Biography
Ted Driver is
the Senior Navigation Engineer at Analytical Graphics Inc. Ted has worked on AGI’s navigation
capabilities for over three years, having previously been the technical lead
for Navigation Tool Kit development at Overlook Systems Technologies. He has led the engineering team in developing
the navigation algorithms and data stream definitions and is currently working
on statistical prediction models for GPS accuracy. He was previously the senior
Abstract
This paper will explore the
obstacles inherent in predicting GNSS-based navigation accuracy, discussing
each obstacle in turn and attempting to bound navigation errors as a function
of time. Depending on your definition of
accuracy – this topic can be relatively easy or terribly difficult. For this analysis, we will focus on major
contributors to navigation positioning errors, such as Dilution of Precision
(DOP), and User Range Error (URE) behaviors.
This paper is born out of the
nascent need by organizations to plan missions based on future navigation
accuracy – a need that is not easily met currently. I will look at the behaviors of each error
contributor to the accuracy prediction problem, and attempt to mathematically
or statistically bound the error produced by each. Complexities such as the non-Gaussian
behavior of the errors must be addressed as well. The predictions will take on two forms:
extrapolated instantaneous errors and statistical errors mapped to specific
confidence levels. We’ll also show how these bounds can be converted to other
accuracy parameters using standard methods.
Some of the elements that
make up the total navigation accuracy picture can be bounded fairly easily,
given a few modeling parameters. Others
are not so tame and will exhibit complex behaviors. I anticipate the results will show large
variances in the different predicted error bounds and that this in turn will
drive future work in the area. I
anticipate writing follow on papers discussing the specific details of the
different error contributors in the near future and encourage others to do so
as well.
Navigation error prediction
is becoming more prevalent and as users get more sophisticated, dilution of
precision predictions are no longer sufficient for their needs. Providing a framework to work from, this
paper will allow users of position error predictions to better understand their
specific problem. This work will also
lead to further research in accuracy prediction, forming a foundation from
which to grow.
Introduction
Two subjective terms are used in the title of this
paper; Long-Term and Accuracy. Differing
individuals or groups will have different definitions for both. I do not have a strict definition myself, so
I’ll let my examination of the data provide some answers.
Radio-navigation position accuracy consists of
several error sources, with error budgets well defined in many texts.
[1][2][3] Finding a complete theory of
navigation error prediction will necessarily require that all error sources be
predictable – to one degree or another.
In theory, once the different error sources can be predicted, the
problem is complete, all that would remain is to combine these error
predictions in some meaningful fashion to determine the entire navigation error
problem at some future time. References
on the propagation of errors are good places to begin that study [10]. Of course when theory meets practice,
problems inevitably arise. Let’s start
with a list of the common error sources and see where we start to have
trouble. Once we find the trouble, let’s
see if we can find ways around the issues or somehow bound the problem.
Navigation Error Sources
This error source list is not
meant to be all-inclusive, but includes those with the largest effects
- Dilution of Precision
- Ephemeris Error
- Clock Error
- Atmospheric Error
- Ionospheric Error
- Tropospheric Error
- User Equipment Error
- Multipath
- Receiver noise
For this paper, I will not look at the atmospheric behaviors. This is a topic in and of itself [9]. Let’s look at each of the remaining errors in
turn.
Dilution of Precision
Many
references provide details regarding what dilution of precision (DOP) is and
how it arises [1] [2] [3]. For this
paper, I’ll assume the reader is familiar with DOP and how it affects
navigation accuracy. For the most part,
DOP is predictable, since it arises from constellation geometry alone. All that’s required to predict DOP is to be
able to predict the GPS satellite orbital positions and even then no great
accuracy is required.
Note
that it is assumed here that there are no obstructions to your obtaining the
predicted DOP value. Physical
obstructions as well as radio frequency obstructions can limit your receiver’s
ability to track all satellites that could otherwise be seen. This paper’s analysis will assume that none
of those obstructions are in place and all satellites predicted to be tracked
are actually tracked.
