Nonconservative forces acting on GFO consist of radiative forces and atmospheric drag. Radiative forces include solar radiation pressure, the Earth's albedo (reflected light) and infrared radiation, and other secondary effects such as thermal imbalance in emission from spacecraft surfaces. The macro model approximates GFO's surface geometry and material properties using eight plates (Figure 2). Each plate has been assigned a body-fixed orientation, area, and specular and diffuse reflectivity coefficients based on pre-launch engineering information. All plate interaction effects, such as shadowing and multiple reflections, are ignored. The total acceleration with respect to the center of mass (CoM) is computed by summing vectorially the force acting on each plate, taking into account each plate's area, angle of incidence and material properties. Throughout the orbit and over a Beta prime cycle, radiation will be incident to a changing orientation of the macro model as computed using an analytical attitude model. Beta prime is the angle to the sun from the orbit plane (Figure 3), and for GFO shows a period of about 336 days (Figure 4). The spacecraft mass is assumed to be 369 kg.

Figure 2: GFO spacecraft macromodel approximation.	Figure 3: Orbit Geometry

Figure 4: GFO Beta Prime Cycle.

As shown in Figure 5, the largest nonconservative force acting on GFO is by far due to solar radiation pressure. Since the solar radiation pressure is so large, even a small error will have a significant impact. The error for the untuned macro model will be 10 to 20 percent of the radiative force. For instance the a priori macro model for TOPEX was meticulously constructed using finite element modelling and could only account for 90% of the radiative forces. However, after tuning, the TOPEX macro model is believed to account for over 95% of the radiative forces.

Figure 5: GFO Surface Force Acceleration.

The measurement model must first be verified before proceeding to refine the force models. Two vectors are involved for defining the Laser Retroreflector Array (LRA) position with respect to the spacecraft CoM: 1) the location of the LRA phase center with respect to the spacecraft body-fixed coordinate system, and 2) the location of the Center of mass in this coordinate system. The LRA is fixed and only the location of the CoM changes with time, based on propellant usage. We estimated the LRA phase center from SLR tracking, and this estimate would accommodate to first order changes in the CoM for which we do not have detailed information. The LRA consists of nine corner cubes arranged hemispherically, and is expected to have a stationary phase center, independent of the tracking geometry. An analysis of measurement biases computed from one month of SLR residuals, does not show any correlation with elevation angle.

Errors in the attitude model will not only affect the computation of the nonconservative forces, but more directly the modelled observation. The LRA orientation changes according to the attitude profile. As in all radar altimeter satellites, GFO is nadir pointing, and with attitude modelling defined only in the yaw. The analytical GFO yaw attitude model follows a cosine function with a period of one orbit revolution and whose amplitude is determined by the Beta prime angle (Figure 6). On several occasions, the implemented analytical model has been checked against especially requested telemetered attitude angles and has shown little discrepancy. For example, over August 8 1998, there is a Root Mean Square (RMS) discrepancy of 0.206°, 0.139°, and 0.248° for the roll, pitch, and yaw axes respectively (Figure 7). This can map to an SLR RMS measurement error of no larger than 0.5 cm, which is near the expected 0.3-.05 cm SLR measurement noise level of third generation systems.

Figure 6: GFO yaw attitude model.

Figure 7: GFO roll, pitch, yaw comparisons (telemetry - macromodel).

TRACKING DATA AND ORBIT ANALYSIS

GFO POD relies on SLR tracking provided by a global network of NASA and foreign stations (Figure 8). Unfortunately the tracking has been sparser than anticipated showing an average of about seven passes per day (Figure 9). Operational tracking Doppler data from the three stations (Guam, Point Mugu California, and Prospect Harbor Maine) although noisy is also abundant, and serves to slightly strengthen the SLR solution. After 40% of the data is edited, we typically use nine Doppler passes per day. The Doppler station positions have been adjusted to the SLR frame using three months of Doppler data and SLR-determined orbits that were held fixed in the solution.

Figure 8: GFO SLR tracking network.

Figure 9: GFO SLR tracking history.

Given the tracking density, an arc length of five days was selected over shorter arcs to increase the dynamic strength of the solution. Arc lengths of nine and ten days would also be suitable, however the frequency of satellite events such as computer resets or maneuvers which are not modelled in GEODYN for POD, have allowed only a few uninterrupted ten day spans.

The solution strategy, with the objective to minimize orbit error, was developed considering the strength of the tracking data. Several parameterization schemes were tested and the one finally selected is summarized in Table 3 . According to this strategy orbit error is minimized by adjusting, in addition to the orbit state, one coefficient (lightly constrained) to scale the solar radiation pressure, a daily atmospheric drag scale coefficient, and an empirical one cycle per revolution (1cpr) acceleration for both the along-track and cross-track components. The empirical terms absorb much of the residual accelerations remaining from the mismodelling of the various forces and greatly reduce orbit error. Since the empirical acceleration terms capture information about the residual accelerations they should be left out of solutions designed to tune say the macro model, but in this case may be used to reveal the characteristics of the mismodelled forces.

