# Assessment of Different Stochastic Models for Inter-System Bias between GPS and BDS
## 1 Introduction  
Along with the modernization of the Global Positioning System (GPS) [1], the revitalization of the Russian Global Navigation Satellite System (GLONASS) [2], the launch of more European Global Navigation Satellite System (Galileo) satellites [3], and the fast development of Chinese BeiDou Navigation Satellite System (BDS) [4; 5], more and more satellites can be observed at one place. Together with the initiation of the Multi-GNSS Experiment (MGEX) in 2012 [6], precise point positioning (PPP) has been stepped up into the stage of multi-system combined mode. However, because of different time systems, reference frames, and hardware delays within receivers used in different systems [7; 8], biases exist between different systems when we are making multi-GNSS combined PPP. If we blindlyapply observations from all systems without any calibration of these biases, a poor accuracy result will be achieved, and sometimes even worse than that of the single-system PPP. Hence, biases between different systems need to be carefully considered and calibrated, which is the basis of multi-system integrated positioning. Often, an extra parameter named inter-system bias (ISB) is introduced to cover the biases between different satellite systems [9; 10; 11]. Currently, the processing strategy for ISB is usually to estimate it together with other unknown items, such as coordination, receiver clock, and tropospheric delay, etc. As for the estimation stochastic models of ISB, normally a strategy of considering ISB as an arc-dependent constant is applied in [PERSON] and [PERSON] [12], [PERSON], et al. [13], [PERSON], et al. [14], and [PERSON], et al. [15]. Besides applying the code ISB estimation, the stochastic model of considering ISB as an arc-dependent constant is also used in the phase ISB estimation for PPP ambiguity resolution (PPP-AR). Because day-to-day variation of phase ISB is less than 0.1 cycles, the phase ISB is usually estimated as an arc-dependent constant every 24 h [16; 17]. [PERSON], et al. [9] regard ISB as a piece-wise parameter, which can increase the sample size and is good for ISB modeling and prediction. Besides these two strategies, another two stochastic models can be used, which are considering ISB as white noise or as random walk. However, up to now, no specific analysis has been made to assess the difference between these stochastic models and their impact on ISB estimation, therefore in this study, we will evaluate and compare the influence of different stochastic models on ISB, positioning accuracy, and processing time. Based on the results, the recommendations on the selections for the ISB stochastic models will be given.  
The paper is organized as follows: Section 2 gives the introductions on the PPP model for ISB estimation and processing strategies. In addition, the detailed descriptions of four different ISB stochastic models and five designed schemes are made. Then, the experimental data used in this study are declared in Section 3. The results and analysis are shown in Section 4, inside pseudorange observation residuals. The estimated ISBs, convergence time, positioning accuracy with GBM and WUM precise products, and accuracy improvement during the convergence time are discussed and compared with five different strategies. Finally, conclusions of this study are made in Section 5.