A report distributed by the Aerospace Corporation [14] has claimed that the correction expressed in Eqs. (38) and (39) would not be valid for a highly dynamic receiver – e.g., one in a highly eccentric orbit. This is a conceptual error, emanating from an apparently official source, which would have serious consequences. The GPS modernization program involves significant redesign and remanufacturing of the Block IIF satellites, as well as a new generation of satellites that are now being deployed – the Block IIR replenishment satellites. These satellites are capable of autonomous operation, that is, they can be operated independently of the ground-based control segment for up to 180 days. They are to accomplish this by having receivers on board that determine their own position and time by listening to the other satellites that are in view. If the conceptual basis for accounting for relativity in the GPS, as it has been explained above, were invalid, the costs of opening up these satellites and reprogramming them would be astronomical.

There has been therefore considerable controversy about this issue. As a consequence, it was proposed by William Feess of the Aerospace Corporation that a measurement of this effect be made using the receiver on board the TOPEX satellite. The TOPEX satellite carries an advanced, six-channel GPS receiver. With six data channels available, five of the channels can be used to determine the bias on the local oscillator of the TOPEX receiver with some redundancy, and data from the sixth channel can be used to measure the eccentricity effect on the sixth SV clock. Here I present some preliminary results of these measurements, which are to my knowledge the only explicit measurements of the periodic part of the combined relativistic effects of time dilation and gravitational frequency shift on an orbiting receiver.

A brief description of the pseudorange measurement made by a receiver is needed here before explaining the TOPEX data. Many receivers work by generating a replica of the coded signal emanating from the transmitter. This replica, which is driven through a feedback shift register at a rate matching the Doppler-shifted incoming signal, is correlated with the incoming signal. The transmitted coordinate time can be identified in terms of a particular phase reversal at a particular point within the code train of the signal. When the correlator in the receiver is locked onto the incoming signal, the time delay between the transmission event and the arrival time, as measured on the local clock, can be measured at any chosen instant.

Let the time as transmitted from the satellite be denoted by . After correcting for the eccentricity effect, the GPS time of transmission would be . Because of SA (which was in effect for the data that were chosen), frequency offsets and frequency drifts, the satellite clock may have an additional error so that the true GPS transmission time is .

Now the local clock, which is usually a free-running oscillator subject to various noise and drift processes, can be in error by a large amount. Let the measured reception time be and the true GPS time of reception be . The possible existence of this local clock bias is the reason why measurements from four satellites are needed for navigation, as from four measurements the three components of the receiver’s position vector, and the local clock bias, can be determined. The raw difference between the time of reception of the time tag from the satellite, and the time of transmission, multiplied by , is an estimate of the geometric range between satellite and receiver called the pseudorange [22]:

On the other hand the true range between satellite and receiver is Combining Eqs. (41)and (42) yields the measurement equation for this experiment:The purpose of the TOPEX satellite is to measure the height of the sea. This satellite has a six-channel receiver on board with a very good quartz oscillator to provide the time reference. A radar altimeter measures the distance of the satellite from the surface of the sea, but such measurements play no role in the present experiment. The TOPEX satellite has orbit radius 7,714 km, an orbital period of about 6745 seconds, and an orbital inclination of 66.06° to earth’s equatorial plane. Except for perturbations due to earth’s quadrupole moment, the orbit is very nearly circular, with eccentricity being only 0.000057. The TOPEX satellite is almost ideal for analysis of this relativity effect. The trajectories of the TOPEX and GPS satellites were determined independently of the on-board clocks, by means of Doppler tracking from 100 stations maintained by the Jet Propulsion Laboratory (JPL).

The receiver is a dual frequency C/A- and P-code receiver from which both code data and carrier phase data were obtained. The dual-frequency measurements enabled us to correct the propagation delay times for electron content in the ionosphere. Close cooperation was given by JPL and by William Feess in providing the dual-frequency measurements, which are ordinarily denied to civilian users, and in removing the effect of SA at time points separated by 300 seconds during the course of the experiment.

