CORRELATION OF ERRORS

This section shows two interesting results concerning the relationship between the errors in measuring position and the time differences (lag times) between the measurements. The first result concerns how errors are related in the long-term; the second concerns relationships in the short-term.

A common question asked is whether there is a diurnal variation of GPS errors in measuring position. By this, we mean whether there is a tendency for errors at certain times of day to follow some pattern. The plot below shows the horizontal position RMS errors obtained from 18 24-hour periods. The error of each plot was truncated at 15 meters to prevent the plots from overlapping. The plots in red were taken on 6 consecutive days in June while those in blue were taken on 12 consecutive days in July. The lower plots were taken on earlier days than those higher in the figure.

One does see some tendency for errors at certain times of day to be greater. The cause for this might be a tendency for HDOP to be larger at those times, certain satellites to perhaps yield larger error results or more likely multipath (reflected signals from objects in the antenna’s vicinity). No day versus night pattern is apparent.

On the other hand, in the short term, GPS errors are strongly correlated. The section on averaging to improve the accuracy of GPS horizontal position measurements presented an equation (Model 2) with two error components of different time constants (correlation lengths) to model the RMS error when position-averaging. The question arises as to whether these time constants and error components are just curve-fit artifacts or do they represent some underlying correlation of errors effect. The natural way to attempt to answer this is to plot the autocorrelation of the horizontal errors. For a given lag time, an autocorrelation of 1 would indicate the error after the lag time is simply a multiple of the original error. An autocorrelation of 0 would indicate that there is no (linear) relationship between the errors, or that the errors are not correlated. Generally, for “small” lag times, we might expect an autocorrelation between 0 and 1 indicating some degree of correlation.

With a Garmin 12XL and a Garmin III+, the shorter time constant was of the order of 1 minute, while the other was much longer--as large as roughly 1 or 2 hours depending on the data session. Both error components contributed significantly to the position-averaging RMS error as indicated by the magnitudes of E₁ and E₂.

The plot below show the measured autocorrelation for 70 minutes of horizontal errors obtained for the Garmin 12XL twenty-day test.

The plot below magnifies the first 10 minutes of the above figure.

The autocorrelation of errors for the Garmin 12XL makes a dramatic change (bend) in its general direction around 1 minute. This is roughly the same as the smaller correlation length L₁ in the modeling of position-averaging horizontal error. This equation to model RMS error when averaging will be called "Model 2". At present, it has not been possible to this error component with known error sources such as receiver hardware or algorithm, multipath, satellite geometry or GPS satellite constellation status.

The figure below shows that the Eagle Explorer autocorrelation of errors behaves differently.

The Eagle Explorer autocorrelation decreases with slight upward concavity until long-term errors dominate causing it to go nearly flat near zero or negative due to random error in the measurement. This would seem to agree with the very small correlation length L₁ and small error coefficient E₁ in the modeling of the position-averaging horizontal error of this receiver having only a very small effect; thus the other component (with much longer correlation length L₂ and much larger magnitude E₂) appears to dominate the autocorrelation throughout and no "bend" in the autocorrelation curve is perceived.

Finally, the figure below shows the autocorrelation of vertical errors for the Garmin eMap and Eagle Explorer.

Note that the "bend" in the Garmin eMap autocorrelation plot around 5 minutes roughly corresponds to the value of L₁ of about 6.6 minutes in the vertical averaging portion of the section on position-averaging. The "bend" in the Eagle Explorer autocorrelation plot corresponding to the value of L₁ of about 11.1 minutes for it in that section is harder to see.

In conclusion, the modeling in the position-averaging section and the approximate time constants used in them appear to be confirmed by autocorrelation of the errors.

( Return to: http://www.erols.com/dlwilson/gps.htm )