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Dust storms in the earth’s major desert regions significantly influence microphysical weather processes, the CO2-cycle and the global climate in general. Recent increases in the spatio-temporal resolution of remote sensing instruments have created new opportunities to understand these phenomena. However, the scale of the data collected and the inherent stochasticity of the underlying process pose significant challenges, requiring a careful combination of image processing and statistical techniques. Using satellite imagery data, we develop a statistical model of atmospheric transport that relies on a latent Gaussian Markov random field (GMRF) for inference. In doing so, we make a link between the optical flow method of Horn and Schunck and the formulation of the transport process as a latent field in a generalized linear model. We critically extend this framework to satisfy the integrated continuity equation, thereby incorporating a flow field with nonzero divergence, and show that such an approach dramatically improves performance while remaining computationally feasible. Effects such as air compressibility and satellite column projection hence become intrinsic parts of this model. We conclude with a study of the dynamics of dust storms formed over Saharan Africa and show that our methodology is able to accurately and coherently track storm movement, a critical problem in this field.