Abstract
Data assimilation in models representing spatio-temporal phenomena poses a challenge, particularly if the spatial histogram of the variable appears with multiple modes. The traditional Kalman model is based on a Gaussian initial distribution and Gauss-linear forward and observation models. This model is contained in the class of Gaussian distribution and is therefore analytically tractable. It is however unsuitable for representing multimodality. We define the selection Kalman model that is based on a selection-Gaussian initial distribution and Gauss-linear forward and observation models. The selection-Gaussian distribution can be seen as a generalization of the Gaussian distribution and may represent multimodality, skewness and peakedness. This selection Kalman model is contained in the class of selection-Gaussian distributions and therefore it is analytically tractable. An efficient recursive algorithm for assessing the selection Kalman model is specified. The synthetic case study of spatio-temporal inversion of an initial state, inspired by pollution monitoring, suggests that the use of the selection Kalman model offers significant improvements compared to the traditional Kalman model when reconstructing discontinuous initial states.
Highlights
Data assimilation in models representing spatio-temporal phenomena is challenging
We focus on a particular smoothing challenge, namely to assess the initial state given observations at later times and we denote the task spatio-temporal inversion
The posterior distributions are analytically tractable for both the selection Kalman model and the traditional Kalman model
Summary
Data assimilation in models representing spatio-temporal phenomena is challenging. Most statistical spatio-temporal models are based on assumptions of temporal stationarity, possibly with a parametric, seasonal trend model [1]. We consider spatio-temporal phenomena where the dynamic spatial variables evolve according to a set of differential equations. Such phenomena will, in statistics, normally be modeled as hidden Markov models [2]. In studies of hidden Markov models, it is natural to distinguish between filtering and smoothing [2]. Filtering entails predicting the spatial variable at a given time with observations up to that point in time. Smoothing entails predicting the spatial variable given observations both at previous and later times. We focus on a particular smoothing challenge, namely to assess the initial state given observations at later times and we denote the task spatio-temporal inversion
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