# A Designed Calibration Approach for the Measurement -While-Drilling Instrument
## 4 The Experimental Verification
### Experimental Process and Result Analysis  
The whole calibration was performed in a laboratory without magnetic interference. In this process, the self-designed MWD instrument needed to be fixed on a tri-axial non-magnetic turntable which could provide reference attitude. We read the attitude angles both from the mechanical encoders of the non-magnetic turntable and the output of the self-designed MWD instrument. Then, we operated the non-magnetic turntable using the method mentioned in Section 2.5 and collected the output data of the MWD instrument at the specified attitude angles. The datasets \(\mathbf{D_{k}}\) (\(N=150\)) were obtained; of course, the data set \(\mathbf{D_{a-x}}\) (\(N=40\)) could be also extracted from \(\mathbf{D_{k}}\). By the calibration approach shown in Figure 3, the calibration matrix \(L_{k}\) and bias vector \(b_{k}\) were determined as shown in Table 6. After that, five sets of random attitude angles were chosen to check the accuracy of the parameters in Table 6. From the results, shown in Table 7, we found that the proposed calibration approach could completely calibrate the system error of MWD instrument but could not eliminate the influence of the random noise; the noise of the sensor could only be suppressed, not eliminated. \begin{table}
\begin{tabular}{c c c} \hline \multicolumn{3}{c}{**Resolved by the Calibration Approach**} \\ \hline
**Sensor Type** & \multicolumn{1}{c}{**The Calibration Matrix \(\mathbf{L_{ks}^{b}}\)**} & \multicolumn{1}{c}{**Bias \(\mathbf{b_{k}}\)**} \\ \hline Accelerometer (\(L_{ns}^{b}\) and \(b_{a}\)) & \(\begin{pmatrix}0.9971&-0.0038&0.0332\\ 0.0264&0.9851&-0.0567\\ 0.0144&0.0553&0.9974\end{pmatrix}\) & \(\begin{pmatrix}-0.0007\\ -0.0178\\ 0.0013\end{pmatrix}\) \\ \hline Magnetometer (\(L_{ns}^{b}\) and \(b_{m}\)) & \(\begin{pmatrix}1.1381&0.0013&-0.0001\\ -0.0058&1.0846&-0.0306\\ 0.0195&0.0396&1.1058\end{pmatrix}\) & \(\begin{pmatrix}0.7778\\ 2.5803\\ 0.6863\end{pmatrix}\) \\ \hline \end{tabular}
\end{table} 
Table 6: The calibration parameters determined by the method. Figure 10: Voltage curve diagram of the signal from HMC1021Z after a positive current pulse. In order to further verify the authenticity of the parameters in Table 6, a field test was conducted in a coal mine to draw the borehole trajectory map. The mathematical method for drawing trajectory is called the balanced tangent method which mainly depends on the attitude angles (pitch angle \(\theta\) and azimuthal angle \(\psi\)) at the measuring point of each drill pipe. Before the test, a pre-designed trajectory should be prepared. Then, we moved the MWD instrument along the prepared path and sampled at each measuring point at the same time. Finally, the measured data were uploaded to the software for drawing the trajectory map, which is shown in Figure 11 in the form of the upper-lower deviations and the left-right deviations. This deviations refer to the distance that the actual drilling trajectory deviates from the straight line of the initial direction (the attitude information of the orifice). The upper-lower trajectory deviations are mainly related to the difference between the pitch angle \(\theta\) at each measuring point and the pitch angle at the orifice position, and the left-right trajectory deviations are mainly related to the difference between the azimuthal angle \(\psi\) at each measuring point and the azimuthal angle at the orifice position. With this trajectory showing method, we can intuitively see how far the actual drilling trajectory deviates from the direction of the orifice. In Figure 11, the black line is the prepared path, and the black dots on the line are the points with known attitude information. The other lines are the trajectories drawn with the data uploaded from the self-designed MWD instrument. We can clearly see that the trajectory drawn with the data from the calibrated MWD instrument roughly coincides with the prepared path, and the trajectory from the un-calibrated MWD instrument deviates far from the prepared one.