## 8 Experiments  
Aiming to further verify the proposal algorithm, a vibration experiment was implemented with a ready-made FOGs-based MWD. The accuracy of FOGs in MWD system is 0.03 \({}^{\circ}\)/h. The vibration experiment is carried out at the high temperature of 65 \({}^{\circ}\)C. The x-axis FOG is mounted along with the vibration direction of the vibration platform. The PSD of the vibration is reported in Figure 11. Its vibration level is high, which simulates the drilling vibration underground. When the vibration frequency is between 700 Hz and 800 Hz, the PSD of the vibration is up to 1 g\({}^{2}\)/Hz. The root-mean-square value of the whole vibration is 13.39 g.  
The vibration experiment was conducted via the following steps. First, the vibration platform was kept static (0-330 s). Secondly, the vibration platform began to vibrate and kept vibrating for 5 min (330 s-630 s). Finally the vibration platform returned to static state (630 s-930 s). The movement of the MWD system was in accordance with the movement of the vibration platform. We collected the output signal of the x-axis FOG in the MWD system. The sampling interval is 2.5 ms. Hence, the total value of this gyro data is \(37.2\times 10^{4}\). The primary signal is shown in Figure 12. It shows that the platform began to vibrate at \(t=330\) s and stopped at \(t=630\) s. In addition, we must pay attention to a piece of lost data when 432 s \(<t<450\) s.  
Figure 11: PSD of the random vibration.  
Figure 10: The fast DAVAR of the discontinuous data. (**a**) The improved fast DAVAR of \(x_{3}[n]\); (**b**) Angle random walk of \(x_{3}[n]\).  
Then the improved fast DAVAR has been applied to analyzing this discontinuous vibration data. Its result is obtained with a truncation window of length \(N_{W}=2000\) samples, and a step width of 1000 samples. The Matlab program calculation is carried out on an Intel (R) Core (TM) i7-3770 CPU with a 3.4 GHz clock. The improved fast DAVAR costs only 28.616520 s to deal with this long time series.  
Figure 13 is the improved fast DAVAR result. It can be seen that the DAVAR surface is stationary at the beginning. Then there appears a large crest that starts at \(t=330\) s and stops at \(t=630\) s. In the end it goes back to being stationary. Obviously, the canyon (430 s \(<t<450\) s) is the graphical representation of the missing data in the time series. The changing process of the DAVAR surface is consistent with Figure 12. In conclusion, the improved fast DAVAR could track and reveal the non-stationary characteristics in a clear way.  
Fitting the double logarithmic curve \(\sigma^{2}(t,\tau)-\tau\) at any given time \(t\), the time-varying coefficients of noise terms can be obtained [27; 28]. Figure 14 shows the changing coefficients of each noise terms evaluated by the improved fast DAVAR.  
Aside from the canyon at \(t=432\) s, it is easy to distinguish the change in the noise. Before vibration (\(t<330\) s), since the FOG is static, the coefficient of each noise is small without evident change. When the platform starts to vibrate (\(t=432\) s), the noise terms of the FOG change dramatically, especially the Rate Random walk, which is affected by the vibration. Meanwhile, the coeffici  
Figure 12: Output signal of x-axis FOG.  
Figure 13: The improved fast DAVAR of the vibration data.  
noise increases when the FOG is vibrating. The data acquisition circuit of FOG has better aseismic performance. After vibration, all confidents of noise terms are back to the previous state. We can make a conclusion that the FOG could endure the drilling vibration. In conclusion, the improved fast algorithm of the DAVAR could successfully analyze the discontinuous gyroscope data. The improved fast DAVAR identifies and reveals the highly dynamic instability in the FOG's discontinuous time series.