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.

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 dynamic 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 dynamic 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, containing an extreme event suggests that the use of the selection Kalman model offers significant improvements compared to the traditional Kalman model when reconstructing discontinuous initial states.

Assimilation of spatio-temporal data poses a challenge when allowing non-Gaussian features in the prior distribution. It becomes even more complex with nonlinear forward and likelihood models. The ensemble Kalman model and its many variants have proven resilient when handling nonlinearity. However, owing to the linearized updates, conserving the non-Gaussian features in the posterior distribution remains an issue. When the prior model is chosen in the class of selection-Gaussian distributions, the selection Ensemble Kalman model provides an approach that conserves non-Gaussianity in the posterior distribution. The synthetic case study features the prediction of a parameter field and the inversion of an initial state for the diffusion equation. By using the selection Kalman model, it is possible to represent multimodality in the posterior model while offering a 20 to 30% reduction in root mean square error relative to the traditional ensemble Kalman model.

Data assimilation in reservoir modeling often involves model variables that are multimodal, such as porosity and permeability. Well established data assimilation methods such as ensemble Kalman filter and ensemble smoother approaches, are based on Gaussian assumptions that are not applicable to multimodal random variables. The selection ensemble smoother is introduced as an alternative to traditional ensemble methods. In the proposed method, the prior distribution of the model variables, for example the porosity field, is a selection-Gaussian distribution, which allows modeling of the multimodal behavior of the posterior ensemble. The proposed approach is applied for validation on a two-dimensional synthetic channelized reservoir. In the application, an unknown reservoir model of porosity and permeability is estimated from the measured data. Seismic and production data are assumed to be repeatedly measured in time and the reservoir model is updated every time new data are assimilated. The example shows that the selection ensemble Kalman model improves the characterisation of the bimodality of the model parameters compared to the results of the ensemble smoother.

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