Modeling of Inhomogeneous Spatio-Temporal Signals by Least Squares Collocation

Through inverse modeling and adjustment techniques, the geodesists try to derive mathematical models from their measurements to get a better understanding of various processes in the system Earth. Sophisticated deterministic and stochastic models are developed to achieve the best possible reflection of reality and the remaining uncertainty. The main focus of this article is on the further development of stochastic model representations, with the capability to switch from the usual assumption of homogeneous (time-stationary) to inhomogeneous (time-variable) stochastic models. To accomplish this we build up and extend a methodical framework to connect the filter and the covariance approach represented by time-variable autoregressive processes (AR) and time-variable (inhomogeneous) covariance models for least squares collocation. We apply these time-variable covariance models to describe the temporal component of a spatio-temporal point stack of surface displacements derived from a DInSAR-SBAS analysis of the ERS1 and ERS2 missions from the Lower-Rhine Embayment in North Rhine-Westphalia. The construction of a time-variable spatio-temporal covariance model allows to use the least squares collocation approach to predict the vertical movements at any location and at any time. Furthermore, a report on the uncertainty of the prediction is provided.