## 3 Redundant Sensors Configuration and Near-Vertical Algorithm
### Near-Vertical Well Attitude Measurement Method  
Theoretically, the deflection angle of the redundant accelerometer is 45\({}^{\circ}\), so the horizontal attitude calculation method can be written as Equation (19) by solving Equation (12). \[\left\{ \begin{array}{l}inc^{\prime}=sin^{-1}\Big{(}-\Big{(}a_{y}^{\prime}+a_{z} ^{\prime}\Big{)}/\sqrt{2}g\Big{)}\\ tf^{\prime}=cos^{-1}\Big{(}\Big{(}a_{y}^{\prime}-a_{z}^{\prime}\Big{)}/\Big{(} \sqrt{2}g\ cos\ inc\Big{)}\Big{)}\end{array} \right. \tag{19}\]  
To overcome the problem of singular values of trigonometric functions, the Equation (19) is refined as  
\[\left\{ \begin{array}{l}L=-\Big{(}a_{y}^{\prime}+a_{z}^{\prime}\Big{)}/\sqrt{ 2}g\\ inc^{\prime}=tan^{-1}\Big{(}L/\sqrt{1-L^{2}}\Big{)}\\ tf^{\prime}=tan^{-1}\Big{(}\sqrt{2}a_{x}/\Big{(}a_{y}^{\prime}-a_{z}^{\prime} \Big{)}\Big{)}\end{array} \right.. \tag{20}\]  
Considering the mounting error, the installation angle of the redundant accelerometers can rarely be equal to 45\({}^{\circ}\); therefore, the accurate angle \(\alpha\) obtained by the aforementioned calibration method is introduced in Equations (19) and (20). Finally, the updated wellbore attitudes calculation simultaneous equations are written as Equation (21). \[\left\{ \begin{array}{l}K=cos(\alpha)/sin(\alpha)\\ P=cos^{2}(\alpha)/sin(\alpha)+sin(\alpha)\\ L=-(Ka_{y}^{\prime}+a_{z}^{\prime})/Pg\\ inc^{\prime}=tan^{-1}\Big{(}L/\sqrt{1-L^{2}}\Big{)}\\ tf^{\prime}=tan^{-1}\Big{(}Pa_{x}/(Ka_{y}^{\prime}-a_{z}^{\prime})\Big{)}\\ az^{\prime}=arctan\big{(}\big{(}m_{x}\ cos\ tf^{\prime}+m_{z}\ sin\ tf^{\prime} \big{)}/\big{(}(m_{x}\ sin\ tf^{\prime}-m_{z}\ cos\ tf^{\prime})sin\ inc^{ \prime}+m_{y}\ cos\ inc^{\prime}\big{)}\big{)}\end{array} \right. \tag{21}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Items**} & \multicolumn{4}{c}{**N100**} & \multicolumn{2}{c}{**AXO315**} \\ \cline{2-5}  & **Accelerometer** & **Gyroscope** & **Magnetometer** & **Accelerometer** \\ \hline Range & \(\pm 16\) g & \(\pm 2000^{\circ}/\mathrm{s}\) & \(\pm 4900\) uT & \(\pm 14\) g \\ Bias instability & \(0.4\) mg & \(5^{\circ}/\mathrm{h}\) & - & \(15\) ug \\ Linearity & \(0.1\%\mathrm{FS}\) & \(0.1\%\mathrm{FS}\) & \(0.1\%\mathrm{FS}\) & \(80\) ppm (\(8\) g) \\ Noise density & \(75\) ug/\(\sqrt{\mathrm{Hz}}\) & \(0.0028^{\circ}/\mathrm{s}/\mathrm{VHz}\) & \(140\) uGauss\(\sqrt{\mathrm{Hz}}\) & \(15\) ug/\(\sqrt{\mathrm{Hz}}\) \\ Axis misalignment & \(\pm 0.05^{\circ}\) & \(\pm 0.05^{\circ}\) & \(\pm 0.05^{\circ}\) & \(20\) mrad \\ Resolution & \(0.5\) mg & \(0.02^{\circ}/\mathrm{s}\) & \(1.5\) Millgauss & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of AXO315 and N100. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Reference (\({}^{\circ}\))**} & \multirow{2}{*}{**Before Calibration**} & \multicolumn{2}{c}{**After Calibration and No NOE**} & \multicolumn{2}{c}{**After NOE Compensation and**} \\  & & & **Compensation (\({}^{\circ}\))** & & **Calibration (\({}^{\circ}\))** \\ \cline{2-5}  & **Pitch (\({}^{\circ}\))** & **Errors (\({}^{\circ}\))** & **Pitch (\({}^{\circ}\))** & **Errors (\({}^{\circ}\))** & **Pitch (\({}^{\circ}\))** & **Errors (\({}^{\circ}\))** \\ \hline
60 & 59.28 & 0.72 & 59.57 & 0.43 & 59.95 & 0.05 \\
65 & 64.52 & 0.48 & 64.74 & 0.26 & 65.03 & \(-0.03\) \\
70 & 69.44 & 0.56 & 69.60 & 0.4 & 69.96 & 0.04 \\
80 & 79.50 & 0.50 & 79.55 & 0.45 & 80.0 The near-vertical approach based on redundant accelerometers can be summarized in the flowchart shown in Figure 11. The main application is that the instrument is dropped to the survey station for measuring the azimuth and inclination of the straight well section. In order to guarantee measurement accuracy, the instrument needs to be kept stationary while measuring. If the pitch is less than 85\({}^{\circ}\), the attitude can be calculated by the conventional method utilizing the standard equipment. If the pitch is above 85\({}^{\circ}\), the redundant accelerometers are employed to calculate the attitude.