## 2 Materials and Methods
\begin{matrix}\bar{\mathbf{r}}_{T+1}\\ \mathbf{v}\end{matrix} \middle|\mathbf{d}_{0:T}\right]\sim f(\bar{\mathbf{r}}_{T+1},\mathbf{v} \middle|\mathbf{d}_{0:T}), \tag{14}\]  
and, based on this ensemble, empirical sampling based inference, see [21], is used to obtain the forecast of interest:  
\[[\mathbf{r}_{T+1}\middle|\mathbf{d}_{0:T}]\sim f(\mathbf{r}_{T+1}\middle|\mathbf{d}_{0:T})=f( \bar{\mathbf{r}}_{T+1}\middle|\mathbf{d}_{0:T},\mathbf{v}\in A). \tag{15}\]  
The SEnKF algorithm is specified in Algorithm A1 in Appendix A. The traditional EnKS algorithm aims at evaluating the interpolation pdf \(f(\mathbf{r}_{0:T}\middle|\mathbf{d}_{0:T})\) with corresponding HM model recursions, see [23]. The algorithm is initiated by  
\[[\mathbf{r}_{0}\middle|\mathbf{d}_{0}]\sim f(\mathbf{r}_{0}\middle|\mathbf{d}_{0})=[f(\mathbf{d}_{0 })]^{-1}f(\mathbf{d}_{0}\middle|\mathbf{r}_{0})f(\mathbf{r}_{0}), \tag{16}\]  
and the recursions for \(t=1,\ldots,T\),  
\[ \begin{split}[\mathbf{r}_{0:t}\middle|\mathbf{d}_{0:t}]& \sim\ f(\mathbf{r}_{0:t}\middle|\mathbf{d}_{0:t})\\ &=\ [f(\mathbf{d}_{t}\middle|\mathbf{d}_{0:t-1})]^{-1}f(\mathbf{d}_{t}\middle| \mathbf{r}_{0:t},\mathbf{d}_{0:t-1})f(\mathbf{r}_{t}\middle|\mathbf{r}_{0:t-1},\mathbf{d}_{0:t-1} )f(\mathbf{r}_{0:t-1}\middle|\mathbf{d}_{0:t-1})\\ &=\ [f(\mathbf{d}_{t}\middle|\mathbf{d}_{0:t-1})]^{-1}f(\mathbf{d}_{t}\middle| \mathbf{r}_{t})f(\mathbf{r}_{t}\middle|\mathbf{r}_{t-1})f(\mathbf{r}_{0:t-1}\middle|\mathbf{d}_{0:t -1}).\end{split}  \tag{17}\]  
The expressions are represented by an ensemble of realizations. Forwarding is made on the ensemble and the conditioning is empirically linearized. Note that the dimension of the model increases very fast, one may therefore only store the interpolation pdf \(f(\mathbf{r}_{s}\middle|\mathbf{d}_{0:T})\) at the time point \(s\) of interest. The SEnKS introduced in this study relies on the relations defined above and uses an extended \((n+n)\)-vector \([\bar{\mathbf{r}},\mathbf{v}]\) as defined in Equation (2). The forward and likelihood models are identical to those defined for the filter. The SEnKS algorithm provides an ensemble representation of  
\[\left[ \begin{matrix}\bar{\mathbf{r}}_{0:T}\\ \mathbf{v}\end{matrix} \middle|\mathbf{d}_{0:T}\right]\sim f(\bar{\mathbf{r}}_{0:T},\mathbf{v} \middle|\mathbf{d}_{0:T}), \tag{18}\]  
and by using empirical sampling based inference, see [21], the interpolation of interest is assessed,  
\[[\mathbf{r}_{0:T}\middle|\mathbf{d}_{0:T}]\sim f(\mathbf{r}_{0:T}\middle|\mathbf{d}_{0:T})=f( \bar{\mathbf{r}}_{0:T}\middle|\mathbf{d}_{0:T},\mathbf{v}\in A). \tag{19}\]  
The SEnKS algorithm is specified in Algorithm A2 in the Appendix A. Both algorithms, SEnKF and SEnKS, contain empirically linearized conditioning and asymptotic results, when the ensemble size goes to infinity, and are consistent only for Gauss-linear forward and likelihood models. Under these assumptions, the model is analytically tractable; however, see [22]. In spite of this lack of asymptotic consistency for general HM models, the ensemble Kalman scheme has proven surprisingly reliable for high-dimensional, weakly nonlinear models even with very modest ensemble sizes [25].