## 4 Doppler Shift Estimation Error and Code Phase Estimation Error
The pseudo-range rate \(\dot{\rho}_{1}^{n}\) from the INS to the satellite can be described by:  
\[\dot{\rho}_{1}^{n} =e_{1}(\dot{x}_{1}-\dot{x}_{\rm s}^{n})\ +\ e_{2}\Big{(}\dot{y}_{1}- \dot{y}_{\rm s}^{n}\Big{)}\ +\ e_{3}(\dot{z}_{1}-\dot{z}_{\rm s}^{n}) \tag{21}\] \[=e_{1}(\dot{x}-\dot{x}_{\rm s}^{n})\ +\ e_{2}\Big{(}\dot{y}-\dot{y}_{ \rm s}^{n}\Big{)}\ +\ e_{3}(\dot{z}-\dot{z}_{\rm s}^{n})\ +\ e_{1}\delta\dot{x}\ +\ e_{2}\delta\dot{y}\ +\ e_{3} \delta\dot{z}\]  
The pseudo-range rate \(\dot{\rho}_{\rm G}^{n}\) from the receiver to the satellite can be described by:  
\[\dot{\rho}_{\rm G}^{n}=e_{1}(\dot{x}-\dot{x}_{\rm s}^{n})\ +\ e_{2}\Big{(}\dot{y}- \dot{y}_{\rm s}^{n}\Big{)}\ +\ e_{3}(\dot{z}-\dot{z}_{\rm s}^{n})+c\delta\dot{t}+\dot{n}_{\rho} \tag{22}\]  
From Equations (15) and (20)-(22), the pseudo-range error \(\delta\rho^{n}\) and pseudo-range rate error \(\delta\dot{\rho}^{n}\) can be described by:  
\[\delta\rho^{n}=\rho_{1}^{n}-\rho_{\rm G}^{n}=e_{1}\delta x\ +\ e_{2}\delta y \ +\ e_{3}\delta z-c\delta t-n_{\rho} \tag{23}\] \[\delta\dot{\rho}^{n}=\dot{\rho}_{1}^{n}-\dot{\rho}_{\rm G}^{n}=e_ {1}\delta\dot{x}\ +\ e_{2}\delta\dot{y}\ +\ e_{3}\delta\dot{z}-c\delta\dot{t}-\dot{n}_{\rho} \tag{24}\]  
where \(\delta x,\delta y,\delta z\) are the position error, and \(\delta\dot{x},\delta\dot{y},\delta\dot{z}\) are the velocity errors of INS in the space rectangular coordinate system. In the geodetic coordinate system, they can be expressed using the following equations:  
\[\left\{ \begin{array}{rl}\delta x\ =\ \delta h\cos L\cos\lambda-(R_{N}+h)\sin L \cos\lambda\delta L-\ (R_{N}+h)\cos L\sin\lambda\delta\lambda\\ \delta y\ =\ \delta h\cos L\sin\lambda-(R_{N}+h)\sin L\sin\lambda\delta L+\ (R_{N}+h)\cos L\cos \lambda\delta\lambda\\ \delta z\ =\ \delta h\sin L+[R_{N}(1-e^{2})+h]\cos L\delta L\end{array} \right. \tag{25}\]  
\[\left\{ \begin{array}{rl}\delta\dot{x}\ =-\delta v_{E}\sin\lambda-\delta v_{N} \sin L\cos\lambda+\delta v_{U}\cos L\cos\lambda\\ \delta\dot{y}\ =\ \delta v_{E}\cos\lambda-\delta v_{N}\sin L\sin\lambda+\delta v_{U} \cos L\sin\lambda\\ \delta\dot{z}\ =\ \delta v_{N}\cos L+\delta v_{U}\sin L\end{array} \right. \tag{26}\]  
where \(L,\lambda,h\) represent latitude, longitude and altitude, respectively. \(\delta L,\ \delta\lambda,\ \delta h\) are the latitude error, longitude error and altitude error, respectively. \(\left[ \begin{array}{cc}\delta v_{E}&\delta v_{N}&\delta v_{U}\end{array}  \right]^{\rm T}\) is the vector of the receiver velocity error in the navigation frame. The code phase estimation error \(\Delta T_{I}\) and the Doppler estimation error \(\Delta f_{I}\) caused by INS velocity error and position error have the following relations with \(\delta\rho^{n}\) and \(\delta\dot{\rho}^{n}\):  
\[\lambda_{\rm code}\Delta T_{I}\ =\ \delta\rho\ =\ e_{1}\delta x\ +\ e_{2}\delta y\ +\ e_{3}\delta z-c\delta t-n_{\rho} \tag{27}\]  
\[\lambda_{\rm carrier}\Delta f_{I}\ =\ \delta\dot{\rho}\ =\ e_{1}\delta x\ +\ e_{2}\delta\dot{y}\ +\ e_{3}\delta\dot{z}-c\delta\dot{t}-\dot{n}_{\rho} \tag{28}\]  
where \(\lambda_{\rm code}\) is the code phase wavelength and \(\lambda_{carrier}\) is the carrier wavelength, \(\delta t\) and \(\delta\dot{t}\) are the clock error and clock shift which can be calibrated locally. INS velocity and position error are mainly determined by the bias and shifts of gyros and accelerometers. The dynamic equations for a strapdown INS are given by:  
\[\delta\dot{v}^{\rm n}=-(\delta\omega_{\rm en}^{\rm n}+2\delta\omega_{\rm ie}^{ \rm n})\ \times\ \mathbf{v}^{\rm n}-(\mathbf{\omega}_{\rm en}^{\rm n}+2\mathbf{\omega}_{\rm ie}^{\rm n})\ \times\ \delta\mathbf{v}^{\rm n}+\mathbf{C}_{\rm b}^{\rm n}\mathbf{\rho}\times\ \mathbf{\varphi}+\mathbf{C}_{\rm b}^{\rm n}\mathbf{\
abla}^{\rm b} \tag{29}\]  
\[\left\{ \begin{array}{rl}\delta\dot{L}=\frac{1}{R_{M}+h}\delta v_{N}\\ \delta\dot{\lambda}=\frac{\rm sec}{R_{N}+h}\delta v_{E}+\frac{v_{E}}{R_{N}+h} \sec L\tan L\delta L\\ \delta\dot{h}=\delta\mathbf{\varphi}_{U}\end{array} \right.