AVERAGING HORIZONTAL POSITION

WITH WEIGHTING BY HDOP

Rather than simple averaging to improve the accuracy of a position measurement, one might consider weighting each measurement by the dilution of precision (DOP) to even better improve the accuracy. This section explores averaging with weighting using the horizontal dilution of precision (HDOP) to attempt to improve the horizontal (latitude and longitude) accuracy when measuring a position. (If one finds the below too technical to read, I suggest going to the conclusion at the end.)

We start by assuming the distribution of latitude and longitude measurements are normally (Gaussian) distributed and unbiased (with mean being the true value). Although measurements close in time are correlated, as equal weighting would be applied to these in what follows, the results from assuming the measurements are independent generally can be applied.

It is easily derived that the “maximum likelihood estimator” for measurements of the mean from unbiased Gaussian distributions having the same mean but possible different standard deviations s_i is given by:

where:

(This is different from the simple average in which all the l_i are 1/n.) The above formulas would both be applied to separately calculate weighted averages for the latitude and longitude.

If we assume that s_i is proportional to HDOP_i (that is, s_i = HDOP×s₁), a little simple algebra will give:

The predicted ratio of the RMS of horizontal error from this weighted average to the RMS error of horizontal error from simple averaging is then:

When the HDOP distribution obtained from 6 days using a Garmin 12XL were inserted into the above formula, the result obtained was 0.89. In other words, under the assumptions, one should obtain an 11% reduction in RMS error of latitude and longitude from averaging if one uses the above HDOP weighted average rather than the simple average. However, in looking at actual data, the author has failed to see evidence of this predicted small improvement by using the weighting described above.

As the relationship between HDOP and error is only approximate, inaccuracies due to that modeling might be the cause of the failure to observe the expected improvement in accuracy by weighting using HDOP in the above effort. One is then tempted to try using the apparently better model, described elsewhere in this work, relating HDOP and error based on measured Garmin 12XL data:

In this case, the desired weighting of coordinates would be given by:

and the predicted ratio of RMS of horizontal error from this new weighted average to the RMS error of horizontal error from simple averaging is given by:

Using the 6 days of Garmin12XL data again, one obtains a result of 0.96. In other words, under the assumptions, one should obtain a 4% reduction in RMS error of latitude and longitude from averaging if one uses the new HDOP weighted average rather than the simple average. Although this is smaller than the earlier 11%, it is believed to more accurately reflect the possible improvement in horizontal accuracy by weighting the average. It is a very small improvement and probably too small to detect.

In conclusion, evidence suggests that in general, weighting horizontal position measurements using HDOP in averaging to improve accuracy is of minimal value. The exception is probably in the case where a few values are being averaged with very different HDOP, rather that the distribution of HDOP usually seen in continuously recording GPS data. Finally, some people eliminate fixes with large HDOP when simple averaging. A better approach in the same vein for large samples would be to eliminate fixes with any coordinate being an outlier (say perhaps more than 3 standard deviations from the sample mean).

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