Differences
in the actual positions of the satellites on the order of tens of kilometers
lead to DOP differences only in the 3rd and 4th decimal
place. This is provided of course that the almanac propagation is close to the
Time of Almanac (TOA) [4] when compared to the precise ephemeris.
Here
again is a subjective term – close.
Let’s look a little closer at DOP values predicted with an almanac
propagated into the future some number of weeks against the actual DOP value
for that time calculated from the National Geospatial-Intelligence Agency (NGA)
precise ephemeris. This analysis will
show us how far we can use a single almanac, before it starts affecting our
accuracy. Figure 6.3 in [2] shows that
position accuracy is linearly dependent on dilution of precision. The dilution of precision error then has a
direct affect on position errors. Figure
1 shows the position DOP residual error plotted against time for 22 weeks of
prediction. Figure 2 is a closer look at
the first 8 weeks.
Figure 1 - PDOP Residuals 22 Weeks
Figure 2 - PDOP Residuals 8 weeks
Upon
inspection, the first two weeks of DOP values have a very small residual error
– roughly 0.003 peak-to-peak (excluding the few spikes present in this
range). As time progresses however the
spikes gain control and eventually dominate the residual plot. However, even at 22 weeks out, the absolute
DOP prediction error is still under 0.6.
Since
the DOP residual error is pretty low, even 22 weeks out, let’s see how well the
predicted DOP values correlate with the actual DOP values.
Figure 3 shows a
cross-correlation plot of the predicted, almanac DOP values to the actual DOP
values. The interesting portions of this
plot are the spikes that occur at daily intervals. One would expect the DOP values to be
correlated at the same time each day (more precisely at the 23:59:56 mark) –
for at least the first few weeks, and this is what we see. Note though that the correlation linearly
decreases until at 22 weeks, where there is little correlation between the DOP
values calculated by the almanac prediction and the actual DOP. The magnitude of the DOP value may only
change by 0.6 peak-to-peak, but the lack or correlation tells us that the
actual DOP data may have spikes where the predicted data showed none.
Figure 3 – Predicted PDOP, Actual PDOP Cross Correlation
Figure 4 - 3 week DOP prediction example
Figure 5 - 21 week DOP prediction example
Figures 4 and 5 show examples of the predicted DOP compared to the actual DOP. Prior to the two week boundary, the predicted DOP is virtually indistinguishable from the actual DOP – the graphs overlay each other.
Figure 6 - Variance of Predicted Position DOP
Figure
6 shows the variance of the predicted position DOP by week of prediction. As the prediction time increases, the
variance of the predicted DOP increases linearly. Another clear marker here is that the
variance is practically zero for the first two weeks. The wise reader will note here that the
almanac can be safely used for two weeks or so before the almanac orbit
predictions start to degrade user accuracy by providing increasingly incorrect
DOP values. So, for DOP analysis, we can
define Long-Term as two weeks.
As
we’ll see further on in the paper, DOP is the defining criteria for
predicting GPS accuracy. While DOP
cannot give you a precise navigation error in and of itself, the extent to
which we can predict DOP directly ties to the extent we can predict navigation
accuracy.
The
SIS ranging error consists of two primary pieces, ephemeris error and clock
error. The ephemeris errors are errors
between the actual GPS satellite position and the satellite position broadcast
to receivers. The clock error is similar
– it’s the difference between the actual clock phase and the clock phase that’s
calculated from parameters sent to the receiver. These errors are typically a few meters but
can be much more, especially in the case of the clock. Ephemeris errors result from unmodeled
perturbations on the satellite and are reduced to almost zero when a new
navigation upload is made to the satellite.
At this point, the age of data (AOD) is zero and the broadcast ephemeris
is at its most accurate. As time
progresses throughout the day, imperfections in the ephemeris prediction slowly
appear, leading to larger ephemeris errors.
The
clock errors act similarly. The clock
errors arise from quantum mechanical fluctuations in the atomic clock itself,
leading the clock phase to exhibit a random walk behavior. This effect is difficult to predict and over
days, and over weeks and months would be impossible to determine.
