## 3 MILs Detection  
Previous studies [PERSON] et al. [2] already show mesospheric temperature inversions with the OHP lidar and the first statistical characteristics of the mesospheric inversion. In the middle and lower mesosphere, MILs occur at altitudes ranging from 55 to 75 km in winter and 70 to 83 km in summer. Some specific characteristics of MILs such as amplitude and thickness can appear to be similar to other temperature perturbation forcing such as gravity waves or planetary waves. However, MILs tend to persist for several hours to a few days, their vertical extent spans several kilometres, and they can have significantly larger amplitudes compared with gravity waves [32]. However, the exact range of these characteristics is not well-defined. In the examples provided in Figure 1, such a complex issue is obvious. In addition, several MILs can be observed simultaneously. The amplitude of an inversions layer was defined by [PERSON] and [PERSON] [5] as the temperature difference between the top and bottom of the inversion layer. The thickness of the inversion layer is the difference between the altitude of the temperature at the top and the altitude of the temperature at the bottom. Therefore, the top and bottom of the inversion layer by the maximum and minimum temperature peaks need to be defined. The bottom part of the  
Figure 1: Selected examples of vertical temperature profiles from the OHP lidar since 1982. Perturbations, potentially identified as MILs, appear at different levels and altitudes of the bottom part of the MIL are indicated with an arrow. According to noise and gravity wave mean amplitude, 2 cases were rejected as significative MILs: 6 January 1984 and 24 August 2001. MIL is defined as the closest minimum temperature \(L_{i}\) of the \(i\)th layer, while the top part is defined by the maximum temperature, \(M_{i}\) as illustrated in Figure 2.  
The retrieval of the characteristics of the temperature inversions consists of identifying the minimum and maximum peaks on the temperature graph named as \(L_{i}\) and \(M_{i}\), respectively. To extract the local extremum, a smooth 'background' temperature profile is subtracted from the measured temperature profile. The method for estimating temperature extremum is using the derivative of the temperature as a function of altitude, \(\overline{T}=\frac{dT}{dz}\) with \(z\) considered as the altitude. By horizontal tangent of the temperature graph, the graph of the derivative of the temperature crossing the zero axis \(\frac{dT}{dz}=0\). In the next step, detecting all the maximum and minimum peaks from a temperature profile, the amplitude values, \(A_{i}\), are calculated from the difference between the temperature at the top, \(M_{i}\), and the temperature at the bottom, \(L_{i}\), of the temperature graph:  
\[A_{i}=M_{i}-L_{i} \tag{1}\]  
The errors of the maximum temperature and minimum temperature can be used to calculate the amplitude error, \(\Delta A\). This estimated error is a quadratic calculation, and accordingly, sigma amplitude is the summation of sigma maximum temperature and sigma minimum temperature. Indeed, the error amplitude, \(\Delta A\), is the square root of the sigma amplitudes, and sigma temperatures are the square of Lidar errors, \(\Delta Lidar\):  
\[\Delta T=(\Delta Lidar)^{2} \tag{2}\]  
\[\sigma_{A}=\sigma_{T_{\rm max}}+\sigma_{T_{\rm min}}=\sigma_{M_{i}}+\sigma_{L _{i}} \tag{3}\]  
Figure 2: A sample middle atmospheric temperature profile obtained at OHP on 16 December 1982, illustrating the detection of MIL and their characteristics. The \(L_{i}\) and \(M_{i}\) correspond to the altitudes of minimum and maximum temperature for the evident \(i\)th structure in the data. The thickness (km), the amplitude (Kelvin), and the temperature gradient of the MIL retained are in italics in this case, with several large structures.