Statistical evaluation of the influence of the uncertainty budget on B-spline curve approximation

  • Xin Zhao, Hamza Alkhatib, Boris Kargoll, Ingo Neumann
  • Journal of Applied Geodesy, January 2017, De Gruyter
  • DOI: 10.1515/jag-2017-0018

Testing the adequacy of a simple vs. a refined uncertainty budget for the variances of measurements

What is it about?

Oftentimes, terrerstrial laser scanning (TLS) measurements are simply assumed to have uniform/identical standard uncertainties or variances, which is known to be unrealistic. To obtain a possibly more adequate description of the uncertainties, the Guide to the Expression of Uncertainty in Measurement (GUM) is used in this paper. It is shown that this approach leads to different standard uncertainties of the individual TLS measurements, by taking into account instrumental, object-related and atmospheric influences. The main research question in this contribution is, how it can be tested statistically whether the use of a certain refined uncertainty budget (based on measurement-specific standard uncertainties) leads to a better observation model than the use of the simple uncertainty budget (based on uniform standard uncertainties). This problem is addressed within the real measurement situation of a surveyed footbridge, which is modeled by means of B-splines. On the one hand, a known statistical test (based on the idea of a score test by the statistician C. R. Rao) is applied under the null hypothesis that the variances of all the TLS measurements are identical. As it might be more adequate to take the assumption of different variances as the null hypothesis, a second statistical test (employing the idea of a generalized likelihood ratio by the statistician D. R. Cox) is derived and applied, too. Whereas the first test rejects the assumption of identical variances in favor of the refined, GUM-based uncertainty budget, the second test suggests in addition that the refined budget is also inadequate. Thus, the suggested test could generally be used in the process of further improving the uncertainty budget of given measurement results.

Why is it important?

The estimation of geodetic model parameters from given measurement results is usually accompanied by a description of the uncertainties of the estimates (e.g., in terms of their standard deviations or variances). If the assumed uncertainty budget of the measurements is unrealistic, then its propagation to the uncertainties of the estimates yields also unrealistic values. Further consequences could then also be, for instance, flawed confidence intervals and wrong decisions resulting from hypothesis tests, which should be avoided. To arrive at a well-founded decision as to whether a given uncertainty budget is adequate or not, a rigororous statistical hypothesis test should be carried out. The current state-of-the-art test emphasizes a null hypothesis in terms of a very simple uncertainty budget that can be expected to be unrealistic in a real measurement situation. Therefore, this study addresses the important question how a given simple (probably unrealistic) and a competing refined (supposedly more realistic) uncertainty budget can be tested against each other without preferring the simple budget at the outset.

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The following have contributed to this page: Prof. Dr.-Ing. Boris Kargoll, xin zhao, and Dr Hamza Alkhatib