Real-Time Estimation of Ionospheric Power Spectral Density for Enhanced BDS PPP/PPP-AR Performance

The undifferenced and uncombined (UDUC) model preserves raw code and carrier-phase observations for each frequency, avoiding differencing or ionosphere-free combinations. This approach enables the direct estimation of atmospheric parameters. However, the stochastic characteristics of these parameters, particularly ionospheric delay, are often oversimplified or based on empirical assumptions, limiting the accuracy and convergence speed of Precise Point Positioning (PPP). To address this issue, this study introduces a stochastic constraint model based on the power spectral density (PSD) of ionospheric variations. The PSD describes the distribution of ionospheric delay variance across temporal frequencies, thereby providing a physically meaningful constraint for modeling their temporal correlations. Integrating this PSD-derived stochastic model into the UDUC framework improves both ionospheric delay estimation and PPP performance, especially under disturbed ionospheric conditions. This paper presents a BDS PPP/PPP-AR method that estimates the ionospheric power spectral density (IPSD) in real time. Vondrak smoothing is applied to suppress noise in ionospheric observations before IPSD estimation. Experimental results demonstrate that the proposed approach significantly improves convergence time and positioning accuracy. Compared to the empirical IPSD model, the PPP mode using the estimated IPSD reduced horizontal and vertical convergence times by 11.1% and 13.2%, and improved the corresponding accuracies by 15.7% and 12.6%, respectively. These results confirm that real-time IPSD estimation, coupled with Vondrak smoothing, establishes an adaptive and robust ionospheric modeling framework that enhances BDS PPP and PPP-AR performance under varying ionospheric conditions.