This Special Study Group was established by the International Association of Geodesy (IAG) during the XXI General Assembly of the International Union of Geodesy and Geophysics (IUGG) held in Boulder, Colorado, in July 1995. Its program of activities and its members are given next.
Program of Activities
The proposed list of activities and research topics is as follows:
1. Modeling and estimation techniques. This includes:
- alternative forms of gravity field representation
2. Improvement on the consideration of the error properties of large data sets used in the development of global gravity models (e.g., consideration of correlated errors among the gravity anomalies in global 30'x30' data bases).
3. Design and setup of a data base that may include:
List of Members
D. Blitzkow (Brazil)
J.Y. Chen (China)
T. Gruber (Germany)
C. Jekeli (USA)
A.H.W. Kearsley (Australia)
J.-M. Lemoine (France)
A.N. Marchenko (Ukraine)
R.S. Nerem (USA)
N.K. Pavlis (USA) Chairman
K. Seltz (Germany)
M.G. Sideris (Canada)
G. Sona (Italy)
H. Suenkel (Austria)
I.N. Tziavos (Greece)
W. Wiejak (Poland)
R. Biancale (France)
W. Bosch (Germany)
H. Denker (Germany)
B. Heck (Germany)
E.C. Pavlis (USA)
R.H. Rapp (USA)
P. Schwintzer (Germany)
Members of SSG 3.165, as well as other interested colleagues, made a significant effort towards accomplishing the program of activities of this SSG. Progress has been made in three major areas of research as it is discussed next.
Global geopotential model development and evaluation
During this reporting period several new models were developed. These include EGM96 [Lemoine et al., 1997; 1998a], TEG-3 [Tapley et al., 1997], GRIM4-S4/C4 [Schwintzer et al., 1997], GFZ96 [Gruber et al., 1997]. EGM96 represents a significant milestone in global gravity field modeling. Its development incorporated a plethora of satellite tracking data, satellite altimetry, and the most up to date surface gravity information that was available at that time. The performance of EGM96 on a wide variety of applications (orbit determination, land and marine geoid modeling) was tested both by its development team, and by an independent, international group of researchers, chaired by M. Sideris and operating under the auspices of IGeS. The findings of this group were published in Bulletin No. 6 of the IGeS [Sansò, 1997].
In addition to the above models Wenzel  presented ultra-high degree solutions (GPM98A, B and C) obtained from orthogonality relations applied on global grids of gravity anomalies. These anomalies result from the merging of terrestrial and altimetry-implied data. Chambers et al.  presented an accuracy assessment of recent global geoid models. Results from the intercomparison and evaluation of five contemporary global geopotential models (JGM-3 [Tapley et al., 1996], GRIM4-C4, TEG-3, EGM96, and GPM98A) were presented by Pavlis et al. [1998a]. Jekeli  presented an analysis of vertical deflections derived from high-degree spherical harmonic models.
Gravity model improvement activities continue within several groups (e.g., GFZ/GRGS, NASA/GSFC, UT/CSR). Such activities include reprocessing of historic tracking data using improved models and analysis techniques, addition of new tracking data types (such as high-low satellite-to-satellite tracking (SST) data from the TDRSS and GPS constellations), and the possible incorporation of Ocean Circulation Model (OCM) information into the development of global geopotential solutions. The German-French team that produced the GRIM series of models is currently working on the development of GRIM5 [Schwintzer et al., 1999]. This model aims to provide the basis for inclusion of tracking data to be acquired from the CHAMP mission [Reigber et al., 1996]. Lemoine et al. [1998b; 1998c] discussed ongoing efforts towards gravity field model improvements within NASA/GSFC. Preliminary results from the introduction of OCM information into global geopotential solutions were presented by Pavlis et al. [1998b; 1998c] and by Tapley et al. .
During this reporting period two missions have been approved for launch, which will have a direct impact on the determination of the global gravity field. The CHAMP mission will carry a GPS receiver and an accelerometer in low Earth orbit, and the GRACE mission which is a low-low SST configuration [NRC, 1997]. CHAMP is currently scheduled for launch in early 2000, and GRACE in 2001. In addition, the GOCE mission has been proposed to the European Space Agency, to carry a gradiometer in low Earth orbit [Balmino et al., 1998]. Both GRACE and GOCE will also include on board GPS receivers, whose tracking data are sensitive to the long wavelength gravitational signal, and thus complement the low-low SST observations (GRACE) and the gradiometer observations (GOCE). The realization of these missions (especially GRACE and GOCE) is expected to enable a "quantum leap" in our global gravity field determination capability. A review of past and future developments in geopotential modeling was presented by Rapp .
Modeling and estimation techniques
Numerous contributions were made in this area. Those mentioned next represent just a sample that covers some of the specific topics within the SSGs program of activities.
