# 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  
The ionosphere-free (IF) pseudorange and phase observation functions between the receiver and a satellite can be written as:  
\[ \begin{array}{l}P=\rho+c\!\cdot\!(dt_{r}-dt^{s})+m\!\cdot\!ZTD+(b_{r}-b^{s})+ \varsigma\\ L=\rho+c\!\cdot\!(dt_{r}-dt^{s})+m\!\cdot\!ZTD+N+(B_{r}-B^{s})+\varepsilon \end{array}  \tag{1}\]  
where \(P\), \(L\) are pseudorange and carrier-phase \(IF\) combined observations, respectively; \(\rho\) is the range between the receiver and the satellite; \(c\) is the light speed; \(dt_{r}\) is the receiver clock error and \(dt^{s}\) is the satellite clock error; \(m\) and \(ZTD\) are mapping function and zenith tropospheric delay; the first-order error of ionospheric delay is eliminated by ionosphere-free combination; \(b_{r}\), \(b^{s}\) and \(B_{r}\), \(B^{s}\) are the pseudorange and carrier-phase hardware delay biases of the ionospheric-free function for receivers and satellites; \(N\) is ambiguity in distance; \(\varsigma\) and \(\varepsilon\) denote unmodeled parameters (e.g., the measurement noise and multipath errors for pseudorange and carrier phase). The pseudorange hardware delay biases \(b_{r}\), \(b^{s}\) in the GNSS observations can be assimilated into the clock errors \(c\!\cdot\!(dt_{r}-dt^{s})\). The carrier-phase hardware delay biases \(B_{r}\), \(B^{s}\) are not considered in most GNSS data processing, since the carrier-phase hardware delay biases is satellite dependent and stable over time; thus, it can be absorbed by the ambiguity [19; 20]. Using the precise satellite orbit and clock products, Equation 1 can be rewritten as:  
\[ \begin{array}{l}P=\rho+c\,d\overline{t_{r}}+m\cdot ZTD+\varepsilon\\ L=\rho+c\,d\overline{t_{r}}+m\cdot ZTD+\overline{N}+\varepsilon\end{array}  \tag{2}\]  
where \(d\overline{t_{r}}\) and \(\overline{N}\) are redefined receiver clock error and ambiguity:  
\[ \begin{array}{l}c\,d\overline{t_{r}}=c\,d{t_{r}}+b_{r}\\ \overline{N}=N+B_{r}-b_{r}\end{array}  \tag{3}\]  
Because the satellite hardware delays are contained in the precise satellite clock and can be eliminated on the user side when applying the precise products in Equation (2) [21]. In Equation (3), the ambiguity term is not an integer property anymore since it contains the bias term, which is the uncalibrated hardware delay biases, showing a fractional cycle property. Thus, the term \(B_{r}-b_{r}\) is called the uncalibrated phase delay [22]. The tight combined PPP model requires the estimation of an extra ISB parameter. Hence, the combined observation model can be expressed as:  
\[ \begin{array}{l}P^{G}=\rho^{G}+c\,d\overline{t_{r}}+m^{G}\cdot ZTD+ \varepsilon^{G}\\ L^{G}=\rho^{G}+c\,d\overline{t_{r}}+m^{G}\cdot ZTD+\overline{N}^{G}+ \varepsilon^{G}\\ P^{O}=\rho^{O}+c\,(d\overline{t_{r}}+ISB)+m^{O}\cdot ZTD+\varepsilon^{O}\\ L^{O}=\rho^{O}+c\,(d\overline{t_{r}}+ISB)+m^{O}\cdot ZTD+\overline{N}^{O}+ \varepsilon^{O}\end{array}  \tag{4}\]  
where in the notation of the receiver clock error, mapping functions, the ambiguity, and some other parameters, we use the superscript \(G\) as GPS to distinguish them from the parameters of other systems, which are denoted with the superscript \(O\). \(d\overline{t_{r}}\) is a common item between different systems, being the receiver clock term relating to the GPS. The \(ISB\) is the inter-system bias parameter between GPS and other systems in the unit of time. It will be modeled as an unknown parameter and estimated together with the other parameters such as coordinates and the receiver clock error. So, the corresponding estimated parameters vector \(\mathbf{X}\) is,  
\[\mathbf{X}=\left[\mathbf{\Delta r},d\overline{t_{r}},ISB,ZTD,\overline{N}^{G},\overline{N}^{O}\right]^{T} \tag{5}\]  
where \(\mathbf{\Delta r}\) denotes the vector of the receiver position increments, \(\overline{N}^{G}\) and \(\overline{N}^{O}\) are the modified ambiguity parameters for different systems.