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Absolute GPS to better than one meter -- [page 4 of 4]

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6. Four-day Residuals

Figure 2 through Figure 5 show pseudo-range residuals of analysis runs for the entire four-day time span, with clock rate corrections every 16 hours (as described above). For these four figures, the horizontal axis is time partitioned into 1-day blocks, and the vertical axis is the pseudo-range residual (in the sense of observed minus computed) in meters, partitioned into 5-meter blocks.

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Figure 2. Pseudo-range residuals for all satellites at Kwajelain.


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Figure 3. Pseudo-range residuals for all satellites at Diego Garcia.  

Figure 2 shows all 19 satellites for Kwajelain, which has the smallest average residuals of the five monitor stations during this particular time period. Figure 3 shows the 19 satellites as observed at Diego Garcia, which has the largest average residuals for this data. The legend in the upper right corner of these two figures shows the different symbols and shading used to identify each of the 19 individual satellites.

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Figure 4. Pseudo-range residuals for all monitor stations for satellite SV 18.


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Figure 5. Pseudo-range residuals for all monitor stations for satellite SV 32.

Figure 4 shows all five monitor stations for satellite SV 18, which has relatively small average residuals among the 19 satellites during this particular time period. Figure 5 shows all five monitor stations observing satellite SV 32, which has relatively large average residuals during this interval. The legend in the upper right corner of these two figures shows the different symbols and shading used to identify each of the five monitor stations.

In all four plots, we have sampled each satellite-monitor station pair once every 15 minutes. There was no averaging or smoothing over the 15-minute block represented. However, we did average the four 1.5-second pseudo-ranges in the 6-second block ending on the exact 15-minute block.

Superimposed on true data noise and smaller systematic trends is a "saw-tooth" effect in the residuals for individual satellites. This effect is most pronounced in Figure 5, where a roughly 12-hour periodicity is likewise evident. Thinking that the 1-2 meters amplitude was too large to be due to errors in the orbits, we considered various exotic mechanisms, such as variations in clock behavior due to high-speed motions through Earth’s magnetic fields, which would reverse polarity in each satellite every six hours as the satellites changed magnetic hemispheres. However, none of the mechanisms considered had good predictive behavior over the entire set of data.

Finally, we tested the hypothesis that the adopted satellite orbital state vectors from JPL might contain errors larger than the advertised 20-30 centimeters. We formed high-order differences, and noted the familiar pattern of large differences with alternating signs that would arise on occasion, often at 12-hour intervals. These are both large enough and have the correct periodicity to account for the saw-tooth effect in the residuals. Further inspection revealed that the discontinuities were most significant in the orbital mean motion parameter, likewise consistent with the observed residuals. By inference, JPL fitted orbital arcs with 12 hours of overlap in such a way as to minimize discontinuities in position. But they perhaps paid no heed to discontinuities in mean motion in this process, which (we now know) would lead to a saw-tooth appearance of residuals.

7. Short-time-span Residuals

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Figure 6. Residuals over a typical 15-minute time span.

Figure 6 shows the behavior of typical residuals over a 15-minute time span. These are shown for SV 37 observed at Kwajelain. In the figure heading, (O-C)s refer to (Observed minus Computed) ranges. PR is pseudo-range (lightly shaded curve), DR = ADR is accumulated delta range (darkly shaded curve), and Avg. is the arithmetic average of the two (curve through middle). The vertical scale runs from -2 to +6 meters. As before, we averaged the four 1.5-second ranges in each 6-second block.

It is readily apparent that highly systematic trends remain in this data. And the largest such effect has a mirror behavior in PR and ADR: when one goes up, the other goes down. The arithmetic average of the two removes most of this variation, whatever its cause. However, closer inspection reveals that even the arithmetic average tends to go up when PR does, and vice versa, but to a greatly reduced extent. Therefore, some weighting of the PR and ADR curves other than 50%-50% would remove almost all this variation.

Most physical effects on the signals between satellite and receiver affect PR and ADR equally. The only correction that affects them oppositely is ionospheric delay, because the PR signal travels with the group velocity of light (always less than or equal to c), and the ADR signal travels with the phase velocity of light (always greater than or equal to c). The denser the ionosphere, the further these two signals will diverge in arrival time. We are therefore led to suspect that the ionospheric corrections applied are not as accurate as they potentially could be.

As a working hypothesis, we assume that the coefficients for the two-frequency ionosphere model (equations [7]) may not be simple functions of the frequencies, and may need empirical corrections. We can solve for only one of the coefficients because any constant offset in range is practically indistinguishable from a clock correction. But one parameter is all that is needed to model the variation in the ionosphere correction we see in the data. When we do this with the entire four-day data set, we arrive at D k1 = -0.556 0.013. This is a correction of substantial size, but one that also has strong statistical significance. It probably represents smaller corrections to both k1 and k2 that our data cannot separate.

Figure 7. Residuals over a 6-hour time span.

Even without having a specific model for the cause of this effect, a simple averaging of the PR and ADR values greatly improves the accuracy of the observed pseudo-ranges. In Figure 7, we show residuals for the same satellite-monitor station pair over a 6-hour time span. We immediately see the great reduction in the amplitude of the scatter. However, a set of residuals near 14.6 hours shows a brief departure from this pattern, clearly discordant with the rest of the data.

