## 5 Statistical Analysis of Temperature Inversion Layers  
This study has used 4366 OHP nightly Rayleigh lidar observations between 1978 and 2019 to detect MILs. This is a much more robust dataset than has been used in any previous studies of MILs using lidar. Following the MIL detection criteria described in the previous section, 3194 significant inversion layers are detected, and 539 of those layers satisfied the gravity wave \(E_{p}\) criteria in this study. The occurrence distributions of the MIL amplitude, bottomside altitude, layer thickness, and temperature gradient were calculated for each three-month period of summer, winter, spring, and autumn. Figure 4 represents four groups of histograms of the distributions of the MIL's values to compare seasonal evolutions from 1978 to 2019. In the first row of Figure 4, we can see that the peak amplitudes of the MILs during winter have a broad, skewed distribution with a maximum near 66 to 70 km. In contrast, the summer maximum is smaller, more narrow, and higher than in the winter, near 69 to 73 km. There does not appear to be a coherent distribution in spring and autumn, giving the reader the visual impression that these periods seem to be transition periods between winter and summer. The absolute number of MIL occurrences measured during the winter months is greater than that measured during the other seasons (see also Figure 5). In the second row of Figure 4, we can see that the amplitude of the temperature deviation of the MIL also has the largest variability in the winter. Most MILs have amplitudes less than 25 K, but a few can grow as large as 50 K. The spring, summer, and autumn show very similar distributions and are notably different from the winter. In the third row of Figure 4, we can see that the thickness of the MIL follows the same winter vs. spring-summer-autumn pattern as for the MIL amplitude. Most MILs are thin structures having average vertical extents of 2.5 km; however, some extreme winter layers have a thickness of 8 to 10 km. In the fourth row of Figure 4, we can see that the gradient of the MIL also follows the same pattern as the previous two MIL variables. If we assume that a typical mesospheric temperature lapse rate might be around 3 K/km, then we can see that summer and spring might have broad normal distributions about the environmental lapse rate. However, it appears that autumn and winter have skewed distributions indicating that complex energy dissipation is happening in these layers. We can compare our results to a similar study based on 119 nights of Rayleigh lidar observations over an equatorial site from March 1998 to February 2000, [PERSON] et al. [36]. They showed that the peak occurrence altitudes were between 73 and 79 km (n.b., these authors defined the altitude of the maximum temperature deviation instead of the altitude of the bottomside of the layer, as has been performed in this study). Adjusted for an average MIL thickness of 3-4 km, this result agrees well with our winter results. These authors found the temperature amplitude of the MILs ranged between 12 and 32 K, which again agrees with our winter results. Figure 4: Histogram of the occurrence of MIL characteristics (amplitude, minimum altitude, thickness, and gradient) by season. The inversion frequency is defined as the fraction of the total number of temperature profiles in the month exhibiting one or more inversions. Considering the whole lidar dataset of observations, the frequency of MIL occurrence in winter months (DJF) is much higher than in other seasons. According to Table 2, the highest MIL detection ratio is observed during the winter months of December, January, and February (19.07%, 29.18%, and 18.71%, respectively). In contrast, the minimum occurs in the late spring and early summer months of April, May, and June (4.33%, 6.25%, and 6.13%, respectively). March as well as late summer and autumn appear more transitional, having neither maxima nor minima.