## 3 Materials and Methods
### Conditions Under Which the Proposed Error Model Is Valid  
The accurate assessment of the validity of the proposed error model at a particular geographic location examines whether the four partial derivatives of the two dependent variables, derived from the system of two implicit functions, exist and exhibit continuity at that specific point. The main mathematical symbols employed in Section 3.2 are presented in Table 1.  
The determination process of a function's differentiability is complex. By analyzing the properties of the four partial derivatives, this study determines whether this process can be simplified. The four partial derivatives consist of elementary functions, which are derived from the sum, difference, product, and quotient of trigonometric functions. Consequently, the four partial derivatives are classified as elementary functions. Further, according to the mathematical principles, elementary functions are continuous and differentiable at every point within their domain. Thus, determining whether the proposed error model is differentiable at a specific point on the Earth's surface (\(\eta\), \(\psi\)) can be reduced to the process of verifying whether the four partial derivatives are defined at that point. If all four partial derivatives are defined at a given point, then the error model is considered differentiable at that point. However, if any one of the four partial derivatives is undefined at a particular point, the model is regarded as non-differentiable at that point.  
To examine whether an elementary function is defined at a specific point, it is necessary to distinguish the type of elementary function. The function types include frac  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\psi\in U_{\psi}\) & \(\eta\in U_{\eta}\) & \(N\) & \(\cos\psi\) & \(M\) \\ \hline \(\psi_{N\
eq 0}\in C_{\psi}\) & \(\eta_{N\
eq 0}\in C_{\eta}\) & \(\
eq 0\) & \(\
eq 0\) \\ \(\psi_{N=0}\in A_{\psi}\) & \(\eta_{N=0}\in A_{\eta}\) & 0 & — & — \\ \hline \hline \end{tabular}
\end{table}
Table 1: The main mathematical symbols in Section 3.2.  
tional forms, radical forms, and logarithmic forms, among others. The four partial derivatives are expressed in fractional form, so they are defined where the denominator is not equal to zero.  
In summary, when the denominators of the four partial derivatives are all non-zero, the derivatives are simultaneously defined at a specific point. In addition, as all these functions are elementary functions, they exist and are continuous at that point. Consequently, at that point, the implicit function system is differentiable, and the total differential can be considered the total increment, indicating that the error model is valid at the given point. In contrast, if any one of the four partial derivatives has a denominator that equals zero, the implicit function system is not differentiable at the given point, and the proposed error model is inapplicable at that point.  
Next, the denominators of the four partial derivatives are studied. According to Equation (8), the denominators of partial derivatives \(\partial\psi/\partial A_{\mathrm{s}}\) and \(\partial\psi/\partial h_{\mathrm{s}}\) are equal, and the denominators of partial derivatives \(\partial\eta/\partial A_{\mathrm{s}}\) and \(\partial\eta/\partial h_{\mathrm{s}}\) are also equal. The two denominators are expressed as follows:  
\[\begin{array}{l}M=\cos A_{\mathrm{s}}\cos h_{\mathrm{s}}\sin\psi-\sin h_{ \mathrm{s}}\cos\psi,\\ N=\sin h_{\mathrm{s}}-\sin\psi\sin\delta.\end{array} \tag{12}\]  
Combining Equations (1), (2), and (12), the relation between \(N\) and \(M\) is expressed as follows:  
\[N=-\cos(\psi)M. \tag{13}\]