Richard H. Rapp

The Development of the NASA GSFC and NIMA Joint Geopotential Model

The NASA Goddard Space Flight Center, the National Imagery and Mapping Agency (NIMA; formerly the Defense Mapping Agency or DMA) and The Ohio State University have collaborated to produce EGM96, an improved degree 360 spherical harmonic model representing the Earth’s gravitational potential. This model was developed using: (1) satellite tracking data from more than 20 satellites, including new data from GPS and TDRSS, as well as altimeter data from TOPEX, GEOSAT and ERS-1. (2) 30’ x 30’ terrestrial gravity data from NIMA’s comprehensive archives, including new measurements from areas such as the former Soviet Union, South America, Africa, Greenland, and elsewhere. (3) 30’ x 30’ gravity anomalies derived from the GEOSAT Geodetic Mission altimeter data, as well as altimeter derived anomalies derived from ERS-1 by KMS (Kort and Matrikelstyrelsen, Denmark) in regions outside the GEOSAT coverage. The high degree solutions were developed using two different model estimation techniques: quadrature, and block diagonal. The final model is a composite solution consisting a combination solution to degree 70, a block diagonal solution to degree 359, and the quadrature model at degree 360. This new model will be used to define an undulation model that will be the basis for an update of the WGS-84 geoid. In addition, the model will contribute to oceanographic studies by improving the modeling of the ocean geoid and to geodetic positioning using the Global Positioning System (GPS).

Gravity model development for TOPEX/POSEIDON: Joint gravity models 1 and 2

The TOPEX/POSEIDON (T/P) prelaunch Joint Gravity Model-1 (JGM-I) and the postlaunch JGM-2 Earth gravitational models have been developed to support precision orbit determination for T/P. Each of these models is complete to degree 70 in spherical harmonics and was computed from a combination of satellite tracking data, satellite altimetry, and surface gravimetry. While improved orbit determination accuracies for T/P have driven the improvements in the models, the models are general in application and also provide an improved geoid for oceanographic computations. The postlaunch model, JGM-2, which includes T/P satellite laser ranging (SLR) and Doppler orbitography and radiopositioning integrated by satellite (DORIS) tracking data, introduces radial orbit errors for T/P that are only 2 cm RMS with the commission errors of the marine geoid for terms to degree 70 being ±25 cm. Errors in modeling the nonconservative forces acting on T/P increase the total radial errors to only 3–4 cm RMS, a result much better than premission goals. While the orbit accuracy goal for T/P has been far surpassed, geoid errors still prevent the absolute determination of the ocean dynamic topography for wavelengths shorter than about 2500 km. Only a dedicated gravitational field satellite mission will likely provide the necessary improvement in the geoid.

Separation between reference surfaces of selected vertical datums

This paper discusses the separation between the reference surface of several vertical datums and the geoid. The data used includes a set of Doppler positioned stations, transformation parameters to convert the Doppler positions to ITRF90, and a potential coefficient model composed of the JGM-2 (NASA model) from degree 2 to 70 plus the OSU91A model from degree 71 to 360. The basic method of analysis is the comparison of a geometric geoid undulation derived from an ellipsoidal height and an orthometric height with the undulation computed from the potential coefficient model The mean difference can imply a bias of the datum reference surface with respect to the geoid. Vertical datums in the following countries were considered: England, Germany, United States, and Australia. The following numbers represent the bias values of each datum after adopting an equatorial radius of 6378136.3m: England (-87 cm), Germany (4 cm), United States (NGVD29 (-26 cm)), NAVD88 (-72 cm), Australia AHD (mainland, -68 cm); AHD (Tasmania, -98 cm). A negative sign indicates the datum reference surface is below the geoid. The 91 cm difference between the datums in England and Germany has been independently estimated as 80 cm. The 30 cm difference between AHD (mainland) and AHD (Tasmania) has been independently estimated as 40 cm. These bias values have been estimated from data where the geometric/ gravimetric geoid undulation difference standard deviation, at one station, is typically ±100 cm, although the mean difference is determined more accurately.The results of this paper can be improved and expanded with more accurate geocentric station positions, more accurate and consistent heights with respect to the local vertical datum, and a more accurate gravity field for the Earth. The ideas developed here provide insight on the determination of a world height system.

Degree variances of the earth's potential, topography and its isostatic compensation

A spherical harmonic expansion of the earths gravitational potential and equivalent rock topography to degree and order 180 is described. The potential implied by the topography considered as uncompensated and with isostatic compensation has been computed. Good agreement with the observed potential field is found when the depth of compensation in the Airy theory is assumed to be 50 km. At the higher degrees the correlation coefficient between the potential expansion and the equivalent rock topography is about 0.5.The Lachapelle equations for the topographic isostatic potential were tested using 1ox1o equivalent rock topography. The degree variances agree at the lower degrees but at degree 36 the Lachapelle results using 5o data underestimate the potential degree variances by about one-third.

