###### Abstract  
Asimilation 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; EnKF; multimodality +
Footnote †: journal: Applied  
0000-0001-5742-4000]H.O.O. model is non-Gaussian, the distribution of the variable of interest conditioned on the data will tend toward Gaussianity as observations are assimilated due to the linear assimilation rule. Non-Gaussian initial distributions may be conserved by using a univariate transform into Gaussian marginals while assuming multi-Gaussianity in the transformed space. A univariate back transform is then used to return to the original space. This approach has a long history in traditional statistics, geostatistics, and more recently in ensemble methods for data assimilation, which is referred to as copulas [7], normal score transform [8], and Gaussian anamorphosis [9], respectively. The latter has shown to improve the performance of the EnKF in many applications [10; 11]. There are however some unresolved issues since Gaussian anamorphosis transforms the marginal distributions rather than the full distribution, and the effect on the resulting variables interdependence is uncertain. The Ensemble Randomized Maximum Likelihood Filter (EnRML) [12] and its close relative the Iterative EnKF (IEnKF) [6] are primarily used to handle nonlinearities in the forward and observation models, but they will also retain certain non-Gaussian features in the filtering distribution. These filters require gradient evaluations to execute the update which can be complicated even if the adjoint state method is used. One alternative is to evaluate the gradient using the ensemble itself [13], but this approach introduces an approximation with unclear consequences, particularly in models with multimodal marginals. Multimodality in the prior model can be represented using categorical auxiliary variables to construct Gaussian mixture prior models [14; 15; 16]. In a spatial setting, these models appear as a combination of Gaussian random fields whose parameters depend on the value taken by the categorical variable, but in order to retain spatial dependence, the categorical variable must also have a spatial dependence. This indicator spatial variable can be modeled as a Markov [17] or truncated pluri-Gaussian [18] random field. For both of these models, there are challenges related to temporal data assimilation, although some encouraging examples have been developed [19]. We define and study an alternative prior model, the selection-Gaussian random field [20; 21], which may represent multimodality, skewness, and peakedness. This random field model is conjugate with respect to Gauss-linear forward and observation models, similarly to the Gaussian random field model. The posterior distribution is therefore analytically tractable under these assumptions [22]. For general forward and observation models, ensemble based algorithms along the lines of the EnKF can be designed. Such selection ensemble Kalman algorithms are the focus of this study, and they are evaluated on a couple of examples. In Section 2, we introduce the selection ensemble Kalman model.