## 4 Global Error Model Verification by Numerical Simulations
### First Case Analysis: \(\Delta A_{\mathrm{s}}\) Equals \(\Delta h_{\mathrm{s}}\)  
In this simulation experiment, first, \(\Delta A_{\mathrm{s}}=\Delta h_{\mathrm{s}}\) was substituted into Equation (9), which yielded  
\[\frac{\mathrm{d}\psi}{\Delta A_{\mathrm{s}}}=\frac{\partial\psi}{\partial A_ {\mathrm{s}}}+\frac{\partial\psi}{\partial h_{\mathrm{s}}}, \tag{18}\]  
\[\frac{\mathrm{d}\eta}{\Delta A_{\mathrm{s}}}=\frac{\partial\eta}{\partial A_ {\mathrm{s}}}+\frac{\partial\eta}{\partial h_{\mathrm{s}}}, \tag{19}\]  
where \(\mathrm{d}\psi/\Delta A_{\mathrm{s}}\) is the total differential ratio of the latitude to the solar azimuth angle (TDRLAT), and \(\mathrm{d}\eta/\Delta A_{\mathrm{s}}\) is the total differential ratio of the longitude to the solar azimuth angle (TDRLON). It should be noted that neither TDRLAT nor TDRLON are a partial derivative or a total differential, but were introduced in this study for the convenience of graphical representation of the error analysis results.  
Considering the same conditions as those in Figures 6-9, the TDRLAT and TDRLON obtained in this experiment are shown in Figures 10a and 11a, respectively. These conditions are \(UTC=21\) h, \(\delta=21.4849^{\circ}\), and \(E_{\omega}=0.0456\) h.  
For \(\Delta A_{\mathrm{s}}=\Delta h_{\mathrm{s}}=0.1^{\circ}\), the global distribution map of the total latitude increment \(\Delta\psi\) was obtained, as shown in Figure 10b; the global distribution map of the total longitude increment \(\Delta\eta\) is displayed in Figure 11b.  
The calculation process was as follows. First, it was assumed that the theoretical solar altitude angle and azimuth angle at a point on Earth (\(\eta\), \(\psi\)) were \(A_{\mathrm{s}}\) and \(h_{\mathrm{s}}\), and their measurement deviations were \(\Delta A_{\mathrm{s}}\) and \(\Delta h_{\mathrm{s}}\), respectively; \(UTC=21\) h, \(\delta=21.4849^{\circ}\), and \(E_{\omega}=0.0456\) h. Then, the measured solar altitude angle and azimuth angle at point (\(\eta\), \(\psi\)) were obtained as (\(A_{\mathrm{s}}+\Delta A_{\mathrm{s}}\)) and (\(h_{\mathrm{s}}+\Delta h_{\mathrm{s}}\)). By substituting the measured solar altitude angle and azimuth angle into Equations (1) and (2), the measured latitude and longitude (\(\eta^{\prime},\psi^{\prime}\)) values were obtained. If the system of equations, composed of Equations (1) and (2), had two sets of solutions, the solution that was closer to the theoretical solution was considered a unique solution, whereas the other solution was excluded. Then, it held that \(\Delta\psi=\psi^{\prime}-\psi\) and \(\Delta\eta=\eta^{\prime}-\eta\).  
In Figures 10a and 11a, in the numerical bar on the right side, the values exhibiting an absolute total differential ratio larger than four (dimensionless) are depicted in black. In Figures 10b and 11b, regions where the absolute values of latitude and longitude deviations surpassed \(0.4^{\circ}\) are also illustrated in black. In Figures 10a,b and 11a,b, all other areas are represented in color.  
Therefore, Figures 10 and 11 depict two distinct regions, namely the colored and black regions, each of which was analyzed independently, as presented in the following sections.