# A Designed Calibration Approach for the Measurement -While-Drilling Instrument
## 3 Analysis and Discussion of the Calibration Approach
In addition, it should be mentioned that a total of 77 sets of data points (N = 77) were used to estimate the error parameters when using the improved EF method on the two conditions of \(\varepsilon_{stop}=10^{-9}\) and \(N_{min}=70\). Secondly, a random noise was added in the collected data and the testing data. Thus, a noise suppression method was also needed. This noise suppression method was realized by obtaining average values of the multiple samples. That is, the MWD instrument continuously sampled several times and outputted the mean of these sampling data when this instrument is stable at a certain attitude. This method was used to improve the calibration approach by changing the numbers of sampling times from a single time to several times at each attitude. Of course, the sampling times of the MWD instrument were also transformed from a single to several times in actual measurement. In the following simulation, the estimated results, by the calibration approach with the noise suppression method and without the suppression method, are compared in Table 4 on the same condition of \(\gamma_{a}=0^{\circ}\). The number of sampling times during the collection of the datasets \(D_{k}\) was set as 16 times at each attitude in the improved calibration approach. Then, the collection of the 15 sets of  
Figure 6: The test results on the condition of \(\gamma_{a}=0^{\circ}\) and \(\gamma_{a}=5^{\circ}\). **(a,b)** The test results of tri-axial accelerometer and tri- axial magnetometer, respectively, on the condition of \(\gamma_{a}=0^{\circ}\). **(c,d)** The test results on the condition of \(\gamma_{a}=5^{\circ}\). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline \multicolumn{10}{c}{**Cleulating Attitude with the Error Parameters in Table 2 (Unit: \({}^{\circ}\))**} \\ \multicolumn{10}{c}{**Clentitic (Unit: \({}^{\circ}\))**} \\ \hline \(\boldsymbol{\psi}\) & \(\boldsymbol{\theta}\) & \(\boldsymbol{\varphi}\) & \(\boldsymbol{x_{m}}\) & \(\boldsymbol{y_{m}}\) & \(\boldsymbol{z_{m}}\) & \(\boldsymbol{x_{e}}\) & \(\boldsymbol{y_{e}}\) & \(\boldsymbol{z_{a}}\) & \(\boldsymbol{\psi}\) & \(\boldsymbol{\theta}\) & \(\boldsymbol{\varphi}\) & \(\boldsymbol{\psi}\) & \(\boldsymbol{\theta}\) & \(\boldsymbol{\varphi}\) \\ \hline
30 & \(-80\) & \(-150\) & \(82.933\) & \(41.3127\) & \(13.4254\) & \(1.262\) & \(-0.1243\) & \(-0.442\) & \(30\) & \(-80\) & \(-140.575\) & \(30\) & \(-80\) & \(-145.6115\) \\
75 & \(-54\) & \(-90\) & \(-61.6012\) & \(-40.3344\) & \(-21.4979\) & \(0.9285\) & \(-0.5982\) & \(-0.1586\) & \(75\) & \(-54\) & \(-80.5751\) & \(75\) & \(-54\) & \(-85.6115\) \\
240 & 18 & 45 & \(-27.6695\) & \(93.0142\) & \(13.2316\) & \(-0.2186\) & \(1.0809\) & \(0.7251\) & \(240\) & \(18\) & \(544.249\) & \(240\) & \(18\) & \(94.3885\) \\
330 & 54 & 120 & \(-16.8204\) & \(66.5614\) & \(-23.4914\) & \(-0.703\) & \(0.6639\) & \(-0.6725\) & \(330\) & \(54\) & \(129.4249\) & \(330\) & \(54\) & \(124.3885\) \\
350 & 75 & 150 & \(-45.071\) & \(33.2847\) & \(-27.6866\) & \(-0.9773\) & \(0.2939\) & \(-0.5187\) & \(350\) & \(75\) & \(159.4249\) & \(350\) & \(75\) & \(154.3885\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The comparison of attitude angles before and after calibration. a given attitude also employed this noise suppression method by increasing the number of sampling times from one to eight times at each attitude, and five sets of them are shown in Table 5. In this Table, the calibrated results after the suppressed noise method and non-suppressed noise method were compared. Therefore, this noise suppression method is effective. It should be stated that such a setting of the sampling times not only meets the operational requirements of the MWD instrument, but also facilitates MCU programming. We also emphasize that if the \(N_{min}\) is set as 90 and \(\varepsilon_{stop}\) is set as \(10^{-4}\), the number of sampling data points may be decreased to 95. This number (less than the total number 150) indicates that the improved EF method is more effective than the original one.