MINIMIZING SELECTIVE AVAILABILITY ERROR ON TOPEX GPS MEASUREMENTS S. C. Wu*, W. I. Bertiger and J. T. Wu Jet Propulsion Laboratory California Institute of Technology Pasadena, California 9119 Abstract* GPS measurements made at Topex/Poseidon and the accompanying ground tracking sites will be affected by the Selective Availability. Although in principle the effects may be removed by differencing between receivers observing the same GPS satellites, this requires accurate synchronization of all receiver clocks. In the case of Topex/Poseidon application, there are two sources of imperfect clock synchronization. The first and larger is due to the constantly drifting clock onboard Topex, which may cause a residual effect as large as 1 cm on Topex carrier phase and 1 m on Topex pseudorange. The second is due to light-time differences between receivers observing the same GPS satellites, which may amount to a few mm error. In this paper a data reduction scheme which incorporates a low-order polynomial interpolation and carrier phase smoothing on pseudorange acquired at Topex and ground receivers is described; a simulation analysis is given demonstrating the effectiveness of the scheme for reducing the GPS S/A effects; and comparison with other schemes is discussed. Introduction The Global Positioning System (GPS) of the US Department of Defense will be in full operation in the early 199s. The system will be able to turn on Selective Availability (S/A) to reduce the positioning accuracy for users who lack access to the S/A information. Topex/Poseidon will be among the users affected by S/A. Topex/Poseidon is a joint US/French Ocean Topography Experiment satellite 1 which will be launched in mid 1992. For its primary scientific mission, it will carry a microwave altimeter that will precisely measure ocean topography. To fully exploit the few-centimeter accuracy of this measurement, the Topex/Poseidon orbit will have to be independently determined to comparable accuracy; the goal is 15 cm or better. Although the baseline tracking system will be ground-based laser ranging, an experimental GPS receiver will also be carried onboard to demonstrate the capability of recently proposed precise GPS tracking techniques. Extensive analysis has been carried out at JPL for precise Topex orbit determination using GPS pseudorange and carrier phase measurements 2 4. In this paper, the S/A effects on Topex GPS * Technical Group Leader, Tracking Systems and Applications Section; senior member AIAA. Member of Technical Staff, Tracking Systems and Applications Section; member AIAA Member of Technical Staff, Tracking Systems and Applications Section. measurements and a scheme to minimize them will be investigated. The S/A reduces the user positioning accuracy in two ways. First, artificial offsets are added onto the broadcast GPS ephemerides and, second, the GPS clocks which generate the carrier phase and coded signals are dithered. Only the second aspect will affect Topex orbit determination using GPS since real-time operation is not needed. The effects of the S/A clock dithering may in principle be removed by differencing between receivers observing the same GPS satellites, or equivalent operation (e.g., solving for white-noise clocks). However, this is possible only if the receiver clocks are sufficiently good to keep the time-tags accurate to better than 1 msec. Although the clock on Topex will be monitored from time to time using onboard real time navigation, it will not be re-aligned to GPS clocks so that continuity in carrier phase can be maintained. The clock information, of course, will be recorded and telemetered back to ground, together with the tracking data. Hence the time-tags of the Topex pseudorange and carrier phase measurements will be drifting away from the correct time. When the discrete timetags are later corrected using the telemetered clock information, the data will be non-simultaneous with ground receivers, leaving GPS clock cancellation imperfect. With the planned 1- second sampling for the carrier phase and 1-second sampling for the pseudorange data, the maximum non-simultaneity from ground receivers is.5 second and 5 seconds respectively for the two data types. Another much lower non-simultaneity arises from the unequal light-time between a given GPS satellite and different receivers. The difference in light-time can be as large as 2 msec. In the following sections, the characteristics of S/A clock dithering is first described. How the S/A effects on differential GPS measurements can be reduced by a simple interpolation scheme is explained. A data reduction scheme for the GPS measurements from Topex and from ground receivers is proposed and analyzed. A simulation analysis is performed to demonstrate the effectiveness of the scheme. A comparison is made between a cubic, a quadratic and a linear interpolation strategies. S/A Effects on GPS Measurements Several Block II GPS satellites are already in orbit and with S/A signal turned on. Carrier phase data from ground receivers observing these satellites clearly show the fluctuation due to S/A signal. A sample of the S/A signal extracted from the carrier phase data taken at Mojave observing GPS 14 on April 16, 199 is shown in Fig. 1. At times the error is as high as 3 m. Earlier analysis 5 on the actual S/A signal indicates that its 1
