Spatio-temporal Inversion using the Selection Kalman Model

The challenge in data assimilation for models representing spatio-temporal phenomena is made harder when the spatial histogram of the variable of interest appears with multiple modes. Pollution source identification constitutes one example where the pollution release represents an extreme event in a fairly homogeneous background. Consequently, our prior belief is that the spatial histogram is bimodal. The traditional Kalman model is based on a Gaussian initial distribution and Gauss-linear dynamic and observation models. This model is contained in the class of Gaussian distribution and is therefore analytically tractable. These properties that make its strenght also render it unsuitable for representing multimodality. To address the issue, we define the selection Kalman model. It is based on a selection-Gaussian initial distribution and Gauss-linear dynamic and observation models. The selection-Gaussian distribution may represent multimodality, skewness and peakedness. It can be seen as a generalization of the Gaussian distribution. The proposed selection Kalman model is contained in the class of selection-Gaussian distributions and therefore analytically tractable. The recursive algorithm used for assessing the selection Kalman model is specified. We present a synthetic case study of spatio-temporal inversion of an initial state containing an extreme event. The study is inspired by pollution monitoring. The results suggest that the use of the selection Kalman model offers significant improvements compared to the traditional Kalman model when reconstructing discontinuous initial states.