In above demostrations, we use a Markovian kernel to establish smooth approximation to physical fields associated with sea ice models. The Markovian kernel is define as \( p_{\epsilon}: \Omega \times\Omega \rightarrow \mathbb R \), $$ p_{\epsilon}(x,y) = \frac{K_{\epsilon}(x,y)}{\int_{\Omega} K(x,s) \, ds } $$ where \( K_\epsilon(x,y) = e^{-\lVert x-y\rVert^2 / \epsilon^2} \). The integration over \( \Omega \) has been implemented using a quadrature scheme that is defined according to the configuration of ice fleos, $$ \int_{\Omega} f(y) \, dy \approx \sum_{j \in \mathcal J} W_j f(y_j) $$ As a result, given a physical field of interest such as mass density, velocity, or stress we obtain the corresponding smooth representation: $$ \tilde{m}(x) \approx \sum_{j \in \mathcal J} p_{\epsilon}(x,y_j) W_j m(y_j) $$ $$ \tilde{v_i}(x) \approx \sum_{j \in \mathcal J} p_{\epsilon}(x,y_j) W_j v_i(y_j) $$ $$ \tilde{\sigma}_{ab}(x) \approx \sum_{j \in \mathcal J} p_{\epsilon}(x,y_j) W_j \sigma_{ab}(y_j) $$