In
this analysis, the job of having to predict the ephemeris and clock errors is
made simpler by the fact that these errors are clamped. The 2nd Space
Operations Squadron (2SOPs) watches both the ephemeris and clock residuals in
near real time and ensures that they stay below certain thresholds by uploading
new navigation data predictions to the satellites. Typical satellites are uploaded once per day;
some more often (usually the older satellites), some less often. This clamping effect on the SIS errors makes
predicting long term behavior easier, in that we do not need to be able to
predict random clock behaviors for weeks at a time. Under nominal conditions, we can assume a
worst case set of errors for the ephemeris and clocks based on an analysis of
the long-term trends of the data. To
that end, I have analyzed over 800 days of ephemeris and clock errors from
2SOPs and looked at the absolute maximum ephemeris and clock errors for each
satellite. See Figures, 7, 8 and 9. Indeed, one can see that the errors do not
run off past certain boundaries. Of course, each PRN exhibits a different
bound. If 2SOPs did not upload the satellites on a regular basis, these plots
would look markedly different. It should
be noted that data for all satellite outages was removed for this analysis.
Figure 7 – Sample of Maximum Clock Error by Day
Figure 8 – Sample of Maximum Ephemeris Error by Day
Figure 9 – Sample of Maximum Global
The
averaged maximum values by satellite [5] are shown in Figures 10, 11 and
12. These plots show the mean value one
could use in a prediction scheme for long-term navigation errors by satellite. For example, when predicting the SISURE
component of the navigation error using PRN 19, one needn’t allow the predicted
error to raise much above 0.75 meters.
The data here has shown that the maximum global URE error for PRN 19 has
consistently been in this range.
This
data is helpful in the long-term prediction regime; prediction times longer
than a day. In fact, since our certainty of the general clock phase state
decreases as time increases (assuming no clamping), this maximum error
information becomes more valuable as time increases. In the next section I’ll discuss
extrapolated predictions in the range from 1 minute to 12 hours to see how to
better describe navigation errors in this regime.
Figure 10 – Averaged Maximum Ephemeris Error
Figure 11 – Averaged Maximum Clock Error
Figure 12 - Averaged Maximum Global
User Equipment Errors
The
User Equipment Errors (UEEs) are the least predictable of the error sources in
our problem. Receiver noise is generated
by the receiver tracking loops as they track code, phase and frequency in a
variety of dynamic, signal-rich environments. Multipath error results from the
receiver receiving multiple signals from the same spacecraft – along different
reflected and refracted paths. The receiver can use only those signals within a
specific time of reception rendering the other signals as noise the receiver
must endure.
Receiver
noise error is dependent on many factors that change as a function of time;
temperature, g-loading, antenna positioning, etc. Multipath error is dependent upon knowing the
exact position of reflective surfaces surrounding the receiver antenna and the
orientation of the antenna itself.
Determining all of these parameters to produce a viable navigation error
prediction is a significant task and presents many challenges – though good
work is being done in this area [11].
One way we may attack this problem currently is to take a lesson from
the physicists of the 19th century.
By then, the laws of classical mechanics could aptly describe all the
motions of the particles of a gas in a box, but the sheer number of gas
particles precluded the scientists of the time from conducting such a
calculation. In their case, they
resorted to using the methods of the Statistical Mechanics branch of the
discipline. With this, they had to be
content with understanding the statistical behaviors of the gas, rather than
the explicit motion of each gas particle.
In our case, we can do something similar. Instead of trying to understand each
multipath reflection, we can create ensemble behaviors for different categories
of environments and use root-mean-square (RMS) values derived from these
environment types as an additional navigation error. This approach could lead to more efficient
calculation schemes than current ray tracing algorithms – though this method
will not be able to provide us with instantaneous errors, only statistical
behaviors. The field of noise and
multipath prediction is nascent and difficult.
For this paper, I’ll suffice to leave it at being able to add an error
value for either noise or multipath or both to the navigation error
prediction. Determining ensemble
behaviors for different classes of environments is beyond the scope of this
paper, but a good topic for a follow-on paper [9].
