The uncertainty in position can be expressed as the probability that the error will not exceed a certain amount. Under the assumption that position errors follow the normal (or Gaussian) error distribution (for arguments sake, there are only random errors being propagated into the position results), this probability can be related to the magnitude of the standard deviation (HARVEY, 1994). For example, in the case of a linear (one-dimensional) accuracy measure, one standard deviation (one sigma) would correspond to a 68.27% confidence interval. That is, it is assumed that: (a) the mean value of an infinitely large sample of position results is the correct result, and (b) the standard deviation of this sample defines the interval on either side of the mean (or correct) quantity that contains 68.27% of all the results. 31.73% of the results will therefore be outside this range, and if the one sigma quantity is taken as the accuracy measure, then 68.27% of the results will be deemed acceptable and the remainder will be outside the accuracy "specification". The probability of the result being in the interval two standard deviations on either side of the mean is 95.45%. In general, the 95% confidence level is taken as the measure of adequate accuracy, and this corresponds to 1.96 standard deviations (but is generally approximated by two standard deviations, or two sigma). (The probability corresponding to three sigma is 99.73%, which is inclusive of almost all position results.) Vertical uncertainty can be expressed in this one-dimensional form.
This concept can be extended to two dimensions, so that areas can be constructed corresponding to distinct error probabilities such as 50%, 95%, etc. These zones are centred at the correct or true position. In general these zones are elliptical in shape, and they are known as error ellipses (see HARVEY, 1994; and section 9.1.3). However, the error ellipse constructed from the standard deviations of the two dimensional quantities (for example, east and north position components), and the correlations between these two quantities, contains only 39% of the position results and the error ellipse's axes are generally inflated by a factor of approximately 2.45 to create the 95% error ellipse. Surveyors are well acquainted with the concepts of error ellipses. (In three dimensions, the error figure is an error ellipsoid.)
Traditionally navigation users have expressed horizontal position uncertainties in the form of circles and 3-D position uncertainties as spheres. This simplification of the error distribution requires the definition of the radial or distance "root mean square" error, which can be determined for the 2-D case from the horizontal component standard deviations (the 3-D case would involve three standard deviation quantities) E and N:
The probabilities described by 1.DRMS and 2.DRMS are defined as the typical 68.27% and 95.44% values respectively, associated with the 1-dimensional distribution. (2.DRMS refers to TWICE the distance root mean square, irrespective of whether it is the 2-D or 3-D case.) These probabilities are not constant, but are dependent on the geometry of the position solution. For example, if the geometry of the solution is very poor then the 95% error ellipse is very elongated, and the probability associated with an error circle of radius 2.DRMS may not be 95%. Conversely, if the error ellipse is almost circular, then the probabilities of the ellipse and the circle are also almost identical. In the case of GPS it has been found that the probability associated with the circle of radius 2.DRMS ranges from 95% to 98.5%.
This accuracy measure is also closely tied to Dilution Of Precision analysis (section1.4.9) by:
|2.DRMSHor = 2.HDOP.o|
|2.DRMSVer = 2.VDOP.o|
|2.DRMS3-D = 2.PDOP.o|
where o is the standard deviation of the range measurement errors, DRMS<Hor refers to the horizontal components, DRMSVer refers to the vertical component, and DRMS3-D refers to the 3-D position. HDOP, VDOP and PDOP refer to Horizontal DOP, Vertical DOP and Position DOP respectively.
An alternate measure is the Circle Error Probable (CEP), sometimes also referred to as the "Circle Error Probability" or "Circle of Equivalent Probability". This defines the radius of a circle inside which there is a 50% probability of the position being located. How is CEP related to DRMSHor ? There is no exact relation, but an approximate relation defined for GPS is:
|2.DRMSHor 2.45 . CEP|
Other CEP's for different probability radii can be obtained by multiplying the above 50% CEP by the appropriate factors. The Figure below illustrates the accuracy circles for 2-D positioning. In the 3-D case the equivalent accuracy measure is the Spherical Error Probable (SEP), and it is 60m and 16m for the SPS and PPS respectively.
If the accuracy performance of a positioning system is not known apriori, then the radius of the 95% confidence interval zone, as well as for other probabilities, has to be determined empirically from a large sample of positioning results (see Figure below). On the other hand, if the positioning system controllers wish to guarantee a certain level of accuracy performance, then the quantity must be specified apriori and all effort applied to reduce the level of total error in the system so that 95% of position results are within the specified zone. Hence, the SPS service is designed to support 2-D (horizontal) positioning so that at least 95% of position results are within a circle of radius 100m centred at the "true" position (which can be determined from the mean of a very large sample of results). In this case o, referred to as the User Equivalent Range Error (UERE), is maintained, on average, to be 32m, while the average HDOP is 1.6.
Rings of accuracy used to describe GPS horizontal point positioning accuracy. (CLARKE, 1994)
Further details of the SPS performance can be found in NAVSTAR (1993).
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© Chris Rizos, SNAP-UNSW, 1999