12.1 EGM96 Solution Achievements

In this report we have described the derivation of the EGM96 geopotential model, including the estimation of the 30´x30´ anomalies, the processing of the satellite tracking data, and the direct altimeter data. The solution methodologies are described in detail for both the low-degree combination model, and the high-degree models. The creation and testing of the intermediate and final solutions are also described. The final solution blends a low degree (to degree 70) combination model (obtained from combining satellite tracking data, surface gravity data, and direct altimeter measurements) which is based on the most complete and rigorous modeling and estimation techniques, with high-degree models (beyond degree 70 to degree 360) that exploit symmetry properties associated with the potential coefficient estimation from regularly gridded 30´x30´ mean gravity anomaly data.

The development of the EGM96 geopotential model was a major undertaking which challenged our current technical and computational capabilities. The three year cooperative effort combined the insights, resources, and data available within NASA and NIMA, and involved more than two dozen participants. The major technical objectives were achieved and an improved high degree gravitational model was delivered to the science, mapping, and navigation communities. Major advancements in gravitational field modeling achieved with EGM96 included: (a) the incorporation of new surface gravity data, satellite-tracking data and altimeter data into a 360x360 geopotential solution, (b) improved model accuracy, (c) the development of important solution by-products including a global topographic model used in reduction of the surface gravity data and the simultaneous estimation of a tidal solution along with the geopotential coefficients, (d) design, testing, and implementation of the block-diagonal method for development of the high-degree solutions.

An important aspect in the development of the EGM96 model was the multiple set of criteria used to test the interim and final project geopotential models. A variety of techniques were used to assess the performance of the models including satellite tracking data fits, GPS/leveling geoid undulation comparisons, dynamic ocean typography comparisons with ocean circulation models, comparisons to altimeter-derived gravity anomalies, and other land and ocean geoid tests. The extensive testing assured not only that the model provided good orbit fits, but that it also performed well for a variety of terrestrial and oceanic applications.

From the early design stages of EGM96 it was recognized that a large amount of new surface gravity data were becoming available due to changes in the international political landscape. A major effort was undertaken by NIMA to process these data and to form 30´x30´ mean anomaly estimates. The 30´ mean values were estimated using a uniformly consistent and rigorous approach (least squares collocation). This was true for all continental areas where detailed gravimetry was available, as well as for those areas covered by airborne gravity surveys. Over most of the Earth’s oceans 30´ mean gravity anomalies were estimated using satellite radar altimeter data acquired by the US Navy’s GEOSAT satellite during its Geodetic Mission. Mean gravity anomalies derived from ERS-1 altimeter data were used in ocean areas not covered by GEOSAT.

Surface Gravity Data

EGM96, through its incorporation of newly available surface gravimetry has significantly improved continental geoid modeling. The new data include contributions over most of Asia and the former Soviet Union, airborne gravity surveys over polar regions including Greenland, surveyed data from South America, Africa, and North America, as well as improvements to the data sets provided by many countries. These data enhancements have all increased the short wavelength global geoid accuracy of the resulting model. Of importance is the progress which was achieved in eliminating a significant level of inconsistency between the geopotential signal sensed by satellite tracking versus terrestrial anomaly data. Earlier combination solutions "required" (given model design considerations) the strong downweighting of surface gravimetry (for example in JGM—2 and JGM—3). EGM96 gave much higher weight to the surface information, yet still performs well on orbital and ocean geoid modeling applications. The more effective use of this unique information resulted in a model which has more realistic error estimates, especially at higher degrees, and spectral error characteristics which are less discontinuous at the degree 70 boundary than earlier "cut and paste" models such as JGM—3/OSU91A. At degree 70, comprehensive solution approaches were abandoned in favor of more computationally efficient block-diagonal and quadrature techniques. Since the surface gravity data are no longer downweighted, stronger information comes from surface gravimetric sources to define the middle degree terms in the model. It is this part of the field (n ³ 40) where satellite tracking information falls off significantly because of the attenuation in the field sensitivity experienced on satellites now used for geodetic purposes.

Satellite-to-Satellite Tracking Data

EGM96 used several new data types to great advantage. The range and range-rate tracking of low Earth orbiting user satellites by the TDRSS geostationary constellation, and the complete 3-D positioning of similar spacecraft achieved using the constellation of 24 GPS satellites, provided precise data not available in previous models. The TDRSS and GPS tracking acquired on the low altitude (525 km), low inclination (28.5° ), EP/EUVE satellite provided a large geopotential modeling improvement in the equatorial regions. While only three satellites tracked by these systems were used in EGM96 (TOPEX/POSEIDON, EP/EUVE, and GPS/MET), these data represent a sizable fraction of the observational data used in EGM96. By providing nearly continuous tracking, these data are sensitive to many of the short period orbit perturbations which are not well sensed by conventional, discontinuous tracking data types (like SLR and ground based Doppler). They improved the separation of harmonic terms in the satellite-only EGM96S model, and provided complementary information to the surface gravimetry and altimeter data sets in the middle degrees of the model.

