## 4 Numerical Simulation Results and Analysis  
### Correlation Properties of TILAS Received Power Signal  
Consider first the correlation properties of the TILAS received power signal \(P_{\mathrm{TILAS}}\) (\(t\)) for the single layer model, where the turbulence layer of a fixed strength is placed at different distances from the TILAS transceiver. Figure 3a shows the time series of \(P_{\mathrm{TILAS}}\) (\(t\)) computed over a 10-ms time interval with 0.1-ms sampling time for two different relative positions of the layer between the TILAS transceiver and the moving target, \(z/L=0.1\) (closer to TILAS) and \(z/L=0.9\) (closer to the target), characterized by the same Fried radius \(r_{0}=4.1\) cm and turbulence outer scale \(L_{0}=4\) m. As can be seen from Figure 3a, the signal obtained for the case of the turbulence layer located closer to the transceiver (blue line) is changing slower than that for the layer located near the target (sky-blue line). This can be explained by the fact that the turbulence-induced refractive index inhomogeneities inside the propagating beam footprint are updated faster when the layer is located farther from the TILAS transceiver aperture, or closer to the moving target. This happens because during the same time interval, the laser beam following the target would scan sections of different length for layers positioned at different distances from TILAS (see Figure 2).  
Typical dependencies of the autocorrelation functions \(B\) (Equation (2)) on the time delay \(\tau\) computed for five different turbulence layer positions ranging from \(z=0.1\)\(L\) to \(z=0.9\)\(L\) with the step of 0.2\(L\) are shown in Figure 3b. Note that the autocorrelation functions \(B(\tau)\) have Gaussian-like shape, which allows us to directly determine their width by the drop in the function level by a factor \(\exp(-1)\approx 0.37\) compared with \(B(\tau=0)\). This width is referred to here as the correlation time, \(\tau_{\mathrm{cor}}\). As seen, the correlation time depends on the position of the turbulence layer along the propagation path from the TILAS transceiver to the moving target. The autocorrelation function corresponded to the turbulence layer located near the moving target is characterized by the smallest correlation time while \(B(\tau)\) related to the layer near the transceiver has the largest \(\tau_{\mathrm{cor}}\).  
Consider now the distributed turbulence model characterized by the same Fried radius equal to 4.1 cm (\(C_{n}^{2}~{}=~{}2\times 10^{-15}\) m\({}^{-2/3}\)). The corresponding autocorrelation function is shown in Figure 3b by the dash-dotted black line. As can be seen from the figure, the distributed turbulence correlation time defined at \(B(\tau)\approx 0.37\) lies between \(\tau_{\rm cor}(z/L~{}=~{}0.7)\) and \(\tau_{\rm cor}(z/L~{}=~{}0.5)\) defined using the single layer model. However, the distributed turbulence autocorrelation function has a unique shape with a specific slow-decline region at the function's tail (a 'pedestal' or a 'base'), which is distinct from the Gaussian-like shape of the autocorrelation function computed for the single layer model. This means that the presence of such a base can serve as an indication of the turbulence being distributed, or at least of the presence of more than one turbulence layer along the propagation path.