Mean sea surface and geoid gradient comparisons with TOPEX altimeter data

Cycles 4 to 54 of TOPEX data have been analyzed through comparisons with the mean sea surface given on the distributed geophysical data record (GDR). Two inverted barometer correction procedures were considered for the data reduction. One used a constant atmospheric pressure for all data while the one adopted for use, for most computations, introduced a cycle average pressure. The maximum difference between the two estimates was 3.0 cm with a clear annual signal. With the modified correction the TOPEX sea surface was compared to The Ohio State University (OSU) mean sea surface, given on the GDR, to estimate three translations (Δx = −2.3 cm; Δy = 25.0 cm; Δz = −0.3 cm) and a bias (43.3 cm) between the two surfaces. The only significant translation is Δy which indicates the reference frame of the TOPEX system differs from that used in the OSU mean sea surface system. The bias between the TOPEX mean sea surface and the OSU mean sea surface was used to estimate an equatorial radius of 6,378,136.55 m based on an 18-cm bias estimate of the TOPEX altimeter. Examination of the average difference, by cycle, between the TOPEX sea surface and the OSU mean sea surface suggested a bias change of 3.1±2.2 mm/yr with a positive sign indicating the average ocean surface is rising or the altimeter measured distance is decreasing. Models were implemented that solved directly for a bias, bias rate, annual/semiannual, and tide correction terms. These latter two terms were represented by a degree 8 spherical harmonic expansion with the tide corrections being to the components of the Cartwright and Ray tide model given on the GDR. The most complete solution gave a bias rate of 3.6±0.6 mm/yr when the OSU mean sea surface was used and 3.2±0.4 mm/yr when the TOPEX data were used to define a mean sea surface. When the bias and bias rate terms were included in the estimation process, the annual/semiannual terms and the tide terms estimate changed slightly. The greatest change (±1 mm) took place in the annual term. The computations indicated that a simultaneous solution for bias, bias rate, and annual/semiannual terms gave the most accurate results. Nonsimultaneous solutions led to slightly different bias rate values. The root mean square difference between the TOPEX sea surface and the OSU mean sea surface, after translation and bias correction, was ±17 cm for a typical cycle. However, some locations were identified where the difference could reach 2.3 m and were repeated over several cycles indicating errors in the mean sea surface. Most of the large differences occur in regions lacking altimeter data prior to the TOPEX/POSEIDON mission and/or areas of significant bathymetrie signature. Geoid gradients are needed for the reduction of individual track data to a reference track. The accuracy of the determination of such gradients was determined through the comparison of predicted (from a mean sea surface) along-track gradients to the observed gradients. Among four mean sea surfaces tested the best agreement (±0.9 cm/km) was found with the OSU mean sea surface placed on the TOPEX geophysical data record.

Undulation and anomaly estimation using Geos-3 altimeter data without precise satellite orbits

The paper describes results obtained from the processing of 53 Geos-3 arcs of altimeter data obtained during the first weeks after the launch of the satellite in April, 1975. The measurement from the satellite to the ocean surface was used to obtain an approximate geoid undulation which was contaminated by long wavelength errors caused primarily by altimeter bias and orbit error. This long wavelength error was reduced by fitting with a low degree polynomial the raw undulation data to the undulations implied by the GEM 7 potential coefficients, in an adjustment process that included conditions on tracks that cross. The root mean square crossover discrepancy before this adjustment was ±12.4 meters while after the adjustment it was ±0.9 m. These adjusted undulations were used to construct a geoid map in the Geos-3 calibration area using a least squares filter to remove remaining noise in the undulations. Comparing these undulations to ones computed from potential coefficients and terrestrial gravity data indicates a mean difference of 0.25 m and a root mean square difference of ±1.92 m.The adjusted undulations were also used to estimate several 5o, 2o, and 1o anomalies using the method of least squares collocation. The resulting predictions agreed well with known values although the 1o x 1o anomalies could not be considered as reliably determined.

