Abstract:
Objectives: Achieving millimeter-level accuracy in short-baseline positioning using the BeiDou Navigation Satellite System (BDS) is critical for high-precision deformation monitoring of infrastructures such as reservoirs and bridges. The traditional double-differenced (DD) model, although widely adopted, introduces observation correlation through the differencing process, has a complicated stochastic model, and handles multi-frequency data inflexibly. The undifferenced (UD) model is theoretically equivalent to the DD model but offers greater flexibility, especially for multi-frequency signals. However, direct UD processing normally requires precise ionospheric and tropospheric products and suffers from a large number of unknown parameters, which reduces computational efficiency. To overcome these limitations, this paper develops a practical undifferenced short-baseline BDS processing method that achieves accuracy comparable to the DD model while simplifying the stochastic model and enabling flexible multi-frequency processing. The specific objectives are to: (1) design a UD strategy that absorbs atmospheric delays into satellite clock parameters without external products; (2) enable UD ambiguity fixing by imposing fixed DD ambiguities as constraints; (3) evaluate positioning accuracy under different observation durations (1 h, 4 h, 8 h, 24 h) and with different frequency strategies (single-frequency B1, dual-frequency B1B3, and the ionosphere-free LC combination); and (4) compare the proposed UD model with the classical DD model (GAMIT) in detecting millimeter-level deformations using a controlled micro-motion platform.
Methods: The proposed approach exploits the near-identical atmospheric delays between reference and monitoring stations in short-baseline scenarios (baselines <5 km). Instead of estimating ionospheric and tropospheric delays separately, we absorb them into satellite clock bias parameters. For each frequency, a separate set of satellite clock parameters is estimated per epoch, which also absorbs residual orbital errors. The geometric range is linearized around the reference station coordinates, and the unknown parameters include: the baseline vector (constant over a static session), differential receiver clocks (white noise per epoch), satellite clock parameters (white noise per epoch per frequency), and UD ambiguities (constant in the absence of cycle slips). All epochs within a given observation duration are processed together using least-squares adjustment with a satellite elevation-angle-based stochastic model. To address the absence of integer properties in undifferenced ambiguities, we first map UD float ambiguities to double-differenced (DD) ambiguities. Once the DD integer ambiguities are fixed, they are introduced as pseudo-observations with very tight constraints into the original UD normal equations, and the solution is re-estimated to obtain fixed UD results. For the ionosphere-free (LC) combination, the DD ambiguity is first decomposed into wide-lane and narrow-lane components to improve the fixing success rate.
Results: Field experiments were conducted using ten days of real BDS monitoring data. The monitoring net comprises five stations (baselines <300 m), and a micro-motion platform attached to each rover station. Three processing strategies were compared: single-frequency B1, dual-frequency B1B3, and the ionosphere-free LC combination. Observation durations of 1 h, 4 h, 8 h and 24 h were tested, with baseline repeatability as the main accuracy measure. Experimental results show that both the single- and dual-frequency strategies outperform the ionosphere-free combination, with the dual-frequency solution exhibiting superior stability in ambiguity fixing. For observation durations ≥4 h, the horizontal accuracy of B1/B1B3 reaches better than 1 mm and the vertical accuracy better than 5 mm. Even with only 1 h of observations, the horizontal precision remains better than 2 mm and the vertical precision better than 10 mm after rejecting outliers. For one baseline, the LC combination suffered from occasional narrow-lane ambiguity fixing failures (fixing rates as low as 68%-75%) due to noise amplification; after removing satellites with abnormally large observation noise (carrier-phase RMS >20 mm, pseudorange RMS >4000 mm), fixing rates increased above 90% and vertical repeatability improved from 10.5 mm to 5.9 mm. For 24-h solutions, all three strategies achieved comparable high precision: horizontal <0.7 mm and vertical <1.7 mm. The micro-motion experiment confirms that the proposed BDS-only undifferenced model achieves positioning accuracy comparable to the double-differenced model (GAMIT) and effectively detects 3 mm-level deformations. The root-mean-square errors of the UD B1B3 solution are 0.85 mm (N), 0.65 mm (E), and 1.44 mm (U), closely matching the GAMIT DD results.
Conclusions: The developed undifferenced BDS short-baseline model achieves millimeter-level precision and demonstrates strong potential for autonomous, high-stability deformation monitoring of engineering infrastructures. The key advantages are: elimination of external atmospheric products, a simpler stochastic model, flexible multi-frequency processing, and accuracy equivalent to the classical double-differenced model. With ≥4 h of observations, the method delivers horizontal <1 mm and vertical <5 mm; even with 1 h, it maintains horizontal <2 mm and vertical <10 mm. The reliable detection of 3 mm deformations under BDS-only standalone positioning makes this method a practical technical support for static infrastructure monitoring.