Testing deformation hypotheses by constraints on a time series of geodetic observations

Abstract

In geodetic deformation analysis observations are used to identify form and size changes of a geodetic network, representing objects on the earth’s surface. The network points are monitored, often continuously, because of suspected deformations. A deformation may affect many points during many epochs. The problem is that the best description of the deformation is, in general, unknown. To find it, different hypothesised deformation models have to be tested systematically for agreement with the observations. The tests have to be capable of stating with a certain probability the size of detectable deformations, and to be datum invariant. A statistical criterion is needed to find the best deformation model. Existing methods do not fulfil these requirements. Here we propose a method that formulates the different hypotheses as sets of constraints on the parameters of a least-squares adjustment model. The constraints can relate to subsets of epochs and to subsets of points, thus combining time series analysis and congruence model analysis. The constraints are formulated as nonstochastic observations in an adjustment model of observation equations. This gives an easy way to test the constraints and to get a quality description. The proposed method aims at providing a good discriminating method to find the best description of a deformation. The method is expected to improve the quality of geodetic deformation analysis. We demonstrate the method with an elaborate example.

Which Are the Effects of Hard and Soft Equality Constraints on the Iterative Outlier Elimination Procedure?

“…In other words, the relaxation of constraints makes it possible to estimate deformation effects that are unmodelled in the deterministic model matrix, for example, deformations which have a spatial and temporal variations. In that case, the deformation can be modelled by covariance functions that determine the cofactors between constraints [28]. This should be applied to control points located in the structure, but not to control points located outside the structure.…”

“…The approach of this example can be applied in the design stage of geodetic network analysis [62,65]. Furthermore, the experiments performed here can also be extended to geodetic deformation analysis when the deformation effects are unmodelled in the sense of deterministic form [28].…”

“…As a consequence of the inequality in (28), the MIB will be larger than MDB, i.e., MIB ≥ MDB. For the special case of having only one single alternative hypothesis, there is no difference between the MDB and MIB [55].…”

“…Soft constraints (or looser constraints) are, however, for which such functional relations can slightly be violated depending on their uncertainty [3,27]. Soft constraints may also be referred to as a pseudo-observation model [28].…”

“…In this paper, we investigate the effects of models subject to constraints (minimum, redundant, hard and soft) on the probability levels associated with IDS. It is important to emphasised that if a standard-deviation of a constraint (or a set of a constraint) is changed from zero to a nonzero value, it is called "relaxation" of the constraint [28]. Here, we also evaluate the effect of relaxing constraints on the MIB and MDB.…”