PREDICTION METHODS
We’ve
looked at the data necessary to perform the predictions, now we need to
understand how to predict navigation errors and within which time regimes the
answers are viable. It’s important to
understand how best to predict errors at different times in the future. One method may lead to more accurate results
in one regime, and another method may work better at a different time.
To
better understand the time regimes, let’s look at the types of data available
to us from which can make predictions.
The analysis of this data should naturally point us to times when it
should be used.
Prediction Input Data
There
are three types of data most likely to be available for predictions:
1) Information about the
navigation errors from the previous time step, either from some differential
network, or by some other means
2) Statistical information on
the errors from previous days
3) The maximum error
information from the previous section of this paper
These
three choices provide different ways to predict the navigation error at a given
time. If all types of data are
available, it may be possible to switch prediction techniques based on the time
of prediction. Let’s look at each of
these data types in turn.
Performance Assessment File
Data (option 1)
I’m
using the term performance assessment file (PAF) because the GPS Operations
Center (GPSOC) produces PAF files containing ephemeris and clock errors for
each satellite in near-real time. Using
this type of standardized data, one can produce the instantaneous navigation
errors for a given time. The GPSOC
performs these calculations daily. To
understand how this data can be used for predictions, we must come up with some
type of extrapolation scheme for the ephemeris and clock errors contained in
the file. Fortunately, the ephemeris and
clock error rates are also included in the PAF file. Using the following simple PAF extrapolation
algorithm, I’ll propagate the ephemeris and clock error states into the future,
and then calculate the user range error and navigation accuracy based on these
propagated errors.
The
notation follows [6].
Here,
is the predicted
ephemeris error, from time N predicted h steps into the future and 
is the time step of the data.
A similar equation holds for the clock data.
Once
we have the predicted ephemeris errors and clock errors, we need to create user
range errors. The user range errors are
created by dotting the predicted ephemeris error vector into the line of sight
vector from the receiver to the GPS satellite.
This dotted quantity then has the predicted clock error subtracted from
it.
Figure 13 –
Figure
13 shows how the user range error prediction residuals behave as the prediction
time h increases from 0 to 12 hours.
Within the first hour, the URE residuals do not vary by more than ±1
meter. These errors grow as the
prediction time increases. To see how
these predicted UREs affect the predicted navigation performance, see Figures
14 and 15.
Figure 14 - Horizontal Error Residual
For
roughly the first 6 hours (360 minutes), the navigation error residuals are
within a few meters. This single
day-single site analysis shows at a rudimentary level, how the PAF
extrapolation method can be used for navigation accuracy prediction. However, a more detailed analysis is needed
here to understand how regional effects and daily effects average out over the
long term. It does appear though that
even with a detailed analysis we are not going to get much better than 6-12
hours of predictability using this method and expect to maintain an accuracy
level expected by the majority of GPS users.
Figure 15 - Vertical Error Residual
Prediction Support File Data
(option 2)
The
GPSOC also produces statistical data for each GPS satellite’s performance. In particular the Prediction Support File (PSF)
contains the 1-sigma errors for the radial, along-track and cross-track
components, as well as for the global user range error and clock error over the
last seven days. The global user range
error is defined as:
Equation 1 - Global URE
Here, UREG is the
7-day 1-sigma global user range error, 
is the 7-day 1-sigma clock error, 
is the 7-day 1-sigma radial error, 
is the 7-day 1-sigma along-track error and
is the 7-day 1-sigma cross-track error. The global user range error equation derives
from integrating the user range error over the entire face of the Earth.
This
statistical data can be used to make statistical predictions of navigation
accuracy. We will not be able to get
instantaneous errors as we did with extrapolated PAF data, but we can predict
navigation errors with a specified confidence level.
