# 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
### Model for Inter-System Bias Estimation and Processing Strategies
Here, we take GPS and BDS systems as an example for data processing, and a Kalman filter is employed to estimate the unknown parameters in the processing [23; 24; 25]. Suppose the initial state value is \(X_{0}=\hat{X}_{0}\), the variance is \(\sum_{X_{0}}\), so the state estimation \(\hat{X}_{i}\) at epoch \(i\) can be obtained recursively by Kalman filter:  
1. One-step state prediction: \[X_{i,i-1}=\Phi_{i,i-1}\hat{X}_{i-1}\] (6)
2. One-step prediction of covariance: \[\sum\
olimits_{X_{i,i-1}}=\Phi_{i,i-1}\sum\
olimits_{X_{i-1}}\Phi_{i,i-1}^{ T}+Q_{i}\] (7)
3. Filtering gain: \[K_{i}=\sum\
olimits_{X_{i,i-1}}A_{i}^{\top}(A_{i}\sum\
olimits_{X_{i,i-1}}A_{ i}^{\top}+R_{i})^{-1}\] (8)4. Status update: \[\hat{X}_{i}=X_{i,i-1}+K_{i}(L_{i}-A_{i}X_{i,i-1})\] (9)
5. State covariance update: \[\sum\
olimits_{X_{i}}=(I-K_{i}A_{i})\sum\
olimits_{X_{i,i-1}}\] (10) where \(\Phi_{i,i-1}\) is transition matrix; \(A_{i}\) is the coefficient matrix of observation equation; \(Q_{i}\) and \(R_{i}\) is the variance matrix of system noise sequence and variance matrix of observation noise, respectively; \(X_{i,i-1}\) and \(\sum\
olimits_{X_{i,i-1}}\) is one-step prediction value and its variance-covariance matrix, respectively; \(K_{i}\) is gain matrix; \(\hat{X}_{i}\) and \(\sum\
olimits_{X_{i}}\) is filtered estimation and its variance-covariance matrix, respectively; \(I\) denotes the unit matrix. In this experiment, during the preprocessing we detect and repair clock jumps to avoid misidentifying an observation jump caused by the receiver clock jumps as the cycle slip. Afterwards, the Geometry-Free (GF) and Hatch-Melbourne-Wubbena (HMW) [26; 27; 28] combinations are used to detect cycle slips. The previously described Kalman filter is employed to estimate the unknown parameters with a sliding window of 3 h station by station, which means the 24-h observation data for each station will be divided into 8 processing sessions. GPS and BDS combined PPP in static and simulated kinematic modes are applied in our study. In static PPP mode, the solution strategy for position coordinates is to consider as contents, while for the kinematic PPP mode, they are estimated as white noise at each epoch. To analyze the impacts of different precise products from different analysis centers (AC) on the ISB estimation, the MGEX precise products from two ACs, GFZ (GBM) and Wuhan University (WUM) are used to fix the satellite orbit and clock in the combined PPP model. The dry component of the tropospheric delay is corrected with the Saastamoinen model, while the residual zenith wet delay is estimated as a random-walk process. We apply the Global Mapping Function (GMF) [29] to convert zenith delay to slant directions. The cutoff elevation angle is set as \(7^{\circ}\) and an elevation-dependent weighting method is used. Moreover, the phase-wind up effects [30], the solid earth tide, the ocean loading tide [31] and relativistic effects are corrected by the model. Up to now, BDS-specific receiver antenna Phase Center Offsets (PCOs) and Phase Center Variations (PCVs) are not provided in the jgs14.atx. In this case, GPS-specific receiver PCOs and PCVs are applied instead. The initial standard deviation for the carrier-phase observations in GPS and BDS systems are set to 0.003 m, while that for the pseudorange observations is set to 0.3 m. The details of models and strategies related to the data processing are listed in Table 1.