NOTE: This study was done at a time when selective availability (SA) was enabled.
Now that SA has been turned off, civilian GPS receivers are much more accurate. For an
excellent analysis of GPS measurement errors in modern times, please visit
David L. Wilson's site.
This information has effectively been superseded by Dr. Wilson's study, but it remains here
for historical interest.
Study of the Accuracy of Averaged Non-Differential GPS Measurements
Dan Charrois, 1999/02/08
Though GPS receivers have opened up a whole new era of accuracy in navigation
and geodetics, there are a multitude of errors inherent in measurement. Left
uncorrected, these errors can result in inaccuracies of the order of a hundred
metres or so. This brief document describes in a quantitative fashion how
averaging positional measurements over a period of time results in improved
accuracy. For those of you interested in the results of the study, feel
free to skip over the background if you already have an idea of how GPS
To get the most out of this document (and GPS usage in general), you are encouraged
to read Trimble's excellent
description about how
standard and differential
GPS measurements are done. It goes into more details there than could
possibly be covered here.
To determine its position on the Earth, a GPS receiver has to know the distance
to the various GPS satellites as accurately as possible. Most of the errors
in taking a standalone GPS measurement affect this distance measurement in
To calculate the distance to the satellite, the GPS receiver determines
the time it took for the radio signal to travel from the satellite. But the
speed of a radio signal, though constant in the vacuum of space, is affected
by the Earth's atmosphere, particularly the ionosphere. Most GPS receivers have
built in an "ionospheric correction", but this is based on a fixed model of the
behaviour of the ionosphere. Since the characteristics of the ionosphere
change, the signal from a satellite may take more or less time to reach the
receiver than anticipated.
If a radio signal is reflected off a nearby object on its way from the
satellite to the receiver, it will have travelled a longer distance than if
it reached the receiver directly. If the receiver locks onto this instead of
the original signal, it will calculate an erroneous position of the satellite.
Unfortunately, there are not many ways to help eliminate or reduce this effect
other than by using more expensive receivers that are less prone to multipath.
Selective Availability, or SA
This is the largest contributor of errors to consumer GPS receivers. When
the U.S. military set up the GPS constellation of satellites, they didn't want
consumers to have access to the same accuracy as the military. So, they
intentionally introduce an error into the public, unscrambled signal. This
error arises from the satellites providing slightly inaccurate orbital data
to the receivers, as well as offsetting the time transmitted from their atomic
clocks slightly. Both effects cause the receiver to calculate an erroneous position
of the satellite
To improve the accuracy of a GPS receiver, several methods can be utilized:
- Real-time differential corrections
- In this configuration, a stationary GPS receiver is told as accurately
as possible where it is actually located. It receives the signals from
the various satellites and calculates the distances to them as a normal
GPS would. But since it knows its true position, it is also capable of
calculating the true distances to the satellites. From this, it can determine
how much the satellite distances are in error, as caused by ionospheric
effects and selective availability. If a roving GPS receiver is within
a few hundred kilometres of the stationary receiver, it is likely affected
by similar phenomenae. So, the distance errors are transmitted via a
real time radio link and the roving receiver can correct its position,
enabling an accuracy of a few metres or better. The major disadvantage of this
system is the requirement of a real-time radio link, which means expensive
transmitters and receivers, as well as the uncertainty in receiving a signal
in hilly terrain.
- Post processing differential corrections
- With this scenario, a stationary and roving GPS receiver are used as
before, but rather than transmit the correction data in real time, the
raw "pseudorange" distances to the satellites are stored at both receivers.
After the mapping excursion is over, the data from the two receivers
is "post processed" in a computer to reduce the effects of the errors.
This has the advantage of not requiring a real time radio link, but its
disadvantages are that the roving receiver does not know an accurate position
until after the excursion (making it unsuitable for high accuracy navigation),
and that GPS receivers capable of storing pseudorange data are traditionally
- This is probably the simplest, and undoubtedly the least expensive way
to get more accuracy out of a standard GPS. A GPS receiver is just
left in a fixed position for as long as possible, and its calculated
position is "averaged out" over time. For obvious reasons, the technique is not
feasible for navigation, but accuracy does improve when measuring the positions
of fixed points. But it is very hard to find an answer to the question of
"how long is long enough", so the rest of this document attempts to quantify
the relationship between accuracy and the length of time the receiver is
left at a fixed point.
Over the period from January 20 to January 29, 1999, 568,830
positional measurements were collected at one second intervals
Garmin GPS 38 (an inexpensive consumer grade receiver). Extensive
statistical analysis was then done with the data, the results of which
The receiver was located in a "real world" position, with nearby buildings
affecting the view of the sky and satellite signals. But overall, the
position was relatively free of interfering structures.
The following image displays the "wandering" of the apparent position of
the fixed GPS receiver. The plot is color coded so that sites visited less
frequently are in red and those more frequently in white. Resolution of the
image is at 1 metre per pixel.
