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 Earths internal structure generates the (interior and exterior) gravity field, this should be considered in gravity field modelling.
This report gives the Chairs 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 Earths 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 Newtons 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 Earths 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.
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.
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 Newtons 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:
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 Earths 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 Earths physical surfaces.|
Balmino G, Barriot J, Koop R, Middel B, Thong N, Vermeer M (1991) Simulation of gravity gradients: a comparison study, Bulletin Géodésique, 65: 218-229.
Barthelmes F (1986) Untersuchungen zur Approximation des ausseren Gravitationsfeldes der Erde durch Punktmassen mit optimierten Positionen. Veroff. d. Zentr. Inst. f. Phys. d Erde, Nr. 92, Potsdam, Germany.
Barthelmes F, Dietrich R, Lehmann R (1991) Use of point masses on optimised positions for the approximation of the gravity field, in: Rapp RH, Sanso F (eds) Determination of the Geoid, Springer, Berlin, 484-493.
Blakely RJ (1995) Potential Theory in Gravity and Magnetic Applications, Cambridge University Press.
Bullen KE (1975) The Earths Density, Chapman-Hall.
Crank J (1984) Free and moving boundary problems, Oxford University Press.
Dampney CNG (1969) The equivalent source technique, Geophysics, 34: 39-53.
Dziewonski AM, Anderson DL (1981) Preliminary reference Earth model. Physics of the Earth Planetary Interiors, 25: 297-356.
Engels J (1991) Eine approximative Loesung der fixen geodaetischen Randwert-aufgabe im Innen- und Aussenraum der Erde. Deutsche Geodaetische Kommission Reihe C Nr. 379, Bayerische Akademie der Wissenschaften, Germany.
Engels J, Grafarend E, Keller W, Martinec Z, Sanso F, Vanicek P (1993) The geoid as an inverse problem to be regularised. in: Inverse Problems: Principles and Applications to Geophysics, Technology and Medicine, Anger et al., (eds), Akademie Verlag, Berlin, 122-167.
Featherstone WE, Olliver JG (1997) A method to validate gravimetric geoid computation software based on Stokes's integral, Journal of Geodesy, 72(3): 154-160.
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.
Feistritzer M (1998) Geoidbestimmung mit geopotentiellen Koten. Deutsche Geodaetische Kommission Reihe C Nr. 486, Bayerische Akademie der Wissenschaften, Germany.
Grafarend E, Engels J (1993) The gravitational field of topographic-isostatic masses and the hypothesis of mass condensation, Surveys in Geophysics, 14: 495-524.
Grafarend EW, Keller W (1995) Setup of observational functionals in gravity space as well as in geometry space, manuscripta geodaetica, 20: 301-325
Grafarend E, Engels J, Sorcik P (1995) The gravitational field of topographic-isostatic masses and the hypothesis of mass condensation - part I and II. Technical Report 1995.1 Schriftenreihe der Institut des Fachbereichs Vermessungswesen, Department of Geodesy, Stuttgart, Germany.
Grafarend E, Engels J, Sorcik P (1996) The gravitational field of topographic-isostatic masses and the hypothesis of mass condensation II - the topographic-isostatic geoid, Surveys in Geophysics, 17: 41-66.
Grafarend E, Engels J, Varga P (1997) The spacetime gravitational field of a deformable body, Journal of Geodesy, 72(1): 11-30.
Heikkinen M (1981) Solving the shape of the Earth by using digital density models, Report 81, Finnish Geodetic Institute, Masala, Finland.
Holota P (1995) Two branches of the Newton potential and geoid, In: Sünkel H, Marson I (eds) Gravity and Geoid, Springer, Berlin, 92-101.
Ihde J, Schirmer U, Stefani F, Toeppe F (1998) Geoid modelling with point masses, Proc Second Continental Workshop on the Geoid in Europe, Budapest, 199-204.
Jordan SK (1978) Statistical model for gravity, topography, and density contrasts in the Earth, Journal of Geophysical Research, 83(B4): 1816-1824.
Kakkuri J, Wang Z (1998) Structural effects of the crust on the geoid modelled using deep seismic sounding interpretation. Geophysical Journal International 135: 495-504.
Katsambolos, KE (1981) Simulation studies on the computation of the gravity vector in space from surface data considering the topography of the Earth, Report 314, Department of Geodetic Science and Surveying, Ohio State University, Columbus.
Keller W (1998) On a scalar fixed gravimetry-altimetry boundary value problem, Journal of Geodesy, 70(8): 459-469.
Klees R (1995) Report SC2-project: comparison of several techniques for solving geodetic boundary value problems by means of numerical experiments in Willis p (ed) Travaux de lassociation internationale de geodesie, paris.
Lambeck K (1988) Geophysical Geodesy, Oxford University Press, Oxford, 718p
Lehmann R (1993a) The method of free-positioned point masses - geoid studies on the Gulf of Bothnia, Bulletin Geodesique, 67: 31-40.
