The next part of the overall system to be discussed is the determination of the satellite's orbital height. Referring to Fig. 6.1 again, we see that without this independent measurement of the orbital height relative to some fixed coordinate frame, that is, relative to the reference ellipsoid, the range measurement would not be very useful. It would actually tell us more about the height of the satellite than the height of the sea surface! To determine the sea surface height relative to the reference ellipsoid, it is necessary to independently determine the height of the satellite, and difference the range h and orbital height R in order to obtain a sea level estimate. If the height of the satellite, R, can be determined in some suitable Earth-centered/Earth-fixed reference frame (relative to a reference ellipsoid that best fits the actual shape of the Earth), then the height of sea level, S, may be computed as

There is an additional complication in that oceanographers would prefer that the reference surface be the oceanic geoid rather than an idealized ellipsoid, but for our purposes this distinction is not important, since we are mainly interested in temporal sea level changes and any reference surface that does not change with time is satisfactory. So how do we independently determine the height of the satellite relative to the height of the reference surface? Satellite geodesy techniques for orbit determination provide the solution.

Orbit determination can be described as a process that combines knowledge of the dynamics of Earth-orbiting spacecraft with very precise observations of the spacecraft. For T/P, measurements of the spacecraft range and/or rangerate are made from both Earth-based tracking stations (Satellite Laser Ranging (SLR) and the DORIS Doppler system) and space-based satellites (the Global Positioning System (GPS) constellation). Differences between the actual tracking observations and predicted observations based on the dynamics are used to adjust various dynamical (e.g., atmospheric drag, gravity) and measurement (e.g., tracking measurement biases) parameters to obtain better agreement. As of this writing, orbit determination methods are shifting from techniques that rely more on accurate modeling of the satellite dynamics (because the tracking data are geographically sparse) to techniques that rely more on the tracking measurements (because new tracking systems such as GPS provide virtually continuous orbital information).

The quality of the T/P precision orbit determination has been a major factor in the overall success of the mission. This success has come about because of several factors. First, the mission was designed with precision orbit determination in mind; thus the satellite is flown at a relatively high 1336-km altitude that reduces non-conservative forces, such as atmospheric drag, which are difficult to model. Second, significant improvements were made in our knowledge of Earth's gravity field, which is important because gravity is the dominant force acting on the satellite. Third, the satellite is tracked by several independent systems: SLR, DORIS, and GPS. Obviously this is a very complex activity and only a brief overview can be offered here. The reader interested in further information on precision orbit determination techniques is referred to Tapley et al. (1994b) and Nouel et al. (1994).

Basically, as the satellite moves about Earth, occasional observations of components of its position (e.g., range by SLR) or velocity (e.g., range-rate by DORIS) are made from ground stations, the positions of which are known accurately relative to the reference ellipsoid (Fig. 6.1). In addition to these point estimates of the satellite's height or speed, a physical model of the orbit is available that enables computation of a continuous estimate of the satellite's position and velocity. This physical model includes the effects of conservative forces (i.e, Earth's gravity field) (Nerem et al, 1994; Tapley et al, 1996) and non-conservative (e.g., atmospheric drag, solar radiation pressure) (Marshall and Luthcke, 1994) forces. The accuracy of the computed orbit is the most severely limited by our knowledge of Earth's gravity field, and to a lesser extent by the parameterizations necessary to model the non-conservative forces acting on the satellite. The model has adjustable parameters, however, that can be used to force the model orbit to conform to the observations provided at various points by the tracking stations. In a sense, we can view the precision orbit determination problem as obtaining a series of fixes on the satellite, and then interpolating between the relatively sparse fixes through the integration of the physical model for the satellite's orbit.

T/P also carries an experimental GPS tracking system (Melbourne et al, 1994) which has produced orbits estimated to be accurate to 20-30 mm radially (Bertiger et al, 1994; Schutz et al, 1994; Yunck et al, 1994). Because the GPS satellite constellation tracks T/P continuously in a three-dimensional fashion, the orbits have been determined using a "reduced-dynamic" orbit determination technique (Yunck et al, 1994) which is less dependent on the dynamical models and thus less susceptible to errors in those models. However, only time periods when the anti-spoofing (A/S) security measure (Melbourne et al, 1994) was not turned on have been processed, because the T/P GPS receiver is not equipped to compensate for A/S degradation. Since A/S-off has only occurred for roughly a dozen 10-day cycles, GPS cannot be used as a primary tracking technique for T/P. However, as the follow-on mission to T/P, Jason-1 will carry a more sophisticated GPS receiver that will work even when A/S is activated, and thus will be one of the primary tracking techniques for this mission.

A final orbit-related issue that warrants discussion is the effect of center-of-mass variations, since the center of mass of the earth (about which the satellite orbits) is the origin for satellite-based sea level measurements. The Cartesian coordinates of the center of mass of the earth with respect to a "crust-fixed" reference frame (defined by the location of the tracking stations) vary due to mass redistribution in the Earth system, primarily in the oceans and atmosphere. All satellites orbit the center of mass of the aggregate solid earth/ocean/atmosphere system. If the center of mass is the reference frame origin for the altimeter measurements of sea level, center-of-mass variations must be properly accounted for in the realization of the tracking station locations within this frame. These "geocenter" variations, which have been measured using precise geodetic satellites such as LAGEOS (e.g., Chen et al, 1999), are generally less than 20 mm. However, most of this variation is at seasonal periods, and significant secular trends in the geocenter components are expected to be quite small. Only recently have time series of geocenter variations been determined with accuracies approaching that needed for mean sea level studies, and thus these effects are not incorporated in current precision orbit solutions. While the effect is not expected to be large (due to averaging over the oceans and the accommodation of the error by other orbit parameters), the monitoring of the geocenter will likely be important for providing a well-defined origin for long-term measurements of sea level change (Nerem et al, 1998) when data from multiple missions are combined over several decades.

What sort of errors might we expect from the orbit determination? The spectrum of the orbit errors tends to have peaks once and twice per orbital revolution in addition to other lower frequency terms. Thus the orbit error tends to be large-scale in comparison with oceanic scales of variability. But over several days, during which time the satellite orbits Earth many times, the temporally varying components of these errors decorrelate (Marshall et al, 1995) and are not so serious for studies concerned with low-frequency variability. Very good estimates of the orbit error for T/P have been determined by comparing orbits computed from SLR/DORIS systems with GPS-determined orbits, which have largely independent sources of error. The RMS height error of the satellite as determined from these comparisons is ~20-30 mm (Marshall et al., 1995), which is fortunate because this is precisely the variable that needs to be well determined so that estimates of sea level trends not be biased by the orbit determination. A significant portion (~10 mm) of the RMS radial orbit error can be attributed to errors in the gravity field model (Nerem et al, 1994; Tapley et al, 1996); these errors are time-invariant at a fixed geographic location (if the ascending and descending tracks are processed separately) and thus do not affect the mean sea level measurements presented here. The remaining error (~10-20 mm) is primarily due to errors in modeling conservative forces, such as gravity effects arising from ocean tides. Measurement errors and reference frame definition (Nerem et al., 1998) are smaller but nevertheless significant components.

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