Investigations related to harmonic analysis and the issues of aliasing and filtering were reported by Jekeli . Sneeuw and Bun  presented a method for performing global spherical harmonic computations using 2-D Fourier transformation methods. Several papers related to estimation problems can be found in the DEOS Progress Letters 97.1 and 98.1, published by the Delft University of Technology. Bouman  presented a report on the quality of regularization methods. Schuh  presented tailored numerical solution strategies for global gravity field determination, addressing specifically the future missions mentioned above. Kim et al.  presented a simulation study for the GRACE mission. Strakhov et al.  investigated a numerical approach (SNAP-approach) applicable to the solution of very large linear systems that are encountered in high-degree spherical harmonic modeling. Freeden and Windheuser  discussed combined spherical harmonic and wavelet expansions.
Rapp  revisited the use of potential coefficients for geoid undulation determination, demonstrated the significance of the systematic difference between height anomalies and geoid undulations, and developed appropriate corrections to account for this difference. Rapp  also investigated some artifacts that are introduced when truncated spherical harmonic series of functions (to degree 360) are used for point function evaluations. Pavlis [1998d] considered the inconsistencies that are (still) observed between satellite-only and surface gravity-only geopotential models. Pavlis [1998e] presented an approach for the modeling and the estimation of long wavelength systematic errors in surface gravimetric data, along with preliminary results from this method.
A relatively new area of investigation is related to the
use of oceanographic information in the development of global geopotential
solutions. Naturally, this approach requires the inter-disciplinary cooperation
between geodesists and oceanographers, and offers the opportunity for inter-comparison
of models that are developed by the respective groups. Pavlis et al. 
reported results from such an inter-comparison between models developed
at NASA/GSFC and corresponding models developed at MIT [Stammer et al.,
Collection of Relevant Data Sets
Two types of relevant information were collected: 1) published sets of spherical harmonic coefficients of the geopotential, and, 2) geoid undulation and height anomaly files obtained from GPS positioning and leveling observations. Additional work needs to be done in order to re-format some of the collected data sets into a consistent format, and to construct user-friendly electronic means for their distribution (e.g., through a web page).
The potential coefficient files were collected from three primary sources: the archives of R.H. Rapp, files provided by H.-G. Wenzel, and files retrieved from the NASA Goddard Space Flight Centers archives. Bouman  presented a survey of global gravity models, where several of the models that were collected here are referenced and briefly described in terms of their development strategy, data used, etc..
Several colleagues provided GPS/leveling data upon request. Unfortunately some of these files were provided with restrictions imposed on their distribution. It is highly desirable that systematic collection and documentation of such data becomes a longer-term activity under the auspices of some IAG body (such as IGeS), which may also ensure their unrestricted distribution. Such a data collection should preferably include data for other functionals of the field (e.g., deflections of the vertical), in addition to geoid undulation information obtained from space positioning (GPS, SLR) and leveling, which is currently the most widely used data type for gravity model evaluations over land areas. It is also important that GPS/leveling data collections are accurately documented, so that model evaluations can be carried out with due care of appropriate corrections. Accurate and complete documentation of these data could also permit their use for investigations related to vertical datum connections.
Conclusions and Recommendations
During the past four years members of this SSG and other colleagues made significant contributions that enabled addressing most of this SSGs objectives and goals. Global gravity field determination and evaluation is an endeavor that is expected to continue into the foreseeable future. The new missions that are currently in preparation promise a wealth of highly precise data that could enable "quantum leap" improvements in our knowledge of both the static (time-averaged) and the time-varying components of the global gravity field. Analysis, validation, and geophysical interpretation of the data from these new missions will present new challenges both for theoretical and for applied/numerical investigations. The combination of these new data with existing, complementary data is another aspect that may play a prominent role in the future. The need for accurate, independent data and information that can be used to calibrate and validate the results from new missions is expected to become ever more critical. It is recommended here that the evolution of this SSG into a new body (SSG or other) should take into account these developments and address these future challenges.
Balmino, G., et al., European Views on Dedicated Gravity Field Missions: GRACE and GOCE, ESA, ESD-MAG-REP-CON-001, May 1998.
Bouman, J., A survey of global gravity models, DEOS Report 97.1, Delft Inst. for Earth-Oriented Space Research, Delft University of Technology, July 1997.
Bouman, J., Quality of regularization methods, DEOS Report 98.2, Delft University Press, 1998.
Chambers, D.P., M.C. Kim, J.C. Ries, and B.D. Tapley, Accuracy assessment of recent global geoid models, presentation at the 1998 Spring AGU Meeting, Boston, May 26-29, 1998.
Freeden, W. and U. Windheuser, Combined spherical harmonic and wavelet expansion a future concept in Earths Gravitational determination, Applied and Computational Harmonic Analysis, 4, 1-37, 1997.
Gruber, T., et al., Improvements in high resolution gravity field modeling at GFZ, in: Gravity, Geoid and Marine Geodesy, J. Segawa, H. Fujimoto, and S. Okubo (eds.), IAG Symposia, Vol. 117, Springer-Verlag, Berlin, Heidelberg, 1997.