Lehmann R (1993b) Nonlinear gravity field inversion using point masses diagnosing non-linearity, in: Montag H, Reigber C (eds) Geodesy and Physics of the Earth, Springer, Berlin, 256-260.
Lehmann R (1996) Information measures for global geopotential models, Journal of Geodesy, 70: 342-348.
Martinec Z, Pec K (1987) Three-dimensional density distribution generating the observed gravity field of planets Part 1, The Earth, Proc Figure of the Earth Moon and other planets, Research Institute of Geodesy, Prague.
Martinec Z (1993) A model of compensation of topographic masses, Surveys in Geophysics, 14: 525-535.
Martinec Z (1994a) The minimum depth of compensation of topographic masses, Geophysical Journal International, 117: 545-554.
Martinec Z (1994b) The density contrast at the Mohorovicic discontinuity, Geophysical Journal International, 117: 539-544.
Marussi A (1980) On the density distribution in bodies of assigned outer Newtonian attraction, Bolletino di Geofisica Teorica ed Applicata, XXII(86): 83-94.
Mooney WD, Laske G, Masters G (1998) CRUST 5.1: A global crustal model at 5°x5°, Journal of Geophysical Research, 103: 727-747.
Moritz H (1968a) Density distributions for the equipotential ellipsoid, Report 115, Department of Geodetic Science and Surveying, Ohio State University, Columbus.
Moritz H (1968b) Mass distributions for the equipotential ellipsoid, Bolletino di Geofisica Teorica ed Applicata, 10: 59-65.
Moritz H (1989) A set of continuous density distributions within a sphere compatible with a given external gravitational potential, Gerlands Beitr. Geophysik, 98: 185-192.
Moritz H (1990) The Figure of the Earth: Theoretical Geodesy of the Earths Interior, Wichman.
Novak P, Vanicek P, Veronneau M, Holmes S, Featherstone WE (1999) On the accuracy of Stokess integration in the precise high-frequency geoid determination, poster presented to the AGU spring meeting, Boston, USA, June.
Okubo S (1991) Potential and gravity changes raised by point dislocations, Geophysical Journal International, 105: 573-586.
Pail R. (1999): Synthetic Global Gravity Model for Planetary Bodies and Applications in Satellite Gravity Gradiometry, Ph.D. Thesis, Technical University of Graz, Austria.
Papp G, Wang, ZT (1996) Truncation effects in using spherical harmonic expansions for forward local gravity field modelling. Acta Geodaetica et Geophysica Hungarica, 31(1-2): 47-66.
Robertson DS (1996) Treating absolute gravity data as a space-craft tracking problem. Metrologia, 33: 545-548
Sanso F, Barzaghi R, Tscherning CC (1986) Choice of norm for the density distribution of the Earth, Geophysical Journal of the Royal Astronomical Society, 87:123-141.
Sanso F, Rummel R (eds) (1997) Geodetic Boundary Value Problems in View of the One Centimeter Geoid, Springer-Verlag.
Strakov VN Schaefer U, Strakhov AV, Luchitsky AI, Teterin DE (1995) A new approach to approximate the Earths gravity field, in Sünkel H, Marson I (eds) Gravity and Geoid, Springer, Berlin, 225-237.
Strykowski G (1995) Geodetic and geophysical inverse gravimetric problem, the most
adequate solution and the information content. In: Sünkel and Marson (eds) Gravity and Geoid, Springer, Berlin, 215-224.
Strykowski G (1996) Borehole data and stochastic gravimetric inversion. PhD-thesis, University of Copenhagen. Publ. 4 series, vol. 3, National Survey and Cadastre - Denmark, ISBN 87-7866-013-0.
Strykowski G (1997) Formulation of the mathematical frame of the joint inverse gravimetric-seismic modelling problem based on the analysis of regionally distributed borehole data, In: Segawa, Fujimoto and Okubo (eds) Gravity, Geoid and Marine Geodesy, Springer, Berlin, 360-367.
Strykowski G (1998a) Geoid and mass density - why and how? In: Forsberg, Feissl and Deitrich (eds) Geodesy on the Move, Springer, Berlin, 237-242.
Strykowski G (1998b) Experiences with a detailed estimation of the mass density contrasts and of the regional gravity field using geometrical information from seismograms, Proc. XXII General Assembly of the European Geophysical Society, Vienna, 1997, Phys. Chem. Earth, 23, 845-856.
Strykowski G, Dahl OC (1998c) The geoid as an equipotential surface in a sense of Newton's integral - ideas and examples, Proc. 2nd Continental Workshop on the Geoid in Europe, Budapest, Hungary, Report 98.4, Finnish Geodetic Institute, 113-116.
Strykowski G (1999a) Some technical details concerning a new method of gravimetric-seismic inversion, Proc. XXIII General Assembly of the European Geophysical Society, Nice, France, 1998, Phys. Chem. Earth, 24, 207-214.
Strykowski G (1999b) Silkeborg gravity high revisited: horizontal extension of the source and its uniqueness. Proc. XXIV General Assembly of the European Geophysical Society, The Hague, The Netherlands.
