GPS ERROR WHEN AVERAGING POSITION

A way to improve GPS measurement of position accuracy without additional equipment is to simply average the coordinates. In this section, we consider only simple averaging (no weighting by DOP) of receiver NMEA position data. As with all pages at this site, the study is for SA off unless explicitly stated otherwise. Not only does simple averaging decrease random errors in the measurement, it also allows interpolation beyond the resolution of the measurement; thus averaging may yield accuracy better than the 0.001 seconds of latitude/longitude resolution or 1 meter of height resolution reported in the NMEA data from some GPS receivers.

As is well known, if the horizontal (latitude and longitude) errors were not correlated, the RMS error would be inversely proportional to the square root of the number of measurements. However, the errors are correlated and this causes the error from averaging to decrease at a slower rate than if the errors were not correlated.

As a first look at the effect of position averaging on horizontal error, consider the plot below of twelve 24-hour averaging sessions - all starting at midnight local times.

From the plot, we see that, as expected, position-averaging tends to decrease the error. However, note the occasional error peaks on about the same time on different days. This may be due to similar larger (poor) HDOP, particular satellites yielding poorer accuracy, multipath at near the same time daily, or some other reason. Clearly, if one is collecting short sessions at the same point on different days, it is best to use different times (as different as possible) on the different days. If not continuously collecting data but doing short sessions on a single day, one should separate the sessions as much as possible in times to attempt to avoid correlated errors.

The points in the plot below are measured RMS errors from position-averaging over the corresponding different periods of every possible interval of each period in the 12 days of data. Only periods up to 8 hours are plotted, as longer periods would mean a "small" number of non-overlapping averaging periods. The separate results from the 6 days in May and 6 days in June as well as the combined 12 days are shown to indicate the robustness or variability of the measurements. The curve fits are explained in what follows.

When SA (Selective Availability) was on, it was noted that if one position-averaged, the error when position-averaging roughly fell by the reciprocal of the square root of the number of fixes divided by a constant. This constant was twice the correlation time of the fixes; this allowed the previous effective fix to "decay" and the next effective fix to "build". (Whether one calls L or 2L the correlation time depends on one's choice of terminology.)

One took into account the first fix (with averaging time of zero) acting as the first "true" measurement by adding a "1" under the radical. Thus we had the following formula for calculating the error from position-averaging from the error of single measurements and the period over which the averaging was done:

This equation to model error will be called "Model 1". When SA was turned off, smaller errors became significant and Model 1 did not work as well for modeling RMS when position-averaging.

Modeling the error as two errors as above added in quadrature seemed to work better when SA was turned off:

This equation to model RMS error when averaging will be called "Model 2". At present, it has not been possible to associate either error component with known error sources such as receiver hardware or algorithm, multipath, satellite geometry or GPS satellite constellation status.

For time in minutes and RMS error in meters, the constants obtained by non-linear least-squares regression are shown in the following table for the above Garmin 12XL test data:

	E1	L1	E2	L2
6 days (May)	4.33	1.54	3.38	69.13
6 days (June)	3.27	0.48	3.50	32.82
12 days (May+June)	3.85	1.06	3.43	48.85

The figures below show three pairs of measured position-averaging RMS errors. In each case, the two receivers antennas were separated by 1.23 meters to avoid interference but attempt similar reception condition geometry.

In each of the above three figures, an early production Garmin 12XL was used as the common comparison receiver. Note how both the Garmin 12XL and Garmin III+ have a significant error component that rapidly falls off in the first few minutes (that is, that error component has a short correlation length). The Eagle Explorer and Garmin eMap also seem to have a similar component but much smaller in size, as is indicated by the relative smallness of E₁ matched with the short L₁ in the table below for these two receivers:

	E1	L1	E2	L2
Simultaneous session
Garmin III+	3.09	1.10	2.92	55.39
Garmin 12XL	4.50	1.25	3.51	106.52
Simultaneous session
Eagle Explorer	0.49	0.07	3.61	36.45
Garmin 12XL	4.80	1.23	2.86	91.57
Simultaneous session
Garmin eMap	1.90	3.17	3.45	117.82
Garmin 12XL	4.11	1.09	3.19	69.33

As can be seen in the table, the Garmin 12XL values varied in the three 48-hours sessions. The figure below shows the results of measuring position-averaging RMS error for the Garmin 12XL in the separate sessions.

It is clear from the above that longer sessions would be required to obtain model parameters accurate for long periods of time. The above plots and formulas would seem to imply that 1 to 2 days might get a position-average RMS error down to the 1-meter level. Note well: this is RMS error--not every measurement error.

The plot below shows 20 1-day (24 hour) horizontal position-averages using the Garmin 12XL (and Micropulse antenna).

For this small sample, the RMS error for 24-hour position-averaging was 0.78 m while the initial Model 2 parameters (based on 12-days) for the Garmin 12XL data predict 0.87 m -- this is fairly good agreement considering the approximation of values due to the relatively short periods of the measurements involved and that the results probably vary with time. In each of the 20 1-day averages, the error of the position-average was less than 2 meters.

The plot below shows averaging results for 30 days using a Garmin eMap. As the plot indicates, the 30-day average is displaced about a meter and a half from the true position. This possible bias indicates that there is a limit in the accuracy that may be obtained from averaging position.

Finally, we take a look at averaging height data with a Garmin eMap and an Eagle Explorer to improve vertical accuracy.

Again we see that averaging improves the accuracy of the vertical measurement. In this case, Model 2 formulas were again used to model the measured vertical error. The values for the constants to model the height measurements from the (non-simultaneous) sessions with these two receivers are:

	E1	L1	E2	L2
Garmin eMap	3.30	6.65	5.07	306.51
Eagle Explorer	2.16	11.07	5.65	107.61

In summary, different GPS receivers perform differently when position-averaging. Several days of position-averaging appear to be needed to obtain 1-meter level horizontal accuracy. High-end (survey-grade) units will do significantly better. Finally, the statistics vary somewhat and extensive measurements may be required to obtain accurate model values. For this reason, predictions should be taken only as approximations. Remember that the analysis done here were for simple position-averaging done on the NMEA data. Any "tricks" or re-configuring of the receiver algorithm for firmware position-averaging have not been analyzed.

As the Rayleigh distribution would approximately model the horizontal position-averaging error distribution, the 95% error bound would be predicted to be approximately 1.73 times the horizontal position-averaging RMS error and other error parameters could be similarly predicted by multiplying the position-averaging RMS by the appropriate factor due to the Rayleigh distribution. The normal distribution should be used to model the height errors when position-averaging to improve vertical accuracy. Thus the 95% error bound would be predicted to be approximately 1.96 times the vertical position-averaging RMS error.

Extreme caution is recommended in applying these results here to other GPS receivers at other times and places, especially considering that there is some variation with latitude, GPS satellite constellation status and local reception of signal effects. See the section on correlation of errors for related material.

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