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The TOPEX/POSEIDON mission objective requires that the radial position of the spacecraft be determined with an accuracy better than 13 cm RMS (root mean square). This stringent requirement is an order of magnitude below the accuracy achieved for any altimeter mission prior to the definition of the TOPEX/POSEIDON mission. To satislfy this objective, the TOPEX Precision Orbit determination (POD) Team was established as a joint effort between the NASA Goddard Space Flight Center and the University of Texas at Austin, with collaboration from the University of Colorado and the Jet Propulsion Laboratory. During the prelaunch development and the post launch verification phases, the POD team improved, calibrated, and validated the precision orbit determination computer software systems. The accomplishments include (1) increased accuracy of the gravity and surface force models and (2) improved peformance of both laser ranging and Doppler tracking systems. The result of these efforts led to orbit accuracies for TOPEX/POSEIDON which are significantly better than the original mission requirement. Tests based on data fits, covariance analysis, and orbit comparisons indicate that the radial component of the TOPEX/POSEIDON spacecraft is determined, relative to the Earth's mass center, with an root mean square (RMS) error in the range of 3 to 4 cm RMS. This orbit accuracy, together with the near continuous dual-frequency altimetry from this mission, provides the means to determine the ocean's dynamic topography with an unprecedented accuracy.

The TOPEX/POSEIDON (T/P) prelaunch Joint Gravity Model-1 (JGM-1) 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 root mean square (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.

An improved model of Earth's gravitational field, Goddard Earth Model T-3 (GEM-T3), has been developed from a combination of satellite tracking, satellite altimeter, and surface gravimetric data. GEM-T3 provides a significant improvement in the modeling of the gravity field at half wavelengths of 400 km and longer. This model, complete to degree and order 50, yields more accurate satellite orbits and an improved geoid representation than previous Goddard Earth Models. GEM-T3 uses altimeter data from GEOS 3 (1975-1976), Seasat (1978) and Geosat (1986-1987). Tracking information used in the solution includes more than 1300 arcs of data encompassing 31 different satellites. The recovery of the long-wavelength components of the solution relies mostly on highly precise satellite laser ranging (SLR) data, but also includes Tracking Network (TRANET) Doppler, optical, and satellite-to-satellite tracking acquired between the ATS 6 and GEOS 3 satellites. The main advances over GEM-T2 (beyond the inclusion of altimeter and surface gravity information which is essential for the resolution of the shorter wavelength geoid) are some improved tracking data analysis approaches and additional SLR data. Although the use of altimeter data has greatly enhanced the modeling of the ocean geoid between 65 deg N and 60 deg S latitudes in GEM-T3, the lack of accurate detailed surface gravimetry leaves poor geoid resolution over many continental regions of great tectonic interest (e.g., Himalayas, Andes). Estimates of polar motion, tracking station coordinates, and long-wavelength ocean tidal terms were also made (accounting for 6330 parameters). GEM-T3 has undergone error calibration using a technique based on subset solutions to produce reliable error estimates. The calibration is based on the condition that the expected mean square deviation of a subset gravity solution from the full set values is predicted by the solutions' error covariances. Data weights are iteratively adjusted until this condition for the error calibration is satisfied. In addition, gravity field tests were performed on strong satellite data sets withheld from the solution (thereby ensuring their independence). In these tests, the performance of the subset models on the withheld observations is compared to error projections based on their calibrated error covariances. These results demonstrate that orbit accuracy projections are reliable for new satellites which were not included in GEM-T3.

The contribution of TOPEX/Poseidon (T/P) TDRSS data to geopotential model recovery is assessed. Simulated TDRSS one-way and Bilateration Ranging Transponder System (BRTS) observations have been generated and orbitally reduced to form normal equations for geopotential parameters. These normals have been combined with those of the latest prelaunch T/P gravity model solution using data from over 30 satellites. A study of the resulting solution error covariance shows that TDRSS can make important contributions to geopotential recovery, especially for improving T/P specific effects like those arising from orbital resonance. It is argued that future effort is desirable both to establish TDRSS orbit determination limits in a reference frame compatible with that used for the precise laser/DORIS orbits, and the reduction of these TDRSS data for geopotential recovery.

