## 5 Conclusions  
Data assimilation of spatio-temporal variables with multimodal spatial histograms is challenging. Traditional ensemble Kalman algorithms enforce a regression towards the mean due to the linearized conditioning on observations, hence the multimodality is averaged out. We introduce the selection ensemble Kalman algorithms, termed SEnKF and SEnKS. These algorithms are based on recursive expressions similar to the ones justifying the traditional ensemble Kalman algorithms, but they are defined in an augmented space including the selection variable. From the two case studies, we conclude  
\begin{table}
\begin{tabular}{l c c} \hline \hline  & **SEnKS** & **ENKS** \\ \hline \(RMSE_{T=50}\) & 2.92 & 3.72 \\ \hline \hline \end{tabular}
\end{table}
Table 4: RMSE comparing the MMAP prediction of the initial temperature field and the initial temperature field at time \(T=50\).  
Figure 21: EnKS approach: Realizations from the posterior distribution of the initial temperature field \(f(\mathbf{r}_{0}|\mathbf{d}_{0:T})\) at time \(T=50\). Upper panels: the colorbar gives the temperature in \({}^{\circ}\)C. Lower panels: the horizontal axes represent the temperature (\({}^{\circ}\)C), the vertical axes represent the relative prevalence of each temperature value for the realization right above.  
that multimodality is much better represented by the selection ensemble Kalman algorithms than by the traditional ones. We obtain RMSE reductions in the range of 20 to 30%.  
The traditional ensemble Kalman algorithms provide an ensemble representation of the posterior model of interest hence making assessment of the posterior pdf simple. The selection ensemble Kalman algorithms are defined in an augmented space and conditioning on the selection variable must be made a posteriori. For this conditioning to be reliable, the ensemble size needs to be much larger than for the traditional algorithms. Hence, there is a trade-off between improved reproduction of multimodal characteristics of the phenomenon under study and the computational demands. In our case study, the ensemble size needed to be increased by approximately a factor of ten.  
We have not fully explored the possibilities of robust estimation of model parameters in the conditioning of the selection variable. This robustification may reduce the ensemble size requirements. Note that parallelization in forwarding of the ensemble is possible and it will reduce the computer demands.  
Conceptualization, M.C. and H.O.; Formal analysis, M.C. and H.O.; Funding acquisition, H.O.; Investigation, M.C. and H.O.; Methodology, M.C. and H.O.; Project administration, H.O.; Resources, H.O.; Software, M.C.; Supervision, H.O.; Validation, M.C. and H.O.; Visualization, H.O.; Writing--original draft, M.C. and H.O. All authors have read and agreed to the published version of the manuscript.  
This research and the APC are funded by the research initiative: 'Uncertainty in Reservoir Evaluation' at Department of Mathematical Sciences, NTNU, Trondheim, Norway.  
The research is a part of the Uncertainty in Reservoir Evaluation (URE) activity at the Norwegian University of Science and Technology (NTNU), Trondheim, Norway.  
The authors declare no conflict of interest.