Analytical Error Bounds for Lunar-Surface Relative Positioning Under Non-Identical Line-of-Sight Geometry

Lunar navigation systems are being developed to support sustained surface operations, but their early deployment phase may involve limited satellite visibility and degraded accuracy in orbit and clock determination compared with terrestrial global navigation satellite systems (GNSS). Relative positioning with a lunar-surface reference station can mitigate common error sources through differenced measurements. In conventional GNSS, this mitigation often relies on the assumption that the line-of-sight (LOS) vectors from the reference and user receivers to the same satellite are nearly identical. Because lunar navigation satellite ranges can be considerably shorter than GNSS ranges, this approximation may be insufficient for kilometer-level surface baselines. This paper analytically quantifies three error mechanisms resulting from non-identical LOS geometries: the deterministic bias of the identical-LOS-vector approximation, the residual projection of broadcast ephemeris errors into single-differenced measurements, and the satellite-position-evaluation-time error caused by receiver clock initialization. For each mechanism, a simple closed-form bound is derived as a function of the baseline length and satellite range and is verified against time-series evaluations for representative lunar orbits. For a 10 km baseline in a low lunar orbit, the approximation alone can bias positioning by several tens of meters, whereas single differencing suppresses broadcast ephemeris errors to the meter level, and the clock-initialization effect can reach the meter level or larger. These results indicate that in lunar-surface relative positioning, the single-differenced range should be evaluated from the exact receiver-dependent geometry rather than from a common LOS vector. Additionally, the derived bounds provide a practical basis for budgeting broadcast ephemeris and receiver clock synchronization requirements.