## 2 Autonomous Global Positioning Method
### Number of Solutions to Global Positioning Method Equation System
In Figure 2, the solar azimuth angle is defined as the angle between the line connecting the Earth's true north point and the observation point and the line connecting the direct point of the sun and the observation point. The direct point of the sun refers to the point on the Earth's surface where the sun's rays are perpendicular; at this point, \(h_{\rm s}\) is \(90^{\circ}\). The line from the observation point to the true north point is taken as a reference direction (i.e., \(Y_{n}\) in the ENU coordinate system), where a clockwise direction indicates a positive direction. The value range of \(A_{\rm s}\) is \([0^{\circ},360^{\circ}]\), as indicated by the color bar on the right side of Figure 2. As shown in Figure 2, the contour lines representing the solar azimuth angle exhibit a ray-like pattern. It should be noted that Figure 2 depicts a projection of the three-dimensional spherical Earth onto a two -dimensional plane, which introduces certain distortion. Further, all contour lines intersect at the geographic poles where the latitude is either \(-90^{\circ}\) or \(90^{\circ}\), which indicates that at these locations, the azimuth angle can assume any value in the range of [\(0^{\circ},360^{\circ}\)]. Because of this distortion, in Figure 2, two geographic poles are illustrated as lines rather than points. This situation is very similar to a world map. Figure 3: The contour color maps of the solar altitude angle (\(h_{\rm s}\)). Figure 2: The contour color maps of the solar azimuth angle (\(A_{\rm s}\)). In Figure 3, the contour lines of the solar altitude angle radiate outward in concentric circles from the direct point of the sun, where the solar altitude angle is \(90^{\circ}\), and the anti-direct point of the sun, which corresponds to the solar altitude angle of \(-90^{\circ}\). The color map presented in Figure 4 is constructed by superimposing the contour lines representing solar altitude angles in Figure 3 onto the contour lines in Figure 2. In Figure 4, the contour lines of \(h_{\rm s}\) are denoted by the black dashed lines. In addition, to minimize clutter in the image annotations, Figure 4 displays only the contour line values corresponding to the azimuth angle, whereas the values for solar altitude angle are omitted. Thus, it can be inferred that the intersection points of the contour lines for solar azimuth angle and solar altitude angle indicate the solutions to the system of equations of the global positioning method. Analysis of Figure 4 reveals that most intersection points of the contour lines yield a single solution, whereas a minority present two solutions. The results of the analysis are consistent with the conclusions of the specific steps for solving the system of equations. Figure 4 illustrates the number of solutions of the autonomous global positioning method at a designated time. The subsequent analysis focuses on the number of solutions to the equations of this method under different temporal conditions. As depicted in Figure 4, the distribution of the contour lines of the two independent variables determines the number of solutions to the equations. The distribution of the contour lines is mainly defined by the direct point of the sun, which is determined by the UTC. According to astronomy, during a 24 h period (i.e., one complete rotation of the Earth), the variations in a solar declination \(\delta\) are very small. Therefore, the latitude of the direct point of the sun remains almost constant, whereas the longitude changes with time. For instance, when UTC changes from 21 h to 15 h, the superimposition of the contour lines of the two independent variables is highly similar to the graph obtained by shifting the map presented in Figure 4 horizontally along the horizontal axis by six hours; their global distribution of the number of solutions is also highly similar.