Reliability in constrained Gauss-Markov models: An analytical and differential approach with applications in photogrammetry

Reliability analysis explains the contribution of each observation in an estimation model to the overall redundancy of the model, taking into account the geometry of the network as well as the precision of the observations themselves. It is principally used to design networks resistant to outliers in the observations by making the outliers more detectible using standard statistical tests. It has been studied extensively, and principally, in (linearized) Gauss-Markov models. We show how the same analysis may be extended to various rank-deficient and constrained (linearized) Gauss-Markov models and present preliminary work for its use in unconstrained Gauss-Helmert models. In particular, we analyze the prominent "reliability matrix" of the constrained model to separate the contribution of the constraints to the redundancy of the observations from the observations themselves.; In addition, we make extensive use of matrix differential calculus to find the Jacobian of the reliability matrix with respect to the parameters that define the network through both the original design and constraint matrices. The resulting Jacobian matrix reveals the sensitivity of elements of the observation reliability matrix (and the redundancy numbers along its diagonal) to particular design parameters and allows the model to identify weak areas in the network where changes in observations may result in unreliable observations.; We apply the analytical framework to photogrammetric networks in which the exterior orientation parameters of images comprising a block are directly observed by calibrated GPS/INS systems. Such directly oriented blocks offer the potential of significantly reduced ground control survey cost but suffer from lack of redundancy. Tie-point observations provide some redundancy (for relative orientation only and even a few collinear tie-point and tie-point distance constraints improve the reliability of these direct observations by as much as 33%. Using the same theory we compare networks in which tie-points are observed on multiple photos (n-fold points and tie-points are observed in photo pairs only (two-fold points. Apparently, the use of n-fold tie-points does not significantly degrade the reliability of the direct exterior observation observations. Coplanarity constraints added to the common two-fold points do not add significantly to the reliability of the direct exterior orientation observations.; The differential calculus results may be used to provide a new measure of redundancy number stability in networks. We show that a typical photogrammetric network with n-fold tie-points was less stable with respect to at least some tie-point movement than an equivalent network with n-fold tie-points decomposed into many two-fold tie-points.