Kriging Based Ionospheric Grid Model and Threat Model for Single-Frequency BDSBAS

The ionosphere poses a formidable challenge to the effective application of satellite-based augmentation system (SBAS), due to its unpredictable variation along the propagation path of satellite single-frequency (SF) signals. To address this challenge, The utilization of multiple signals from core satellite constellations emerges as a potential solution for monitoring ionospheric activity. Nonetheless, the grid model founded on planar fitting or employing the Kriging method, along with the threat model considering the imperfect observation from limited ground stations, remains pivotal for current SBASs. To model the accurate ionosphere in China, a spatiotemporal Kriging involving the time information is introduced to generate ionospheric vertical delays for BeiDou SBAS (BDSBAS) ionospheric grid points (IGPs). Characteristics of the regional ionosphere are dynamically constructed and expressed by the fourdimensional variables, i.e., the time and space coordinates. Gaussian functions are optimally designed to create the correlation model and the covariance matrix of the stochastic process. Covariances of the measurement noise are modified by the time difference in the new method. The weight matrix comprising of both matrices is used for the Kriging estimator coefficients and the formal error variances. To model the flexible undersampled threat in China, an extended annular scheme of data deprivation under the three-dimensional spherical shell assumption is developed to simulate the extreme observation condition. Residuals between the spatial Kriging estimates and virtual user measurements are counted to form the safety margin, which is an alternative to deal with the possible undersampling risk. Ionospheric delays and their error variances still rely on the combined matrix while ignoring the time factor in the noise covariance. An undersampled threat model is established using the spatial distance among ionospheric pierce points (IPPs), since the temporal influence has been already considered in the grid model. Both the grid model and the threat model are built with the real observation data from 27 Crustal Movement Observation Network of China (CMONOC) stations. The differential code biases (DCBs) for 92 satellites and 27 stations are estimated to show their features of short-term stability from November 1st, 2021 to November 7th, 2021. Results on November 4th, 2021, the date with the most active ionosphere during 2021 and 2022, shows that the spatio-temporal Kriging after removing DCBs can predict vertical ionospheric delays with a formal error standard deviation of 0.23m for 117 BDSBAS IGPs. There are 97.46% of undersampled threats less than 0.5m under our scheme to compensate the capability of monitor stations at 14:00 local time. Daily threats are 0.42m on average and have a standard deviation of 0.22m. Combining the inflated formal error and the variable threat, final total uncertainties are around 0.65m for the provability of the feasibility and stability of proposed algorithms. Methodologies in this paper have the potential to establish the real-time grid model and the near real-time threat model, to guidance the construction of SBAS, and to improve the systematic service capability.