## 3 Materials and Methods
### Conditions Under Which the Proposed Error Model Is Valid
#### 3.2.1 Denominator \(N\) Equals Zero  
First, the case when the denominator \(N\) of a partial derivative equals zero is examined. By considering UTC a constant and treating longitude and latitude as two independent variables for further analysis, the following expression can be obtained:  
\[g(\eta_{N=0},\psi_{N=0})=N=0. \tag{14}\]  
According to the definition of longitude, the value range of \(\eta\) is from \(-180^{\circ}\) to \(180^{\circ}\). The set of all longitude (\(\eta\)) values is defined as \(U_{\eta}\), and the set of all \(\eta\) values, where \(N=0\), is denoted as \(A_{\eta}\); set \(A_{\eta}\) denotes a subset of \(U_{\eta}\). Next, let set \(A_{\eta}\) be a universal set; then, the complement of \(A_{\eta}\) is \(C_{\eta}\). Similarly, according to the definition of latitude, the value range of \(\psi\) is from \(-90^{\circ}\) to \(90^{\circ}\). The set of all latitude (\(\psi\)) values is denoted by \(U_{\psi}\), and the set of all \(\psi\) where \(N=0\) is labeled as \(A_{\psi}\) and represents a subset of \(U_{\psi}\). Further, let set \(U_{\psi}\) be a universal set; then, the complement of \(A_{\psi}\) is \(C_{\psi}\). A summary of the set labels is provided in Table 1.  
Based on the latitude (\(\psi\)) value, there are two possible cases: (1) \(\psi\) equals \(90^{\circ}\) or \(-90^{\circ}\), and (2) \(\psi\) is in the interval (\(-90^{\circ}\), \(90^{\circ}\)).  
When \(\psi\) equals \(90^{\circ}\) or \(-90^{\circ}\), \(\cos\psi=0\), as shown in row 2 of Table 2. According to Equation (13), regardless of the denominator (\(M\)) value, the denominator (\(N\)) is equal to zero. The (\(\eta_{N=0}\),\(\psi_{N=0}\)) set of row 2 in Table 2 corresponds to the geographic locations of the South and North Poles. Every meridian passes through both the South and North Poles; the latitude of the South Pole is \(-90^{\circ}\), and its longitude can be any value from the [\(-180^{\circ}\), \(180^{\circ}\)] range. The latitude of the North Pole is \(90^{\circ}\), and its longitude can also be any value from the [\(-180^{\circ}\), \(180^{\circ}\)] range.  
When \(\psi\) is in the \((-90^{\circ},90^{\circ})\) interval, \(\cos\psi\
eq 0\), as shown in row 3 of Table 2. Further, since \(N\) is zero, according to Equation (13), \(M\) is also zero. Substituting Equations (2) and (12) into Equation (14) yields  
\[\cos\psi\cos\delta\cos\omega=0. \tag{15}\]  
Because the value range of solar declination (\(\delta\)) is \([-23^{\circ}27^{\prime},\,23^{\circ}27^{\prime}]\), then \(\cos\delta\
eq 0\); therefore, \(\cos\omega=0\). Combining this with Equation (2) yields  
\[\eta_{N=0,\cos\omega=0}=\begin{cases}270^{\circ}-(UT\mathrm{C}+E_{\omega}) \times 15^{\circ},&\omega=90^{\circ},\\ 450^{\circ}-(UT\mathrm{C}+E_{\omega})\times 15^{\circ},&\omega=270^{\circ}, \end{cases} \tag{16}\]  
where the latitude (\(\psi_{N=0}\)) value is in the \((-90^{\circ},90^{\circ})\) range.  
By combining the two above-presented conclusions, it can be found that \((\eta_{N=0}\),\(\psi_{N=0})\) satisfies Equation (14), as shown in Table 2. In other words, sets \(A_{\eta}\) and \(A_{\psi}\) are obtained as shown in row 3 of Table 1. The three-dimensional spatial distribution of sets \(A_{\eta}\) and \(A_{\psi}\) forms a circular ring, which consists of two lines of longitude on Earth. At this ring, the denominator of the partial derivative (\(N\)) is zero, and the implicit function system is non-differentiable. Consequently, the proposed error model is invalid at \((\eta_{N=0}\),\(\psi_{N=0})\).