power spectrum contains only low frequency components. A similar spectral analysis on the S/A signal in Fig. 1 further confirms its low-frequency characteristics, as shown in Fig. 2. In other words, S/A signal behaves as if it has a relatively slowly time-varying frequency offset from an otherwise undithered clock. This allows a low-order interpolation process to be performed on the tracking data so that data received at Topex and different ground sites can be made simultaneous, i.e., interpolated so that all signals have a common transmit time. Fig. 3 illustrates how differential S/A effects can be reduced by a first-order (linear) interpolation process. In this figure, the residual pseudorange (with satellite dynamics removed by a priori models) due to S/A clock is plotted as a function of time. Data are sampled consecutively at time t 2 t and t 2. Let t 1 be the time at which the data should be taken so that it is simultaneous with data received at other sites. Ideally t 1 is chosen so that all receivers are sampling the same GPS wavefront. Thus t 1 would be different for each receiver. A linear interpolation from the two adjacent received data at t 2 t and t 2 to the time t 1 is performed. The error due to time-tag S/A CLOCK ERROR (m) 4 2-2 -4 6 Mojave observing GPS 14 12 18 24 TIME PAST 199/4/16 17:3 UT (min) Fig. 1. A sample of S/A effects extracted from actual carrier phase data Fig. 2. Power spectral density of the S/A signal in Fig. 1 3 PSEUDORANGE interpolation error linear interpolation S/A dithered clock interpolated data t ²t 2 t 1 t 2 offset is now traded for the interpolation error. Note that, when the absolute frequency offset (the slope of the curve) is large compared to the frequency change (the change in slope), the interpolation error is always smaller than the original error due to time-tag offset. The error is further reduced if higher-order (quadratic or cubic) interpolation is used. S/A on Topex GPS Measurements error due to time-tag offset t 2 : actual receiving time t 1 : time at which data should be received (also actual time-tag) ²t : data interval TIME Fig. 3. Reduction of S/A effects on pseudorange by linear interpolation The GPS receiver onboard Topex will make carrier phase measurement at a rate of once every second, and P-code pseudorange at once every 1 seconds. When later brought back to the central processing site, these data will then be compressed to 5-minute data points for Topex orbit determination in the following filtering process. At the maximum time-tag offset of.5 second for the 1-sec carrier phase data received at Topex, the differential (from a ground receiver free of time-tag error) S/A effect is estimated to be about 1 cm. Due to its low-frequency characteristics, this error is highly correlated over the expected data compression interval of 5 minutes and hence will not average down. With the 1-sec sampling of Topex pseudorange data, the maximum time-tag offset is 5 seconds; the residual error due to the differential S/A effects is estimated to be 1 m. These levels of error are grossly unacceptable for precision Topex orbit determination. Based on the characteristics of S/A signal, a two-order-ofmagnitude reduction of its effects on differential measurement can be expected. When a simple linear interpolation as outlined in the previous section is carried out to remove the time-tag offset, not only is the 1-cm differential S/A effect on carrier phase reduced to only about 1 mm, the correlation of this residual error is also weakened, allowing further average down with data compression to 5-minute interval. Similarly, the 1-m effect on differential pseudorange is reduced to about 1 cm by a simple interpolation, with similar de-correlation expected over the 5-minute compression interval. When a smoothing of pseudorange using carrier phase 6 is applied before interpolation, the residual S/A effect on pseudorange is further reduced to the level of carrier phase error. This will be explained in a latter section. 2
