## 2 Principle of the FOG-Based MWD
At this time, the dynamic error navigation error model should be established. The navigation error differential equations include three basic navigation parameters errors, namely the attitude error angles (2), the velocity errors (3) and the position errors (4) [23]. \[\dot{\mathbf{\psi}}^{l}=\delta\mathbf{\omega}_{ie}^{l}+\delta\mathbf{\omega}_{el}^{l}- \left(\mathbf{\omega}_{ie}^{l}+\mathbf{\omega}_{el}^{l}\right)\times\mathbf{\psi}^{l}-C_{ b}^{l}\Delta\mathbf{\omega}_{ib}^{b} \tag{2}\]  
\[\delta\dot{\mathbf{V}}^{l}=\mathbf{f}\times\mathbf{\phi}^{l}-\left(2\mathbf{\omega}_{ie}^{l}+ \mathbf{\omega}_{el}^{l}\right)\times\delta\mathbf{V}^{l}+\mathbf{V}^{l}\times\left(2 \delta\mathbf{\omega}_{ie}^{l}+\delta\mathbf{\omega}_{el}^{l}\right)+C_{b}^{l}\Delta \mathbf{f}_{ib}^{b} \tag{3}\]  
\[\left[ \begin{array}{c}\delta\dot{L}\\ \delta\dot{\lambda}\\ \delta\dot{\mu}\end{array} \right]=\left[ \begin{array}{ccc}0&\frac{1}{(R+h)}& 0\\ \frac{\cos L}{(R+h)}&0&0\\ 0&0&1\end{array} \right]\times\left[ \begin{array}{c}\delta V_{x}^{l}\\ \delta V_{y}^{l}\\ \delta V_{z}^{l}\end{array} \right]+\left[ \begin{array}{ccc}0&0&-\frac{V_{y }^{l}}{(R+h)^{2}}\\ \frac{V_{x}^{l}\tan L\sec L}{(R+h)}&0&-\frac{V_{x}^{l}\sec L}{(R+h)^{2}}\\ 0&0&0\end{array} \right]\times\left[ \begin{array}{c}\delta L\\ \delta\lambda\\ \delta h\end{array} \right] \tag{4}\]  
\[\text{where }\mathbf{\psi}^{l}=\left[ \begin{array}{cc}\delta I&\delta T&\delta A \end{array} \right]^{T}\text{is the attitude error, }\mathbf{V}^{l}=\left[ \begin{array}{cc}V_{x}^{l}&V_{y}^{l}&V_{z}^{l} \end{array} \right]^{T}\text{is the velocity of the FOG-based MWD and }\delta\mathbf{V}^{l}=\left[ \begin{array}{cc}\delta V_{x}^{l}&\delta V_{y}^{l} &\delta V_{z}^{l}\end{array} \right]^{T}\text{is the velocity error. }L\text{, }\lambda\text{, and }h\text{ is the latitude, longitude and depth of the FOG-based MWD. }\delta L\text{, }\delta\lambda\text{, and }\delta h\text{ is the latitude error, longitude error, and depth error. }\mathbf{\omega}_{ie}^{l}=\left[ \begin{array}{cc}\omega_{lex}^{l}&\omega_{ ley}^{l}&\omega_{iez}^{l}\end{array} \right]^{T}\text{is the projection of the earth rotation velocity in l-frame. }\mathbf{\omega}_{el}^{l}=\left[ \begin{array}{cc}\omega_{elx}^{l}&\omega_{ely}^{l}& \omega_{elz}^{l}\end{array} \right]^{T}\text{is the angular velocity of the l-frame with respect to e-frame, expressed in the l-frame. \(\mathbf{\omega}_{il}^{l}=\left[ \begin{array}{cc}\omega_{lix}^{l}&\omega_{ily}^{l} &\omega_{liz}^{l}\end{array} \right]^{T}\) is the angular velocity of the l-frame with respect to i-frame, expressed in the l-frame. R is the earth's diameter. \(\Delta\mathbf{\omega}_{ib}^{b}\) is the error vector of the three FOGs, where \(\Delta\mathbf{\omega}_{ib}^{b}=\widetilde{\mathbf{\omega}}_{ib}^{b}-\mathbf{\omega}_{ib}^{b}\), \(\Delta\mathbf{f}_{ib}^{b}\) is the error vector of the three ACCs, where \(\Delta\mathbf{f}_{ib}^{b}=\widetilde{f}_{ib}^{b}-f_{ib}^{b}\). In Equations (2) and (3), it can be seen that the measurement error of the ACC would influence the velocity and the measurement error of the FOG influence the attitude. When compensated the sensors' errors online or offline, the sensor error model should be established ahead. The coefficients in the error model should be calibrated by special instrument. Once the error coefficients have been figured out, the error model of the sensor has been obtained. With this error model, the error in the sensor could be compensated. Generally, the linear error model is sufficient for most applications [24]. However, under the drilling conditions, because of the high level of vibration, a linear error model could no longer meet the requirements.