Critical Limitation in Use of $\tau$ Test for Gross Error Detection

In the computation of a geodetic network, it is well known that the least-squares adjustment may not be the ideal solution when gross errors appear. In general there are two fundamental approaches to determining gross errors: the application of the detection tests (Baarda’s data snooping, Pope’s $\tau$ test, etc.) over the least-squares solution, and the use of robust estimation methods. It has been shown that in the case of more than one gross error, these detection tests become inefficient. Moreover, even for the case of single error detection, these tests may be questionable. Hence, data snooping relies on a good a priori precision estimation, and, regarding Pope’s $\tau$ test, it is the purpose of this paper to analyze the limitations in its use with correlated observations, even for a single gross error detection. Conclusions will be drawn by means of a generalization of a result mentioned in Pope’s work (originally restricted to the case of uncorrelated data) and then applied in an example.