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 This information has effectively been superseded by Dr.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

Introduction

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 receivers work.

Background

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 various ways:

Ionospheric errors
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.
Multipath
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 quite expensive.
Averaging
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.

The Experiment

Over the period from January 20 to January 29, 1999, 568,830 positional measurements were collected at one second intervals from a Garmin GPS 38 (an inexpensive consumer grade receiver). Extensive statistical analysis was then done with the data, the results of which appear here.

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.

Frequency plot

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:
  1. 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 greater.
  2. 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.
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 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.

Animated plot

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:

  1. East-West deviation, using all available data
  2. North-South deviation, using all available data
  3. Altitude deviation, using all available data
  4. East-West deviation, for all data where D.O.P. < 2
  5. North-South deviation, for all data where D.O.P. < 2
  6. Altitude deviation, for all data where D.O.P. < 2
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 how much.

Standard deviation vs. bin size plot

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 dan01@syz.com. 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!