Models

Advances in satellite tracking techniques, satellite altimetry, and gravity solution techniques along with the availability of surface gravity data in previously uncovered regions have led to significant improvements in the model of the Earth's gravity field during the last four years. While the 1980s saw the development of long wavelength gravity models mainly from satellite tracking data; models developed in the 1990s have increasingly been based on a combination of satellite tracking, satellite altimeter, and surface gravity data. This in turn has led to a marriage of techniques used to compute long wavelength (spherical harmonic degree 70) gravity models and high resolution (degree 360) gravity models [ Rapp, 1993a] such that future models will represent the best in each (the half-wave-length resolution of a spherical harmonic model is determined by dividing the circumference of the body by 2N, where N is the maximum degree of the spherical harmonic expansion).

The development of high resolution spherical harmonic models of the gravity field has been pursued in the U.S. primarily by Richard Rapp and his colleagues at the Ohio State University. The most recent of these models is Ohio State University (OSU)-91A [ Rapp et al., 1991] which is a comprehensive model complete to degree 360 in spherical harmonics. This model has become a standard for high resolution spherical harmonic representations, as no comparable model has been published since, although this will almost certainly change in the near future as will be discussed later. The 1 sigma () errors in the geoid defined by this model are described by Rapp [1993b], and are estimated at 28 cm rms over the oceans and 46 cm rms over the continents with significantly larger errors seen in regions lacking available surface gravimetry (Asia, polar regions). This accuracy is only sufficient to determine the ocean dynamic topography at wavelengths shorter than about 2500 km, thus significant improvement in the geoid is sought for oceanographic investigations [ Nerem and Koblinsky, 1993].

In the mid-1980s, an effort was initiated by the National Aeronautics and Space Administration's Goddard Space Flight Center (NASA/GSFC) and the Center for Space Research at the University of Texas to substantially improve the Earth's gravity field model and provide enhanced radial orbit accuracies for the TOPEX/POSEIDON altimeter mission. A series of models were developed in support of this effort including Goddard Earth Model (GEM)-T3 [ Lerch et al., 1994] and Texas Earth Gravity model (TEG)-2B [ Tapley et al., 1991], with both models complete to degree 50. These models are based on a combination of satellite tracking, satellite altimeter and surface gravity data. The use of altimetry to measure the geoid also requires a simultaneous solution for the ocean dynamic topography as described for GEM-T3 by Nerem et al. [1994c]. The launch of the French SPOT-2 satellite and its DORIS tracking system provided the best globally distributed set of precise tracking data prior to the GPS experiment on TOPEX/POSEIDON. Nerem et al. [1994a] describe the addition of SPOT-2/DORIS data to GEM-T3, resulting in a substantially improved model called GEM-T3A. The largest improvements were in the definition of the gravity field in the polar regions and the representation of the gravitational perturbations for sun-synchronous orbits. The development of improved supercomputers and ancillary orbit models led to a complete reprocessing of the available tracking data, altimeter data, and surface gravity data to produce Joint Gravity Model (JGM) 1 [ Nerem et al., 1994b], a 70 x 70 model which was the final prelaunch model for TOPEX. The improvement of JGM-1 over previous models was largely obtained through the improvement of background models and constants and the iteration of the linearized estimation process, since the data employed were largely the same as in GEM-T3A. The inclusion of SLR and DORIS TOPEX/POSEIDON tracking data resulted in the post-launch model, JGM-2 [ Nerem et al., 1994b]. The improvement in the RMS radial orbit accuracy for TOPEXPOSEIDON (Table 1) is evidence of the progress which has been made beginning with GEM-L2 at 65 cm and progressing to JGM-2 at 2 cm. However, in addition to benefiting TOPEX/POSEIDON, these gravity models have also improved the definition of the global gravity field for geophysical studies, as indicated by the global geoid error predictions shown in Table 2 [ Rapp, 1993b, Rapp and Wang, 1993].

Pre-launch simulations indicated that the experimental GPS receiver on TOPEX/POSEIDON would provide substantial improvements in the long wavelengths of the gravity model [ Bertiger et al., 1992; Wu and Yunck, 1992]. These simulations were based on the errors in the GEM-T2 gravity model, thus the actual improvement in the JGM-2 gravity model obtained with TOPEX/POSEIDON GPS data has been modest [ Schutz et al., 1994; Tapley et al., 1994a]. Nevertheless, the results that have been achieved are important and demonstrate the potential for a GPS receiver flown on a lower altitude satellite. The radial orbit errors for TOPEX/POSEIDON due to errors in the JGM-3 gravity model [ Tapley et al., 1994a] have been reduced to the 1 cm level (Table 1), with virtually no geographical correlation [ Christensen et al., 1994]. The improvements in the long wavelengths of the gravity model are in reasonable agreement with the simulations, which lends confidence to additional simulations which indicate that substantial improvements could be gained for satellites at lower altitudes.

These global gravity models are often used as a reference for computing detailed regional gravity models [ Wang, 1993b]. Recent examples of this include the detailed geoid models for the U.S. denoted GEOID90 [ Milbert, 1990] and GEOID93 [ Milbert and Schultz, 1993] which have become popular for various GPS applications. Rapp and Wang [1994] and Rapp and Smith [1994] have similarly developed a gravimetric geoid for the Gulf Stream region. Refinements in data processing and solution techniques [ Wang, 1993a; 1993b] have led to improvements in the calculation of geoid undulations. These global models have also been used for a number of statistical studies of the gravity field [ Kaula et al., 1993; Jekeli, 1991].

A number of improvements have been made to supporting models which have improved the gravity field models. Ries et al. [1992] determined an improved value of the product of the gravitational coefficient and the mass of the Earth (GM = 398600.4415 0.0008 km/sec) which is generally used as a background constant for many of these models. They achieved nearly identical results processing only Lageos SLR data as well as from a multi-satellite solution. Improvements have also been gained in modeling the effects of relativity for orbit determination [ Ries et al., 1991]. Satellite altimetry has also provided substantial improvements to the ocean tide models [ Ray, 1993], with TOPEX/POSEIDON providing some of the best models to date [ Egbert et al., 1994; Ma et al., 1994; Schrama and Ray, 1994; Wagner et al., 1994]. Marshall et al. [1995, this issue] review many of these orbit modeling improvements.

One of the most difficult aspects of any model development is the estimation of its accuracy. The TOPEX/POSEIDON effort to produce an improved gravity model has also led to improvements in the techniques for weighting individual data sets contributing to the model, resulting in more realistic assessments of the model errors. Lerch [1991], Lerch et al. [1991], Yuan [1991], and Lerch et al. [1993a; 1993c] have developed procedures for optimally determining the weights of the data sets used in comprehensive gravity solutions containing satellite tracking data, altimeter data, and surface gravity data. As a result, error estimates for these models, such as given in Table 1, have been shown to be quite reliable [ Nerem et al., 1994b].