# Atmospheric Turbulence Study with Deep Machine Learning of Intensity Scintillation Patterns
## 3 Wave-Optics Numerical Simulations of Atmospheric Sensing Trials: SIM Datasets  
In the machine learning experiments described in the proceeding sections we utilized several auxiliary datasets that were obtained using numerical simulations (SIM datasets). The simulations were performed using GPU-enhanced WONAT software [18]. A schematic illustration of the numerical simulation setting is illustrated in Figure 3. In the simulations we mimic the experimental trial propagation geometry and major parameters of the laser beacon and optical receiver modules. Propagation of a monochromatic collimated Gaussian laser beam (beacon beam) over 7 km was simulated using the conventional split-step operator (also referred to as wave-optics) technique [19; 20]. The turbulence-induced refractive index random fluctuations were assumed to be statistically homogeneous, isotropic and obey one of the following refractive index power spectrum models (see Appendix A): Kolmogorov, Tatarskii, [PERSON], Andrews and non-Kolmogorov. The corresponding SIM datasets are referred to as SIM-KO, SIM-TA, SIM-VK, SIM-AN and SIM-NK. The refractive index structure parameter \(C_{n}^{2}\) was assumed to be constant along the propagation path in most of the machine learning experiments described. To match parameters of the experimental measurement trials, the physical pixel size of the numerical grid (\(1024\times 1024\)) was set to either \(\Delta_{1}~{}=~{}1.02\) mm (SIM#1) or \(\Delta_{2}~{}=~{}1.41\) mm (SIM#2). These pixel sizes correspond to the pixel sizes in the scintillation images in the ATM#1 and ATM#2 datasets collected using optical receivers with two different aperture diameters (\(D\) = 11 cm and \(D\) = 14.4 cm). Each SIM dataset was composed of \(M_{SIM}\) = 15,000 (15K) scintillation images \(\left\{I_{SIM}(\mathbf{r}_{i,j},m)\right\}\) and the corresponding refractive index structure parameter values \(\left\{C_{n}^{2}(m)\right\}\), where \(\left\{\mathbf{r}_{i,j}\right\}\) (\(i,j\) = 1,..., 128) are coordinates of numerical grid pixels, and \(m\) =1,..., \(M_{SIM}\) is the image frame stamp. The scintillation images were obtained by computing intensity distributions \(\left\{I(\mathbf{r}_{i,j},m)\right\}\) (\(i,j\) =1,..., 1024) at the optical receiver plane and selecting the central \(128\times 128\) pixel numerical grid area. To mimic refractive index structure parameter diurnal variation with a simple model, the following dependence of \(C_{n}^{2}(m)\) (ground truth curve) on the scintillation frame number \(m\) was used:  
\[C_{n}^{2}(m)~{}=~{}C_{n,0}^{2}[\cos(3\pi m/M_{SIM})+1]+b, \tag{1}\]  
where parameters \(C_{n,0}^{2}~{}=~{}2.0\cdot 10^{-14}\) m\({}^{-2/3}\) and \(b~{}=~{}1.0\cdot 10^{-17}\) m\({}^{-2/3}\) were selected to match the overall range of \(C_{n}^{2}\) change observed during the experimental trials (see Figure 2). For each \(C_{n}^{2}(m)\) value the turbulence-induced refractive index variations along the propagation path were represented by a set of \(N_{\varphi}~{}=~{}10\) thin, equidistantly distributed along the path, statistically independent random phase screens \(\left\{\varphi_{l}(\mathbf{r}_{i,j},m)\right\}\), where \(l~{}=~{}1,\ldots,N_{\varphi}\) and \(i\); \(j\)=1,..., 1024. The phase screens were generated using one or another of the turbulence power spectrum models mentioned  
Figure 3: Notional schematic of the numerical simulation setting used for computation of laser beam intensity scintillation images and corresponding refractive index structure parameter \(C_{n}^{2}\) (simulation (SIM)-datasets) over 7 km under different atmospheric turbulence conditions. above. To simplify notations, a single set of \(N_{\varphi}\) phase screens \(\left\{\varphi_{l}(\tau_{i,j},m)\right\}\) is referred to here as the \"\(m\)-th turbulence realization.\" The phase screens in all \(M_{SIM}\) turbulence realizations were statistically independent.