An improved model for the Earth's gravity field, TEG-1, was determined using data sets from fourteen satellites, spanning the inclination ranges from 15 to 115 deg, and global surface gravity anomaly data. The satellite measurements include laser ranging data, Doppler range-rate data, and satellite-to-ocean radar altimeter data measurements, which include the direct height measurement and the differenced measurements at ground track crossings (crossover measurements). Also determined was another gravity field model, TEG-1S, which included all the data sets in TEG-1 with the exception of direct altimeter data. The effort has included an intense scrutiny of the gravity field solution methodology. The estimated parameters included geopotential coefficients complete to degree and order 50 with selected higher order coefficients, ocean and solid Earth tide parameters, Doppler tracking station coordinates and the quasi-stationary sea surface topography. Extensive error analysis and calibration of the formal covariance matrix indicate that the gravity field model is a significant improvement over previous models and can be used for general applications in geodesy

Ries, J. C.

Schutz, B. E.

Shum, C. K.

Tapley, B. D.

Yuan, D. N.

English

NASA Technical Reports Server

c N90-20515
An Improved Model for the Earth's Gravity Field
B. D. TAPLEY, C. K. SHUM, D. N. YUAN, J. C. RIES AND B. E. SCHUTZ
Center for Space Research, The University of Texas at Austin, Austin, Texas
ABSTRACT
An improved model for the Earth's gravity field, TEG-1, has been determined using data sets
from fourteen satellites, spanning the inclination ranges from 15 ° to 115 °, and global surface gravity
anomaly data. The satellite measurements include laser ranging data, doppler range-rate data, and
satellite-to-ocean radar altimeter data measurements, which include the direct height measurement
and the differenced measurements at ground track crossings (crossover measurements). Also
determined was another gravity field model, TEG-1S, which included all the data sets in TEG-1 with
the exception of direct altimeter data. The effort has included an intense scrutiny of the gravity field
solution methodology. The estimated parameters included geopotential coefficients complete to
degree and order 50 with selected higher order coefficients, ocean and solid Earth tide parameters,
doppler tracking station coordinates and the quasi-stationary sea surface topography. Extensive error
analysis and calibration of the formal covariance matrix indicate that the gravity field model is a
significant improvement over previous models and can be used for general applications in geodesy.
1. INTRODUCTION
Significant progress has been achieved during the last decade in the determination of the spherical
harmonic coefficients of the Earth's external gravitational potential. A substantial portion of this
progress can be directly attributed to the advent of Earth-orbiting artificial satellites and to the ability
to observe their motion from either ground-based or satellite-originated tracking data. While the
satellite data primarily resolve the long and intermediate wavelengths (> 1500 km), global surface
gravity measurements and the altimeter data are capable of recovering the shorter wavelength
components of the Earth's gravity field. Recent trends in gravity model improvement have been
driven, in part, by requirements for more accurate satellite orbits to achieve the objectives of the
Crustal Dynamics Project and the recently approved NASA/CNES Topex/Poseidon mission. A joint
effort to develop an improved model for the Earth's gravity field has been undertaken to develop a
gravity model to meet the orbit accuracy requirement of the Topex/Poseidon mission. The gravity
field solution will represent the first complete reiteration of the historical tracking data used to define
the NASA Goddard Space Flight Center (GSFC) Earth Model series. The GSFC GEM-T1 [Marsh et
al., 1988] and the University of Texas (UT) TEG-I fields, described in this paper, are preliminary
versions of the Topex gravity field solution.
2. THEORY AND METHOD
The gravitational potential, U, due to the Earth's nonspherical mass distribution can be expressed
as follows
Iq EU- GM _ k /_(sin*) (_m+ACt")cosm_.+(Sll"+ASl")sinmk
r l=:O"=O
where GM is the product of the gravitational constant and the total mass of the Earth and the
atmosphere; R e is the mean equatorial radius of the Earth; /_t" are the normalized Legendre
associated function of degree l and order m; Ctm ,St," are the the normalized spherical harmonic