Using
the constant 1-sigma value for the UREG for each satellite, we can
calculate the 1-sigma value for our navigation error into the future. If we predict the vertical error or the time
error, our prediction will have a confidence level of 68.27% since both time
and vertical errors are 1-dimensional quantities. If however, we predict horizontal error, a
2-dimensional quantity; our 1-sigma prediction will have a confidence level of
39.35%. The predicted position error, a
3-dimensional value, will have a 1-sigma confidence level of 19.9%. To be able to measure the effectiveness of
these predictions, we need to convert these 1-sigma error predictions to some
standard confidence interval. Typically,
the confidence levels used are 50% and 95%.
Standard conversion multipliers exist [7] [12], however these standard
multipliers are derived from normal, Gaussian processes. Unfortunately, GPS errors are not well
represented by Gaussian statistics in the long term. Referring to Figure 16, I’ve created a
histogram of roughly 20,000 position errors, and then plotted several best-fit
distributions against the position error data.
It’s obvious that the Normal distribution is not well-suited; however,
the usually quoted Rayleigh distribution is not the best fit either. The Weibull fit is roughly equivalent to the
Rayleigh fit (the Weibull distribution is a generalization of the Rayleigh
distribution [8]) but the Gamma fit seems to be the best.
Figure 16 - Position Error Distribution
Table
1 lists the best fit parameters for these position error data distributions.
|
Distribution |
1st parameter |
2nd parameter |
|
Rayleigh |
1.18535 |
N/A |
|
Weibull |
1.64471 |
1.85666 |
|
Gamma |
3.14733 |
0.462432 |
|
|
1.45543 |
0.831808 |
Table 1 - Position Error Distribution Parameters
Rather
than deriving the multipliers using an analytic distribution [9], I’ll use the
PAF data provided by the GPSOC to derive the multipliers empirically. For each
of the days analyzed, I’ll calculate the navigation error at 1 minute time intervals,
then sort the position, horizontal, vertical and time accuracy data and find
the 50th and 95th percentile errors.
Dividing
these errors by the root-mean-square error for the day provides an estimate of
the one, two and three dimensional multipliers for that site for the day. To get an accurate picture of the global
distribution of the multipliers, I’ll repeat this analysis over the globe using
a 5 degree grid. The values for each
grid site are then averaged on a daily basis.
The results of this analysis are plotted in Figure 17. It’s interesting to see that over the 600+
days of analysis, there appear to be no trending behaviors in the multiplier
data, though daily variations are quite apparent. Tables 2 and 3 compare the empirically
derived multiplier values by dimension and confidence percentage to the
theoretical (Gaussian-based) values.
Figure 17 - Empirical Confidence Interval Multipliers
|
Dimensions |
Empirical
Value / Standard Deviation |
Theoretical
Value |
|
1 – Vertical |
0.6323/0.0223 |
0.6745 |
|
1 – Time |
0.6084/0.0220 |
0.6745 |
|
2 –
Horizontal |
0.7824/0.0236 |
0.8326 |
|
3 – Position |
0.7551/0.0236 |
0.8880 |
Table 2 - 50% Confidence Multiplier Values
|
Dimensions |
Empirical
Value / Standard Deviation |
Theoretical
Value |
|
1 – Vertical |
2.0096/0.0316 |
1.960 |
|
1 – Time |
2.0230/0.0281 |
1.960 |
|
2 –
Horizontal |
1.8109/0.0431 |
1.731 |
|
3 – Position |
1.8433/0.0380 |
1.614 |
Table 3 - 95% Confidence Multiplier
Values
The
variability of the empirically derived multiplier values as seen in figure 17
suggest that on a daily basis, the confidence values will not be identically
95% and 50%, but will vary slightly. For
example, see Figure 18 where I compare actual position errors to 50% and 95%
confidence predicted position errors.
For
this particular day, the percent of actual errors outside of the 95% confidence
level is 6.8% - not the 5.0% we would expect.
Similarly, for the 50% confidence level data, this particular day saw
51.8% of the actual errors above the predicted errors.
The
next question is then, how long can I use a single 7-day PSF file to represent
navigation errors accurately, within a given confidence level? To decide this, we must take into account the
variability of the empirical multipliers we use to arrive at a given confidence
level – and look for excursions beyond this inherent variability.