Figure 8. Residuals during a peculiar 6-minute time span.

In Figure 8, we show the pre-averaged PR and DR = ADR values separately in high time resolution so as to learn more about the nature of this anomaly. Once again, it is evident that the PR and ADR excursions are mirror images of one another, except that the amplitude of the PR excursions slightly exceeds the amplitude of the ADR excursions. So even for anomalous events such as this of unknown origin, our altered ionosphere corrections would almost completely eliminate the effect of the anomaly from the observed pseudo-ranges.

What might cause such an event? Multi-path signal confusion has been suggested – for example, when the signal is reflected from a building or a passing object to the receiver. However, this ignores the coincidence that PR and ADR continue to mirror one another as if the anomaly is entirely in the ionosphere. Taking the data literally, relative to its quiescent state, the ionosphere first thins for about a minute, then thickens rapidly for another minute, then thins and thickens several more times with rapidly decreasing amplitude over the next couple of minutes. This behavior suggests a shock or pressure wave propagating through the upper atmosphere that initially sweeps away many ions, causing an intense rebound effect that then exponentially decays.

The event just described is not of a completely unexpected variety. It is often said that the Earth’s upper atmosphere experiences one meteorite explosion with the force of a small atomic bomb every day, on average. However, we cannot theorize with any confidence about the cause of this event until our analysis investigates similar events in depth for other satellite-monitor station pairs. For example, it is reasonable to ask if other receivers or satellites experienced anomalies at around this same time. This is one of many lines of inquiry needing additional research that is still in progress as of the writing of this report. The same remark can be made of our conclusion that adjustments appear needed to the standard two-frequency ionosphere model.

8. Conclusions

Our analyses are based primarily on pseudo-range observations, with accumulated delta ranges enhancing the strength of certain parameters such as ionosphere corrections. All solutions reported here included clock offset and rate corrections for 19 satellites and 4 of 5 monitor stations (Colorado Springs excepted), plus 3 coordinate corrections and 2 demodulator delay corrections for each of the 5 monitor station. When we solved for only a single offset and a single rate correction for each clock over the entire four-day span of data, the rms of the residuals is 2.3 meters. Almost all of that is due to clock rate instabilities, however. So when we allow one rate correction (with forced join-on) per clock every 16 hours, the rms drops down to 1.2 meters, and is then dominated by the errors of the satellite orbital state vectors. Although we did not attempt to improve the orbits, we estimate that doing so would lower the rms to at most 0.8 meters. Inspection of the systematic nature of the trends that remain over short time spans when elements corrections are essentially constant indicates that the intrinsic error of this observation type is only about 0.2 meters.

The appearance of a saw-tooth effect in our residuals has apparently been traced to the procedure used to overlap fits of orbits for each GPS satellite to minimize discontinuities in position. We recommend that future orbit-fitting procedures for all purposes consider that discontinuities in any orbital element are equally undesirable in certain applications. Orbit analysts should do their best to minimize both position and rate discontinuities.

We have proposed empirical corrections to the ionosphere model that work well during our entire data span. Although these corrections have small formal errors, the two-frequency ionosphere model is supposed to be more accurate than this would suggest, and the possibility remains that we may be modeling some other effect that imitates an ionosphere correction. Speculations on the causes of sudden anomalous apparently ionospheric events also needs additional work. Only tests of similar data over different time spans, and entirely different kinds of data, can shed further light on the interpretation of these corrections and their robustness for making predictions.

There is clearly still much room for improvement in the analysis of this data, and in the consequent predictability and accuracy of the entire GPS system. We hope that a comparison of simultaneous laser ranging and pseudo-range data for SV 35 and 36 will shed additional light on sources of smaller errors, and on open issues involving relativity.

9. References

Klobuchar, J.A., "Ionospheric effects on GPS", in Global Positioning System: Theory and Applications, Volume I, B.W. Parkinson and J.J. Spilker Jr., eds., Amer. Inst. of Aeronautics and Astronautics, Washington, pp. 485-515 (1996).

McCarthy, D.D., IERS Technical Note 13: IERS Standards (1992), Cen. Bur. of IERS, Paris, ch. 7: "Solid Earth tides", pp. 52-61 (1992).

Navstar GPS Volume 2, Technical Proposal, Appendix 1, Analytic Task 12, IBM FSD, "Tropospheric corrections", 18 Sept. (1978).

Robertson, H.P. & Noonan, T.W., Relativity and Cosmology, W.B. Saunders Co., Philadelphia, p. 45 (1968).

Seidelmann, P.K., ed., Explanatory Supplement to the Astronomical Almanac, University Science Books, Mill Valley, CA, pp. 111-116 (1992).

Spilker, J.J., "Tropospheric effects on GPS", in Global Positioning System: Theory and Applications, Volume I, B.W. Parkinson and J.J. Spilker Jr., eds., Amer. Inst. of Aeronautics and Astronautics, Washington, pp. 517-546 (1996).

Wells, D., ed., Guide to GPS Positioning, U. of New Brunswick Graphic Services, Fredericton, New Brunswick, Canada, sections 9.06 & 9.07 (1987).


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