coefficients whose values are functions of the mass distribution within the Earth and the atmosphere;
A_,", ASt," are the time-varying components of Ct,, and St," caused by tides; also are functions of the
tidal coefficients, Ct_ and St_; and r,tL_, are the Earth-fixed spherical coordinate system; r is the
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TAPLEYETAL: ANIMPROVED MODEL FOR THE EARTH'SGRAVITY FIELD
radial distance, # is the geocentric latitude and _. is the longitude measured from the Greenwich
meridian.
The estimation of Ct,,,,_,,,, ± ±Ctm,Slm and other orbit and geophysical parameters can be
accomplished using a modified least-squares estimation procedure. This estimation procedure, which
provides adjustments to satellite orbit-dependent parameters and other geophysical and geodetic
parameters, is given by Tapley 11973] and modified to include the simultaneous estimation of the
relative weights for the individual satellite information arrays [Yuan et al., 1988]:
= (HTI_-IH)-IHTI_-Iy ; l_i = 1/ki(Yi-Hi;_)-l(yi-Hi_)'I
where _ is the state parameter;/_ is the weighting matrix; I is the identity matrix; H i is the partial
derivative with respect to x for the i th data set; and k i is the number of observations for i th dataset.
The system of equations given above can be solved iteratively using orthogonal transformation
techniques [Gentleman, 1973]. The estimation process has been implemented in the University of
Texas Orbit Processor (UTOPIA) software system [Schutz and Tapley, 1980]. The optimal weighting
algorithm to combine satellite and nonsatellite information equations was installed in the Large
Linear System Solver (LLISS). Vectorized versions of UTOPIA and LLISS are operational on the
University of Texas System Center for High Performance Computing Cray X-MP/24 supercomputer.
Reference orbits for each of these satellites were computed using UTOPIA with the best a priori
gravity model and gravity field information equations were generated for each data set. The
combination solution was performed using LLISS.
3. DATA AND MODELS
Fourteen satellites were selected for the current gravity model solution. Their orbital
characteristics and data types are summarized in Table 1. The inclinations of these satellite orbits
vary from 15° for Peole to 115 ° for Gets-3. The solution includes data at 90 ° for Oscar-14 and
Nova-1. Data types include laser range, one-way range-rate, altimeter, altimeter crossover and
surface gravity data. Detailed descriptions of the gravitational and nongravitational force models, the
Earth orientation and time model, laser, doppler, direct altimeter and surface gravity measurement
models are summarized in Tapley et al. [1987].
4. SOLUTION
The list of parameters which are simultaneously estimated with a relative weighting factor for
each data set include: (1) geopotential complete to degree and order 50, plus selected coefficients; (2)
GM, (3) ocean tides which include long period tides (m = 0, l = 2,3): Ssa, Sa, Mm, and Mf; diurnal
tides (m = 1, l = 2,3,4,5): Q1, O1, P1, and K1; semi-diurnal tides (m = 2, l = 2,3,4,5): N2, M2, $2,
K2 and T2 (l = 2); (4) quasi-stationary sea surface topography, complete to degree and order 15; (5)
equipotential surface, W o, or altimeter biases; (6) correction to significant wave height, Hlr3; (7)
doppler and low inclination satellite laser station coordinates; (8) arc parameters for satellite orbits,
which include position and velocity vectors, drag and solar radiation pressure coefficients, density
correction parameters for selected satellites, and pass-dependent frequency biases for doppler
satellites. Kaula's constraint equation [Kaula, 1966], which was inferred from surface gravity
anomaly data, was used as an a priori constraint for degrees 19-50 of the geopotential. Two gravity
models, TEG-1 and TEG-1 S, were generated. TEG-1S did not include direct altimeter data.
5. ACCURACY EVALUATION
Efforts to evaluate and calibrate the accuracy of the UT gravity models were performed.
Comparison of orbit fits using different gravity fields for Starlette, Ajisai, Seasat and Geosat were
performed. It is shown that using TEG-1, a Starlette five-day orbit fit is at the -20 cm level, Ajisai
five-day orbit fit is at the -15 cm level, and that a Seasat six-day orbit and a Geosat 17-day orbit have
Table1. SatelliteDatafortheUniversityofTexas
GravityModel,TEG-1
Satellite Launch Data Inclination Eccentricity Altitude
Date (kin)
Vanguard- 1
Vanguard-2RB
Courier- 1B
Geos- 1
BE-C
DI-C
DI-D
Oscar- 14
Geos-2
Peole
Geos-3
Starlette
Lageos
Seasat
Nova- l
Geosat
Ajisai
1958
1959
1965
1965
1966
1967
1967
1967
1968
1971
1975
1975
