AbstractsMathematics

Dust storms emerging in the Earth's major desert regions significantly influence weather processes, the CO2-cycle and the climate on a global scale. Their effects on organisms range from providing nutrition to vegetation and microbes to direct impact on human settlements, transportation and health. The detection of dust storms, the prediction of their development, and the estimation of sources are therefore of immediate interest to a wide range of scientific disciplines. Recent spatio-temporal resolution increases of remote sensing instruments have created new opportunities to understand these phenomena. The scale of the data and their inherent stochasticity, however, pose significant challenges. This thesis develops a combination of methods from statistics, image processing, and physics that paves the way for efficient probabilistic dust assessment using satellite imagery. As a first step, we propose a BHM that maps SEVIRI measurements to a predictor of the dust density. Case studies demonstrate that, as compared to linear methods, our LSM approach mitigates effects of signal intrinsic noise on further processing steps. Furthermore, an extensive cross-validation study is employed to show that LSM successfully adapts to intra-daily changes of the infrared data and yields outstanding dust detection accuracy. Physically, the dust density and its transport process are tied together by the continuity equation. A traditional approach to determine the flow field for a given density is the variational method of Horn and Schunck (HS), which simplifies the equation to compression free motion. We characterize the equation's solution as a GMRF and introduce compressible dynamics. This link between probabilistic and variational perspectives leads to applied and theoretical advances. It enables us to employ the INLA technique for computationally efficient inference and integration over hyper-parameters. The importance of allowing for compressible motion and treating the problem in a statistical manner is emphasized by simulation and case studies showing a significant reduction in errors of the estimated flow field. In addition, we demonstrate how our methodology provides uncertainty quantification, dust storm forecasts and estimation of emission sources. The thesis is concluded by examining the analytical properties of our approach. It is shown that, under mild restrictions on an underlying Sobolev space, existence and uniqueness of the compressible flow can be guaranteed on a continuous domain and a well-posed discretization exists. Lastly, our variational calculations point to an interpretation of the density as a solution to flow-parameterized SPDE naturally extending Matern fields to non-isotropy, which provides a further step towards a joint model of dust density and flow field.