Incomplete Cancellation of S/A Effects Due to Light-Time Delay Measurements made by the Topex GPS receiver and most ground receivers will be tagged by receiving time. On the other hand, S/A effects are common only between measurements with the same transmitting time. The difference in light-time between different receivers observing a given GPS satellite can be a large as 2 msec. Hence, an offset as large as 2 msec in the transmitting time exists in data received with perfect timetags. Since the cancellation of GPS clock errors with double differencing relies on data simultaneity in transmitting time, any non-simultaneity will result in incomplete cancellation. With a 2-msec offset in transmitting time, the difference in S/A effect is estimated to be a few mm for both pseudorange and carrier phase data. Although such error is negligible for the pseudorange (with an expected data noise of 5 1 cm) it is significant to carrier phase data. This error can also be reduced by interpolation to common transmitting time. Shifting the time-tags from common receiving time to common transmitting time, in fact, trades the errors due to the GPS clocks (with S/A) for those due to the far better receiver clocks (the receiving clocks are now non-simultaneous, but they will have far smaller errors than the S/A effects). Data Compression Scheme P-code pseudorange is a measurement of the absolute range between a GPS satellite (transmitter) and a receiver, corrupted by a data noise and other error sources such as clock bias, transmission media delays, multipath, etc. The pseudorange data noise of a well-designed GPS receiver is of the order of a few centimeters when averaged to 5-minute points 7. On the other hand, carrier phase is a precise measurement of range change, with a data noise and multipath effects typically two orders of magnitude lower 8. To keep the data noise low, a compression scheme should take advantage of smoothing over the entire 5-minute integration (averaging) time. On the other hand, for proper smoothing the signature of the satellite dynamics should be removed with a reasonably good model, which is not always conveniently available. A judicious compression scheme which removes the dynamics without requiring a dynamic model is recommended in the following. Since carrier phase data noise is intrinsically low, smoothing over the entire 5-minute integration time does not buy much. Instead of removing the satellite dynamics with a good model, a low-order polynomial interpolation over a short time period can be used for the compression of carrier phase. The simulation analysis in the following section indicates that for Topex data at 1-sec intervals, a cubic interpolation over four points every 5 minutes is appropriate even with the strong Topex dynamics. For ground data a compression scheme with a quadratic interpolation over three 1-sec data points every 5 minutes can be adopted. This is based on the fact that the dynamics are much lower and the corrected time-tag for the ground data will always be less than.1 sec away from the nearest raw data point, hence a quadratic fit would be appropriate; the corrected time-tag for Topex could be as large as.5 sec away from the nearest raw data point and a cubic interpolation is recommended to assure a low interpolation error. The following simulation analysis will demonstrate that the differential S/A effects between Topex and ground data are small despite different interpolation polynomials used. The higher data noise in pseudorange discourages the use of any data decimation. However, since the precise carrier phase has identical satellite dynamics as the pseudorange, it can be treated as a dynamic model and subtracted from the pseudorange. Then the dynamic-removed pseudorange can be compressed with a simple averaging over the entire 5-minute period. The dynamics are later recovered by adding the compressed carrier phase to the compressed pseudorange. This scheme is called smoothing of pseudorange using carrier phase and has been used in the Rogue GPS receiver 6. Since the removal of dynamics using carrier phase also removes the S/A effects, the compression through averaging of pseudorange does not introduce any additional interpolation error as outlined in Fig. 3. Hence, the compressed differential pseudorange will have the same residual S/A effects as the compressed differential carrier phase, which are low due to the small (1-sec) data intervals. Note that it is important to maintain identical S/A effects in pseudorange and carrier phase data types since they are later removed as common clock error in the filtering process (which is comparable to differencing). The data compression scheme is outlined in the following. Steps 1 5 apply to ground data and Steps 6 1 apply to Topex data. It is assumed that carrier phase has been converted to the same unit (length) as pseudorange, and that all ground receivers' clocks are synchronized to better than 1 msec. For ground data: 1. For each GPS/ground receiver pair, define a new time-tag, T new, to be the receiving time corresponding to a transmitting time which is an integer multiple of 5 minutes in UT. (The new time-tags are no longer at integer 5- minute marks but are 66 to 86 msec greater.) 