# Atmospheric Turbulence Study with Deep Machine Learning of Intensity Scintillation Patterns
## 5 DNN-Inspired Atmospheric Turbulence Characterization: Cn^2 Net Scintillometer  
The ability of Cn^2 Net to operate with experimentally captured data (ATM datasets) was evaluated in a set of machine learning experiments. Prior to DNN-based data processing, each ATM dataset was subdivided into two subsets composed of scintillation images and \(C_{n}^{2}\) measurements representing the full range of \(C_{n}^{2}\) values observed during the atmospheric trials as described in Section 2. This subdivision was performed by assigning odd and even scintillation image frames to correspondingly training (ATM-T) and validation (ATM-V) data subsets. Recall that during collection of the ATM datasets, the camera of the optical receiver was capturing \(N_{\Delta T}\) short exposure scintillation images (\(N_{\Delta T}~{}=~{}60\) for ATM#1 and \(N_{\Delta T}~{}=~{}120\) for ATM#2) between each two subsequent \(C_{n}^{2}\) measurements. Half of these images were used for Cn^2 Net model training while the remaining images were utilized for \(C_{n}^{2}\) prediction (validation). Note that the validation data subsets (ATM#1-V and ATM#2-V) were composed of images never seen during the Cn^2 Net model training. The results of DNN processing using the preliminary trained Cn^2 Net models (\(N_{model}\) =20) are presented in Figure 7. The \(C_{n}^{2}\) \"measurement-vs.-prediction\" plots in Figure 7a,c provide a comparison of measured and predicted \(C_{n}^{2}\) values for the entire range of turbulence conditions observed during the experimental trials. The prediction error standard deviations \(\sigma_{val}(c_{n}^{2})~{}=~{}\varepsilon_{val}(c_{n}^{2})/c_{n}^{2}\) for both ATM#1-V and ATM#2-V datasets are shown in Figure 7b,d as functions of the normalized refractive index structure parameter \(c_{n}^{2}~{}=~{}C_{n}^{2}/C_{n,0}^{2}\). For each \(C_{n}^{2}\) measurement, the Cn^2 Net processed \(N_{\Delta T}\)/2 images and computed equal number of the averaged over DNN models \(C_{n}^{2}\) predictions that are shown in Figure 7 by dots (prediction dots). The density of the prediction dots is noticeably higher for the ATM#2 dataset (Figure 7c,d) that has twice as many scintillation images per each \(C_{n}^{2}\) measurement and almost three times more \(C_{n}^{2}\) data points compared with the ATM#1 dataset. The density of prediction dots in Figure 7b,d reflects the frequency of \(C_{n}^{2}\) data point occurrence in the corresponding dataset. Note that the prediction error standard deviation \(\sigma_{val}\) is significantly higher for relatively weak turbulence (\(C_{n}^{2}<0.2\)-\(C_{n,0}^{2}\)). Under these conditions the characteristic size of turbulence-induced speckles approaches or even exceeds the optical receiver aperture size (see Figure 4, 1 eft column) leading to a reduction in spatial features manifold of the scintillation images processed by DNN and resulting in \(\sigma_{val}\) increase. A similar effect was observed with DNN processing of the SIM dataset (see Figure 6b). The prediction error standard deviation inside the weak turbulence region can in principal be reduced by increasing the optical receiver aperture size. However, under weak turbulence and/or for relatively short propagation distances (even in strong turbulence), intensity scintillations diminish and hence cannot be utilized for accurate characterization of turbulence strength. The intensity scintillation level is commonly characterized by the Rytov variance \(\sigma_{R}^{2}=1.23C_{n}^{2}\)\(k^{7/6}\)\(L^{11/6}\), where \(k=2\pi/\lambda\)[29]. For the experimental setting used for ATM dataset collection (see Section 2), \(\sigma_{R}^{2}=10.9C_{n}^{2}\). The results in Figure 7b,d suggest that accurate (within 10-15%) \(C_{n}^{2}\) prediction using Cn^2 Net can only be achieved under conditions of relatively strong scintillations with Rytov variance ranging from \(\sigma_{R}^{2}\sim 1\) to \(\sigma_{R}^{2}\sim\) 20.