Jekeli, C., Spherical harmonic analysis, aliasing, and filtering, J. Geod., 70, 214-223, 1996.
Jekeli, C., An analysis of vertical deflections derived from high-degree spherical harmonic models, J. Geod., 73, 10-22, 1999.
Kim, J.-R., P.J. Roesset, S.V. Bettadpur, and B.D. Tapley, Simulation of the Gravity Recovery and Climate Experiment (GRACE) mission, Paper AAS 99-144, presented at the AAS/AIAA Space Flight Mechanics Meeting, Breckenridge, Colorado, February, 1999.
Lemoine, F.G., et al., The development of the NASA GSFC and NIMA joint geopotential model, in: Gravity, Geoid and Marine Geodesy, J. Segawa, H. Fujimoto, and S. Okubo (eds.), IAG Symposia, Vol. 117, Springer-Verlag, Berlin, Heidelberg, 1997.
Lemoine, F.G., S.C. Kenyon, J.K. Factor, R.G. Trimmer, N.K. Pavlis, D.S. Chinn, C.M. Cox, S.M. Klosko, S.B. Luthcke, M.H. Torrence, Y.M. Wang, R.G. Williamson, E.C. Pavlis, R.H. Rapp, and T.R. Olson, The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96, NASA/TP-1998-206861, Goddard Space Flight Center, Greenbelt, MD, July, 1998a.
Lemoine, F.G., E.C. Pavlis, N.K. Pavlis, C.M. Cox , D.S. Chinn, M.H. Torrence, R.G. Williamson, and Y.M. Wang, Gravity Field Improvement Activities at NASA GSFC, International Association of Geodesy Symposia, Vol. 119, Forsberg et al. (eds), Geodesy on the Move, Springer-Verlag, Berlin, Heidelberg, 1998b.
Lemoine, F.G., C.M. Cox, D.S. Chinn, N.K. Pavlis, Y.M. Wang, M.H. Torrence, R.G. Williamson, and E.C. Pavlis, Improved Earth Gravity Solutions Derived From TDRSS, presentation at European Geophysical Society XXIII EGS98, Nice France, April 20-24, 1998c.
NRC (National Research Council), Satellite Gravity and the Geosphere, National Academy Press, Washington D.C., 1997.
Pavlis, N.K., C.M. Cox , E.C. Pavlis, and F.G. Lemoine, Intercomparison and evaluation of some contemporary global geopotential models, presentation at the 2nd Joint Meeting of the International Gravity Commission (IGC) and the International Geoid Commission (IGeC), Trieste, Italy, September 7-12, 1998a.
Pavlis, N.K., Y.M. Wang, D.S. Chinn, C.M. Cox, and F.G. Lemoine, Introduction of Ocean Circulation Model Information in Global Geopotential Solutions: Preliminary Results, presentation at European Geophysical Society XXIII EGS98, Nice France, April 20-24, 1998b.
Pavlis, N.K., C.M. Cox , Y.M. Wang, and F.G. Lemoine, Further analyses towards the introduction of ocean circulation model information into geopotential solutions, presentation at the 2nd Joint Meeting of the International Gravity Commission (IGC) and the International Geoid Commission (IGeC), Trieste, Italy, September 7-12, 1998c.
Pavlis, N.K., Observed Inconsistencies Between Satellite-only and Surface Gravity-Only Geopotential Models, International Association of Geodesy Symposia, Vol. 119, Forsberg et al. (eds.), Geodesy on the Move, Springer-Verlag, Berlin, Heidelberg, 1998d.
Pavlis, N.K., Modeling of Long Wavelength Systematic Errors in Surface Gravimetric Data, presentation at European Geophysical Society XXIII EGS98, Nice France, April 20-24, 1998e.
Pavlis, N.K., C.M. Cox, and F.G. Lemoine, Comparison of dynamic ocean topography solutions combining geodetic and oceanographic information, Eos Trans., AGU, Vol. 80, No. 17, 1999.
Rapp R.H., Use of potential coefficient models for geoid undulation determinations using a spherical harmonic representation of the height anomaly/geoid undulation difference. J. Geod., 71, 282-289, 1997.
Rapp, R.H., Past and future developments in geopotential modeling, International Association of Geodesy Symposia, Vol. 119, Forsberg et al. (eds.), Geodesy on the Move, Springer-Verlag, Berlin, Heidelberg, 1998.
Rapp, R.H., Artifacts introduced in the point evaluation of functions expanded into a degree 360 spherical harmonic series, NASA/CR-1999-209229, Goddard Space Flight Center, Greenbelt, MD, May, 1999 (in press).
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Sansò, F., The Earth Gravity Model EGM96: Testing Procedures at IGeS, in International Geoid Service Bulletin No. 6, Politecnico di Milano, Milano, Italy, 1997.
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