## 2 Method
The differences between the measured signatures and the expected values can be expressed as:  
\[\Delta\mathbf{y}_{i}\approx\mathbf{J}(\mathbf{x}\star,\mathbf{a}_{i})\cdot \Delta\mathbf{x}+\epsilon_{i}. \tag{5}\]  
To determine the degree of independence among \(N\) measurements, Equation (5) is multiplied by an arbitrary factor \(\eta_{i}\) and is summed over all \(i\), to give:  
\[\sum\limits_{i=1}^{N}\eta_{i}\Delta\mathbf{y}_{i}=\sum\limits_{i=1}^{N}\eta_{ i}\mathbf{J}(\mathbf{x}\star,\mathbf{a}_{i})\cdot\Delta\mathbf{x}+\sum\limits_{i=1 }^{N}\eta_{i}\epsilon_{i}, \tag{6}\]  
where, \(\sum\limits_{i=1}^{N}\eta_{i}^{2}=1\). Then, the difference \(\Delta\mathbf{y}_{j}\) for a certain measurement configuration \(j\) can be expressed as [23]:  
\[\Delta\mathbf{y}_{j}=-\eta_{j}^{-1}\sum\limits_{i\
eq j}\eta_{i}\Delta\mathbf{ y}_{i}+\eta_{j}^{-1}\left\{\sum\limits_{i=1}^{N}\eta_{i}\mathbf{J}(\mathbf{x} \star,\mathbf{a}_{i})\cdot\Delta\mathbf{x}+\sum\limits_{i=1}^{N}\eta_{i} \epsilon_{i}\right\}. \tag{7}\]  
The first term of Equation (7) on the right side is completely dependent on the other differences and represents the predictable part of \(\Delta\mathbf{y}_{j}\); the bracketed expression on the right side is unpredictable and independent of the other measurements. When the first term in the bracketed expression does not exceed the second random measurement error part, Equation (7) provides a way for predicting \(\Delta\mathbf{y}_{j}\) from other measurements, in other words, the \(\Delta\mathbf{y}_{j}\) is redundant information and can be obtained by others; this is based on:  
\[\left|\sum\limits_{i=1}^{N}\eta_{i}\mathbf{J}(\mathbf{x}\star,\mathbf{a}_{i}) \cdot\Delta\mathbf{x}\right|\leq\left|\sum\limits_{i=1}^{N}\eta_{i}\epsilon_{ i}\right|. \tag{8}\]  
It follows that the smaller the left term of Equation (8), the better the prediction. The question of mutual independence can thereby be examined by considering the minimal value of \(\left|\eta^{\mathrm{T}}\cdot\mathbf{J}(\mathbf{x}\star,\mathbf{a})\right|^{2}\), which can be achieved by a suitable selection of \(\eta_{i}\). Here, \(\eta\) is a \(N\) dimensional vector consisting of elements \(\eta_{i}\) (\(i=1,2,...,N\)). According to the Schwartz inequality, the minimum value of \(\left|\eta^{\mathrm{T}}\cdot\mathbf{J}(\mathbf{x}\star,\mathbf{a})\right|^{2}\) is equal to the smallest \(\zeta_{\min}\) of the eigenvalues of the matrix \(\mathbf{J}(\mathbf{x}\star,\mathbf{a})\cdot\mathbf{J}^{\mathrm{T}}(\mathbf{x} \star,\mathbf{a})\) when the \(\eta_{i}\) values are equal to the \(N\) components of the corresponding eigen-vector. If the condition by the Equation (8) holds, the difference \(\Delta\mathbf{y}_{i}\) that corresponds to the smallest \(\eta_{i}\) of the eigen-vector for \(\zeta_{\min}\) is the most linearly dependent. Consequently, the value of \(\Delta\mathbf{y}_{i}\) can be calculated by the first term on the right side of Equation (7) from the other measurement differences with no loss of information. Therefore, we can identify the redundant measurement configuration by:  
\[\mathbf{a}_{i}^{\mathrm{redundant}}=\text{ argmin}[\eta(\mathbf{a}_{i})], \tag{9}\]  
here, \(P\) is the associated configuration domain.