Illustrative presentation of some basic concepts of network internal reliability with comments as regards engineering surveys <s> Level 3. <s> An approach to internal reliability analysis is presented which, compared to existing approaches, offers further insight into the system responses to observation gross errors. The proposed reliability measures are defined on the diagonal and non-diagonal elements of the modified reliability matrix being an oblique projector. It maps the observations into the least squares residuals, both expressed as multiples of the a priori standard deviations of the observations. The system responses to a single gross error, such as the local, the quasi-global or the global response, are discussed and their consistency with non-correlation case is proved. Also, the reliability criteria interpretable in terms of those responses are proposed. They enable one to evaluate the observation systems with correlated and uncorrelated observations with respect to internal reliability. An attempt is also made to relate the global responses to reliability numbers that are the basis for determining the magnitude of the minimal detectable bias. Some differences of the new reliability measures in relation to existing measures are indicated. A numerical example is provided, showing the reliability analysis for a simulated horizontal network (distances, angles and GPS vectors), with given covariance matrix for GPS-vector components. In addition, a reliability analysis is presented for a simple leveling network being a study case for other authors referred to in this paper. <s> BIB001 </s> Illustrative presentation of some basic concepts of network internal reliability with comments as regards engineering surveys <s> Level 3. <s> The concept of minimal detectable bias (MDB) as initiated by Baarda (Publ Geod New Ser 2(5), 1968) and later developed by Wang and Chen (Acta Geodaet et Cartograph Sin Engl Edn 42–51, 1994), Schaffrin (J Eng Surv 123:126–137, 1997), Teunissen (IEEE Aerosp Electron Syst Mag 5(7):35–41, 1990, J Geod 72:236–244 1998, Testing theory: an introduction. Delft University Press, Delft, 2000) and others, refers to the issue of outlier detectability. A supplementation of the concept is proposed for the case of correlated observations contaminated with a single gross error. The supplementation consists mainly of an outlier identifiability index assigned to each individual observation in a network and a mis-identifiability index being the maximum probability of identifying a wrong observation. To those indices there can also be added the MDB multiplying factor to increase the identifiability index to a satisfactory level. As auxiliary measures there are indices of partial identifiability concerning pairs of observations. The indices were derived assuming the generalized outlier identification procedure as in Knight et al. (J Geod. doi: 10.1007/s00190-010-0392-4, 2010), which with one outlier case being assumed is similar to Baarda’s w-test (Baarda in Publ Geod New Ser 2(5), 1968). The following two options of identifiability indices and partial identifiability indices are distinguished: I. the indices related to identification of a contaminated observation within a set of observations suspected of containing a gross error (identifiability), II. the indices related to identification of a contaminated observation within a whole set of observations (pseudo-identifiability). To characterize the proposed approach in the context of the existing solutions of similar topic being the separability testing, the properties of both types of identifiability indices are discussed with reference to the concept of Minimal Separable Bias (Wang and Knight in J Glob Position Syst 11(1):46–57, 2012) and a general approach in Yang et al. (J Geod 87(6):591–604, 2013). Numerical examples are provided to verify the proposed approach. <s> BIB002
A basic factor that decides about the existence of identifiable and the unidentifiable gross errors in an individual model is whether the model contains the Region(s) of Unidentifiable Errors (RUE). The RUE can be defined descriptively as follows: -a subset of network observations ), such when contaminated with any combination of gross errors that do not form a vector belonging to the space U, always yields: Unidentifiable gross error -a gross error residing in any of the observations which form RUE. Such an error is unidentifiable within a particular RUE. The RUE takes place specially in leveling networks of low redundancy, and in any type of network with 1 redundant observation where it covers the whole network. The conditions for non-existence of RUE in a network are as follows: where i h and i * h , i k are indices of response-based internal reliability for the i-th observation BIB001 . Therefore, we can see that in high-redundancy networks, being the networks of high internal reliability, the detectable gross errors are always identifiable. To emphasize that the RUE regions that allow the occurrence of unidentifiable gross errors and should thus be avoided in a network, the corresponding box in a diagram has been deliberately shaded. Level 4. The degree of identifiability can be characterized by the i ID index BIB002 . As a supporting quantity one can use the maximum value of misidentifiability index, i.e. 