## 4 MTF Model and Numerical Analysis
Figure 4 shows the long-exposure MTF curves as a function of the normalized spatial frequency \(\mu\) for Gaussian-beam waves as the beam curvature parameter at transmitter \(\Theta_{0}\) varying under the terrestrial and maritime atmospheric turbulence. Here, the power law exponent \(\alpha=11/3\); the Fresnel ratio \(\Lambda_{0}=1\). The beam curvature parameter \(\Theta_{0}=0.5\) and \(\Theta_{0}=0.7\) correspond to the convergent Gaussian-beam wave; \(\Theta_{0}=1\) corresponds to the collimated Gaussian-beam wave; and, \(\Theta_{0}=2\) corresponds to the divergent Gaussian-beam wave. One can observe that the higher value of \(\Theta_{0}\) shows a lower degradation effect on the MTF of the imaging system. Additionally, similar to the results that are illustrated in Figure 3, the optical waves can undergo slightly more degradation under the maritime turbulence than that of terrestrial one. Figure 3: Long-exposure modulation transfer function (MTF) as a function of normalized spatial frequency for collimated Gaussian beam waves with various (**a**) power law exponent \(\alpha\) and (**b**) anisotropic parameter \(\zeta\)  
In addition, more simulation results are shown in Figure 5 in order to compare the MTF properties among different form of optical waves. First, the three long-exposure MTF curves for the plane, spherical, and Gaussian-beam waves are shown in Figure 5a, when \(\alpha=3.4\), \(\Lambda_{0}=1\) and \(\Theta_{0}=1\). Particularly, Figure 5b,c contain the curves of MTFs for the plane waves and the spherical waves with respect to power law exponent \(\alpha\) and anisotropic parameter \(\zeta\), respectively. One clearly can distinguish that the degrading effect for the spherical waves is smaller than that for Gaussian-beam waves; the degradation effect for the Gaussian-beam waves is less than that for the plane waves. Furthermore, the behaviors of the long-exposure MTFs for the plane and spherical waves are similar to the one for the Gaussian-beam waves with increasing \(\alpha\) and \(\zeta\). The curves of the plane waves and spherical waves are consistent with the study result of the optical wave modulation transfer function models in anisotropic marine turbulence in [33]. Finally, Figure 6 shows the behavior of the long-exposure MTF with respect to the outer scale of the turbulence \(L_{0}\). For these curves, the simulations are performed with the anisotropic factor \(\zeta=3\), the power law exponent \(\alpha=11/3\), and the outer scale of the turbulence cell equals to 1 m, 10 m, and 1000 m. The numerical results show that the long-exposure MTFs for all of the optical waves are significantly influenced by the outer scale of the turbulence cell, and the degradation effect increases as \(L_{0}\) increases. Moreover, the spherical waves have least effect in conjunction to the imaging system in comparison to other optical waves; these results are consistent with Figure 5.  
Figure 4: Long-exposure MTF as a function of normalized spatial frequency for collimated (\(\Theta_{0}=1\)), divergent (\(\Theta_{0}>1\)), and convergent (\(0.5<\Theta_{0}<1\)) Gaussian beam waves in (**a**) terrestrial atmospheric turbulence and (**b**) maritime atmospheric turbulence  
Figure 5: Long-exposure MTF as a function of normalized spatial frequency for (**a**) plane, spherical, and Gaussian beam wave; (**b**) plane and spherical wave with various \(\alpha\); and, (**c**) plane and spherical wave with various \(\zeta\)