## 3 Results
### Model  
Consider a discretized spatio-temporal random field, \(\{r_{t}(\mathbf{x}),\mathbf{x}\in\mathcal{L}_{r}\subset\mathbb{R}^{2}\}\) where \(t\in\mathcal{L}_{t}:\{0,1,\dots,T\}\) and \(r_{t}(\cdot)\in\mathbb{R}\) that represents temperature (\({}^{\circ}\)C). Let a discretized spatial random field, \(\{\lambda(\mathbf{x}),\mathbf{x}\in\mathcal{L}_{r}\subset\mathbb{R}^{2}\}\); with \(\lambda(\cdot)\in\mathbb{R}_{\oplus}\) representing diffusivity (\(\text{m}^{2}\,\text{s}^{-1}\)). Let \(\mathbf{x}\) be the spatial reference on the regular spatial grid \(\mathcal{L}_{r}\) on the domain \(D\), while \(t\) is the temporal reference on the regular temporal grid \(\mathcal{L}_{t}\). The number of spatial grid nodes is \(n=21\times 21\), and they are placed every 10 cm vertically and horizontally. The discretized temperature field at time \(t\) may be represented by the \(n\)-vector \(\mathbf{r}_{t}\) and the diffusivity field by the \(n\)-vector \(\mathbf{\lambda}\). Both are assumed to be unknown. Note that the Kalman models are defined on the joint variable \([\mathbf{r}_{t},\mathbf{\lambda}]\).  
Assume that, given the initial temperature field, the field evolves according to the diffusion equation:  
\[\frac{\partial r_{t}(\mathbf{x})}{\partial t}-\
abla\cdot(\lambda( \mathbf{x})\
abla r_{t}(\mathbf{x})) = q \tag{20}\] \[\
abla r_{t}(\mathbf{x})\cdot\mathbf{n} = 0,\]  
with \(\mathbf{n}\) the outer normal to the domain and \(q\) a source term. The expression in Equation (20) is discretized using finite differences and the forward model is defined as  
\[[\mathbf{r}_{t+1}|\mathbf{r}_{t},\mathbf{\lambda}]= \omega^{*}(\mathbf{r}_{t},\mathbf{\lambda}), \tag{21}\]  
with \(\omega^{*}(\cdot,\cdot)\in\mathbb{R}^{n}\). Convergence and stability of the numerical method are easily ensured for the finite difference scheme that is used. The initial temperature field \(\mathbf{r}_{0}\) is considered unknown in the test cases.  
The forward model is assumed to be perfect in the sense that there is no model error. The forward model in Equation (6) then takes the form,  
\[\omega([\mathbf{r}_{t},\mathbf{\lambda}],0\mathbf{i}_{n})=\begin{bmatrix}\omega^{*}(\mathbf{r }_{t},\mathbf{\lambda})\\ \mathbf{\lambda}\end{bmatrix}. \tag{22}\]  
This forward model is nonlinear due to the product of \(\mathbf{r}_{t}\) and \(\mathbf{\lambda}\) in Equation (20). Consequently, the assumption of Gauss-linearity required for both the traditional Kalman model [1] and the selection Kalman model [22] is violated and necessitates ensemble based algorithms.  
The observations are acquired in a \(m=5\) location pattern on the spatial grid \(\mathcal{L}_{r}\) at each temporal node in \(\mathcal{L}_{t}\), providing the set of observations \(m\)-vectors \(\mathbf{d}_{t}\), \(t\in\mathcal{L}_{t}\). The corresponding likelihood model is defined as  
\[[\mathbf{d}_{t}|\mathbf{r}_{t}]=\psi_{t}(\mathbf{r}_{t},\mathbf{e}_{t}^{d}) = \mathbf{H}\mathbf{r}_{t}+\mathbf{e}_{t}^{d} \tag{23}\] \[f(\mathbf{d}_{t}|\mathbf{r}_{t}) = \varphi_{m}(\mathbf{d}_{t};\mathbf{H}\mathbf{r}_{t},\mathbf{\Sigma}_{\mathbf{d}|r}),\]  
where the observation \((m\times n)\)-matrix \(\mathbf{H}\) is a binary selection matrix, while the centered Gaussian \(m\)-vector \(\mathbf{e}_{t}^{d}\) with the covariance \((m\times m)\)-matrix \(\mathbf{\Sigma}_{\mathbf{d}|r}=\sigma_{\mathbf{d}|r}^{2}\mathbf{I}_{m}\), and \(\sigma_{\mathbf{d}|r}^{2}=0.1\) represents independent observation errors. This likelihood model is in Gauss-linear form.