Figure 18 - Actual versus Statistically Predicted Errors
Figure
19 shows one way to visualize these excursions.
A statistical prediction of GPS accuracy was made each day for 155 days
past the prediction epoch. Each day, the
actual navigation accuracy at a specific site was calculated, and then the
predicted accuracy was calculated using the 7-day PSF file from the prediction
epoch only. The 7-day PSF file was not
updated as the prediction day advanced.
This figure shows the percent of actual navigation errors that are
greater than the 95% confidence level predicted navigation error. I’m using the term excursions for this
quantity. This is the interesting
behavior we are interested in – we want to know the actual navigation errors
that are greater than our prediction and, hopefully to be able to minimize
them.
Figure 19 - Confidence level prediction stability using last 7 day statistics
Figure
19 has a few interesting points:
a) There is no apparent
decrease in confidence in this graph as the prediction time increases. This would be signaled by an increasing trend
in the data from left to right.
b) There is much variability in
the 95% confidence predictions. At a 95%
confidence level, based on the variability in the multiplier calculations of
Figure 17, we’d expect a smaller variation of the actual percentage about the
5% line. Instead, we see a larger
variation.
The
preliminary conclusion to draw from this data is that there appears to be no
time dependence on the use of the 7-day PSF file for prediction purposes. When using the 7-day PSF file for predicting
navigation accuracy, and then converting to a specific confidence level, one
must not expect that exact confidence level to be strictly upheld, even in the
shortest prediction times. The inherent
variability of GPS statistics precludes us from being able to precisely
determine statistical predictions with great confidence. Standard error theory
procedures do not appear to hold well when applied to GPS error measurements
and further study on this topic is warranted. [9]
Figure
20 shows that 60% of the actual error excursions are within the 5%
boundary. In fact, 90% of the excursions
are below 10.25% (89.75% confidence).
With no time-dependent behavior to rely on, using a given multiplier to
predict accuracy with a certain confidence is not for the faint of heart.
Figure 20 - Cumulative probability 95% error excursions using last 7 day error statistics
This
analysis suggests that we look further into the generation of the multipliers
used to satisfy our confidence interval analysis criteria. Is there a spread of the multipliers as we
predict further in time? Is there a
better way to derive the multipliers? We
can actually find the multiplier that will
satisfy our 95% confidence (or any confidence level for that matter) by
iterating over different multipliers and counting the excursions for each. To do this analysis, I iterated over
multiplier values from 0 to 2.5, for each day in my 155 day sample (Jan 1, 2007
to Jun 4, 2007). The results are plotted
in figure 21.
Figure 21 – Multiplier analysis for
actual error excursions using last 7 day error statistics
This figure shows a familiar looking curve [9], with
several lines. I’ve highlighted the
lines for the multipliers for the number of days averaged into the prediction.
This graph shows that after the first few days of
prediction, the multipliers settle down to a fairly small range. The horizontal blue line signifies the 50%
excursion criteria (50% confidence) and the horizontal red line signifies the
5% excursion criteria (95% confidence)
I’ve also highlighted the width of the multiplier values for these two
confidence levels. This shows that the
multiplier values are:
1) Different for this regime
than those obtained using the empirical method above
2) Have a fairly large spread
for 95% confidence.
The shape of the curve shows us why the 95% spread
is so large – the multiplier lines are almost tangential to the 5% line.
Reference [12] has a good explanation for this.
The multipliers have a spread of 0.75 to 0.90 for the 50% confidence
level and 1.68 to 2.08 for the 95% confidence level. It appears that averaging the global multipliers
derived empirically then using that single mean multiplier value may not be the
best method to use.
Maximum Error
Data (option 3)
The maximum errors derived in the Signal-In-Space
section above could also be used to create statistical predictions of
navigation accuracy. Instead of the global
URE derived from the last 7 days of data for each satellite, I’ll now use the
maximum global URE data as the 1-sigma error in the PSF prediction scheme. To create the maximum global URE, I’ll use
the global URE equation (Equation 1), and use the maximum radial, along-track,
cross-track and clock error statistics.
