## 3 Uncertainty Evaluation
To determine the elements in vector \(\mathbf{a}\), the general linear least-squares method [20] is used by defining a merit function, as follows:  
\[\chi^{2}=\sum_{l=1}^{MN}\left[\frac{\eta_{l}-\sum_{k=0}^{K-1}a_{k}\Xi_{k}(\xi_{l })}{\sigma(\eta_{l})}\right]^{2}, \tag{22}\]  
where \(\eta_{l}\) corresponds to the value \(E[e_{Z}(x_{i},y_{j})]\) given in Equation (9) and \(\sigma(\eta_{l})\) is corresponding standard deviation \(\sigma[e_{Z}(x_{i},y_{j})]\), as obtained by Equation (11). In addition, an \(MN\times K\) matrix \(D\), which is also called the design matrix of the fitting problem, is defined with elements given as follows:  
\[D_{lk}=\frac{\Xi_{k}(\xi_{l})}{\sigma(\eta_{l})}. \tag{23}\]  
Also, define a vector \(\mathbf{b}\) of length \(MN\) with elements, as follows:  
\[b_{l}=\frac{\eta_{l}}{\sigma(\eta_{l})}. \tag{24}\]  
The minimum of Equation (22) occurs where the derivative of \(\chi^{2}\) with respect to all \(K\) parameters \(a_{k}\) vanish. This condition yields the following matrix equation:  
\[(\mathbf{D}^{\mathrm{T}}\mathbf{D})\mathbf{a}=\mathbf{D}^{\mathrm{T}}\mathbf{b}. \tag{25}\]  
According to Equation (25), the least-squares estimation of vector a can be obtained, as follows:  
\[\mathbf{\hat{a}}=\left(\mathbf{D}^{\mathrm{T}}\mathbf{D}\right)^{-1}\mathbf{D}^{ \mathrm{T}}\mathbf{b}=\mathbf{D}^{+}\mathbf{b}, \tag{26}\]  
where \(\mathbf{D}^{+}=(\mathbf{D}^{\mathrm{T}}\mathbf{D})^{-1}\mathbf{D}^{\mathrm{T}}\) is the Moore-Penrose pseudo-inverse of \(\mathbf{D}\). Let's denote the matrix \((\mathbf{D}^{\mathrm{T}}\mathbf{D})^{-1}\) as \(\mathbf{C}\), namely, \(\mathbf{C}=(\mathbf{D}^{\mathrm{T}}\mathbf{D})^{-1}\), which is also called the covariance matrix and it has a close relation with the uncertainties of the estimated parameters \(a_{k}\) (\(k\) = 0, 1, 2,..., \(K-1\)) by Equation (26). Specifically, the variance in \(a_{k}\) can be evaluated by  
\[\sigma^{2}(\lambda_{k})=C_{kk}, \tag{27}\]  
where \(C_{kk}\) is the \(k\)-th diagonal element of the matrix \(\mathbf{C}\). According to Equations (26) and (27), the least-square estimation \(A_{pq}\) (\(p_{d}\) = 0, 1, 2,..., \(n\)) of all elements in matrix \(\mathbf{A}\) as well as their uncertainties \(u(A_{pq})\) can be finally obtained. With the estimated elements \(A_{pq}\) and their uncertainties \(u(A_{pq})\), the uncertainty of \(e_{Z}(x,y)\) at any pixel position (\(x_{i},y_{j}\)) can be further evaluated. Specifically, according to Equation (14), there is \(e_{Z}(x_{i},y_{j})\) = \(\mathbf{x}_{i}^{\mathrm{T}}\mathbf{A}\mathbf{y}_{j}\), which can be regarded as a linear function of the coefficients \(A_{pq}\). The evaluation of the upper and lower bounds of \(e_{Z}(x_{i},y_{j})\) can be transformed into a simple linear programming problem [21], described as follows:  
\[\left\{ \begin{array}{rl}\max e_{Z}(x_{i},y_{j})=\mathbf{x}_{i}^{\mathrm{T} }\mathbf{A}\mathbf{y}_{j}\\ s.t.\quad A_{pq}\in\left[\hat{A}_{pq}-u(\hat{A}_{pq}),\,\hat{A}_{pq}+u(\hat{A} _{pq})\right]\\ p,\,q=0,\,1,\,2,\,\ldots,n\end{array} \right., \tag{28a}\] \[\left\{ \begin{array}{rl}\min e_{Z}(x_{i},y_{j})=\mathbf{x}_{i}^{\mathrm{T} }\mathbf{A}\mathbf{y}_{j}\\ s.t.\quad A_{pq}\in\left[\hat{A}_{pq}-u(\hat{A}_{pq}),\,\hat{A}_{pq}+u(\hat{A} _{pq})\right]\\ p,\,q=0,\,1,\,2,\,\ldots,n\end{array} \right., \tag{28b}\]  
where Equations (28a) and (28b) yield the upper and lower bounds of \(e_{Z}(x_{i},y_{j})\), respectively.