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.

Most common precision metrics are defined below:

Measure Formula Confidence region probability
2D 2DRMS \(2\sqrt{\sigma_E^2+\sigma_N^2}\) 95 %
2D DRMS \(\sqrt{\sigma_E^2+\sigma_N^2}\) 65 %
2D CEP \(0.62\sigma_N+0.56\sigma_E\), if \(\frac{\sigma_N}{\sigma_E}>0.3\) 50 %
3D 99 % SAS \(1.122 \left(\sigma_E^2+\sigma_N^2+\sigma_U^2\right)\) 99 %
3D 90 % SAS \(0.833 \left(\sigma_E^2+\sigma_N^2+\sigma_U^2\right)\) 90 %
3D MRSE \(\sqrt{\sigma_E^2+\sigma_N^2+\sigma_U^2}\) 61 %
3D SEP \(0.51 \left(\sigma_E^2+\sigma_N^2+\sigma_U^2\right)\) 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:

\[\begin{equation} \sigma_{E}^{(precision)} = \sqrt{\frac{1}{L-1}\sum_{l=1}^L \left(E[l]- \hat{E}\right)^2} \end{equation}\]

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:

\[\begin{equation} \sigma_{N}^{(precision)} = \sqrt{\frac{1}{L-1}\sum_{l=1}^L \left(N[l]- \hat{N}\right)^2}, \end{equation}\]

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

\[\begin{equation} \sigma_{U}^{(precision)} = \sqrt{\frac{1}{L-1}\sum_{l=1}^L \left(U[l]- \hat{U}\right)^2}, \end{equation}\]

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 Repeatability

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.

References

  1. 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. 

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