## 3 Materials and Methods
### Global Positioning Distance Error Expression Derivation  
For the implicit function equation system, this study computes the partial derivatives with respect to the two independent variables: the solar azimuth angle (\(A_{\text{s}}\)) and the solar altitude angle (\(h_{\text{s}}\)). During the calculation process of partial derivatives, the solar declination (\(\delta\)) is considered constant, resulting in four partial derivatives, which can be expressed as follows:  
\[ \begin{array}{l}\frac{\partial\varphi}{\partial A_{\text{s}}}=-\frac{\sin A _{\text{s}}\cos h_{\text{s}}\cos\psi}{\cos A_{\text{s}}\cos h_{\text{s}}\sin \psi-\sin h_{\text{s}}\cos\psi},\\ \frac{\partial\varphi}{\partial h_{\text{s}}}=-\frac{\sin h_{\text{s}}\cos A _{\text{s}}\cos\psi-\cos h_{\text{s}}\sin\psi}{\cos A_{\text{s}}\cos h_{\text {s}}\sin\psi-\sin h_{\text{s}}\cos\psi},\\ \frac{\partial\eta}{\partial A_{\text{s}}}=-\frac{\cos A_{\text{s}}\cos h_{ \text{s}}\cos\psi}{\sin h_{\text{s}}-\sin\psi\sin\psi},\\ \frac{\partial\eta}{\partial h_{\text{s}}}=\frac{\sin A_{\text{s}}\sin h_{ \text{s}}\cos\psi}{\sin h_{\text{s}}-\sin\psi\sin\psi\sin\vartheta}.\end{array}  \tag{8}\]  
To ensure an initial understanding of the global distribution of partial derivatives, it is essential to further describe Equation (8) through visual representations. According to Equations (1), (2), and (8), the calculation equation of the partial derivative includes three variables: longitude (\(\eta\)), latitude (\(\psi\)), and time (\(UTC\)). Because a two-dimensional image can illustrate only the influence of two variables on a partial derivative, in this study, two-dimensional images are created by setting time to a specific value, using longitude as a horizontal axis and latitude as a vertical axis to illustrate the global distribution of partial derivatives. The specific calculation steps of a partial derivative \(\partial\psi/\partial A_{\text{s}}\) are as follows:  
1. Define \(f\) (\(\eta\), \(\psi\), \(UTC\)) = \(\partial\psi/\partial A_{\text{s}}\), where \(\eta\), \(\psi\), and \(UTC\) are the three calculation variables. Set the time to be consistent with the time in Figure 2, which is 27 May 2024, \(UTC\) = 21 h. At this time, \(\delta\) = 21.4849\({}^{\circ}\) and \(E_{\omega}\) = 0.0456 h. Consequently, express \(f\) (\(\eta\), \(\psi\), 21) = \(\partial\psi/\partial A_{\text{s}}\). 2. Calculate the value of a function \(f\) (\(\eta\), \(\psi\), 21) using the two independent variables, namely \(\eta\) and \(\psi\). The value range of the Earth's latitude (\(\psi\)) is [\(-90^{\circ}\), \(90^{\circ}\)], with an interval of 0.01\({}^{\circ}\), resulting in a total of 18,000 data points. The value range of the Earth's longitude (\(\eta\)) is [\(-180^{\circ}\), \(180^{\circ}\)], also with an interval of 0.01\({}^{\circ}\), resulting in 36,000 data points. Therefore, the total number of calculations amounts to 648,000,000. For each calculation, first, the values of solar altitude angle (\(h_{\text{s}}\)) and azimuth angle (\(A_{\text{s}}\)) are calculated for each geographical location using Equations (1) and (2). Next, the values of the solar declination, geographical latitude, solar altitude angle, and solar azimuth angle of each geographical location are substituted into Equation (8) to calculate the partial derivative \(f\) (\(\eta\), \(\psi\), 21) value. However, in Figures 6-9, considering the drastic changes in the partial derivatives' values at certain locations, areas where the absolute value exceeds four are illustrated in black. This approach is employed to ensure that most regions can accurately show the values of the four partial derivatives. Further, assuming that the implicit function equations are differentiable at all locations globally, the total differential (d\(\psi\)) can be regarded as a total increment (\(\Delta\psi\)), and the same  
Figure 8: Partial derivative of the longitude angle with respect to the solar azimuth angle. Figure 9: Partial derivative of the longitude angle with respect to the solar altitude angle.