# A Designed Calibration Approach for the Measurement -While-Drilling Instrument
## 3 Analysis and Discussion of the Calibration Approach
Figure 7, which shows the comparison of the deviation of the 15 sets of data in three cases, gives us a clear description of the improved calibration approach. These three cases are divided according to the different number of sampling times in the noise suppression method. The three cases are represented by C, D and E, respectively. In case C, we collected samples 16 times continuously at each attitude in the datasets \(\mathbf{D}_{k}\) collection; meanwhile, we collected the other samples eight times continuously at each attitude in the 15 sets of testing data. In case D, we collected samples at the same times as in the case of C at each attitude in the datasets \(\mathbf{D}_{k}\) collection, but we abandoned the noise suppression method in the collection of the 15 sets of testing data. In case E, we abandoned the noise suppression method in the collection of both the datasets \(\mathbf{D}_{k}\) and the 15 sets of testing data. Figure 7a,b are the diagrams of deviations between the calibrated attitude angles and the given attitude angles. Figure 7c,d are the diagrams of the standard deviation (_abbr_. STD) for the corresponding deviation. Figure 7a is the pitch angle deviation and Figure 7b is the azimuthal angle deviation. Figure 7 shows that the case C is optimal and can be applied to the real MWD instrument. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Condition**} & \multicolumn{8}{c}{**The Error Matrices and Biases on the Condition of \(\mathbf{\gamma_{a}}\) = 0\({}^{\circ}\)**} \\ \hline \multicolumn{2}{c}{**Random Noise**} & \multicolumn{2}{c}{\(\mathbf{K}_{ab}^{s}\)} & \multicolumn{2}{c}{\(\mathbf{b_{a}}\)} & \multicolumn{2}{c}{\(\mathbf{K}_{mb}^{s}\)} & \multicolumn{2}{c}{\(\mathbf{b_{m}}\)} \\ \hline \multicolumn{2}{c}{Without suppression} & \multicolumn{2}{c}{\(\left(\begin{array}{cc}1.1728&0.182&-0.1765\\ -0.1082&1.1103&0.125\\ -0.03&0.1241&1.4854\end{array}\right)\)} & \multicolumn{2}{c}{\(\left(\begin{array}{c}0.1\\ 0.1205\\ -0.1986\end{array}\right)\)} & \multicolumn{2}{c}{\(\left(\begin{array}{c}1.4849&0.2892&-0.1636\\ -0.0727&1.833&-0.2268\\ 0.0701&0.2756&0.713\end{array}\right)\)} & \multicolumn{2}{c}{\(\left(\begin{array}{c}1.5429\\ 4.0864\\ 0.9022\end{array}\right)\)} \\ \hline \multicolumn{2}{c}{With suppression} & \multicolumn{2}{c}{\(\left(\begin{array}{cc}1.1763&0.184&-0.1739\\ -0.1065&1.1111&0.1252\\ -0.0309&0.1243&1.4813\end{array}\right)\)} & \multicolumn{2}{c}{\(\left(\begin{array}{c}0.099\\ 0.1199\\ -0.1994\end{array}\right)\)} & \multicolumn{2}{c}{\(\left(\begin{array}{c}1.4872&0.2956&-0.1557\\ -0.0788&1.8259&-0.2232\\ 0.0661&0.2716&0.7154\end{array}\right)\)} & \multicolumn{2}{c}{\(\left(\begin{array}{c}1.4674\\ 4.133\\ 0.9021\end{array}\right)\)} \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The solved error matrixes and biases on the condition of adding the random noise. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Given Attitude (Unit: \({}^{\circ}\))**} & \multicolumn{8}{c}{**The Mean of the Output by the Two Sensor Triplet. (Unit: \({}^{\circ}\))**} & \multicolumn{8}{c}{**Calculating Attitude with the Calibration Parameters in Table 4** (Unit: \({}^{\circ}\))**} \\ \cline{3-13} \multicolumn{2}{c}{} & \multicolumn{2}{c}{**Without Suppression**} & \multicolumn{2}{c}{**With Suppression**} & \multicolumn{2}{c}{**With Suppression**} \\ \hline \(\mathbf{\psi}\) & \(\mathbf{\theta}\) & \(\mathbf{\varphi}\) & \(\mathbf{x_{m}}\) & \(\mathbf{y_{m}}\) & \(\mathbf{z_{m}}\) & \(\mathbf{x_{s}}\) & \(\mathbf{y_{s}}\) & \(\mathbf{z_{s}}\) & \(\mathbf{\psi}\) & \(\mathbf{\theta}\) & \(\mathbf{\varphi}\) & \(\mathbf{\psi}\) & \(\mathbf{\theta}\) & \(\mathbf{\varphi}\) \\ \hline \(30\) & \(-80\) & \(