Then, proceeding as above, I’ll predict 155 days out and determine the
95% excursions as a function of the multiplier value required to meet that criteria. Figure 22 shows a plot identical to Figure
21, but using the maximum error statistics instead. Notice the difference in the width of the
spread for the 95% confidence multipliers (along the red 5% line). This spread is much less than with the 7-day
PSF file prediction. These multipliers
have a spread of 0.55 to 0.68 for the 50% confidence level and 1.2 to 1.43 for
the 95% confidence level.
Figure 22 - Multiplier analysis for
actual error excursions using maximum error statistics
To see
how the excursions behaved, Figure 23 was created. This figure looks quite similar to Figure 19,
in fact it’s difficult to glean any new information by studying these two
graphs alone.
Figure 23 - Confidence level prediction stability using maximum error statistics
Looking
now to see if this new maximum error approach is any better or worse, I created
Figure 24, the cumulative probability plot, similar to Figure 20. With this new prediction scheme, I still have
60% of my errors within the 95% confidence level, and 90% of my errors are now
within 89% confidence. The results here
are not statistically significant.
Figure 24 - Cumulative probability of actual error excursions using maximum error statistics
One
final piece of analysis we can perform on these two types of predictions data
is to scatter plot them. We’ll look for
any deviations that may show us one type of prediction method is better than
another.
In
Figure 25, the black diagonal line is the y = x line, the 7-Day error
excursions are plotted along the x axis and the maximum error excursions are
plotted along the y axis. Each blue dot
represents the error excursions for one day of prediction, with the first 14
days of predictions highlighted in orange.
I determined a least squares fit to the 155 days of data and found the
slope of the best-fit line equal to one.
Essentially, the y = x line is the best fit line in the
graph. Thus, there appears to be no
significant difference between these prediction methods when used to try to
reduce the number of excursions beyond the 95% confidence level.
Figure 25 - Maximum Error vs. 7-Day prediction methods
We
know from the previous section on DOP prediction, that we can successfully
predict DOP two weeks with an almanac.
With this in mind, I highlighted the first 14 days of statistical
prediction in Figure 25, wondering if there was some pattern that these
excursions took to lie about the best fit line.
As is apparent from the graph, the pattern of the first 14 days is not
different from that of the whole 155 day dataset. In fact, though not plotted, each successive
month of predictions was analyzed, and found to have this same general
behavior, leading to a final conclusion – there is no time dependence to
statistical error excursions when predicting GPS accuracy.
Time Regimes
Now
that we’ve explored the prediction behaviors in different data regimes, how can
we make use of this information when we need to predict in a particular time
regime? Should I use PAF data and
extrapolate to get my navigation errors at some future time? Should I use the statistical prediction
method? The following are my
recommendations based on this analysis.
My recommendations are presented with the understanding that as further
analysis is completed, these prediction recommendations may change.
If
one has access to PAF type data, it’s best to use that as far as possible. This is because the PAF type data will allow
you to predict instantaneous, signed errors for times in the future, providing
a specific error vector. The data
analyzed above shows that PAF based extrapolations can be used for roughly 6
hours with a meter or two of error. After six hours the navigation errors begin
to grow and may no longer be acceptable.
The choice of how long you use the PAF extrapolation technique is
directly related to how much error you can stand. In this type of prediction, I would denote long-term as 6 hours.
If
only statistical data is available, use that as a second choice. While statistical error predictions can be
made for any time in the future, the nature of the 1-Sigma prediction technique
does not allow for signed errors. Thus
only an error ellipsoid can be generated from this type of data, instead of an
error vector.
For
the statistical predictions, I would use these for what ever time span you
have. Since there appears to be no
time-dependence on the length of prediction time with this type of data, I can
recommend its use for at least several months in advance. The tricky part of this prediction is using
the correct multiplier to achieve the level of confidence you want. Using Figures 21 and 22 as guides, select the
multiplier value appropriate for your desired confidence level and type of PSF
data, then apply to your predicted error values. For this type of navigation, I’ll define long-term as 5+ months.