Altimeter Data

EGM96 incorporated altimeter data in two distinct forms: (1) as 30’x30’ mean altimeter-derived anomalies in the high-degree models, and (2) as direct tracking data in the low-degree (to degree 70) combination model.

The 30´x30´ mean altimeter-derived anomalies used in the development of EGM96 were obtained from GEOSAT and ERS—1. The major source for these anomaly data was the GEOSAT Geodetic Mission altimeter data, where the oceanic gravity anomalies were produced using a rigorous least squares collocation process. The Danish National Survey and Cadastre or Kort-og Matrikelstyrelsen (KMS) contributed to the anomaly data sets by collaborating in the development of the collocation procedure and by providing ERS—1 gravity anomalies [Andersen et al., 1996; Forsberg, 1987]. The ERS-1 data made an important contribution by extending the coverage in the near-polar areas and a few near-shore areas. Tilo Schoene (of the Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany) provided gravity anomaly values for the Weddell Sea area near Antarctica. These data were derived from a combination of GEOSAT and ERS—1 altimetry [Schoene, 1996].

In addition, altimeter data from GEOSAT, TOPEX/POSEIDON, and ERS—1 were used as direct tracking information in the low degree (n £ 70) combination model, improving both the orbit accuracies of ERS—1 and ocean surface mapping from these systems. Concurrent altimeter data provided by TOPEX and ERS—1 were used to define a consistent dynamic ocean topography (DOT), extending to the high latitudes, where two years of data allowed simultaneous solution for a mean dynamic topography model augmented by both annual and semi-annual terms.

Conventional Tracking Data

Data from conventional tracking, including observations acquired by SLR, TRANET, and DORIS systems were upgraded for inclusion in EGM96. Of special interest was the addition of data from several new laser (LAGEOS—2, Stella, GFZ—1) and Doppler (HILAT and RADCAL) tracked satellites. These data added strength to the solution and filled several important inclination and altitude gaps in the JGM—2S satellite orbit distribution.

Improved Model Accuracy

EGM96 represents a significant model improvement over recent available models such as JGM—2 and JGM—3. This improvement is seen at the lowest degrees, in improved orbital fits to precise SLR data sets and in the improved modeling of the ocean geoid for ocean circulation studies. Through the middle and high degrees, the uncertainty improvements are more than a factor of two over both JGM—2 and OSU91A, which are its major predecessors. Results of the calibration of the satellite-only model foundation and tests of the combination model covariance indicate that the predicted uncertainties are well calibrated and represent reasonable, if somewhat conservative error predictions. Most striking is the elimination of areas with large geoid uncertainties, which was seen in earlier models where accurate surface gravity information was lacking, for instance over large sections of Asia, Africa, and South America.

EGM96 Solution By-Products

Important ancillary products were developed contemporaneously with the EGM96 solution. Along the static geopotential, the combination component of the solution to 70x70 included estimates of dynamic tide parameters, dynamic ocean topography solutions to 20x20 for TOPEX/ERS—1 and GEOSAT, station coordinates, and a pole position time series.

In addition, for the accurate evaluation of 30' mean anomalies and associated terrain reductions, a 5´x5´ global topographic model (JGP95E) was developed. The JGP95E model used previously unavailable terrestrial data as well as topographic information obtained from satellite altimeter measurements acquired over Antarctica.

EGM96 provided an improved dynamic tide model for orbital applications. A select subset of tidal terms, representing the resonant portion of the tidal spectra for the major tide lines, was estimated simultaneously with the static geopotential harmonics. These tidal parameters improve the modeling of lower altitude orbits, provide GSFC’s first estimates for the Q1 tide line, and fully exploited the capability of simultaneously modeling the complete tidal family (mainline and sideband tide lines) to eliminate much of the aliasing arising from lack of sideband modeling in earlier recovery efforts.

Advances in Solution Design and Methodology

A new method of developing high-degree geopotential solutions was designed, tested, and implemented. The block-diagonal technique is computationally efficient, yet allows the preservation of the most important correlative effects found within the high degree model and permits a much smoother transition at degree 70 between solution methodologies.

