Stochastic mirage phenomenon in a random medium

In the framework of geometric optics, we consider the problem of characterizing the ray trajectory in a random medium with a mean refractive index gradient. Such a gradient results in the mirage phenomenon where an object's observed location is displaced from its actual location. We derive formulas for the mean ray path in both the situation of isotropic stochastic fluctuations and an important anisotropic case. For the isotropic model, the mean squared displacement is also given by a simple formula. Our results could be useful for applications involving the propagation of electromagnetic waves through the atmosphere, where larger-scale mean gradients and smaller-scale stochastic fluctuations are both present.