1976
1978
1980
1985
1986
Opticalt
Opticalt
Opticalt
Laser
Laser
Laser, Opticalt
Laser, Opticalt
Doppler
Laser
Laser, Opticalt
Laser
Laser
Laser
Laser, Doppler,
Altimeter and
Crossover
Doppler
Doppler, Altimeter
and Crossover
Laser
34 °
33 °
28 °
59 °
41 °
40 °
39 °
89 °
106"
15°
115 °
50 °
110 °
108 °
90 °
108 °
50 °
0.190
0.183
0.016
0.072
0.026
0.053
0.085
0.005
0.033
0.015
0.002
0.020
0.004
0.002
0.002
0.000
0.001
t Optical data currently withheld from gravity field solution
2318
2318
1100
1600
1130
1000
1200
1100
1400
650
830
90O
5900
800
1200
800
1500
Surface Gravity Data
1° x 1° terrestrial mean gravity anomaly from
Ohio State University [Rapp, 1986]
crossover residuals at the -25 cm level. Table 2 shows the summary for the Geosat orbit fits. Gravity
field comparison using surface gravity data and a comparison of estimated TEG-1 ocean tidal
parameters with solutions derived by other studies were also performed. Covariance matrices for
TEG-1 and TEG-1S were calibrated to obtain estimates of errors associated with the gravity field
using the consider covariance calibration technique [Yuan et al., 1988]. The predicted radial orbit
errors using TEG-1 gravity field covariance matrix for Topex and Geosat are 13 cm and 24 cm,
respectively (Table 3).
6. CONCLUSION
In this investigation, two gravity models, TEG-1 and TEG-1S, each complete to degree and order
50 plus resonant coefficients, were generated. Ground-based tracking data collected by 14 satellites,
altimeter crossover and surface gravity data were used to determine the TEG-1S gravity field model.
TEG-1 contains Seasat and Geosat direct altimeter data in addition to all the data in TEG-1S. The
gravity field models were derived simultaneously with orbit, ocean tides, quasi-stationary sea surface
topography, and other geophysical parameters as well as the relative weights for each data set. The
fields were evaluated using both data included and data withheld from the solution. Formal
covariance matrices were calibrated to reflect realistic error estimates of the gravity field. Evaluations
based on orbit fits and gravity anomaly residuals indicate that the gravity models have achieved a
significant advancement over previously existing gravity models.
10
TAPLEY ET AL: AN IMPROVED MODEL FOR THE EARTH'S GRAVITY FIELD
Table 2. Gravity Field Accuracy Evaluation
Using Geosat Orbit Fits
_o, Fo, CR, daily Co, density correction parameters adjusted
Epoch TEG-1S TEG-I
17-day orbits (rms) (rms)
86/12/7 Doppler (cm/sec) 0.67 0.62
Crossover (cm)t 25 22
Altimeter (cm)t 180 32
87/01/7 Doppler (cm/sec) 0.62 0.61
Crossover (cm)t 24 25
Altimeter (cm)i 180 32
t Data types used for residual prediction only; altimeter data smoothed to
represent gravity spectrum to (50 _. 50)
Table 3. Gravity Field Accuracy Evaluation
Using Covariance Analysis
Model Predicted Topex Radial Predicted Geosat Radial
Orbit Error (cm) Orbit Error (era)
GEM-T1 25 54
TEG-I 13 24L
Topex orbit: 65 ° inclination, 1354 km altitude
Geosat orbit: 108 ° inclination, 800 km altitude
Acknowledgments. This research was supported by the NASA/Jet Propulsion Laboratory under
Contract No. 956689. Additional computing resources were provided by the University of Texas
System Center for High Performance Computing.
REFERENCES
Gentleman, W. M., Least square computations by Givens transformation without square roots, J. Inst.
Math. Applic., 12, 1973.
Kaula, W. M., Test and combination of satellite determinations of the gravity field with gravimetry, J.
Geophys. Res., 71(22), 5304-5314, November 1966.
Marsh, J. G., et al., A new gravitational model for the Earth from satellite tracking data: GEM-T1, J.
Geophys. Res., 93(B6), 6169-6215, June 10, 1988.
Schutz, B. E. and B. D. Tapley, UTOPIA: University of Texas Orbit Processor, IASOM TR-80-1,
Center for Space Research, The University of Texas at Austin, 1980.
Tapley, B. D, Statistical orbit determination theory, Recent Advances in Dynamical Astronomy,
396-425, B. D. Tapley and V. Szebehely, Eds., D. Reidei Publ. Co., Holland, 1973.
Tapley, B. D., B. E. Schutz, C. K. Shurn, J. C. Ries and D. N. Yuan, An improved model for the
Earth's gravity field, Proc. IUGG 19th General Assembly, Vancouver, Canada, August 1987.
Yuan, D. N., C. K. Shum and B. D. Tapley, Gravity field determination and error assessment
techniques, presented at the Chapman Conference for Progress in the Determination of the Earth's
Gravity Field, Ft. Lauderdale, Florida, September 12-16, 1988.
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An improved model for the Earth's gravity field

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