Analysis of dynamic ocean topography using TOPEX data and orthonormal functions

The representation of dynamic ocean topography (ζ) through spherical harmonic (SH) and orthonormal (ON) expansions was studied using TOPEX altimeter data, three potential coefficient models used to define geoid undulations, and three estimates of ζ from oceanographic data and global circulation models (GCMs). The ON expansions are desirable when one wishes to study the spectral characteristics of a function in a defined domain such as the ocean. The potential coefficient models tested were JGM-2, JGM-3, and GRIM4_C4b. Each model was augmented with the OSU91A potential coefficients from degree 71 to 360. The ζ models were those of Levitus [1982] and values implied by the POCM_4B (Semtner/Chervin) model and a Los Alamos National Laboratory Model POP(96) (Malone, Smith, Dukowicz). The latter two models were defined over a 2-year time period. Values of ζ were computed from 2 years of TOPEX data using the three potential coefficient models. The ON expansions of ζ from the TOPEX data were then compared to the estimates from the oceanographic data. The differences, to ON degree 14, with the POCM_4B model and the TOPEX results were ±14.0 cm (JGM-2), ±12.4 cm (JGM-3), and ±14.4 cm (GRIM4_C4b). A comparison with the other ζ estimates using TOPEX/JGM-3 gives differences of ±14.3 cm (Levitus) and ±13.3 cm (POP (96)). The comparisons were made only to degree 14 because (1) the correlation between the ζ coefficients from TOPEX data and POCM_4B fell off beyond degree 14 and (2) the geoid undulation accuracy, in the ocean region, was equal to the ζ signal near degree 14. These results suggest ζ estimates made above degree 14 may be contaminated by geoid undulation errors. Also suggested from the comparisons was that the TOPEX/JGM-3 estimates of ζ were more reliable than those from oceanographic data to degree 8 (2500-km resolution). The ζ estimates from the POCM_4B and POP(96) models, 2-year averages, agreed well north of 40°S. Below this the differences could reach 40 cm in the Antarctic Circumpolar Current (60°S, 215°). The differences between the TOPEX/JGM-3 and POCM_4B ζ estimates exceeded 20 cm in a number of places (e.g., (20°N, 140°), (5°S, 130°), (60°S, 220°), (45°N, 320°)). The largest differences (−62 cm) occurred in the Banda Sea. The ζ representations were used to calculate upper ocean geostrophic velocities in the east/west and north/south directions. Excluding a 10° band on either side of the equator, the difference (TOPEX versus POCM_4B) was ±2.5 cm/s with the magnitude of the total velocity being 4.8 cm/s. The difference was consistent with the error estimates of the velocities implied by the errors in the JGM-3 coefficients to degree 14. The ζ estimates were also determined from four recent mean sea surface grids and the results compared to the POCM_4B model through the ON representation. The MSS grids used were the OSUMSS95, the UTCSRMSS95, the GFZ/D-PAF MSS95A, and the CNES/GRGS MSS95. The best agreement, to degree 14, was found with the OSUMSS95 (±11.1 cm) and the CSRMSS95 (±11.5 cm). The comparisons were poorer (±15 cm) when a mean sea surface was used where no mean inverted barometer correction had been applied to the gridded data. Although substantial progress has been made in the past 10 years in the determination of the Earths gravitational potential, the accuracy limitations of geoid undulation determination still hinder the comparison and assimilation of altimeter data and oceanographic data. The need for a dedicated gravity satellite mission, to yield improved geoid undulation determinations, is clearly seen.

Geoid undulation accuracy

For use in oceanographic applications the geoid is ideally needed to a high accuracy and to a high resolution. In 1979 the cumulative geoid undulation error to spherical harmonic degree 20 was +or-1.4 m for the GEM10 potential coefficient model. Today the corresponding value has been reduced to +or-25 cm for GEM-T3 or +or-11 cm for the OSU91A model. Similar improvements are noted in harmonic degree (wavelength) and in resolution. The accuracy of the determination of the geoid is discussed from several points of views. Some comparisons are made with information available 12 years ago and that now available in the time framework of the TOPEX launch. It is noted that the accuracies described are subject to constant improvement. This is especially true as new satellite tracking data (e.g. DORIS data) are used in the potential coefficient models and more satellite altimeter data become available from the ERS-1 and TOPEX/Poseidon missions. >

Methods for the computation of geoid undulations from potential coefficients

The undulations of the geoid may be computed from spherical harmonic potential coefficients of the earth’s gravitational field. This paper examines three procedures that reflect various points of view on how this computation should be carried out. One method requires only the flattening of a reference ellipsoid to be defined while the other two methods require a complete definition of the parameters of the ellipsoid. It was found that the various methods give essentially the same undulations provided that correct parameters are chosen for the reference ellipsoid. A discussion is given on how these parameters are chosen and numerical results are reported using recent potential coefficient determinations.

A comparison of altimeter and gravimetric geoids in the Tonga Trench and Indian Ocean areas

A gravimetric geoid computed using different techniques has been compared to a geoid derived from Geos-3 altimeter data in two 30°×30° areas: one in the Tonga Trench area and one in the Indian Ocean. The specific techniques used were the usual Stokes integration (using 1°×1° mean anomalies) with the Molodenskii truncation procedure; a modified Stokes integration with a modified truncation method; and computations using three sets of potential coefficients including one complete to degree 180. In the Tonga Trench area the standard deviation of the difference between the modified Stokes’ procedure and the altimeter geoid was ±1.1 m while in the Indian Ocean area the difference was ±0.6 m. Similar results were found from the 180×180 potential coefficient field. However, the differences in using the usual Stokes integration procedure were about a factor of two greater as was predicted from an error analysis.We conclude that there is good agreement at the ±1 m level between the two types of geoids. In addition, systematic differences are at the half-meter level. The modified Stokes procedure clearly is superior to the usual Stokes method although the 180×180 solution is of comparable accuracy with the computational effort six times less than the integration procedures.