Improved models of the Earth's gravitational field have been developed from conventional tracking data and from a combination of satellite tracking, satellite altimeter and surface gravimetric data. This combination model represents a significant improvement in the modeling of the gravity field at half-wavelengths of 300 km and longer. Both models are complete to degree and order 50. The Goddard Earth Model-T3 (GEM-T3) provides more accurate computation of satellite orbital effects as well as giving superior geoidal representation from that achieved in any previous GEM. A description of the models, their development and an assessment of their accuracy is presented. The GEM-T3 model used altimeter data from previous satellite missions in estimating the orbits, geoid, and dynamic height fields. Other satellite tracking data are largely the same as was used to develop GEM-T2, but contain certain important improvements in data treatment and expanded laser tracking coverage. Over 1300 arcs of tracking data from 31 different satellites have been used in the solution. Reliable estimates of the model uncertainties via error calibration and optimal data weighting techniques are discussed.

Models of the TOPEX/Poseidon spacecraft are developed by means of finite-element analyses for use in generating acceleration histories for various orbit orientations which account for nonconservative radiation forces. The acceleration profiles are developed with an analysis based on the use of the 'box-wing' model in which the satellite is modeled as a combination of flat plates. The models account for the effects of solar, earth-albedo, earth-IR, and spacecraft-thermal radiation. The finite-element analysis gives the total force and induced accelerations acting on the satellite. The plate types used in the analysis have parameters that can be adjusted to optimize model performance according to the micromodel analysis and tracking observations. Acceleration related to solar radiation pressure is modeled effectively, and the techniques are shown to be useful for the precise orbit determinations required for spacecraft such as the TOPEX/Poseidon.

An improved model of the Earth's gravitational field is developed from a combination of conventional satellite tracking, satellite altimeter measurements, and surface gravimetric data (GEM-T3). This model gives improved performance for the computation of satellite orbital effects as well as a superior representation of the geoid from that achieved in any previous Goddard Earth Model. The GEM-T3 model uses altimeter data directly to define the orbits, geoid, and dynamic height fields. Altimeter data acquired during the GEOS-3 (1975-1976), SEASAT (1978), and GEOSAT (1986-1987) missions were used to compute GEM-T3. In order to accommodate the non-gravitational signal mapped by these altimeters, spherical harmonic models of the dynamic height of the ocean surface were recovered for each mission simultaneously with the gravitational field. The tracking data utilized in the solution includes more than 1300 arcs of data encompassing 31 different satellites. The observational data base is highly dependent on SLR, but also includes TRANET Doppler, optical, S-Band average range-rate and satellite-to-satellite tracking acquired between ATS-6 and GEOS-3. The GEM-T3 model has undergone extensive error calibration.

Computation of precise orbits for the TOPEX/Poseidon (T/P) spacecraft is analyzed focusing on gravity field modeling, nonconservative force modeling, satellite tracking technologies, and orbit determination software. It was found that the radial orbit error budget for T/P allows 10 cm rms error due to gravity field mismodeling, 3 cm due to solid earth and ocean tides, and 6 cm due to radiative forces, and 3 cm due to atmospheric drag. It is concluded that the current models are capable of achieving the radial orbit error requirements.

The GEM-T2 is the latest in a series of Goddard Earth Models of the terrestrial field. It was designed to bring modeling capabilities one step closer towards ultimately determining the TOPEX/Poseidon satellite's radial position to an accuracy of 10-cm RMS (root mean square). It also improves models of the long wavelength geoid to support many oceanographic and geophysical applications. The GEM-T2 extends the spherical harmonic field to include more than 600 coefficients above degree 36 (which was the limit for its predecessor, GEM-T1). Like GEM-T1, it was produced entirely from satellite tracking data, but it now uses nearly twice as many satellites (31 vs. 17), contains four times the number of observations (2.4 million), has twice the number of data arcs (1132), and utilizes precise laser tracking from 11 satellites. The estimation technique for the solution has been augmented to include an optimum data weighting procedure with automatic error calibration for the gravitational parameters. Results for the GEM-T2 error calibration indicate significant improvement over previous satellite-only models. The error of commission in determining the geoid has been reduced from 155 cm in GEM-T1 to 105 cm for GEM-T2 for the 36 x 36 portion of the field, and 141 cm for the entire model. The orbital accuracies achieved using GEM-T2 are likewise improved. Also, the projected radial error on the TOPEX satellite orbit indicates 9.4 cm RMS for GEM-T2, compared to 24.1 cm for GEM-T1.

