# A Designed Calibration Approach for the Measurement -While-Drilling Instrument
## 3 Analysis and Discussion of the Calibration Approach  
The calibration approach was numerically simulated by the MATLAB program. The collected data in Figure 5 are used to verify the feasibility of the calibration approach. The other 15 sets of attitude data were randomly given to test the accuracy of this approach. The simulation also compared the calibration results with and without random noise. Firstly, the collected data and the testing date (15 sets of random attitude data) did not contain random noise. The calibration approach (in Figure 3) was examined by the collected data, on the condition of the given values of \(\gamma_{a}\). Table 2 lists the estimated error matrixes and biases based on the condition of \(\gamma_{a}=0^{\circ}\) and \(\gamma_{a}=5^{\circ}\). By comparing Tables 1 and 2, we found that, the bias \(\mathbf{b_{k}}\) (\(\mathbf{b_{a}}\), \(\mathbf{b_{m}}\)) and the diagonal elements of the error matrixes \(K_{kb}^{\mathrm{s}}\) (\(K_{ab}^{\mathrm{s}}\) and \(K_{mb}^{\mathrm{s}}\)) were fully determined by the calibration approach. The non-diagonal elements of the error matrixes \(K_{kb}^{\mathrm{s}}\) (\(K_{ab}^{\mathrm{s}}\) and \(K_{mb}^{\mathrm{s}}\)), however, were not. After that, the testing data (15 sets of given attitude data) were employed to verify the accuracy of the estimated error parameters. The test results are shown in Figure 6, and five sets of them were picked out and listed in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Condition** & & & **The Error Matrices and Biases** & \\ \hline
**Pitch \(\theta=-45^{\circ}\)** & & \(\mathbf{K_{ab}^{\mathrm{s}}}\) & \(\mathbf{b_{a}}\) & \(\mathbf{K_{mb}^{\mathrm{s}}}\) & \(\mathbf{b_{m}}\) \\ \hline \(\gamma_{a}=0^{\circ}\) & \(\left(\begin{array}{ccc}1.1766&0.1837&-0.1754\\ -0.1067&1.1104&0.1248\\ -0.0301&0.124&1.4818\end{array}\right)\) & \(\left(\begin{array}{ccc}0.1\\ 0.12\\ -0.2\end{array}\right)\) & \(\left(\begin{array}{ccc}1.4871&0.2952&-0.1569\\ -0.0775&1.8264&-0.2229\\ 0.066&0.2722&0.7155\end{array}\right)\) & \(\left(\begin{array}{ccc}1.5\\ 4.13\\ 0.9\end{array}\right)\) \\ \hline \(\gamma_{a}=5^{\circ}\) & \(\left(\begin{array}{ccc}1.1766&0.1984&-0.1586\\ -0.1067&1.0951&0.2218\\ -0.0301&-0.0066&1.487\end{array}\right)\) & \(\left(\begin{array}{ccc}0.1\\ 0.12\\ -0.2\end{array}\right)\) & \(\left(\begin{array}{ccc}1.4871&0.3078&-0.1304\\ -0.0775&1.8389&-0.0617\\ 0.066&0.2083&0.7366\end{array}\right)\) & \(\left(\begin{array}{ccc}1.5\\ 4.13\\ 0.9\end{array}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The resolved error matrixes and biases by the calibration approach. Figure 5: The collected sampling datasets. (**a**) and (**b**) 150 sets of sampling points to simulate the output of tri-axial accelerometer and tri-axial magnetometer without noise. (**c**) and (**d**) 40 sets of sampling points following the rule of pitch angle \(\theta=-45^{\circ}\) without noise. Figure 6a,b show the test results of the tri-axial accelerometer and tri-axial magnetometer, respectively, on the condition of \(\gamma_{a}=0^{\circ}\) for the MWD instrument. Figure 6c,d put forward the test results on the condition of \(\gamma_{a}=5^{\circ}\). As shown in Figure 6, a fixed deviation in the direction of around the \(x_{b}\)-axis existed between the calibrated data points and the given data points, although the two types of the data points should coincide with each other. This argument is further illustrated in Table 3 by the value of the testing attitude angles. In Table 3, the values of the tool-face angle \(\varphi\) were different, but the calculation of the pitch angle \(\theta\) and azimuthal angle \(\psi\) would not be affected by different values of \(\gamma_{a}\) in the calibration approach. However, this case can be perfectly compatible with the MWD instrument, because the drawing of the borehole trajectory does not need the participation of the tool-face angle \(\varphi\).