# Inter-System Differencing between GPS and BDS for Medium-Baseline RTK Positioning
## 1 Introduction  
With the existing Global Navigation Satellite Systems (GNSS) being modernized or new being ones developed (i.e., GPS, GLONASS, BDS, Galileo), many more satellites and frequencies are available for precise positioning. Combined use of these satellite constellations can enhance the geometric strength of the GNSS positioning model, and thus has been proven beneficial for improving GNSS precise positioning performance in terms of accuracy, reliability, availability [1; 2; 3], and initialization time [4; 5], especially for applications in obstructed environments [6; 7; 8]. As a commonly used precise positioning technology, real-time kinematic (RTK) positioning based on the short baselines (less than 20 km) or network reference stations has proven its efficiency and reliability during the past few years. However, medium-baseline (20-100 km) RTK positioning still has a broad range of applications, such as providingprecise positioning information in a sparse reference network or in marine areas [9]. With multi-GNSS, the positioning performance of medium-baseline RTK has the potential to be further improved. When using the double differenced (DD) observations for RTK relative positioning, mainly two models can be used. One is the classical loose combination in which each system uses its own reference satellite with no double differences being formed across systems. This model will be called classical differencing in this paper. The other one is the tight combination in which two systems use the common reference satellite and thus utilizes double differencing across different systems [10]. This model will be called inter-system differencing in this paper. Using the inter-system differencing model can help to maximize the redundancy if the inter-system biases (ISBs) in range and phase observations can be handled properly [11]. This is essential for positioning in severe observation environments, such as urban areas where signals are easily blocked by high buildings or trees [8; 12]. In this case, the number of observed satellites of each single system can be very low. A typical situation is that when only one satellite is visible for a system, it cannot be used in the classical differencing since no double differencing can be formed. However, in the inter-system differencing model, it can still be used when a priori ISB information is available, thus the positioning model can be strengthened. In relative positioning, only the differenced inter-system biases (DISBs) need to be considered. For the code DISB, it can be easily parameterized or corrected with a priori calibration due to the simplicity of the pseudorange equation. For the phase DISBs, if the a priori phase ISB information is precisely known, the inter-system DD ambiguities can also be fixed to integers for the overlapping frequencies between two systems (e.g., GPS L1 and Galileo E1) [13; 14; 11]. [PERSON] and [PERSON] [13; 15] found that the code and phase DISBs between two systems could be cancelled out for baselines with receiver pairs of the same manufacturer. [PERSON] and [PERSON] [14] also reached the same conclusion. For receivers of different manufacturers, although DISBs cannot be eliminated, their stability in time has been proven, and thus can be calibrated in advance. Using these characteristics, the obvious improvement of ambiguity resolution (AR) and positioning with calibrated DISBs for short and long baseline RTK have been verified, the details of which can be seen in [13; 14; 11; 15; 16]. For the real-time estimation of DISBs and application for AR, [PERSON] et al. [12] proposed a particle filter-based method to estimate GPS L1-Galileo E1 phase DISBs based on the fact that the accuracy of a given DISB value can be qualified by the related fixing ratio. From the above analysis, the current research mainly concerns dealing with the phase DISBs of overlapping frequencies between different GNSS constellations. However, for GPS and the current operational BDS-2, there are no overlapping frequencies.