# Characterization of Inter-System Biases in GPS + BDS Precise Point Positioning
## 2 Methods  
This section reviews the full-rank undifferenced GPS/BDS model and gives insight into the estimable ISBs in the model. In addition, it also presents the processing strategy in detail. _The Undifferenced GPS/BDS PPP_  
Let us start with the original undifferenced observation equations. For a receiver-satellite combination \(r-s\) at frequency \(j\) in epoch \(i\), the undifferenced code \(P_{r,f}^{T,s}(i)\) and phase \(L_{r,f}^{T,s}(i)\) observations are defined as [22]:  
\[ \begin{array}{l}P_{r,f}^{T,s}(i)=\rho_{r}^{T,s}(i)+c\Big{(}\Delta t_{r}^{T}(i) -\Delta t^{T,s}(i)\Big{)}+\mu_{f}^{T}l_{r,1}^{T,s}(i)+T_{r}^{s}+b_{r,f}^{T}+b_{ f}^{T,s}+\varepsilon_{P}^{T}(i)\\ L_{r,f}^{T,s}(i)=\rho_{r}^{T,s}(i)+c\Big{(}\Delta t_{r}^{T}(i)-\Delta t^{T,s}(i )\Big{)}-\mu_{f}^{T}l_{r,1}^{T,s}(i)+T_{r}^{s}+\lambda_{f}^{T}N_{r,f}^{T,s}+B_{ r,f}^{T,s}+B_{f}^{T,s}+\varepsilon_{L}^{T}(i)\end{array}  \tag{1}\]  
where superscript \(T\) represents the system (G for GPS and C for BDS). \(\rho_{r}^{T,s}(i)\) is the geometric distance; \(c\) is the speed of light; \(\Delta t_{r}^{T}(i)\) and \(\Delta t^{T,s}(i)\) are the receiver and satellite clock offsets; \(l_{r,1}^{T,s}(i)\) the (first-order) slant-ionosphere delay on the first frequency with its coefficient (\(\mu_{f}^{T}=\lambda_{f}^{2}/\lambda_{1}^{2}\), in which \(\lambda_{f}\) is the wavelength); \(T_{r}^{s}\) is the tropospheric slant delay; \(b_{r,f}^{T}\) and \(b_{f}^{T,s}\) are the receiver and satellite code biases; \(N_{r,f}^{T,s}\) is the integer ambiguity; \(B_{r,f}^{T}\) and \(B_{f}^{T,s}\) are receiver and satellite phase biases; \(\varepsilon_{P}(i)\) and \(\varepsilon_{L}(i)\) are noise in code and phase observation. Except \(N_{r,f}^{T,s}\)(unit: cycle), \(\lambda_{f}\)(unit: m/cycle) and \(\mu_{f}^{T}\)(without unit), all quantities are expressed in units of range. Note that, the effects of phase windup, solid tide, ocean loading, relativistic effects, earth rotation have been corrected by corresponding models [23]. Conventionally, the IGS precise clock products absorb the ionosphere-free (IF) combination of satellite code biases as following [24]:  
\[\hat{\Delta}t^{T,s}(i)=\Delta t^{T,s}(i)-(\frac{\mu_{2}^{T}}{\mu_{2}^{T}-1}b_ {r,1}^{T}-\frac{1}{\mu_{2}^{T}-1}b_{r,2}^{T}) \tag{2}\]  
Equation (1) represents a rank-deficient system. In order to deal with the linear correlation between parameters (the receiver clock offsets with the receiver-code bias, the slant-ionosphere delay with the receiver-code bias, the ambiguities with the receiver and satellite phase biases, hardware biases between frequencies), the idea is to lump some of them together [25]. The reparameterization process is as below:  
\[ \begin{array}{l}P_{f}^{T,s}(i)=\rho^{T,s}(i)+c\Big{[}\Delta t_{r}^{T}(i)- \Delta t^{T,s}(i)\Big{]}+\mu_{f}^{T}l_{r,1}^{T,s}(i)+T_{r}^{s}+b_{r,f}^{T}+b_{ f}^{T,s}+\varepsilon_{P}\\ =\rho^{T,s}(i)+c\underbrace{[\Delta t_{r}^{T}(i)+\frac{\mu_{2}^{T}}{\mu_{2}^{ T}-1}b_{r,1}^{T}-\frac{1}{\mu_{2}^{T}-1}b_{r,2}^{T}}_{\Delta t_{r}^{T}(i)}-\hat{ \Delta}t^{T,s}(i)]\\ +\underbrace{\frac{c\mu_{f}^{T}}{\mu_{2}^{T}-1}(b_{r,2}^{T}-b_{r,1}^{T}+b_{2 }^{T,s}-b_{1}^{T,s})+\mu_{f}^{T}l_{r,1}^{T,s}(i)}_{\mu_{f}^{T}l_{r,1}^{T,s}(i )}+T_{r}^{s}+\varepsilon_{P}(i)\end{array}  \tag{3}\]  
\[ \begin{array}{l}L_{f}^{T,s}(i)=\rho^{T,s}(i)+c[\hat{\Delta}t_{r}^{T}(i)-\hat {\Delta}t^{T,s}(i)]+\mu_{f}^{T}l_{r,1}^{T,s}(i)+T_{r}^{s}+\\ \lambda_{f}^{T}\underbrace{\Bigg{\{}N_{r,f}^{T,s}+B_{r,f}^{T}+B_{f}^{T,s}+ \frac{c}{\lambda_{f}^{T}(\mu_{2}^{T}-1)}[(\mu_{f}^{T}+1)(b_{r,2}^{T}+b_{2}^{T,s})-(\mu_{f}^{T}+\mu_{2}^{T,s})(b_{r,1}^{T}+b_{1}^{T,s})]\Bigg{\}}+\varepsilon _{L}\end{array}  \tag{4}\]where \(\hat{}\) presents that the parameter after recombination. The full-rank GPS/BDS PPP observation equations can be rewritten as  
\[