Improved a priori constraint models of the expected power in the gravitational field and dynamic ocean topography models were used. The a priori power law constraint used in the satellite-only geopotential solutions was derived from the coefficients of a quadrature combination solution. The Kaula-type power law constraint used in previous models, such as JGM—1S and JGM—2S, underestimated the power and consequently the predicted error at the higher degrees of the satellite-only solutions. A power law fit to the spherical harmonic spectrum of the POCM—4B ocean circulation model was used to better condition the solutions for dynamic ocean topography for solutions that included direct altimetry.

The prediction of gravity anomalies from altimetry was advanced through the incorporation of dynamic ocean topography modeling and improvements in covariance functions. Fitting the GEOSAT GM mean sea surface to that of TOPEX removed a large part of the long wavelength errors in the altimeter-derived gravity anomalies.

Structural and procedural changes in the GEODYN and SOLVE programs were implemented to improve computational efficiencies and eliminate "bottlenecks" in the development of the 70x70 satellite-only and combination model portions of EGM96. These modifications included changes to improve the I/O for the manipulation of numerous large matrices, and recoding to take advantage of multiple processors on the CRAY J932 supercomputer.

International Cooperation

Another key element of the joint project was the contribution made during the testing of various geopotential models by the Special Working Group of the International Geoid Service, chaired by Michael Sideris. Their testing [Sideris, 1997], performed independently, and with out knowledge of the make up of the models provided to them, yielded valuable information that helped to determine the best estimation and solution development strategy for EGM96. The international cooperation that occurred during the EGM96 project represents a first in the development of a major geopotential model.

12.2 Future Challenges

The process of finalizing EGM96, calibrating its errors, and determining the optimal data weights for its diverse sets of data, revealed many areas for future investigation. In some cases, clear deficiencies in current methodologies, or understanding of model properties, were discovered. While EGM96 represents a major milestone, significant efforts are still needed to take full advantage of existing data, and to prepare for future gravity missions. Some of the most important subjects which need to be studied include improved calibration techniques, improved methods for ocean tidal recovery, alternative representations of the dynamic ocean topography, as well as the incorporation of new satellite tracking data and new surface gravity data into future solutions.

Improved Calibration Techniques

Our objective calibration techniques produced unexpected results when applied to the strong data obtained from the continuous tracking of low Earth orbiting satellites by either the GPS or the TDRSS constellations. The non-linear behavior of the deduced calibration factors, described in Section 6.4, which was the basis for the determination of the data weights, is a concern. It both forced us to adopt weights which could not be objectively determined, and to rely on performance metrics against independent data (e.g. tests against altimeter-derived gravity anomalies and GPS/leveling traverses), to determine final weights. While we have several ideas about the cause of this behavior, improved calibration methods are needed as additional data sources like these come to be dominant within geopotential solutions. Indeed, continuous tracking data geometries will be the basis for the upcoming CHAMP and GRACE geopotential missions.

Improved Methods for Ocean Tidal Recovery:

Unlike the static geopotential, the recovery of dynamic tidal terms is critically dependent on the nature of the tidal resonances experienced by a given satellite, and the temporal distribution of the data included in the recovery. Our calibration methods, which focused on the static geopotential model, yielded poor calibration results for the tidal terms. There are additional challenges with tidal recovery:

Alternative tidal recovery strategies need to be investigated. Recovery of larger tide models should also be considered.

Alternative Representation of the Dynamic Ocean Topography

The dynamic ocean topography models recovered as part of EGM96 are represented as spherical harmonics. This representation has certain limitations. First, by being global, it requires definition of the dynamic ocean topography over the continents which is both meaningless, and subject to poor behavior given the non-existence of information over these regions. Secondly, at the ocean/land boundaries, given that the altimeter mapped ocean surface abruptly ends at this interface, it is common to see the implied flow deduced from the dynamic ocean topography going into or out of land. Alternative representations are free of many of these shortcomings. Consideration is being given to using orthonormal functions, defined only over the ocean surface for dynamic ocean topography representation. This includes use of: (1) height functions [Sanchez et al., 1997], (2) Empirical Orthogonal Functions (EOF) [Rapp et al., 1996; Hwang, 1991], and/or (3) Proudman functions [Rao et al., 1987; Sanchez and Pavlis, 1995] to improve the separation of geoidal and dynamic ocean topography signals, and improve the modeling characteristics.

Additional Tracking Data

There are a number of sources of additional tracking data which were not included in EGM96. These include TDRSS tracking of CGRO, RXTE, and ERBS [Luthcke et al., 1998]. In addition, the TDRSS constellation will provide data for future missions in unique orbits and inclinations, such as the TRMM mission (350 km altitude, 35° inclination) which was launched on November 27, 1997, from Tanegashima, Japan. GPS data from other satellite missions, such as OERSTED and the GEOSAT Follow-On (launched on February 10, 1998) will become available in the near future, even prior to the launch of CHAMP.

