Nonstationary stochastic processes in least squares collocation --- NonStopLSC

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Through inverse modeling and adjustment techniques, the geodesists try to derive mathematical models from their measurements to get a better understanding of the processes in the system Earth. Sophisticated deterministic and stochastic models are developed to achieve the best possible reflection of reality and the remaining uncertainty. While deterministic modeling has been improved by much effort, there are still serious weaknesses in the applied stochastic models and representations. Especially in the collocation approach a remove-restore technique is often used to, on the one hand side, guarantee stationarity and, on the other side, get better access to the different frequency contents, which is often hidden in the empirical covariance sequence. Within this project, the representation of stochastic signals with autoregressive processes is proposed which are able to describe the complete frequency spectrum. It shall be highlighted that this is not only the case for stationary processes, but also for time-variable signals. But in theory, this process representation is restricted to equispaced infinite measurement series. Therefore, both representations - the covariance and the process representation - have their pros and cons. A framework for the fusion of the pros is proposed here. The main focus of this proposal is a further development of stochastic model representations, which can reflect the full signal content and have the capability to switch from the usual assumption of time-stationary to time-variable stochastic models. We want to build up and extent a methodical framework to connect the filter and the covariance approach represented by autoregressive processes and least squares collocation. We do this in a strictly formalized way using the 'Magic Square' mechanism which opens the possibility to switch between these two approaches. The proposed extension from the time-discrete processes to stochastic ordinary differential equations opens the way to derive continuous covariance functions from discrete covariance sequences. As a result a family of covariance functions will be established, which will be able to describe the entire signal content as well as the time-variability of stochastic processes.To study this methodical framework with real applications we will refine the analysis of real measurement series from geodetic data sets (from dedicated geodetic satellite missions GOCE, GRACE, GRACE/GRACE-FO and GRAV-D airborne gravity data) especially with respect to their time-variable stochastic signal characteristics.