2. For each new time-tag, T new, the compressed carrier phase Φ comp is computed by a quadratic interpolation using only three adjacent 1-sec data points. 3. For each pair of GPS/ground receiver, form the difference between the raw pseudorange and raw carrier phase, R Φ, for all 1-sec points. 4. For each new time-tag, T new, compute (R Φ) ave, the average of all R Φ points within ±2.5 minutes of T new. 5. The compressed pseudorange at T new is computed as R comp = (R Φ) ave + Φ comp. For Topex data: 6. Compute the daily continuous piecewise linear correction to Topex clock by fitting to the telemetered clock offset. A continuous clock correction is used to maintain continuity in the data. The time-tags and the observables are then corrected* accordingly to remove the error. * For a clock error t, the time-tags are to be corrected by t and the observables are to be corrected by c t where c is the speed of light. 3
7. For each GPS satellite, define new time-tags, T new, for Topex data as in Step 1. ground data transmitting time @ 5-min intervals Topex data 8. For each new time-tag, T new, the compressed Topex carrier phase Φ comp is computed by a cubic interpolation using only four adjacent 1-sec data points. 9. Compress the 1-sec raw carrier phase to 1-sec point using a quadratic interpolation. This must be the same as the pseudorange smoothing in the Topex receiver to preserve common S/A and satellite dynamics between pseudorange and carrier phase. Φcomp = quadratic fit over 3 points of Φ around Tnew compute receiving time, Tnew continuous piecewise linear fit to Topex clock offset correct Topex time-tags and data Φcomp = cubic fit over 4 points of Φ around Tnew 1. For each T new, compute the compressed Topex pseudorange following Steps 3 to 5. A flow diagram of the data compression scheme is as shown in Fig. 4. Simulation Analysis compute R Φ for 2.5 min < T Tnew < 2.5 min (R Φ)ave = average of R Φ compress Φ to 1-sec points with quadratic fit over 1 1-sec points To assess the effectiveness of the proposed data compression scheme in reducing the S/A effects, a simulation analysis was carried out. The orbits of Topex and a constellation of 18 GPS satellites in 6 orbital planes were computed over a 2-hour period. Simulated raw data, at 1-sec intervals for Topex carrier phase and 1-sec intervals for Topex pseudorange and both data types for six globally distributed ground receivers, were generated. The S/A effects for the 18 GPS satellites were computed and added to the simulated data. All ground receivers' clocks were assumed to be perfect whereas different levels of Topex clock offsets were considered. The gross differential S/A effects without correcting for the time-tag offset were first assessed. The raw data without including satellite dynamics were generated and then decimated to 5-minute data points. Singly differenced data were formed between Topex and ground receivers observing the same GPS satellites. The RMS effects of 18 passes of differenced data observing 1 different GPS satellites having the longest common view periods, over the entire 2-hour orbit cycle were computed and plotted in Fig. 5 for three different Topex clock offsets:.5 sec, 2.5 sec and 4.5 sec. The residual effects on carrier phase for the three Topex clock offsets are the same as those on pseudorange at.5-sec offset since Topex carrier phase is sampled at 1-sec intervals. Without proper correction for the time-tag offsets, the residual S/A effects on the differenced data are indeed large: 6 cm for carrier phase and about 12 cm per sec of Topex clock offset for pseudorange, up to 6 cm at the maximum offset of 5 sec. Note that these are the RMS values over the 18 passes; at times the errors are a factor of 3 higher. The residual S/A effects without time-tag correction for a typical pass of differential pseudorange between Topex and a ground receiver are shown in Fig. 6. Next, we investigated the effectiveness of the data compression scheme for reducing the S/A effects using different interpolation strategies (linear, quadratic and cubic). Satellite dynamics were now included in the raw data. The data were compressed and then differenced between Topex and ground receivers. To separate the residual S/A errors from the satellite dynamics a second difference is taken between these results and those with S/A effects turned off. The RMS residual errors are Rcomp = (R Φ)ave + Φcomp Fig. 4. A flow diagram of GPS data compression schemes for Topex and ground receivers summarized in Table 1. A linear interpolation results in a large error (75 cm) due to