## 3 Calculated and Measured Seeing: WRF Model and Solar Image Motion Observations  
In the paper, we estimate the structure constant of the air index refraction fluctuations \(C_{n}^{2}\) along the line-of-sight of the Large Solar Vacuum Telescope (LSVT), which is the main astronomical instrument of the observatory (Figure 7). We should recall that a detailed characterization of the optical turbulence profile permits realistic modeling, performance prediction and real-time validation and optimization of adaptive optics instruments [26; 27]. The integrated value of the constant allows us to estimate the image quality in terms of seeing. Atmospheric seeing represents the full width at half-maximum of the point spread function (PSF), the best angular resolution that an optical telescope can achieve in a long exposure image [28]. In order to verify the simulated characteristics, we compared the temporal changes of the seeing values derived from the WRF model and the measured values. Below, we present the main mathematical expressions used for calculation of the seeing parameter on the basis of WRF simulation data. The seeing parameter associated with the full width at half-maximum of PSF may be estimated using the formula [29]:  
\[seeing=\frac{0.98\lambda}{\left(0.423 seca\left(\frac{2\pi}{\lambda}\right)^{2 }\int_{0}^{H}C_{n}^{2}(z)dz\right)^{-3/5}}, \tag{3}\]  
where \(H\) is the height of the optically active atmosphere, \(\alpha\) is the zenith angle, \(\lambda\) is the light wavelength (\(\lambda\) = 500 nm). Parameter \(C_{n}^{2}\) is related to meteorological characteristics of the atmosphere [30; 31]:  
\[C_{n}^{2}(z)=aL_{0}(z)^{4/3}M(z)^{2}, \tag{4}\]  
where  
\[M=\left(\frac{-79\cdot 10^{-6}P}{T}\right)\frac{\partial ln\theta}{\partial z}, \tag{5}\]  
Figure 7: Large Solar Vacuum Telescope. \[L_{0}^{4/3}= \begin{cases}0.1^{4/3}\cdot 10^{1.64+42.5},&\text{troposphere}\\ 0.1^{4/3}\cdot 10^{0.506+50.5},&\text{stratosphere}\end{cases} , \tag{6}\]  
\[S=\left[\left(\frac{\partial u}{\partial z}\right)^{2}+\left(\frac{\partial v} {\partial z}\right)^{2}\right]^{0.5}, \tag{7}\]  
where \(L_{0}\) is the outer scale of turbulence, \(z\) is the height, \(P\) is the atmospheric pressure, \(T\) is the air temperature, \(\theta\) is the air potential temperature, \(a\) is the constant, \(u\) and \(v\) are the horizontal components of wind speed \(V(u,v,w)\). Thus, the vertical profiles of \(C_{n}^{2}\) are calculated using the classical Dewan model, which takes into account the outer scale of turbulence and the vertical gradients of air refractive index \(M\). Thus, the Dewan model is a convenient method for converting WRF-derived data into vertical profiles of \(C_{n}^{2}\), the refractive index structure constant, which is the critical parameter for describing optical turbulence. For the estimation of seeing, we also performed measurements of sunspot image motion at the Large Solar Vacuum Telescope (LSVT). The LSVT is a 760 mm telescope with elements of adaptive optics. Figure 8 shows a picture of the adaptive optics system of the Large Solar Vacuum Telescope. The optical layout of the system is simple. The system includes tip/tilt and deformable mirrors as well as a Shack-Hartmann wavefront sensor with different numbers of subapertures. In the Shack-Hartmann sensor, the incident wavefront is segmented by a microlens array and the formed spot displacements are measured. The measurements of image motion were taken using a Shack-Hartmann sensor sampled with six subapertures across the pupil. The main parameters of the Shack-Hartmann wavefront sensor are given in Table 2. \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline Aperture diameter & 760 mm \\ Used aperture diameter & 600 mm \\ Focal length & 40 m \\ Radiation wavelength & 535 nm \\ Number of subapertures & 6 \(\times\) 6 \\ Equivalent size of subaperture & 10.0 cm \\ Angular pixel size & 0.3 ′′/pix \\ Frame frequency & 100 Hz \\ Exposure & 30 ms \\ \hline \hline \end{tabular}
\end{table}