## 1 Introduction
(2018) introduced a non-real-time autonomous global positioning method using underwater polarized light [16]. Their research involved the development of an innovative biomimetic underwater polarized light detection camera, which was validated for its positioning capabilities across various geographical locations on the Earth. [PERSON] et al. further advanced this field by employing the time differential polarization field to minimize errors in underwater polarization detection [17]. [PERSON] et al. (2021) developed a non-real-time global positioning method [18], which determines global positions based on the distribution of the polarization degree of sky polarized light; in contrast, the previous studies were based on the polarization angles of sky polarized light. [PERSON] et al. (2023) suggested a method that uses the temporal characteristics of the Earth's sky polarized light and employs a polarization camera to measure this light [19,20]. This method calculates the geographic north direction and an observer's latitude. When combined with additional navigation data, this method can facilitate the global positioning process. [PERSON] et al. (2024) developed an integrated navigation system combining multi-directional polarization sensors [21], a sun sensor, and an inertial measurement unit (IMU). [PERSON] et al. (2024) designed a publicly available detection device computer program [22], thus promoting international collaboration in advancing polarized navigation technology. Based on their fundamental principles, global positioning methods can be roughly classified into two distinct categories. The first category uses sky scattering light detection devices and geographic true north information detection devices simultaneously. First, the solar azimuth and elevation angles, defined in the body coordinate system, are derived from the polarization information [23; 24], including the angle or degree of polarization of the scattered light. This information is then combined with the heading information provided by a geographic true north information detection device to determine the solar elevation and azimuth angles at the carrier's location. Finally, the geographic location of the carrier is obtained. This approach has real-time capability and necessitates only a single detection of sky-scattered light. The second category relies only on the sky-scattered light detection device, and the solar azimuth and elevation angles, defined in the body coordinate system, are determined based on the polarization information of the scattered light. However, due to the inability to obtain the true north heading of the carrier from a single observation, this category requires at least two observations to be performed to determine the carrier's geographic location. Consequently, these methods are classified as non-real-time methods, and if the time interval between the two measurements is very short, the precision of the geographic localization can decrease. Existing research suggests that the global positioning methods based on polarized light both provide global positioning functionality and maintain the autonomy of methods, meaning that they do not rely on any artificial signals. However, research on the positioning errors of these methods has been relatively scarce [25], particularly regarding the global distribution patterns of positioning errors at different times and under different error conditions. Aiming to address the current shortcomings, this paper designs a global error model for the real-time global positioning methods. The proposed model can rapidly and accurately provide the global distribution patterns of positioning errors at different times and under different error conditions. It can also serve as a theoretical framework for navigation task planning, thus enabling the avoidance of non-linear regions where minor measurement inaccuracies might result in substantial positioning discrepancies. Furthermore, the design and development of polarized light sensors and cameras can be determined by the proposed error model.