*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This information has effectively been superseded by Dr.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.*

Dan Charrois, 1999/02/08

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:

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 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.

- 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.
- 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.

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:

- 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

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!