RETURNING TO A POSITION

It is often stated that when attempting to return to a previously measured location that the error is multiplied by the square root of 2, or about 1.41. The information presented here is more general - it includes the case where the initial measurement by GPS and returning measurement by GPS have different error specifications due to differences in measuring equipment or procedure. GPS data used below is non-differential SPS (Standard Positioning Service) with SA (Selective Availability) turned off.

It follows from assuming that the GPS measured positions have approximately a bivariate normal distribution that the error distribution in returning to a position is approximately given by the following Rayleigh distribution:

where:

where RMS_Error₁ is the RMS (Root-Means-Squared) error of the initial measurement and RMS_Error₂ is the RMS error of the measurement during the attempted return to the same position. In words, the RMS error of the return is the "RSS" (Root-Sum-of-Squares) of the individual RMS errors. If one (usually the initial fix) is averaged, then the RMS error for that measurement should be the RMS error for the averaging period.

When both measurements are made with the same measuring procedure with the same errors, RMS_Error₁ = RMS_Error₂. Letting RMS_Error denote the common RMS error in this case, the above equations and some simple algebra give RMS_Returning_Error = 1.41 x RMS_Error, which is the commonly quoted error in returning to a position.

The following plot shows that this relationship works well when compared with actual measurements:

As a second example, we consider returning to a position that was initially measured by simple averaging for 15 minutes or 1 hour. As would seem most likely, in the return attempt to find the position, we assume single measurements (no averaging) as being used. The plot below shows the measured error distribution. For comparison, the measured error distribution where no averaging is used for either measurement is also shown for the same set of data.

The table below summarizes the measured errors of for position averaging on the first visit but no averaging on the return:

Measured Error	No averaging on initial visit	15 min. averaging on initial visit	1 hr. averaging on initial visit
RMS	7.7 m	6.5 m	6.0 m
50% (CEP)	5.7 m	4.9 m	4.5 m
95%	13.7 m	11.3 m	10.3 m

In conclusion, in returning to a location using a measurement of the same accuracy, the error specifications are multiplied by 1.41 (the square root of 2). Otherwise, the RMS error in returning is the RSS of the two separate RMS errors. The above example shows that only a small advantage is achieved by averaging the initial visit for a period up to an hour; however, it is wise to carefully evaluate single measurements for their validity.

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