Geometric Characterization on GNSS Direct Position Estimation in Navigation Domain

Direct position estimation (DPE) is a one-step positioning method that enables direct estimation of the GNSS receiver's position, velocity, and timing (PVT) within the navigation domain. This paper aims to investigate the geometric relationship between the correlation domain and the navigation domain, focusing on the projection direction, least square equivalence, noise characteristics, and navigation solution ambiguity. To achieve this, a comprehensive geometric model for a single satellite signal is established firstly, establishing the correlation peak distribution relationship with respect to azimuth and elevation angles. The correlation peak and noise boundary can be theoretically determined based on the satellite's line-of-sight (LOS) direction and carrier-to-noise-density ratio ( $\mathrm{C/N_{0}}$ ). Subsequently, the spatial mapping and superposition of multiple satellites are analyzed, specifically examining the overlapping area to determine navigation solution ambiguity. Furthermore, the equivalence between the traditional least squares (LS) and DPE solutions from a geometric perspective is confirmed. To validate the effectiveness of the proposed geometrical DPE theory, Monte Carlo simulations and field tests are conducted. The results reveal that in the DPE navigation domain, the projection slope only depends on the azimuth, while the width is only determined by the elevation. The center-boundary interception is related to both azimuth and elevation angles. In addition, noise or interference can distort the correlation shape, and affect the noise boundary oscillation, which leads to the navigation ambiguity in DPE PVT solutions.