# Atmospheric Turbulence Study with Deep Machine Learning of Intensity Scintillation Patterns
## 7 Turbulence Models Evaluation via Machine Learning
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Inference Dataset** & **SIM-TA** & **SIM-VK** & **SIM-AN** & **SIM-NK** \\  & & & & **\(\alpha\) = 3.2/\(\alpha\) = 3.9** \\ \hline \(\overline{\varepsilon}_{\text{val}}\) & 0.077 & 0.091 & 0.093 & 1.16/0.45 \\ \(\sigma_{\varepsilon}\) & 0.079 & 0.083 & 0.091 & 0.54/0.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Averaged prediction errors \(\overline{\varepsilon}_{\text{val}}\) and their standard deviations \(\sigma_{\varepsilon}\) computed in the cross-dataset Cn’2 Net training-inference experiments with SIM-KO dataset used for training, and SIM-Tatarskii (TA), SIM-Von Karman (VK), SIM-Andrews (AN) and SIM-NK datasets for inference (\(C_{n}^{2}\) prediction). Figure 12: Comparison of \(C_{n}^{2}\) ground truth vs. prediction in the cross-dataset training-inference machine learning experiments with the Cn’2 Net training performed using the SIM-KO dataset corresponding to the Kolmogorov turbulence model and inference (\(C_{n}^{2}\) prediction) using the SIM- non-Kolmogorov (NK) dataset obtained for the non-Kolmogorov power spectrum model with \(\alpha\) = 3.2 (**a**) and \(\alpha\) = 3.9 (**b**). Grey-scale images in (**b**) illustrate exemplary scintillation patterns corresponding to \(C_{n}^{2}\) = 1:\(10^{-14}\)m\({}^{-2/3}\). Index \(m\) corresponds to scintillation frame stamp number \(m\) in the datasets. At the same time, DNN processing of scintillation images corresponding to the non-Kolmogorov turbulence power spectrum model (images from SIM-NK inference datasets) resulted in enormously large (\(>\)600%) errors. This mismatch between the ground truth and predicted \(C_{n}^{2}\) values based on processing of the SIM-NK inference datasets is clearly seen in Figure 12.  
The representative scintillation images in Figure 12 additionally demonstrate the clearly visible distinction in speckle pattern structures in the scintillation images corresponding to the Kolmogorov (\(\alpha~{}=~{}11/3\approx 3.66\)) and non-Kolmogorov (\(\alpha~{}=~{}3.2\) and \(\alpha~{}=~{}3.9\)) power spectrum models. In addition to comparison and evaluation of different atmospheric turbulence models, the machine learning concept can also be applied to analyze the impact of the parameters upon which these models may depend. Consider as an example DNN-based analysis of the turbulence inner scale impact on the spatial features of speckles in the scintillation images. The turbulence inner \(l_{0}\) and outer \(L_{0}\) scales are key parameters for the Von Karman and [PERSON] turbulence models (see Appendix A). For this analysis we generated a set of SIM-VK datasets corresponding to the Von Karman turbulence model with varied inner scale values ranging from \(l_{0}~{}=~{}1.0\) mm to \(l_{0}~{}=~{}16.0\) mm, and a fixed outer scale parameter \(L_{0}\) = 1.0 m. Cn'2 Net was trained using the SIM-KO dataset while the generated SIM-VK datasets with different \(l_{0}\) were used for the \(C_{n}^{2}\) true value predictions (inference). Results of the corresponding machine learning experiments are summarized in Table 5 by averaged prediction errors \(\overline{\epsilon}_{val}\). As expected, the smallest difference between ground-truth and predicted \(C_{n}^{2}\) values was achieved with the SIM-VK inference dataset computed for the smallest turbulence inner scale \(l_{0}~{}=~{}1.0\) mm corresponding to the numerical grid pixel size. This result is not a surprise since the DNN was trained with the SIM-KO dataset, while both the Von Karman and Kolmogorov spectra coincide for \(l_{0}~{}=~{}0\) and \(L_{0}~{}=~{}\infty\) (first row in Table 5). When \(l_{0}\) was increased, the accuracy in \(C_{n}^{2}\) prediction declined and the corresponding averaged prediction error \(\overline{\epsilon}_{val}\) increased. The rate in prediction error increase characterizes the sensitivity of spatial features in scintillations to the turbulence inner scale.