# Assessment of Different Stochastic Models for Inter-System Bias between GPS and BDS
## 2 Model for Inter-System Bias Estimation and Descriptions of the Stochastic Models
### The Different Stochastic Models  
To not only verify the importance of ISB, and its influence on the accuracy of tight combined PPP model, but also compare the impacts of different ISB estimation stochastic models on the positioning, five different schemes are proposed in this study. The differences between different schemes mainly focus on the configurations on the ISB. Here, we will take one 3-h processing-arc-window containing 360 observation epochs as an example.  
1. Neglecting inter-system bias between GPS and BDS is named ISB-OFF, which means the positioning model of BDS and GPS is separated, and the two systems are processed without any relation. The BDS observations are processed the same as GPS ones. However, the ISB between GPS and BDS will remain in the BDS processing model and will be reflected in the pseudorange residuals. The configuration equation can be expressed as: \[ISB_{OFF}=\left[\begin{array}{cccc}NULL_{1}&NULL_{2}&\ldots&NULL_{359}&NULL_{ 360}\end{array}\right]^{T}\] (11) where \(ISB_{OFF}\) is the ISB in the scheme ISB-OFF. \(NULL_{1}\) to \(NULL_{360}\) is an empty matrix which means without considering ISB.
2. Estimating the ISB as a piece-wise constant every 30 min (ISB-PW). ISB parameter is initialized at the first epoch, then re-initialized, and updated every 30 min [9]. The configuration equation can be written as: \[ISB_{PW}=\left[\begin{array}{cccc}ISBP_{1}&ISBP_{2}&ISBP_{3}&ISBP_{4}&ISBP_{5} &ISBP_{6}\end{array}\right]^{T}\] (12) where \(ISB_{PW}\) means the ISB in the scheme ISB-PW. The ISB in one 3-h processing-arc-window can be divided into six parts according to size of piece-wise (every 30 min), which are denoted as \(ISBP_{1}\) to \(ISBP_{6}\). \(ISBP_{i}\) is one piece-wise ISB, and \(i\) is from 1 to 6.
3. Estimating the ISB as a random-walk processing (ISB-RW). With this scheme, ISB is initialized at the first epoch, and then with a time-related spectral density of 0.001 \(m/\sqrt{s}\). The configuration equation can be formed as: \[ISB_{RW}=\left[\begin{array}{cccc}ISBR_{1}&ISBR_{2}&\ldots&ISBR_{359}&ISBR_{ 360}\end{array}\right]^{T}\] (13) \[ISBR_{i}=ISBR_{1}+sd\times 30\times(i-1)\,\ i=1,2,\ldots,359,360\] where \(ISB_{RW}\) is the ISB in the scheme ISB-RW. \(ISBR_{i}\) is the ISB for one epoch, with \(i\) from 1 to 360. \(sd\) means the time-related spectral density.
4. Estimating the ISB as a processing-arc-dependent constant (ISB-AD). During the whole processing-arc period, ISB is only initialized at the first epoch. If the processing-arc window is one day, this means the ISB is estimated as a constant each day. In this paper, we make the period of our processing tests as 3 h, so the ISB is considered to be a 3-h arc-dependent constant. The configuration equation can be indicated as: \[ISB_{AD}=\left[\begin{array}{ccccc}&&&&&&\\ &&&&&&\\ \hline ISB&ISB&\ldots&ISB&ISB\\ \hline ISB&ISB&\ldots&ISB&ISB\\ \hline\end{array}\right]^{T}\] (14) where \(ISB_{AD}\) means the ISB in the scheme ISB-AD, which is estimated as a constant during the 3-h processing-arc window.
5. Estimating the ISB as white noise (ISB-WN). In this way, the ISB is initialized for each epoch, which is the same as the normal processing mode as the receiver clock error. The configuration equation can be shown as: \[ISB_{WN}=\left[\begin{array}{ccccc}ISB_{1}&ISB_{2}&\ldots&ISB_{359}&ISB_{360 }\\ \end{array}\right]^{T}\] (15) where \(ISB_{WN}\) means the ISB in the scheme ISB-WN. The ISB in each epoch will be initialized and estimated.  
For convenience, the five schemes mentioned above marked as ISB-OFF, ISB-PW, ISB-RW, ISB-AD, and ISB-WN, respectively, are summarized in Table 2.