11.- Repeatability

Repeatability is related to the spread of a measure, also referred to as precision. It refers to how close a position solution is to the mean of all the obtained solutions, in a static location scenario.

The difference between accuracy and precision is shown below:

Accuracy vs Precision Although the two words precision and accuracy can be synonymous in colloquial use, they are deliberately contrasted in the context of the scientific method. Source: Wikipedia.

The most common precision metrics are defined below:

Measure	Formula	Confidence region probability
2D 2DRMS	$2 \sqrt{σ_{E}^{2} + σ_{N}^{2}}$	95 %
2D DRMS	$\sqrt{σ_{E}^{2} + σ_{N}^{2}}$	65 %
2D CEP	$0.62 σ_{N} + 0.56 σ_{E}$ , if $\frac{σ_{N}}{σ_{E}} > 0.3$	50 %
3D 99 % SAS	$1.122 (σ_{E}^{2} + σ_{N}^{2} + σ_{U}^{2})$	99 %
3D 90 % SAS	$0.833 (σ_{E}^{2} + σ_{N}^{2} + σ_{U}^{2})$	90 %
3D MRSE	$\sqrt{σ_{E}^{2} + σ_{N}^{2} + σ_{U}^{2}}$	61 %
3D SEP	$0.51 (σ_{E}^{2} + σ_{N}^{2} + σ_{U}^{2})$	50 %

which are the same expressions as those defined for accuracy, but now the standard deviations are not referred to a reference value but to the mean of the obtained results:

\begin{matrix} (1) & σ_{E}^{(p r e c i s i o n)} = \sqrt{\frac{1}{L - 1} \sum_{l = 1}^{L} {(E [l] - \hat{E})}^{2}}, \end{matrix}

where $\hat{E} = \frac{1}{L} \sum_{l = 1}^{L} E [l]$ is the mean of all the $E$ coordinates of the obtained positioning solutions, $E [l]$ are the East coordinates of the obtained positioning solutions, and $L$ is the number of available position fixes. Similar expressions can be defined for the North and Up coordinates:

\begin{matrix} (2) & σ_{N}^{(p r e c i s i o n)} = \sqrt{\frac{1}{L - 1} \sum_{l = 1}^{L} {(N [l] - \hat{N})}^{2}}, \end{matrix}

where $\hat{N} = \frac{1}{L} \sum_{l = 1}^{L} N [l]$ , and

\begin{matrix} (3) & σ_{U}^{(p r e c i s i o n)} = \sqrt{\frac{1}{L - 1} \sum_{l = 1}^{L} {(U [l] - \hat{U})}^{2}}, \end{matrix}

where $\hat{U} = \frac{1}{L} \sum_{l = 1}^{L} U [l]$ .

Example:

2D scatter plot 2D position scatter plot and the circles containing 50%, 65%, and 95% of position fixes (corresponding to the CEP, DRMS, and 2DRMS precision errors, respectively)¹.

Indicators of RepeatabilityPermalink

It follows a list of possible repeatability indicators for a software-defined GNSS receiver:

Stand-alone receiver’s static positioning precision.
Differential GNSS static positioning precision.
Average convergence times to sub-metric precision.

ReferencesPermalink

C. Fernández-Prades, J. Arribas and P. Closas, Turning a Television into a GNSS Receiver, in Proc. of the 26th International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS+ 2013), Nashville, TN, Sep. 2013, pp. 1492 - 1507. ↩