It’s
apparent from Figures 21 and 22, that the larger the multiplier value used will
result in fewer excursions above my desired confidence level. We could use a multiplier of 10.0 say to make
all the excursions in figure 23 lie within the 95% confidence level. The problem then is that my predicted errors
are so large that I don’t really have insight into my problem. Judgment is required here and hopefully the
analysis presented here will allow the user to make better informed decisions.
COMPLETING THE PREDICTION PICTURE
Most of the analysis in the
paper has focused on Dilution of Precision and Signal-In-Space errors and their
prediction, either extrapolated or statistical.
These errors are always present in the GPS error budget and warrant the
type of analysis seen in this paper. The
other errors in the GPS error budget are also deserving of analysis and must be
included to complete the prediction picture [9]. I have purposely not analyzed atmospheric
errors and have only touched on how the multipath and receiver errors can be
modeled. Standard error propagation
models [7] can be used to add differing error sources into a single combined
error prediction statistic. These
methods though can only be used to provide statistical error predictions.
SUMMARY
In this paper, I have analyzed the techniques
necessary to predict navigation errors using data available to the typical GPS
user. I’ve shown that almanacs can be
used to predict dilution of precision values for two weeks with little
difference in PDOP values. I then went
on to show how the signal in space user range error values are clamped by the
fact that the 2nd Space Operations Squadron uploads the GPS
satellites on a regular basis. Following
that I investigated an extrapolation technique useful when predicting for up to
six hours in the future. Statistical
prediction techniques were then addressed, first by considering a 7-day
statistical strategy then by using the maximum error method. Both methods are very sensitive to the
multiplier values needed to assess the predicted errors at a specific
confidence level.
This analysis was performed and is applicable to
only those errors that seen on a routine basis in the GPS system. The techniques discussed here will not hold
when there are clock jumps, or other perturbing forces that cause the
navigation errors to be significantly larger than normal. For a clearer understanding of these modes,
see [10].
Several topics have been raised that are good topics
for follow on papers, including:
- Determining the correct distribution for the GPS errors and using
that to derive theoretical multipliers.
- Develop a standard theory for GPS measurements, along the lines of
[7]
- The curve presented in figures 21 and 22 look like normal
distributions – how do these arise?
- Can PAF-based URE extrapolations be clamped by the maximum errors
derived here? What do the 1-12 hour
predictions look like then?
- Develop multipath and atmospheric error prediction models and fit
them into this prediction strategy.
ACKNOWLEDGMENTS
Thanks to
REFERENCES
[1] Parkinson, Bradford W.; Global Positioning System: Theory and Applications, Volume 1, Chapter 11
[2] Misra, Enge, Global Positioning System Signals, Measurements and Performance, Chapter 5
[3] Kaplan, Hegarty, Eds., Understanding GPS Principles and Applications 2nd edition, Chapter 7
[4] IS-GPS-200D, section 20.3.3.5.1.2, Almanac Data
[5] Note that the averages are over a smaller number of days for the following satellites: PRN 12:163 days, PRN 15:550 days, PRN 17:500 days, PRN 31:237 days
[6] Chatfield, Chris; The Analysis of Time Series, An Introduction 6th edition, Chapter 5
[7] Principles of Error Theory and Cartographic Applications ACIC Technical Report No. 96, February 1962
[8] Rayleigh (
) = Weibull (2,
)
[9] An excellent topic for a follow on paper.
[10] GPS
Integrity Failure Modes and Effects Analysis: K. Van Dyke, J. Kraemer,
DOT/Volpe Center; K. Kovach, ARINC; J. Lavrakas, Overlook Systems Technologies,
Inc.; J.P. Fernow, MITRE CAASD; J. Reese, GPS JPO; N. Attallah, B. Baevitz,
SAIC
[12]
http://www.gpsworld.com/gpsworld/Feature/GNSS-Accuracy-Lies-Damn-Lies-and-Statistics/ArticleStandard/Article/detail/395779