Surface Gravity Data

Despite the significant advances made in terms of both the coverage and the accuracy of terrestrial/airborne gravity data for EGM96, many geographic regions are still poorly surveyed (e.g. western China), have very sparse data (e.g. Antarctica), or are completely void of terrestrial anomaly data. Continuation of the collection efforts in these areas will definitely yield future model improvements. Future work is needed to identify problems in model performance over certain regions that have been noted in the literature for instance over the Foxe Basin, Ungava Bay, and Lake Superior (cf. Sansò, [1997]). Not only the gravity data availability, but also their modeling and weighting within combination solutions require additional study. Analytical continuation techniques require more careful examination both from a theoretical and computational standpoint.

Long wavelength systematic errors in terrestrial gravity anomaly data bases require special consideration in the analysis of surface gravity data. It is becoming more evident that a better approach is needed to account for these systematics and preserve the strengths of these unique data over significant bandwidths of the model.

Finally, despite the significant advances in terrestrial gravity anomaly information over land areas, the marine surface gravimetry has not been significantly upgraded or re-examined for EGM96 since the development of The Ohio State University database in 1990. A major effort will be required to improve the quality and coverage of the marine gravimetry for future gravity solutions. Release of additional marine gravimetric holdings would improve this situation and provide additional information for the needed separation of dynamic ocean topography and ocean geoid signals from their aggregate effect sensed by satellite radar altimeter data.

Therefore, while EGM96 has reached several milestones, efforts continue to improve the model for both specialized and multi-purpose applications. In preparation for the CHAMP and GRACE dedicated geopotential missions, better modeling of the ocean geoid to more fully exploit the 3—4 cm accuracy achieved with synoptic TOPEX/POSEIDON altimetry is needed to continue improving our understanding of ocean circulation and also to baseline temporal geopotential effects. We look forward to challenging activities in gravitational field modeling in the years ahead.

12.3 References

Andersen, O., P. Knudsen, and C.C. Tscherning, Investigation of Methods for Global Gravity Field Recovery from the Dense ERS—1 Geodetic Mission Altimetry, in Global Gravity Field and Its Temporal Variations, IAG Symposia, 116, 218—226, Springer—Verlag, Berlin, 1996.

Forsberg, R., A New Covariance Model for Inertial, Gravimetry and Gradiometry, J. Geophys. Res., 90, B2, 1305—1310, 1987.

Hwang, C., Orthogonal Functions Over the Oceans and Application to the Determination of Orbit Error, Geoid and Sea Surface Topography From Satellite Altimetry, Rep. 414, Dept. of Geod. Sci. and Surv., Ohio State Univ., Columbus, OH, 1991.

Luthcke, S.B., J.A. Marshall, C.M. Cox, F.G. Lemoine, D.D. Rowlands, R.G. Williamson, D.E. Pavlis, T.R. Olson, W.F. Eddy, S.C. Rowton, Precision Orbit Determination Using TDRSS, J. Astro. Sci., in review, 1998.

Rao, D.B., S.D. Steenrod, and B.V. Sanchez, A method of calculating the total flow from a given sea surface topography, NASA Tech. Memo. TM—87799, Goddard Space Flight Center, 1987.

Rapp, R.H., C. Zhang, and Y. Yi, Analysis of Dynamic Ocean Topography Using TOPEX data and Orthonormal Functions, J. Geophys. Res., 101, C10, 22583—22598, 1996.

Sanchez, B.V., and N.K. Pavlis, Estimation of main tidal constituents from TOPEX altimetry using Proudman function expansion, J. Geophys. Res., 100, C12, 25229—25248, 1995.

Sanchez, B.V., W.J. Cunningham, and N.K. Pavlis, The Calculation of the Dynamic Sea Surface Topography and the Associated Flow Field From Altimetry Data: A Characteristic Function Method, J. Phys. Ocean., 27, 7, July, 1997.

Sansò, F., The Earth Gravity Model EGM96: Testing Procedures at IGeS, in International Geoid Service Bulletin No. 6 , Politecnico di Milano, Milano, Italy, 1997.

Sideris, M., International tests of the new GSFC/DMA geopotential models, in Gravity, Geoid, and Marine Geodesy, International symposium, Tokyo, September 30-October 5, 1996, Segawa, Fujimoto, and Okubo (ed.), International Association of Geodesy Symposia, Vol. 117, Springer-Verlag, 1997.

Schoene, T., The gravity field in the Weddell Sea Antarctica by radar altimetry from ERS-1 and GEOSAT, Reports on Polar Research, 220/96 Alfred Wegener Institute, Bremerhaven, 1996.