## 2 Problem Formulation  
The CEM method is formulated as a quadratic programming problem [49; 50] as follows:  
\[ \begin{split}\min&\|A(t)\mathbf{u}(t)+\mathbf{s}(t )\|_{2}^{2}/2,\\ \text{s.t.}& B(t)\mathbf{u}(t)=\mathbf{z}(t),\end{split}  \tag{1}\]  
where \(\|\cdot\|_{2}\) denotes the Euclidean norm; \(A(t)\in\mathbb{R}^{j\times m}\) and \(B(t)\in\mathbb{R}^{n\times m}\) are the full-rank matrices; \(\mathbf{s}(t)\in\mathbb{R}^{j}\) and \(\mathbf{z}(t)\in\mathbb{R}^{n}\) are the coefficient vectors; and \(\mathbf{u}(t)\in\mathbb{R}^{m}\) is the unknown vector to be solved. To further illustrate the CEM method (1), let \(f(\mathbf{u})=\|A(t)\mathbf{u}(t)+\mathbf{s}(t)\|_{2}^{2}\), \(C(t)=A^{\text{T}}(t)A(t)\in\mathbb{R}^{m\times m}\), and \(\mathbf{p}(t)=-A^{\text{T}}(t)\mathbf{s}(t)\in\mathbb{R}^{m}\), with superscript \({}^{\text{T}}\) being the transposition operation. The Lagrange formula is as follows:  
\[L(\mathbf{u}(t),\mathbf{r}(t))=f(\mathbf{u}(t),t)+\mathbf{r}(t)^{\text{T}}(B( t)\mathbf{u}(t)-\mathbf{z}(t)), \tag{2}\]  
where \(\mathbf{r}(t)\) equals to \([r_{1}(t),\,\cdots,\,r_{n}(t)]^{\text{T}}\!\!\in\!\mathbb{R}^{n}\), meaning the Lagrange-multiplier operator vector. The optimal solution of the CEM method (1) satisfies the following equation:  
Figure 1: Technical route of Arctic sea ice identification. \[\left\{ \begin{array}{l}\frac{\partial L(\mathbf{u}(t),x(t),t)}{\partial\mathbf{u}( t)}=\frac{\partial f(\mathbf{u}(t),t)}{\partial\mathbf{u}(t)}+B^{\mathsf{T}}(t) \mathbf{r}(t)=0,\\ \frac{\partial L(\mathbf{u}(t),x(t),t)}{\partial\mathbf{r}(t)}=B(t)\mathbf{u}( t)-\mathbf{z}(t)=0.\end{array} \right. \tag{3}\]  
Subsequently, for simplicity, let  
\[N(t)=\left[ \begin{array}{cc}C(t)&B^{\mathsf{T}}(t)\\ B(t)&\mathbf{0}_{n\times n}\end{array} \right]\in\mathbb{R}^{(m+n)\times(m+n)},\] \[\mathbf{x}(t)=\left[ \begin{array}{c}\mathbf{u}(t)\\ \mathbf{r}(t)\end{array} \right]\in\mathbb{R}^{m+n},\] \[\mathbf{g}(t)=\left[ \begin{array}{c}\mathbf{p}(t)\\ \mathbf{z}(t)\end{array} \right]\in\mathbb{R}^{m+n}.\]  
Therefore, (3) is summarized as  
\[N(t)\mathbf{x}(t)=\mathbf{g}(t). \tag{4}\]  
For measuring the accuracy of the computing solution, an error function can be defined as follows:  
\[\varepsilon(t)=N(t)\mathbf{x}(t)-\mathbf{g}(t). \tag{5}\]  
Then, an ND formula [46] with an error integration term making the error function approach zero is designed as  
\[\dot{\varepsilon}(t)=-q\varepsilon(t)-\varpi\int_{0}^{t}\varepsilon( \varepsilon)\mathrm{d}\varepsilon, \tag{6}\]  
where \(\varphi>0\) and \(\varpi>0\). Then, associating (5) and (6) yields  
\[N(t)\dot{\mathbf{x}}(t)= \dot{\mathbf{g}}(t)-\dot{N}(t)\mathbf{x}(t)-\varphi\Big{(}N(t) \mathbf{x}(t)-\mathbf{g}(t)\Big{)}- \tag{7}\] \[\varpi\int_{0}^{t}\Big{(}N(\varrho)\mathbf{x}(\varrho)-\mathbf{ g}(\varrho)\Big{)}\mathrm{d}\varrho,\]  
where \(\dot{N}(t)\), \(\dot{\mathbf{x}}(t)\), and \(\dot{\mathbf{g}}(t)\) are the derivatives of \(N(t)\), \(\mathbf{x}(t)\), and \(\mathbf{g}(t)\), respectively. Then, it can be rearranged as  
\[\dot{\mathbf{x}}(t)= N^{-1}(t)\Big{(}\dot{\mathbf{g}}(t)-\dot{N}(t)\mathbf{x}(t)- \varphi\Big{(}N(t)\mathbf{x}(t)-\mathbf{g}(t)\Big{)}- \tag{8}\] \[\varpi\int_{0}^{t}\Big{(}N(\varrho)\mathbf{x}(\varrho)-\mathbf{ g}(\varrho)\Big{)}\mathrm{d}\varrho\Big{)}.\]  
Next, in order to approximate the first-order derivatives with high computational accuracy, a three-step forward difference formula [51] employed to discrete model (8) is as follows:  
\[\dot{\mathbf{x}}_{k}=\frac{2\mathbf{x}_{k+1}-3\mathbf{x}_{k}+2\mathbf{x}_{k-1 }-\mathbf{x}_{k-2}}{2\sigma}, \tag{9}\]  
where \(k\) denotes the updating index at time instant \(t=k\sigma\) and \(\sigma\) is the sampling period.