The GEM-T2 is the latest in a series of Goddard Earth Models of the terrestrial field. It was designed to bring modeling capabilities one step closer towards ultimately determining the TOPEX/Poseidon satellite's radial position to an accuracy of 10-cm RMS (root mean square). It also improves models of the long wavelength geoid to support many oceanographic and geophysical applications. The GEM-T2 extends the spherical harmonic field to include more than 600 coefficients above degree 36 (which was the limit for its predecessor, GEM-T1). Like GEM-T1, it was produced entirely from satellite tracking data, but it now uses nearly twice as many satellites (31 vs. 17), contains four times the number of observations (2.4 million), has twice the number of data arcs (1132), and utilizes precise laser tracking from 11 satellites. The estimation technique for the solution has been augmented to include an optimum data weighting procedure with automatic error calibration for the gravitational parameters. Results for the GEM-T2 error calibration indicate significant improvement over previous satellite-only models. The error of commission in determining the geoid has been reduced from 155 cm in GEM-T1 to 105 cm for GEM-T2 for the 36 x 36 portion of the field, and 141 cm for the entire model. The orbital accuracies achieved using GEM-T2 are likewise improved. Also, the projected radial error on the TOPEX satellite orbit indicates 9.4 cm RMS for GEM-T2, compared to 24.1 cm for GEM-T1.

Major new computations of terrestrial gravitational field models were performed by the Geodynamics Branch of Goddard Space Flight Center (GSFC). This development has incorporated the present state of the art results in satellite geodesy and have relied upon a more consistent set of reference constants than was heretofore utilized in GSFC's GEM models. The solutions are complete in spherical harmonic coefficients out to degree 50 for the gravity field parameters. These models include adjustment for a subset of 66 ocean tidal coefficients for the long wavelength components of 12 major ocean tides. This tidal adjustment was made in the presence of 550 other fixed ocean tidal terms representing 32 major and minor ocean tides and the Wahr frequency dependent solid earth tidal model. In addition 5-day averaged values for Earth rotation and polar motion were derived for the time period of 1980 onward. Two types of models were computed. These are satellite only models relying exclusively on tracking data and combination models which have incorporated satellite altimetry and surface gravity data. The satellite observational data base consists of over 1100 orbital arcs of data on 31 satellites. A large percentage of these observations were provided by third generation laser stations (less than 5 cm). A calibration of the model accuracy of the GEM-T2 satellite only solution indicated that it was a significant improvement over previous models based solely upon tracking data. The rms geoid error for this field is 110 cm to degree and order 36. This is a major advancement over GEM-T1 whose errors were estimated to be 160 cm. An error propagation using the covariances of the GEM-T2 model for the TOPEX radial orbit component indicates that the rms radial errors are expected to be 12 cm. The combination solution, PGS-3337, is a preliminary effort leading to the development of GEM-T3. PGS-3337 has incorporated global sets of surface gravity data and the Seasat altimetry to produce a model complete to (50,50). A solution for the dynamic ocean topography to degree and order 10 was included as part of this adjustment.

A computation of a terrestrial gravitational field model called the Goddard Earth Model GEM-T1 is discussed and compared to previous models, including the GEM-L2. The software tools were redesigned for the model, allowing for the optimization of the technique of relative data weighting and model estimation used in GEM solutions. The GEM-T1 model provides a simultaneous solution for a gravity model in spherical harmonics complete to degree and order 36, a subset of 66 ocean tidal coefficients for the long-wavelength components of 12 major tides, and 5-day averaged earth rotation and polar motion parameters for the 1980 period on. GEM-T1 was derived from satellite tracking data acquired on 17 different satellites whose inclinations ranged from 15 degrees to polar. A simulation of the TOPEX/POSEIDON orbit using the covariances of the GEM-T1 model was made. Estimated radial error for the simulation was reduced to less than 30 cm rms.

Goddard Earth Model T1 (GEM-T1), which was developed from an analysis of direct satellite tracking observations, is the first in a new series of such models. GEM-T1 is complete to degree and order 36. It was developed using consistent reference parameters and extensive earth and ocean tidal models. It was simultaneously solved for gravitational and tidal terms, earth orientation parameters, and the orbital parameters of 580 individual satellite arcs. The solution used only satellite tracking data acquired on 17 different satellites and is predominantly based upon the precise laser data taken by third generation systems. In all, 800,000 observations were used. A major improvement in field accuracy was obtained. For marine geodetic applications, long wavelength geoidal modeling is twice as good as in earlier satellite-only GEM models. Orbit determination accuracy has also been substantially advanced over a wide range of satellites that have been tested.

