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