## 3 Numerical Simulation Approach and Settings  
Our approach is based on direct numerical integration of the parabolic wave equation (the so-called wave-optics technique). This integration is performed using the split-step operator method [15; 16; 17], where the turbulence-induced refractive index perturbations along the propagation path of length \(L\) are represented by a set of thin, two-dimensional (2D), statistically independent phase screens. In our simulations, the number of phase screens \(M\) is varied from one for the single layer model to five for the distributed turbulence model. For the single layer model, the screens are positioned at five selected distances from the TILAS transceiver, while for the distributed turbulence model,  
Figure 1: (**a**) Schematic of the TILAS transceiver; (**b**) schematic illustration of the TILAS system operating with a target with a glint, moving with velocity \(\mathbf{v}_{T}.\)  
they are distributed equidistantly along the propagation path at distances \(z_{j}\ =\ \Delta z(2j-1)/2\), where \(\Delta z\ =\ L/M\) is the layer width and \(j=1,\ldots,M\). The latter case is illustrated in Figure 2 for \(M=5\). The turbulence strength is defined by the refractive index structure parameter \(C_{n}^{2}(z)\); the corresponding plane-wave Fried parameter computed over the distance \(L\) is given by \(r_{0}\ =\ \left\{0.423k^{2}\int_{0}^{L}C_{n}^{2}(z)dz\right\}^{-3/5}\)[18], where \(k\ =\ 2\pi/\lambda\) is the wavenumber. For the phase screen located at \(z\ =\ z_{j}\), the Fried parameter is correspondingly defined as \(r_{0}^{(j)}\ =\ \left\{0.423k^{2}\int_{z_{j}-\Delta z/2}^{z_{j}+\Delta z/2}C_{n}^{2}(z)dz \right\}^{-3/5}\)[19]. For the single layer model, we suppose that \(C_{n}^{2}(z)\ =\ \text{const}\) withing the layer of width \(\Delta z\) and zero otherwise, i.e., \(r_{0}\ =\ 1.68(k^{2}C_{n}^{2}\Delta z)^{-3/5}\ =\ r_{0}^{(j)}\). For the distributed turbulence model, we assume that the turbulence is homogeneous, which is a commonly used assumption for relatively short or nearly horizontal propagation paths [18]. In this case, \(C_{n}^{2}(z)\ =\ \text{const}\) along the entire path and \(r_{0}\ =\ 1.68(k^{2}C_{n}^{2}L)^{-3/5}\ =\ r_{0}^{(j)}M^{-3/5}\). For fair comparison of the results obtained for the single layer model with those for the distributed turbulence model, which corresponds to the same value of \(r_{0}\) for both models, the refractive index structure parameter values for the single layer model are \(M\)-fold higher than those for the distributed turbulence model. In our simulations of the single layer model, the layer width is fixed at \(\Delta z\ =\ L/5\) and the turbulence strength \(C_{n}^{2}\) is varied in the interval \(10^{-16}\,\text{m}^{-2/3}\leq C_{n}^{2}\leq 10^{-14}\,\text{m}^{-2/3}\). The Fried parameter is respectively ranging from 65 cm to 4.1 cm. For the distributed turbulence model, the corresponding \(C_{n}^{2}\) values are decreased five times. To simulate a target moving with the speed \(v_{T}\ =\ |\textbf{v}_{T}|\) relative to a stationary laser platform with the TILAS sensor, the position of the target is fixed but the \(j\)th phase screen is shifted in the opposite direction with the speed \(v_{j}\ =\ v_{T}z_{j}/L\). Over the time interval \(\Delta t\ =\ t^{\prime\prime}-t^{\prime}\), this corresponds to the shift distance \(\Delta s_{j}\ =\ v_{j}\Delta t\), as illustrated in Figure 2. Thus, during the same time, screens located at different positions along the path are shifted different distances. To model engagements involving moving targets, the technique for generation of infinitely-long phase screens [20] is used. The phase screens are assumed to obey the von [PERSON] power spectrum law; it allows us to vary the value of the turbulence outer scale \(L_{0}\). Note that the use of infinitely-long phase screens permits extending the range of \(L_{0}\) values beyond the numerical simulation area size (up to four times the size).