poor modeling of the dynamics, and thus should not be adopted. With the proposed data compression scheme the effects are reduced to.4 mm using a quadratic interpolation and to <.1 mm using a cubic interpolation for both pseudorange and carrier phase. Although the difference is negligible for pseudorange, cubic interpolation clearly is superior over quadratic for carrier phase, which has a low data noise comparable to the.4 mm level. Hence a cubic interpolation of carrier phase is recommended for Topex data compression. A similar comparison was also made (Table 2) for ground data assuming a 2-msec clock offset due to light-time difference. Due to lower dynamics and small time offset, the RMS PSEUDORANGE ERROR (cm) 6 4 2 1 2 3 4 TOPEX CLOCK OFFSET (sec) Fig. 5. RMS S/A effects on differential pseudorange between Topex and a ground receiver 5 4
error is only 7 mm with a linear interpolation. It reduces to below.1 mm with either a quadratic or a cubic interpolation. Hence a quadratic interpolation of carrier phase is appropriate for ground data compression. Conclusions GPS pseudorange and carrier phase measurements are corrupted by S/A clock dithering on GPS signals. When Topex and ground measurements are differenced, the error will not completely cancel out due to imperfect simultaneity in data acquisition. A simulation analysis indicates that the differential S/A effects can be as high as 1 m on pseudorange and 1 cm on carrier phase. A data compression scheme to minimize the differential S/A error has been proposed and described. It combines a low-order polynomial interpolation strategy and smoothing of pseudorange by carrier phase, thus removing the need of a satellite dynamic model. With this scheme, the residual S/A error on the compressed differential pseudorange and carrier phase is reduced to <.1 mm with a cubic interpolation on Topex carrier phase and a quadratic interpolation on ground carrier phase. A quadratic interpolation on both Topex and ground carrier phase would increase the error to a level comparable to the carrier phase data noise, and is not recommended. Another much lower non-simultaneity (2 msec or lower) arises from the unequal light-time between a given GPS satellite and different receivers. The difference in S/A effect is estimated to be a few mm for both pseudorange and carrier phase data. The proposed data compression scheme also reduces this error by at least an order of magnitude through a quadratic interpolation to common transmitting time. Acknowledgment The authors would like to thank Dr. Y. Bar-Sever for performing spectral analysis on the Block II GPS signal. The work described in this paper was carried out by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Table. 1. RMS differential pseudorange error using different interpolation strategies on Topex carrier phase with.5-sec time offset Administration. Interpolation Polynomial Linear Quadratic Cubic 74.5 cm.4 mm <.1 mm Table. 2. RMS differential pseudorange error using different interpolation strategies on ground carrier phase with 2-msec time offset Interpolation Polynomial Linear Quadratic Cubic 7. mm <.1 mm <.1 mm References 1. G. H. Born, R. H. Stewart and C. A.Yamarone, TOPEX A Spaceborne Ocean Observing System, in Monitoring Earth's Ocean, Land, and Atmosphere from Space Sensors, Systems, and Applications, A. Schnapf (ed.), AIAA, Inc., New York, NY, 1985, pp. 464 479. 2. T. P. Yunck and S. C. Wu, Ultra-Precise Orbit Determination by GPS, paper 83-315, AAS/AIAA Astrodynamics Specialist Conf., Lake Placid, NY, Aug. 1983. 3. S. C. Wu, T. P. Yunck and G. A. Hajj, Toward Decimeter Topex Orbit Determination Using GPS, paper AAS 89-359, AAS/AIAA Astrodynamics Specialist Conf., Stowe, VT, Aug. 1989. PSEUDORANGE ERROR (cm) 12 8 4 4 8 1 2 3 TIME (min) 4.5 sec offset 2.5 sec offset.5 sec offset Fig. 6. S/A effects on a pass of differential pseudorange between Topex and a ground receiver 4 5 4. T. P. Yunck, S. C. Wu, J. T. Wu and C. L. Thornton, Precise Tracking of Remote Sensing Satellites With the Global Positioning System, IEEE Trans. Geoscience and Remote Sensing, Vol 28, No. 1, Jan. 199, pp. 18 116. 5. M. S. Braasch, A Signal Model for the NAVSTAR Global Positioning System, Inst. of Navigation Annual Meeting, Colorado Springs, CO, Sept. 1989. 6. J. B. Thomas, Functional Description of Signal Processing in the Rogue GPS Receiver, JPL Publication 88-15, June 1988. 7. Meehan, T., et al., Rogue: A New High Accuracy Digital GPS Receiver, International Union of Geodesy and Geophysics, XIX General Assembly, Vancouver, BC, Canada, Aug. 1987. 8. Meehan, T., et al., Baseline Results of the ROGUE Digital GPS Receiver, presented at the 1988 American Geophysical Union spring Meeting, Baltimore, MD, May 1988. 5
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