## 3 MILs Detection
\[\Delta A_{i}=\sqrt{\sigma_{A}}=\sqrt{\sigma_{M_{i}}+\sigma_{L_{i}}} \tag{4}\]  
The thickness values, \(H_{i}\), is the altitude difference between \(L_{i}\) and \(M_{i}\):  
\[H_{i}=Z_{M_{i}}-Z_{L_{i}} \tag{5}\]  
For the present study, the nightly mean profiles from 30 to 90 km smoothed over 3 km and interpolated every 1 km are used. Thickness error is the summation of the square root of the squares of maximum altitude and squares of minimum altitude:  
\[\Delta H=\sqrt{(Z_{max})^{2}+(Z_{min})^{2}} \tag{6}\]  
Understanding MIL phenomena is important to the understanding of middle-atmosphere dynamics for two primary reasons: stability and energy transfer. At the bottom side of the thermal layers, where the temperature profile is increasing in altitude, the positive temperature gradient with increasing altitude signifies an increase in atmospheric stability and reduction in vertical mixing. In contrast, at the top side of the thermal layer, the negative temperature gradient with decreasing altitude implies a reduction in atmospheric stability to the point that the atmosphere may become convectively unstable, thus possibly supporting the development of turbulence. Therefore, the temperature gradient \(G_{i}\) with amplitude divided by the thickness was calculated, and the gradient standard error \(\Delta G_{i}/G_{i}\) is the square root of the squared amplitude error, \(\Delta A_{i}\), divided by amplitude, \(A_{i}\), and squared thickness error \(\Delta H_{i}\) divided by thickness \(H_{i}\):  
\[G_{i}=\frac{A_{i}}{H_{i}} \tag{7}\]  
Figure 2.0  
Our primary focus here is detecting the geometric characteristics of individual MILs from the vertical temperature profiles given by lidar observations. An example of determining MIL characteristics from the temperature profile is shown in Figure 3 and Figure 2.1 This example provides the different characteristics of describing inversions observed on 4 January 2000. Figure 3 shows a clear vertical temperature profile in the 30-90 km altitude range with detected MILs that are marked by red and pink triangles. The amplitude, thickness, and gradient has been calculated according to the mentioned formulas. Based on Table 1, more details of the calculation performed are available. For this case, the maximum value of the amplitude is 14.61 K, and the minimum one is 0.31 K. Importantly, the temperature profiles may be too noisy for considering the detected inversions as significant. In the following sections, the temperature data have been filtered to a vertical resolution of 2 km according to the techniques described in [24]. This has been performed to distinguish between significative MILs and perturbations associated with small-scale GWs and noise. Based on measurement noise, only the inversions from 62 to 66 km are very significative, while the one between 57 and 59 km is just above the threshold given by the noise-estimated level Figure 2.2 \begin{table}
\begin{tabular}{c c c c c c c} \hline \(Z_{max}\) (km) & \(M_{i}\pm\Delta M_{i}\) (K) & \(Z_{min}\) (km) & \(L_{i}\pm\Delta L_{i}\) (K) & \(A_{i}\pm\Delta A_{i}\)(K) & \(H_{i}\pm\Delta H_{i}\)(km) & \(G_{i}\pm\Delta G_{i}\) (K/km) \\ \hline
43.85 & 259.80 \(\pm\) 0.49 & - & - & - & - & - \\
59.37 & 225.48 \(\pm\) 1.84 & 57.40 & 221.49 \(\pm\) 1.45 & 3.98 \(\pm\) 2.35 & 1.96 \(\pm\) 0.71 & 2.03 \(\pm\) 2.80 \\
66.05 & 232.00 \(\pm\) 4.13 & 62.02 & 218.55 \(\pm\) 2.39 & 13.44 \(\pm\) 4.77 & 4.02 \(\pm\) 0.71 & 3.33 \(\pm\) 3.42 \\
69.48 & 225.27 \(\pm\) 6.06 & 68.50 & 224.95 \(\pm\) 5.36 & 3.01 \(\pm\) 8.09 & 0.98 \(\pm\) 0.71 & 0.32 \(\pm\) 2.34 \\
71.94 & 223.66 \(\pm\) 8.25 & 71.15 & 223.19 \(\pm\) 7.44 & 0.46 \(\pm\) 11.11 & 0.78 \(\pm\) 0.71 & 0.59 \(\pm\) 4.17 \\
78.13 & 227.51 \(\pm\) 19.81 & 74.89 & 212.89 \(\pm\) 11.63 & 14.61 \(\pm\) 22.97 & 3.24 \(\pm\) 0.71 & 4.50 \(\pm\) 8.50 \\ \hline \end{tabular}
\end{table} 
Table 1: Potential MIL anomalies on the vertical temperature profile derived from lidar observation on 4 January 2000.