# Atmospheric Turbulence Study with Deep Machine Learning of Intensity Scintillation Patterns
## 5 DNN-Inspired Atmospheric Turbulence Characterization: Cn^2 Net Scintillometer
Note that this physics-imposed limitation can be overcome by utilizing for DNN-based data processing optical sensing characteristics that are more  
Figure 7: Results of the \(C_{n}^{2}\) prediction with deep machine learning performed for ATM#1-V (validation) (**a**,**b**) and ATM#2-V (**c**,**d**) datasets using preliminary trained Cn^2 Net models (\(N_{model}=20\)). The plots (**a**) and (**c**) compare measured (solid lines) and predicted (dots) \(C_{n}^{2}\) values dependent on the frame stamp number \(m\) in the corresponding datasets. The scatter plots in (**b**,**d**) characterize the standard deviation of prediction error for the entire range of \(C_{n}^{2}\) values in the corresponding ATM datasets (\(c_{n}^{2}=C_{n}^{2}/C_{n,0}^{2}\) and \(C_{n,0}^{2}=1\cdot\)10\({}^{-14}\)m\({}^{-2/3}\)). affected by atmospheric turbulence under weak scintillation conditions, such as short-exposure focal plane intensity patterns, turbulence-degraded images and wavefront sensing data. However, under strong scintillations (\(\sigma_{R}^{2}>1\)) an optical system (similar to as in Figure 1) composed of a remotely located laser beacon and optical receiver equipped with a camera capturing short-exposure intensity scintillation patterns, and DNN-based data processing hardware, can be utilized as a \(C_{n}^{2}\) sensor for atmospheric turbulence characterization, similarly to conventional scintillometers. For convenience, such an optical system with DNN processing of scintillation images is referred to here as a Cn^2 Net scintillometer. Prior to measurements (\(C_{n}^{2}\) prediction based on captured scintillation images), the Cn^2 Net scintillometer should be side-by-side trained and validated with a calibrated (\"trusted\") conventional \(C_{n}^{2}\) sensor. This DNN training-validation phase also provides estimation of \(C_{n}^{2}\) prediction error and defines the range of turbulence conditions for operation with an acceptable \(C_{n}^{2}\) prediction accuracy. Besides solely economic motivations (the Cn^2 Net scintillometer may cost a fraction of the cost of current commercial instruments), this DNN-based \(C_{n}^{2}\) sensor could also provide important new capabilities for atmospheric turbulence characterization. To further explore both capabilities and limitations of atmospheric turbulence characterization with DNN processing of intensity scintillation images, we compared the defined by Equation (3) root mean square errors (RMSE) \(\varepsilon_{val}(c_{n}^{2})\) for the datasets obtained in numerical simulations (SIM-KO-V) and atmospheric trials (ATM#1-V and ATM#2-V). The corresponding scatter plots are shown in Figure 8.  
The prediction errors averaged across the entire \(C_{n}^{2}\) range \(\overline{\varepsilon}_{val}\) and their standard deviations \(\sigma_{\varepsilon}\) computed using the data in Figure 8 are presented in Table 2.  
Both \(\overline{\varepsilon}_{val}\) and \(\sigma_{\varepsilon}\) are the smallest for the SIM-KO dataset. The prediction error \(\overline{\varepsilon}_{val}\) for this dataset is correspondingly 2.7-fold and 1.25-fold smaller in comparison with the ATM#1 and ATM#2 datasets. The fact that \(C_{n}^{2}\) prediction was more accurate for the ATM#2 dataset than the ATM#1 dataset is not surprising since the ATM#2 dataset contains significantly more instances (both \(C_{n}^{2}\) values and scintillation images) used for Cn^2 Net training than does the ATM#1 dataset. A different question is why Cn^2 Net training using a relatively small (10K) SIM-KO dataset resulted in a prediction RMSE that is 1.25-fold smaller in comparison with training on the ATM#2-T dataset that has a significantly larger number (56.4 K) of scintillation images? To address this question first notice that a single scintillation image of the SIM-KO dataset corresponds to a single \(C_{n}^{2}\) value, while in the ATM#1 and ATM#2 datasets a single measured \(C_{n}^{2}\) value corresponds, respectively, to 30 and 60 scintillation images that were captured between sequential \(C_{n}^{2}\) measurements.