## 5 Results
### Error Model Spatial Properties  
During the computational process, the \(\Delta A_{\rm s}\) and \(\Delta h_{\rm s}\) values were both expressed in degrees (\({}^{\circ}\)), and the \(\Delta L\) value was quantified in kilometers (km). At the time of 27 May 2024, \(UTC\) = 21 h, based on the UTC and the astronomical calendar, \(\delta=21.4849^{\circ}\), and \(E_{\omega}=0.0456\) h. Subsequently, the four partial derivatives of all positions on the Earth's surface were calculated according to Equations (1), (2), and (8).  
When the measurement biases of the independent variables were \(\Delta A_{\rm s}=\Delta h_{\rm s}=1^{\circ}\), \(\Delta A_{\rm s}=\Delta h_{\rm s}=0.1^{\circ}\), and \(\Delta A_{\rm s}=\Delta h_{\rm s}=0.0001^{\circ}\), all \(\Delta A_{\rm s}\) and \(\Delta h_{\rm s}\) satisfied the condition in Section 4.3 that \(\mid\Delta A_{\rm s}\mid\leq 5^{\circ}\) and \(\mid\Delta h_{\rm s}\mid\leq 5^{\circ}\). Consequently, the proposed error model was valid for most global locations under these conditions. It could be inferred that \(\Delta\eta\) was approximately equal to d\(\eta\) and \(\Delta\psi\) was approximately equal to d\(\psi\). The corresponding d\(\psi\) and d\(\eta\) values were obtained by Equation (9). Subsequently, the corresponding positioning errors (\(\Delta L\)) were calculated using Equation (11), as illustrated in Figures 14-16. Because Equation (11) was used instead of Equation (10), the calculation time was significantly decreased. This reduction in time is the main contribution of the proposed error model.  
In Figure 14, the horizontal axis represents longitude, and the vertical axis indicates the latitude value. The contour lines represent the values of the positioning error. Most contour lines intersect at two points, both of which are on the invalid circular ring mentioned in Section 3.2.3. The results indicate that the positioning error in most regions was within 140 km. The black area in Figure 14 indicates an error larger than 400 km, with the central points exhibiting errors of positive infinity. This black area could be divided into two parts, as mentioned in Section 4.1.2: the region where the proposed error model was not valid and the region where the partial derivatives gradually approached positive infinity. The former region is a circular ring; the latter region is referred to as the valid black area.  
When \(\Delta A_{\rm s}=\Delta h_{\rm s}=0.1^{\circ}\), the global positioning error \(\Delta L\) was generally within 14 km, as shown in Figure 15.  
When \(\Delta A_{\rm s}=\Delta h_{\rm s}=0.0001^{\circ}\), as shown in Figure 16, the distribution pattern of \(\Delta L\) remained the same as that shown in Figure 15, with the global positioning error mostly within 14 m. The positioning accuracy of GPS typically falls within the meter range. Therefore, if an autonomous global positioning system using the Earth's sky polarized light  
Figure 16: The global positioning error distribution for \(\Delta A_{\rm s}=\Delta h_{\rm s}=0.0001^{\circ}\) and \(UTC=21\) h.  
Figure 15: The global positioning error distribution for \(\Delta A_{\rm s}=\Delta h_{\rm s}=0.1^{\circ}\) and \(UTC=21\) h.  
and true north measurement instruments can achieve performance comparable to that of the GPS, the accuracy of the polarized light sensors should be 0.0001\({}^{\circ}\).