## 4 MTF Model and Numerical Analysis  
Turbulent atmosphere has serious degrading effects on the quality of imaging system, and this impact can be described by the long-exposure MTF. The MTF under atmospheric turbulence can be derived from the mutual coherence function [30]. It is evaluated as the product of the MCF at \(\rho=\lambda Fv\), where \(v\) is the spatial frequency and F is the focal length of the receiver aperture[1]. The function \(D(\lambda Fv,L)\) represents the wave structure function that is given by Equation (24). It is given, as follows:  
\[MTF_{turb}(\mu,\alpha,\zeta)=\exp[-\frac{1}{2}D_{\omega}(\mu Fv,L)], \tag{25}\]  
With analysis purpose, we normalized the spatial frequency by the aperture diameter D in the form of \(\mu=\lambda Fv/D\). The MTF model has been studied in previous works by [PERSON] [30] and [PERSON] [26]. The model uses the normalized spatial frequency \(\mu\) to analyze the degrading effects of the turbulent atmosphere on the imaging system. The MTF model for the turbulent atmosphere is defined, as follows:  
\[MTF_{turb}(\mu,\alpha,\zeta)=\exp[-\frac{1}{2}D_{\omega}(\mu D,L)], \tag{26}\]  
where \(D_{\omega}(\mu D,\alpha,\zeta)\) is the WSF of Gaussian-beam waves in Equation (24); D is the aperture diameter at the receiver plane. Using Equation (25), we perform the numerical analysis to obtain the MTF of Gaussian-beam waves propagation through the turbulent terrestrial and maritime atmosphere. The Figure 3 shows the  
Figure 1: Wave structure function (WSF) of a collimated beam at transmitter as a function of \(\Lambda_{0}\) for varying: (**a**) power exponent \(\alpha\) and (**b**) anisotropic parameter \(\zeta\). Figure 2: WSF of a collimated beam as a function of increasing turbulence \(\sigma_{R}^{2}\) for varying: (**a**) power exponent \(\alpha\) and (**b**) anisotropic parameter \(\zeta\). long-exposure MTFs as a function of normalized spatial frequency for the collimated Gaussian-beam waves (\(\Theta_{0}=1\)) with the propagation path length \(L=1\) km and the aperture diameter \(D=30\) cm are used. This set of simulations is conditioned on the WSF that is illustrated in Figures 1 and 2. Figure 3 shows the behavior of the long-exposure MTF on the normalized spatial frequency \(\mu\) with various power law exponent \(\alpha\) and anisotropic parameter \(\zeta\) under the terrestrial and maritime atmospheric turbulence. From the study of propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence, the power law exponent \(\alpha=3.4\) corresponds to the free troposphere layer, \(\alpha=11/3\) corresponds to the boundary layer, and \(\alpha=3.8\) denotes the lower stratosphere layer under the stable environment [31]. The power law exponent is one of major influencing factor that can affect the imaging system, as shown in Figure 3a. We infer that the turbulence effects get stronger when the Gaussian-beam waves propagate through the free troposphere layer, thus we observe a stronger degradation in Figure 3a. The MTF curves as a whole can be viewed as that of a low-pass filter and they show a better transfer characteristic as the power law exponent \(\alpha\) increases. Figure 3b indicates the influence of the anisotropic parameter. As it shows the anisotropic parameter is also another factor affecting the MTF of the optical waves used in the imaging system. As the anisotropic parameter increases, the imaging system can maintain a smaller degradation than the one in isotropic turbulence. The result comes from the mechanism of eddies that work as lenses with a larger radius of curvature in the anisotropic turbulence. The large curvature lenses with the bigger anisotropic factors can focus the optical wave better than the one in isotropic turbulence [32]. Figure 3 also shows the MTFs under the maritime atmospheric turbulence. The maritime MTF has slightly more degrading effects in the maritime atmospheric turbulence affecting MTFs more adversely than those of the terrestrial atmospheric turbulence.