## 3 Calculated and Measured Seeing: WRF Model and Solar Image Motion Observations
The complex allows us to measure \(C_{n}^{2}\) and vertical gradients of the horizontal components of wind speed. Using formulas (1)-(4), we compared the values of \(C_{n}^{2}\) calculated on the basis of the weighted vertical wind speed gradients and the measured values of \(C_{n}^{2}\). Figure 12 shows the temporal changes in the calculated and measured values of \(C_{n}^{2}\). The calculated values of \(C_{n}^{2}\) are obtained with the coefficients proposed by [PERSON] (Figure 12a). For convenience, we denote the coefficients as \(a_{Ltsp}=1.64\) and \(b_{Ltsp}=42\) s. Table 3 shows the statistics, including the mean absolute error (\(MAE\)), the standard deviation (\(STD\)) and the linear Pearson correlation coefficient (\(K_{corr}\)). As we can see, we obtain the Pearson correlation coefficient of 0.88, indicating a high degree of linear correlation between calculated and measured values of \(C_{n}^{2}\) for coefficients \(a_{Ltsp}=2.5\) and \(b_{Ltsp}=8\) s.  
The seeing values estimated from the WRF model are also shown in this table. The minimum values of \(MAE\) and \(STD\) between calculated and measured \(C_{n}^{2}\) correspond to coefficients \(a_{Ltsp}=2.5\) and \(b_{Ltsp}=8\) s. The calculated values of \(C_{n}^{2}\) are obtained with the coefficients proposed by [PERSON] (Figure 12a) and new coefficients (Figure 12b) selected by minimizing \(MAE\) and \(STD\). In the troposphere, the analysis of statistical characteristics shows that with an increase in \(a_{Lisp}\) and a decrease in \(b_{Lisp}\), the reproducibility of \(C_{n}^{2}\) variations improves (the correlation coefficient increases; \(MAE\) and \(STD\) decrease). At the same time, we must emphasize that an increase in the coefficient \(a_{Lisp}\) leads to deformations of the vertical profiles of optical turbulence: optical turbulence strength in the upper atmosphere increases significantly. Thus, the calculated profiles of \(C_{n}^{2}(z)\) were obtained with new coefficients chosen by minimizing the mean absolute error and the standard deviation between the calculated and measured values \(C_{n}^{2}\). We propose to use the following model to determine the vertical profiles of the outer scale of turbulence at the BAO site:  
\[L_{0}^{4/3}= \begin{cases}0.1^{4/3}\cdot 10^{2.5+85},\,AtmosphericBoundaryLayer\\ 0.1^{4/3}\cdot 10^{1.64+425},\,FreeAtmosphere\\ 0.1^{4/3}\cdot 10^{0.506+505},\,Stratosphere\end{cases} , \tag{12}\]  
Figure 13 demonstrates averaged daytime profiles of optical turbulence at the BAO site. The profile shown by the orange line is obtained from the well-known Hufnagel-Valley model, which is determined by the average surface value of \(C_{n}^{2}\) and by pseudovelocity calculated for the atmospheric layer from 5 km to 20 km [36; 37]. To estimate the surface values of \(C_{n}^{2}\), we used data from mast measurements of turbulent characteristics (sonic  
Figure 12: Temporary changes of calculated and measured values of \(C_{n}^{2}\). The orange markers correspond to the measured values of \(C_{n}^{2}\). The blue markers correspond to the calculated values of \(C_{n}^{2}\). (**a**) \(a_{Lisp}=1.64\), \(b_{Lisp}=42\) s. (**b**) \(a_{Lisp}=2.5\), \(b_{Lisp}=8\) s.  
anemometers). Pseudovelocity was estimated according to Era-5 reanalysis data. The mean value of the pseudovelocity used in simulations was 23.7 m/s. Analyzing \(C_{n}^{2}\) vertical profiles, we can note that the image quality is largely determined by the lower atmospheric layer and the turbulent layer at heights of 10-12 km. Compared to the Hufnagel-Valley model, the calculated profile \(C_{n}^{2}\) is characterized by suppressed atmospheric turbulence in the lower layer up to a height of about 2 km. The WRF-derived profile \(C_{n}^{2}\) also demonstrates a significant increase in the energy of optical turbulence at altitudes of 10-12 km.