The purpose of the GEODYN Orbit Determination and Parameter Estimation, the SOLVE and ERODYN Programs is to recover geodetic and geophysical parameters from satellite and other data in a state-of-the-art manner. Continued solutions for gravity field, pole positions, Earth rotation, GM, and baselines were made as part of the Crustal Dynamics Project. Some tidal parameters were recovered as well. The eight digit station identification number was incorporated in the software and new techniques for constraining monthly station parameters to each other are being developed. This is allowing the analysts even more flexibility in the shaping of solutions from monthly sets of normal equations and right-hand sides.

Laser range observations taken on the near-earth satellites of Lageos (a = 1.92 e.r.), Starlette (a = 1.15 e.r.), BE-C (a = 1.18 e.r.), and Geos-3 (a = 1.13 e.r.) have been combined to determine an improved value of the geocentric gravitational constant (GM). The value of GM is 398600.61 cu km/sec per sec, based upon a speed of light, c, of 299792.5 km/sec. Using the IAG-adopted value of c equalling 299792.458 km/sec scales GM to 398600.44 cu km/sec per sec. The uncertainty in this value is assessed to be plus or minus 0.02 cu km/sec per sec. Determinations of GM from the data taken on these four satellites individually show variations of only .04 cu km/sec per sec from the combined result. The Lageos information dominated the combined solution, and gave the most consistent results in its data subset solutions. The value obtained for GM from near-earth laser ranging compares quite favorably with the most recent results of the lunar laser and interplanetary experiments.

A recursive algorithm was designed to invert large, dense, symmetric, positive definite matrices using small amounts of computer core, i.e., a small fraction of the core needed to store the complete matrix. The described algorithm is a generalized Gaussian elimination technique. Other algorithms are also discussed for the Cholesky decomposition and step inversion techniques. The purpose of the inversion algorithm is to solve large linear systems of normal equations generated by working geodetic problems. The algorithm was incorporated into a computer program called SOLVE. In the past the SOLVE program has been used in obtaining solutions published as the Goddard earth models.

This report explains the data reduction and spacecraft position determination used in conjunction with two ATS experiments - Trilateration and Turnaround Ranging - and describes in detail a multilateration program that is used for part of the data reduction process. The process described is for the determination of the inertial position of the satellite, and for formating input for related programs. In the trilateration procedure, a geometric determination of satellite position is made from near simultaneous range measurements made by three different tracking stations. Turnaround ranging involves two stations; one, the master station, transmits the signal to the satellite and the satellite retransmits the signal to the slave station which turns the signal around to the satellite which in turn retransmits the signal to the master station. The results of the satellite position computations using the multilateration program are compared to results of other position determination programs used at Goddard. All programs give nearly the same results which indicates that because of its simplicity and computational speed the trilateration technique is useful in obtaining spacecraft positions for near synchronous satellites.

A preliminary Goddard Space Flight Center (GSFC) geopotential and center of mass station coordinate solution was obtained from satellite orbital data using numerical integration theory. This geodetic solution is a prelude to a more general solution which will combine the 1971 International Satellite Geodesy Experiment (ISAGEX) laser data with the present data being employed. The present GSFC geopotential solution consists of the spherical harmonic coefficients through degree and order eight with higher order satellite resonant coefficients. The solution represents a first iteration result from 17 satellites with approximately 150 weekly orbital arcs containing some 40,000 optical observations. The GSFC preliminary result is compared with final results from the Smithsonian Astrophysical Observatory (SAO) solutions including the 1969 SAO Standard Earth II solution. One aspect of interest for the comparison is that SAO uses an analytic theory for the orbital solution whereas GSFC uses a numerical integration theory. The comparison of geopotential results shows that good agreement exists in general but that there are some areas of minor differences.

Goddard range and range rate system measurements and data processing techniques for orbit calculation

A refinement in the satellite geopotential solution for a Goddard Earth Model (GEM 3) was obtained. The solution includes the addition of two low inclination satellites, SAS at 3 deg and PEOLE at 15 deg, and is based upon 27 close earth satellites containing some 400,000 observations of electronic, laser, and optical data. In addition, a new combination satellite/gravimetry solution (GEM 4) was derived. The new model includes 61 center of mass tracking station locations with data from GRARR, Laser, MOTS, Baker-Nunn, and NWL Tranet Doppler tracking sites. Improvement was obtained for the zonal coefficients of the new models and is shown by tests on the long period perturbations of the orbits. Individual zonal coefficients agree very closely among different models that contain low inclination satellites. Tests of models with surface gravity data show that the GEM 3 satellite model has significantly better agreement with the gravimetry data than the GEM 1 satellite model, and that it also has better agreement with the gravimetry data than the 1969 SAO Standard Earth 2 model.