# A Joint Dual-Frequency GNSS/SINS Deep-Coupled Navigation System for Polar Navigation
## 3 Deep-Coupled Navigation System Design
### Design of the Integrated Navigation Filter
#### 3.4.2 Measurement Model  
Before we derive the relations between the outputs of the joint prefilters and the residual errors of the integrated navigation system, the inherent relationships between replica code phase and carrier frequency biases and the SINS navigation solution have been provided here [11]:  
\[ \begin{array}{ll}\delta\tau^{j}\cdot\frac{c}{f_{\text{code0}}}&=\rho_{\text{I }}^{j}-\rho_{G}^{j}\approx r^{j}+e_{x}^{j}\delta x+e_{y}^{j}\delta y+e_{x}^{j} \delta z-\left(r^{j}+\delta t_{\text{u}}-\delta t_{\text{sv}}^{j}-I^{j}-T^{j} \right)-\epsilon_{\rho^{j}}\\ &=e_{x}^{j}\delta x+e_{y}^{j}\delta y+e_{x}^{j}\delta z-\delta t_{\text{u}}+ \delta t_{\text{sv}}^{j}+I^{j}+T^{j}-\epsilon_{\rho^{j}}\\ \\ \delta f^{j}\cdot\frac{c}{f_{\text{carrier0}}}&=\dot{\rho}_{\text{I}}^{j}-\dot{ \rho}_{G}^{j}\approx r^{j}+\dot{e}_{x}^{j}\delta x+e_{y}^{j}\delta y+e_{x}^{j} \delta z-\left(\dot{r}^{j}+\delta f_{\text{u}}-\delta f_{\text{sv}}^{j} \right)-\epsilon_{\rho^{j}}\\ &=e_{x}^{j}\delta x+e_{y}^{j}\delta y+e_{x}^{j}\delta z-\delta f_{\text{u}}+ \delta f_{\text{sv}}^{j}-\epsilon_{\rho^{j}}\end{array}  \tag{22}\]  
where \(\delta\tau\) and \(\delta f\) are the code phase error and carrier frequency error coming from the prefilter, respectively; \(f_{\text{carrier0}}\) and \(f_{\text{code0}}\) denote the normalized carrier frequency and the ranging code chipping rate of GNSS signals; \(c\) is the speed of light; \(\rho_{\text{I}}^{j},\rho_{G}^{j}\) and \(\dot{\rho}_{\text{I}}^{j},\dot{\rho}_{G}^{j}\) denote the pseudorange and pseudorange rate calculated by the SINS and GNSS receiver, respectively; \(r^{j}\) and \(\dot{r}^{j}\) denote the actual range and range rate between the \(j\)th satellite and the receiver, respectively; \(\left(e_{x}^{j},e_{y}^{j},e_{z}^{j}\right)\) are the components of the unit vector in the line-of-sight direction from the user navigation solution to the \(j\)th GNSS satellite; \(I^{j}\) and \(T^{j}\) denote ionospheric and tropospheric propagation delay, respectively; and \(\epsilon_{\rho^{j}}\) and \(\epsilon_{\rho^{j}}\) denote the noise terms. The outputs of the joint prefilters are taken as measurements of the integrated filter based on the fact that the errors of replica code and carrier signals have relationships with the residual errors of the SINS, as shown in Equations (22) and (23). In order to further reduce the ionospheric influence in polar areas, the ionosphere-free combination of GPS L1 C/A and L2 CM signals is used in the proposed system. Firstly, the ionosphere-free pseudorange \(\rho^{j}\) between the \(j\)th satellite and the receiver is calculated by [25]  
\[\rho^{j}=\rho_{\text{LICA}}^{j}-l_{\text{LICA}}^{j}=\rho_{\text{L2 CM}}^{j}-l_{ \text{L2 CM}}^{j}=\frac{f_{\text{LICA0}}^{2}\cdot\rho_{\text{LICA}}^{j}}{f_{ \text{LICA0}}^{2}-f_{\text{L2 CM0}}^{2}}-\frac{f_{\text{L2 CM0}}^{2}\cdot\rho_{ \text{L2 CM}}^{j}}{f_{\text{L2 CM0}}^{2}-f_{\text{L2 CM0}}^{2}} \tag{24}\]where \(\rho_{\rm LICA}^{j}\) and \(\rho_{\rm L2 CM}^{j}\) are the pseudoranges calculated by L1 C/A and L2 CM signals, respectively; \(I_{\rm LICA}^{j}\) and \(I_{\rm L2 CM}^{j}\) are the ionospheric delays for L1 C/A and L2 CM signals; and \(f_{\rm L1 CA0}\) and \(f_{\rm L2 CM0}\) are the normalized carrier frequencies of L1 C/A and L2 CM signals. Then, according to Equations (22) and (23), the \(j\)th pseudorange error \(\delta\rho^{j}\) can be calculated by  
\[\delta\rho^{j}=\rho_{1}^{j}-\rho_{C}^{j}=\rho_{1}^{j}-[\rho^{j}-\delta t_{\rm u }+\delta t_{\rm sv}^{j}-T^{j}+\varsigma_{\rho^{j}}] \tag{25}\]  
where \(\varsigma_{\rho^{j}}\) is the noise term of the ionosphere-free combination. Besides this, a linear combination of carrier frequency errors \(\delta f_{\rm L1 CA}\) and \(\delta f_{\rm L2 CM}\) coming from the prefilter is used to calculate the \(j\)th pseudorange rate error \(\delta\dot{\rho}^{j}\):  
\[\delta\dot{\rho}^{j}=\alpha_{1}\cdot\delta\dot{\rho}_{\rm LICA}^{j}+\alpha_{2} \cdot\delta\dot{\rho}_{\rm L2 CM}^{j} \tag{26}\]  
where  
\[\left\{