The initial orbit solutions show a large SLR residual mean, suggesting a measurement modelling error. One month (June 1998) of data was sufficient to estimate the LRA offset (Table 5) . It is not known which portion of the adjustment is due to an error in the LRA offset or to an error in the specification of the CoM in the spacecraft coordinate system, but application of the estimated offset reduces the SLR fits and the residual mean over several months tested (Tab.5, Figure 10).

Figure 10: GFO SLR residual mean fits and SLR offset.

No single test can uniquely gauge orbit accuracy. This analysis uses SLR residuals, or the misclosure between the highly precise observed and computed ranges, and orbit differences between arcs sharing one day of overlapping data, to indicate the level of orbit error. Since orbit error is typically minimized in the middle of an arc, overlap orbit differences computed over the ends of the arcs should represent a reasonable estimate of error. GFO altimeter crossover residuals may also be used to test for orbit error.

We see considerable variation in the SLR fits from arc to arc with an overall fit of about 10-cm (Figure 11). Gravity error is unlikely to explain the variation in orbit fits, as the four gravity fields tested show similar fits per arc, and more importantly manifest the same trend over time. The change in the GFO orbit fits (for all gravity fields tested) correlates with an increase in the absolute value beta prime angle. The orbit fits peak at the largest absolute beta prime values. This trend in SLR orbit residuals is consistent with a mismodelling of the radiative forces acting on the GFO spacecraft. The radial orbit differences between adjacent overlapping arcs shows a similar degradation at high (in absolute value) beta prime angles (Figure 12 and Table 6).

Figure 11: GFO SLR Fit.

Figure 12: GFO radial orbit difference for overlapping arcs.

As Beta prime decreases from zero to -84 degrees the solar radiation pressure (and the mismodelled effect) will change its projection from the predominately along-track and radial directions to the cross-track (Figure 13). The adjusted empirical accelerations should thus decrease in magnitude in the along-track component, and increase in the cross-track. This is just what we see (Figure 14). Not only does the along-track 1cpr empirical acceleration magnitude decrease, but the associated phase (with respect to orbit angle) remains constant from arc to arc until the spacecraft enters the full sunlight regime (Figure 15). The observed phase coherence indicates that the force error preserves the same orientation with respect to orbit plane from arc to arc, which in fact solar radiation pressure does (Figure 13). As the spacecraft reaches full sunlight (near —65° beta prime), the recovered along-track acceleration magnitude becomes very small, for which the phase is probably not well determined. In another study tuning the TDRSS macro model, a continuous phase was also observed in the recovered 1cpr along-track acceleration prior to tuning. After tuning, the recovered acceleration magnitudes were small and the phases showed no coherence.

Figure 13: GFO solar radiation pressure acceleration (along-track).	Figure 14: GFO estimated 1cpr along-track amplitude

Figure 15: GFO estimated 1cpr along-track phase.

Since nonconservative force model error dominates, tuning the macro model will precede tuning of the gravity field. Use of a macro model tuned over the beta prime cycle of about 336 days should minimize nonconservative force model error contamination in gravity field tuning.

MACRO MODEL TUNING

The macro model represents the GFO spacecraft as an eight surface composite (Fig 2). It approximates the spacecraft geometry and surface material properties to better model the surface force effects due to solar and terrestrial radiation pressure, and due to atmospheric drag. Each surface (or plate) had been assigned an orientation with respect to the satellite fixed frame, an area, and a specular and diffuse reflectivity coefficient based on pre-launch engineering specifications. The material properties of each plate are assumed to be homogenous, representing an average value. In tuning, these average values are adjusted to best fit the GFO tracking data using an orbit determination (OD) solution strategy to insure the mismodelled nonconservative forces are not absorbed in empirical parameter adjustments. Therefore the macro model is tuned to the residual satellite acceleration history which is based on orbit errors sensed from the spacecraft tracking data.

OD parameterization suitable for macro model tuning adjusts the orbit state, the solar radiation pressure coefficient (C_R), and one drag coefficient (C_D). Upon solution convergence for each arc, GEODYN writes out the normal equations for the orbit (state, C_R, C_D) and panel (area, specular, diffuse) parameters. These normal equations will be combined from arcs sampled over the beta prime cycle and the selected panel parameters estimated using Bayesian least squares. The tuning procedure at GSFC, refined with the experience gained tuning the TOPEX (Ref 9) and TDRSS (Ref 11) macro models, has been previously described in Ref 11.

A preliminary sensitivity study was performed using the combined normal matrix from four well-spaced arcs to help identify panel parameters that are to be estimated. Assuming a specified allowed percent change in each respective panel parameter a priori value, and using only the left-hand side diagonal (variance) terms of the normal matrix, the resulting "uncorrelated weighted variance" is computed in order to compare parameter sensitivity or change in residual variance with respect to parameter adjustments. The a priori surface area assigned to each plate is believed to be relatively well determined with about a 10% error. There is much greater uncertainty for the a priori specular and diffuse reflectivity coefficients, computed as an aggregate average of these properties for each surface. The area is allowed to change by 10% and the reflectivity coefficients by 100% for the sensitivity analysis. As shown in Figure 16, specular coefficients for four parameters representing the solar array, the bottom plate (+z facing Earth), and the top and bottom sides of the altimeter antenna reflector, are likely candidates for the macro model tuning adjustment.