Sünkel H (1982) Point mass models and the anomalous gravitational field, Report 327, Department of Geodetic Science and Surveying, Ohio State University, Columbus.
Sünkel H (1985) An isostatic earth model, Report 367, Department of Geodetic Science and Surveying, Ohio State University, Columbus.
Sünkel H (1986) Global topographic-isostatic models, in Mathematical and Numerical Techniques in Physical Geodesy, Sunkel H (ed), Springer, Berlin, Germany: 417-462.
Toth G (1996) Topographic-isostatic models fitting to the global gravity field, Acta Geodaetica et Geophysica Hungarica, 31(3-4): 411-421.
Toth G (1998) Topographic isostatic models fitted to geopotential, Proc. 2nd Continental Workshop on the Geoid in Europe, March, Budapest, Hungary.
Timmen L, Wenzel H-G (1995) Worldwide synthetic gravity tide parameters, in Sünkel and Marson (eds) Gravity and Geoid, Springer, Berlin, 92-101.
Tscherning CC, Sünkel H (1981) A method for the construction of spheroidal mass distributions consistent with the harmonic part of the Earths gravity potential, manuscripta geodaetica, 6: 131-156.
Tscherning CC (1985) On the use and abuse of Molodenskys mountain, In: Schwarz K-P, Lachapelle G (eds) Geodesy in Transition, University of Calgary, 133-147.
Tziavos IN (1996) Comparisons of spectral techniques for geoid computations over large regions, Journal of Geodesy, 70: 357-373.
Tziavos IN, Sideris MG, Sünkel H (1996) The effect of surface density variations terrain modelling - a case study in Austria. Report 96.2, Finnish Geodetic Institute, 99-110.
Vajda P, Vanicek P (1997) On gravity inversion for point mass anomalies by means of the truncated geoid. Studia Geophysica et Geodaetica, 41: 329-344.
Vajda P, Vanicek P (1998a) On the numerical evaluation of the truncated geoid. Contributions to Geophysics and Geodesy, 28(1): 15-27.
Vajda P, Vanicek P (1998b) A note on spectral filtering of the truncated geoid. Contributions to Geophysics and Geodesy, 28(4): 253-262.
Vajda P, Vanicek P (1999a) Truncated geoid and gravity inversion for one point-mass anomaly. Journal of Geodesy, 73: 58-66.
Vajda P, Vanicek P (1999b) The instant of the dimple onset for the high degree truncated geoid. submitted to Contributions to Geophysics and Geodesy.
Vanicek P, Christou N (eds) (1993) Geoid and its Geophysical Interpretations, CRC Press, Boca Raton.
Vanicek P, Kleusberg A (1985) What an external gravitational potential can really tell us about mass distribution. Bolletino Geofisica Teoretica et Applicata, XXCII(108): 243-250.
Vermeer M (1982) The use of point mass models for describing the Finnish gravity field, Proc. Ninth Nordic Geodetic Commission, Sept. 13-17, Gavle, Sweden.
Vermeer M (1989a) Modelling of gravity gradient tensor observations using buried shells of mass quadrupoles, In: Andersen OB (ed) Modern Techniques in Geodesy and Surveying, Kort-og Matrikelstyrelsen, Copenhagen, 461-478.
Vermeer M (1989b) Global geopotential inversion from satellite gradiometry using mass point grid techniques. In: Sacerdote F, Sansó F (eds) Proc. II Hotine-Marussi Symposium on Mathematical Geodesy, June 5-8, Pisa, 325-368.
Vermeer M (1989c) FGI studies on satellite gravity gradiometry. 1. Experiments with a 5-degree buried masses grid representation, Report 89:3, Finnish Geodetic Institute, Helsinki.
Vermeer M (1992) Geoid determination with mass point frequency domain inversion in the Mediterranean, Mare Nostrum 2, GEOMED report, Madrid, 109-119.
Vermeer M (1994) STEP inversion simulations by a spectral domain, buried masses method. In: Rummel R, Schwinzler P (eds) A major STEP for Geodesy. Report 1994, STEP Geodesy Working Group, 75-83.
Vermeer M (1995) Mass point geopotential modelling using fast spectral techniques; historical overview, toolbox description, numerical experiment, manuscripta geodaetica, 20: 362-378.
Wang Z (1998) Geoid and crustal structure in Fennoscandia. Report 126, Finnish Geodetic Insitute. PhD thesis, 118 pp.
Werner RA (1997) Spherical harmonic coefficients for the potential of a constant-density polyhedron, Computers and Geosciences, 23(10): 1071-1077
Xia J, Sprowl DR (1991) Correction of topographic distortions in gravity data, Geophysics, 56: 537-541.
Xia J, Sprowl DR, Adkins-Heljeson (1993) Correction of topographic distortions in potential-field data: A fast and accurate approach, Geophysics, 58: 515-523.
Zhang C, Blais JAR (1993) Recovery of gravity disturbances
from satellite altimetry by FFT techniques: a synthetic study, manuscripta
geodaetica, 18: 158-170.