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

Figure 2. Pseudorange residuals for all
satellites at Kwajelain. 


Figure 3. Pseudorange 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.

Figure 4. Pseudorange residuals for all monitor
stations for satellite SV 18. 


Figure 5. Pseudorange 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
satellitemonitor station pair once every 15 minutes. There was no averaging or smoothing
over the 15minute block represented. However, we did average the four 1.5second
pseudoranges in the 6second block ending on the exact 15minute block.
Superimposed on true data noise and
smaller systematic trends is a "sawtooth" effect in the residuals for
individual satellites. This effect is most pronounced in Figure 5, where a roughly
12hour periodicity is likewise evident. Thinking that the 12 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 highspeed 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 2030 centimeters. We formed highorder differences, and noted the familiar
pattern of large differences with alternating signs that would arise on occasion, often at
12hour intervals. These are both large enough and have the correct periodicity to account
for the sawtooth 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 sawtooth appearance of residuals.
7.
Shorttimespan Residuals

Figure 6. Residuals over a typical 15minute time
span. 
Figure 6 shows the
behavior of typical residuals over a 15minute time span. These are shown for SV 37
observed at Kwajelain. In the figure heading, (OC)s refer to (Observed minus Computed)
ranges. PR is pseudorange (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
1.5second ranges in each 6second 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 twofrequency 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 fourday data set, we arrive at D k_{1} = 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 k_{1}
and k_{2} that our data cannot separate.

Figure 7. Residuals over a 6hour 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 pseudoranges. In Figure 7, we show residuals for the same
satellitemonitor station pair over a 6hour 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 6minute
time span. 
In Figure 8, we show the
preaveraged 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 pseudoranges.
What might cause such an event? Multipath
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 satellitemonitor
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 twofrequency ionosphere model.
8. Conclusions
Our analyses are based primarily on
pseudorange 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 fourday 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 joinon) 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 sawtooth 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
orbitfitting 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 twofrequency 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 pseudorange 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. 485515 (1996).
McCarthy, D.D., IERS Technical Note 13:
IERS Standards (1992), Cen. Bur. of IERS, Paris, ch. 7: "Solid Earth tides",
pp. 5261 (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.
111116 (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. 517546 (1996).
Wells, D., ed., Guide to GPS Positioning,
U. of New Brunswick Graphic Services, Fredericton, New Brunswick, Canada, sections 9.06
& 9.07 (1987).
1997/02/12
