# Estimation of Inter-System Biases between BDS-3/GPS/Galileo and Its Application in RTK Positioning
## 2 RTK Positioning Models
### ISB and TCM RTK  
Different from LCM RTK, TCM RTK only needs one common pivot satellite for all the GNSS applied. However, the restriction is that only in the observation of overlapping frequency can the inter-system DD be completed. In addition, extra parameters known as code and phase ISBs appear in the inter-system DD observation. Prior estimation of the ISBs is required, which can be corrected as a constant to enhance our TCM model. To estimate the ISBs, prior position information about the receivers should be known, and then the equation to estimate ISB can be given as:  
\[\begin{split}\phi^{1_{GS}}_{rb,f}&=\rho^{1_{GS}}_{ rb}+\lambda_{f}a^{1_{GS}}_{rb,f}+\varepsilon^{1_{GS}}_{rb,f}\\ \phi^{1_{GS}*}_{rb,f}&=\rho^{1_{GS}*}_{rb}+\lambda_ {f}\left(a^{1_{GS}*}_{rb,f}+\delta^{C*}_{rb,f}\right)+\varepsilon^{1_{GS}*}_{rb,f}\\ P^{1_{GS}*}_{rb,f}&=\rho^{1_{GS}*}_{rb}+e^{1_{GS} ^{GS}}_{rb,f}\\ P^{1_{GS}*}_{rb,f}&=\rho^{1_{GS}*}_{rb}+d^{C*}_{rb,f}+\varepsilon^{1_{GS}*}_{rb,f}\end{split} \tag{4}\]  
where \(\delta^{C*}_{rb,f}\) and \(d^{C*}_{rb,f}\) denote the code and phase ISB, respectively. In Equation (4), there is a rank deficiency between the vector of ambiguities and the phase ISB. To solve this rank deficiency and make the equations solvable, the usual method is to fix the phase ISB to one float ambiguity. By choosing \(1*\) as the satellite to absorb phase ISB, the estimable form of Equation (4) is:  
\[\begin{split}\phi^{1_{GS}*}_{rb,f}&=\rho^{1_{GS}}_{ rb}+\lambda_{f}a^{1_{GS}}_{rb,f}+\varepsilon^{1_{GS}}_{rb,f}\\ \phi^{1_{GS}*}_{rb,f}&=\rho^{1_{GS}*}_{rb}+\lambda_ {f}\left(a^{1_{LS}*}_{rb,f}+\overline{\delta}^{C*}_{rb,f}\right)+\varepsilon^ {1_{GS}*}_{rb,f}\\ P^{1_{GS}*}_{rb,f}&=\rho^{1_{GS}*}_{rb}+e^{1_{GS} ^{GS}}_{rb,f}\\ P^{1_{GS}*}_{rb,f}&=\rho^{1_{GS}*}_{rb}+d^{C*}_{rb,f}+\varepsilon^{1_{GS}*}_{rb,f}\end{split} \tag{5}\]  
where \(\overline{\delta}^{C*}_{rb,f}=\delta^{C*}_{rb,f}+a^{1_{CL}*}_{rb,f}\) denotes the float ambiguity that absorbs the phase ISB. After the estimation of Equation (5), the phase ISB can be extracted through integer rounding:  
\[\widehat{\delta}^{C*}_{rb,f}=\overline{\delta}^{C*}_{rb,f}-Round(\widehat{ \delta}^{C*}_{rb,f}) \tag{6}\]  
where \(Round(\cdot)\) denotes the integer rounding and \(\delta^{C*}_{rb,f}\) is the estimation of phase ISB that needs to be corrected in the inter-system DD observation.  
Now, with the estimated ISBs, the inter-system DD can be conducted with the ISB corrected as:  
\[\begin{split}\phi^{1_{GS}*}_{rb,f}&=\rho^{1_{GS}}_{ rb}+\lambda_{f}a^{1_{GS}}_{rb,f}+\varepsilon^{1_{GS}}_{rb,f}\\ \overline{\delta}^{1_{GS}*}_{rb,f}&=\rho^{1_{GS}*}_{ rb}+\lambda_{f}a^{1_{GS}*}_{rb,f}+\varepsilon^{1_{GS}*}_{rb,f}\\ P^{1_{GS}*}_{rb,f}&=\rho^{1_{GS}*}_{rb}+e^{1_{GS} ^{GS}}_{rb,f}\\ \overline{P}^{1_{GS}*}_{rb,f}&=\rho^{1_{GS}*}_{rb}+e^{1 _{GS}*}_{rb,f}\end{split} \tag{7}\]  
where \(\overline{\phi}^{1_{GS}*}_{rb,f}=\phi^{1_{GS}*}_{rb,f}-\lambda_{f}\delta^{C*}_ {rb,f}\) and \(\overline{P}^{1_{GS}*}_{rb,f}=\overline{P}^{1_{GS}*}_{rb,f}-d^{C*}_{rb,f}\) denotes the inter-system DD observation with phase and code ISBs corrected, respectively; \(1_{G}\) is the unique pivot satellite for all the GNSSs; and \(s_{*}\) represents BDS-3/GPS/Galilee satellite. In the inter-system DD observations, the ISBs should be calibrated in advance and then be corrected so the stability of ISBs can be assured. Specifically, the ISBs between the two systems (BDS-3/GPS/Galilee system) should be corrected in the inter-system DD observations by subtracting the calibrated ISBs. Then, the ambiguity of the inter-system DD obtains its integer nature and can be fixed by LAMBDA [31]. Thus, for each two overlapping frequencies in two different GNSSs, an extra redundant observation can be obtained, which improves the RTK function model. Table 1 shows the frequency and wavelength of BDS-3/GPS/Galileo. It can be observed that B1C, L1, and E1 share the same frequency, 1575.42 MHz, whereas B2a, L5, and E5a share the same frequency, 1176.45 MHz, among BDS-3, GPS, and Galileo.