The following data were provided through the courtesy of Yoaz Bar-Sever of JPL for October 22–23, 1995:

- ECI center-of-mass position and velocity vectors for 25 satellites, in the J2000 Coordinate system with times in UTC. Data rate is every 15 minutes; accuracy quoted is 10 cm radial, 30 cm horizontal.
- ECI position and velocity vectors for the TOPEX antenna phase center. Data rate is every minute in UTC; accuracy quoted is 3 cm radial and 10 cm horizontal.
- GPS satellite clock data for 25 satellites based on ground system observations. Data rate is every 5 minutes, in GPS time; accuracy ranges between 5 and 10 cm.
- TOPEX dual frequency GPS receiver measurements of pseudorange and carrier phase for 25 satellites, a maximum of six at any one time. The data rate is every 10 seconds, in GPS time.

During this part of 1995, GPS time was ahead of UTC by 10 seconds. GPS cannot tolerate leap seconds so whenever a leap second is inserted in UTC, UTC falls farther behind GPS time. This required high-order interpolation on the orbit files to obtain positions and velocities at times corresponding to times given, every 300 seconds, in the GPS clock data files. When this was done independently by William Feess and myself we agreed typically to within a millimeter in satellite positions.

The L1 and L2 carrier phase data was first corrected for ionospheric delay. Then the corrected carrier phase data was used to smooth the pseudorange data by weighted averaging. SA was compensated in the clock data by courtesy of William Feess. Basically, the effect of SA is contained in both the clock data and in the pseudorange data and can be eliminated by appropriate subtraction. Corrections for the offset of the GPS SV antenna phase centers from the SV centers of mass were also incorporated.

The determination of the TOPEX clock bias is obtained by rearranging Eq. (43):

Generally, at each time point during the experiment, observations were obtained from six (sometimes five) satellites. The geometric range, the first term in Eq. (44), was determined by JPL from independent Doppler tracking of both the GPS constellation and the TOPEX satellite. The pseudorange was directly measured by the receiver, and clock models provided the determination of the clock biases in the satellites. The relativity correction for each satellite can be calculated directly from the given GPS satellite orbits. Because the receiver is a six-channel receiver, there is sufficient redundancy in the measurements to obtain good estimates of the TOPEX clock bias and the rms error in this bias due to measurement noise. The resulting clock bias is plotted in Figure 3.The rms deviation from the mean of the TOPEX clock biases is plotted in Figure 4 as a function of time. The average rms error is 29 cm, corresponding to about one ns of propagation delay. Much of this variation can be attributed to multipath effects.

Figure 3 shows an overall frequency drift, accompanied by frequency adjustments and a large periodic variation with period equal to the orbital period. Figure 3 gives our best estimate of the TOPEX clock bias. This may now be used to measure the eccentricity effects by rearranging Eq. (43):

Strictly speaking, in finding the eccentricity effect this way for a particular satellite, one should not include data from that satellite in the determination of the clock bias. One can show, however, that the penalty for this is simply to increase the rms error by a factor of 6/5, to 35 cm. Figure 4 plots the rms errors in the TOPEX clock bias determination of Figure 3. Figure 5 shows the measured eccentricity effect for SV nr. 13, which has the largest eccentricity of the satellites that were tracked, . The solid curve in Figure 5 is the theoretically predicted effect, from Eq. (39). While the agreement is fairly good, one can see some evidence of systematic bias during particular passes, where the rms error (plotted as vertical lines on the measured dots) is significantly smaller than the discrepancies between theory and experiment. For this particular satellite, the rms deviation between theory and experiment is 22 cm, which is about 2.2% of the maximum magnitude of the effect, 10.2 m. UpdateSimilar plots were obtained for 25 GPS satellites that were tracked during this experiment. Rather than show them one by one, it is interesting to plot them on the same graph by dividing the calculated and measured values by eccentricity , while translating the time origin so that in each case time is measured from the instant of perigee passage. We plot the effects, not the corrections. In this way, Figure 6 combines the eccentricity effects for the five satellites with the largest eccentricities. These are SV’s nr. 13, 21, 27, 23, and 26. In Figure 6 the systematic deviations between theory and experiment tend to occur for one satellite during a pass; this “pass bias” might be removable if we understood better what the cause of it is. As it stands, the agreement between theory and experiment is within about 2.5%.

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