## 3 Establish of the Nonlinear Error Model  
In this paper, only the second-order error is taken into account when establishing the nonlinear error model. Because the third and higher-order error are one-millionth of the linear error coefficient, which could then be ignored [25]. The nonlinear error model of the FOG is given by (5) and the output error of the FOG is given by (6) [23]. \[\widetilde{\mathbf{\omega}}_{ib}^{b}=\mathbf{B}_{g}+(1+\mathbf{K}_{g}+\mathbf{M}_{g})\mathbf{\omega}_ {ib}^{b}+\mathbf{\epsilon}\mathbf{\omega}_{ib}^{b}+\mathbf{d}_{g}\cdot(\mathbf{\omega}_{ib}^{b} )^{2} \tag{5}\]  
\[\Delta\mathbf{\omega}_{ib}^{b}=\mathbf{B}_{g}+(\mathbf{K}_{g}+\mathbf{M}_{g})\mathbf{\omega}_{ib}^{ b}+\mathbf{\epsilon}\mathbf{\omega}_{ib}^{b}+\mathbf{d}_{g}\cdot(\mathbf{\omega}_{ib}^{b} )^{2} \tag{6}\]  
In the Equations (5) and (6), subscript g stands for FOG, and superscript b stands for b-frame. \(\widetilde{\mathbf{\omega}}_{ib}^{b}=\left[ \begin{array}{cc}\widetilde{\omega}_ {ibx}^{b}&\widetilde{\omega}_{iby}^{b}&\widetilde{\omega}_{ibz}^{b}\end{array}  \right]^{T}\) is the original output of the FOGs in the b-frame (\({}^{\circ}\)/h). The subscript x, y, and z represent the \(X\)-axis, \(Y\)-axis, and \(Z\)-axis. \(\mathbf{\omega}_{ib}^{b}=\left[ \begin{array}{cc}\omega_{ibx}^{b}&\omega_{iby}^ {b}&\omega_{ibz}^{b}\end{array} \right]^{T}\) is the ideal output of the FOGs (\({}^{\circ}\)/h). \(\Delta\mathbf{\omega}_{ib}^{b}=\left[ \begin{array}{cc}\Delta\widetilde{\omega}_ {ibx}^{b}&\Delta\widetilde{\omega}_{iby}^{b}&\Delta\widetilde{\omega}_{ibz}^{b} \end{array} \right]^{T}\) is the error between the original output and ideal output. \(\mathbf{B}_{g}=\left[ \begin{array}{cc}B_{gx}&B_{gy}&B_{gz}\end{array} \right] ^{T}\) is the bias error of the FOGs (\({}^{\circ}\)/h). \(\mathbf{K}_{g}=\left[ \begin{array}{cc}k_{gxx}&0&0\\ 0&k_{ggy}&0\\ 0&0&k_{gzz}\end{array} \right]\) is the factor error of the three FOGs. \(\mathbf{M}_{g}=\left[ \begin{array}{cc}0&k_{gxy}&-k_{gxz}\\ -k_{gyx}&0&k_{gyz}\\ k_{gxx}&-k_{gzy}&0\end{array} \right]\) is the misalignment matrix, \(k_{gmn}\) (m = x.y.z, n = x.y.z, \(m\
eq n\)) is the FOGs's instrument angle projection of m axis with respect to n axis (rad). \(\mathbf{\epsilon}\mathbf{\omega}_{ib}^{b}=\left[ \begin{array}{cc}\mathbf{\epsilon}\mathbf{ \omega}_{ibx}^{b}&\mathbf{\epsilon}\mathbf{\omega}_{iby}^{b}&\mathbf{\epsilon}\mathbf{\omega}_{ ibz}^{b}\end{array} \right]^{T}\) stands for the standard deviation of random white noise in FOGs original output (\({}^{\circ}\)/h). \(\mathbf{d}_{g}=\left[ \begin{array}{cc}d_{gx}&d_{gy}&d_{gz}\end{array} \right]^{T}\) stands for the second-order error coefficient of the three FOGs (\({}^{\circ}\)/h). In the same way, the nonlinear error model of the ACCs in the FOG-based MWD is given by (7), and the output error of the ACCs is given by (8)  
\[\hat{\mathbf{f}}_{ib}^{b}=\mathbf{B}_{a}+(1+\mathbf{K}_{a}+\mathbf{M}_{a})\hat{\mathbf{f}}_{ib}^{b} +\hat{\mathbf{g}}_{ib}^{b}+\mathbf{d}_{a}\cdot(\hat{\mathbf{f}}_{ib}^{b})^{2} \tag{7}\]  
\[\Delta\hat{\mathbf{f}}_{ib}^{b}=\mathbf{B}_{a}+(\mathbf{K}_{a}+\mathbf{M}_{a})\hat{\mathbf{f}}_{ib}^ {b}+\mathbf{\epsilon}\hat{\mathbf{f}}_{ib}^{b}+\mathbf{d}_{a}\cdot(\hat{\mathbf{f}}_{ib}^{b})^ {2} \tag{8}\]  
In the Equations (7) and (8), subscript a stands for ACC and superscript b stands for b-frame. The parameters in the Equations (7) and (8) are defined analogously with Equations (5) and (6). \[\text{In Equation (6), }\mathbf{K}_{g}+\mathbf{M}_{g}=\delta\mathbf{k}_{g}=\left[ \begin{array}{ cc}k_{gxx}&k_{gxy}&k_{gxz}\\ k_{gyx}&k_{gyy}&k_{gyz}\\ k_{gzz}&k_{gzy}&k_{gzz}\end{array} \right]\text{, the equation can be simplified}\]  
\[\Delta\mathbf{\omega}_{ib}^{b}=\mathbf{B}_{g}+\delta\mathbf{k}_{g}\mathbf{\omega}_{ib}^{b}+ \mathbf{\epsilon}\mathbf{\omega}_{ib}^{b}+\mathbf{d}_{g}\cdot(\mathbf{\omega}_{ib}^{b})^{2} \tag{9}\]  
\(\delta\mathbf{k}_{g}\) is a matrix merged by the factor error matrix \(\mathbf{K}_{g}\) and the misalignment matrix \(\mathbf{M}_{g}\).