It is clear that the "calculated" position of the receiver was frequently near its "true position",
but that errors of the order of 100 to 150 metres were apparent from time to time.
Also noticeable is the fact that the east-west deviation is less than the
north-south deviation. Subsequent analyses of the standard deviations in x and y
showed that the east-west deviation is approximately 0.6 times that in the
north-south direction. Though the true reason for this effect is unknown,
a few possibilities are:
From the plot, it can also be seen that some positions (in a regular pattern)
were never visted by the GPS. This is related to the number of significant
digits to which the GPS 38 relates its position. Since it is never intended
for centimetre-level surveying, the manufacturers appear to have decided that
significant digits implying an accuracy of a couple of metres were sufficient.
In any case, this does not affect the analysis which follows.
- The horizon was blocked more in the north and south directions than in the
east and west directions. Perhaps since the receiver could not lock onto
satellites as easily in these directions, the north-south deviation became
- Similarly, due to the orbits of the satellites, there is a fairly large "hole" in the sky
from which no satellites will ever be observed. From my location, this hole
extends from nearly overhead to almost the northern horizon. Because of this, satellites
are rarely located to the north. Thus, I expect that the north/south positional
measurement would be significantly more in error than the east/west positional measurement.
The data was then "binned" in an attempt to illustrate how averaging over
various lengths of time affected the calculation of the receiver's position.
The following animated plot shows the effect quite succinctly.
Original, unbinned data (in red), is overlaid with binned data (in blue). The
different frames in the animation indicate varying bin sizes
(essentially, the number of seconds per averaged sample), as indicated by
the numerical display near the upper left.
On the right side of the plot is a frequency distribution graph of altitude.
The apparent "noise" in the plot arises from a fixed number of significant digits
in the data returned from the GPS 38 which "quantized" the results at the sub-metre level.
Again, red refers to the original data whereas blue indicates binned data. The
yellow ellipse on the positional plot and yellow lines on the altitude plot
indicate a 1 standard deviation interval for the binned data.
Since GPS positional measurements averaged over a long period essentially
follow a Gaussian distribution, this means that for a given bin size (number
of seconds per averaged sample), 68.27 percent of the positional measurements
fall within the yellow ellipse, and 68.27 percent of the altitude measurements
fall within the yellow lines at right.
The final plot helps to answer the question of "how long is long enough" when
averaging GPS measurements to attain a given level of accuracy. It shows the
standard deviation of the data vs. the averaging period (on a logarithmic
scale). Six plots are superimposed:
D.O.P. is an abbreviation for "Dilution of Precision", and essentially is a
numerical indicator of the geometric placement of the satellites used at the
time the position was calculated. Essentially, measurements with a smaller
D.O.P. were taken with satellites in a more favorable geometric configuration,
and can be treated as more accurate data (though selective availability and
other effects continue to degrade the accuracy of the calculation). For the
data set used in this study, selecting measurements with a D.O.P. < 2 resulted in
150,102 readings, or approximately 26% of the original data. Clearly, the
data with a D.O.P. < 2 is more accurate, and the plot helps to show you by
- East-West deviation, using all available data
- North-South deviation, using all available data
- Altitude deviation, using all available data
- East-West deviation, for all data where D.O.P. < 2
- North-South deviation, for all data where D.O.P. < 2
- Altitude deviation, for all data where D.O.P. < 2
Several interesting effects of averaging GPS positions can be seen from the
chart. For averaging periods of less than approximately a minute, little
effect is had on the accuracy of a positional determination. For averaging periods
between a minute and an hour or two, a fairly consistent improvement in accuracy is
obtainable. Averaging for longer periods naturally continues to improve accuracy,
though the rate of improvement decreases.
The graph can be used to determine "how long is long enough" when averaging
GPS positions in the following manner. For example, assume that you want
to be accurate to within 10 metres in an east-west direction
68.27% of the time (1 standard deviation), using all available data (not necessarily
waiting for a more optimal D.O.P. in satellite configuration). From the pink
line, we can readily read that approximately 15 minutes of data are required
(gathered once per second, or generally as quickly as possible).
Keep in mind though that this data is by definition only absolutely valid for
the time and geographical position for which it was taken. Different geographical
positions may have varying effects on GPS measurement, and in particular
the military could increase, reduce, or generally modify the effects of selective availability
(the major source of error) at any time. But even though differential GPS solutions (where available)
are preferable for navigation and mapping purposes, this study should give a very good ballpark
idea of the accuracy obtainable when averaging GPS positions for lengths of time.
Questions or comments would be greatly appreciated!
Please contact Dan Charrois at firstname.lastname@example.org. And incidentally,
if you are looking for a connector to build a cable to hook up your Garmin GPS to
a computer and/or power supply, check out this page!