## 1. Introduction

[2] The technique of Markov's parabolic equations for the moments of a stochastic field is one of the classical approaches in the problem of wave propagation in random media [*Rytov et al.*, 1978; *Ishimaru*, 1978]. Despite many years of exploitation of the method, there still exist unsolved problems in the scope of Markov's approximation. In particular, a general analytic solution has not yet been constructed for the fourth moment, and some questions arise with respect to the two-frequency, two-position coherence function.

[3] As far as the second-order coherence function is concerned, the exhaustive solution has been constructed for the single-frequency (pure spatial) coherence function [*Rytov et al.*, 1978]. The spaced position and frequency coherence function was studied numerically by *Lin and Yeh* [1975], who investigated this function in a fading ionospheric communication channel numerically for power-law and Gaussian spectra of ionospheric fluctuations. For the two-frequency coherence function the closed form exact analytic solution was only constructed in the case of a plane wave propagating in a random medium with the quadratic structure function of fluctuations [*Sreenivasiah et al.*, 1976]. *Knepp* [1983] generalized this solution to the case of spherical wave propagation. The technique of separation of variables was employed by *Oz and Heyman* [1996, 1997a, 1997b, 1997c] to construct the second-order coherence function for an arbitrary structure function of the fluctuations of the properties, but in a medium with a homogeneous background. It implies the expansion of the solution into the series of the transversal eigenfunctions of the problem, and, in this way, requires additional quantification of the number of terms needed to achieve the necessary accuracy for a given distance of propagation and spaced frequency. The series fails to converge for some types of initial conditions (e.g., an incident plane wave) as the distance and difference frequency tend to zero. Additional constraints in the separation variables technique may also arise when considering fluctuations with the structure function tending to a constant as the difference variable tends to infinity. In this case the continuous spectrum may likely occur in the spectrum of the transversal operator of the problem, which makes expansion of the solution in terms of the transversal eigenfunctions much more complicated.

[4] In the present paper the asymptotic technique is developed to construct the solution to Markov's parabolic equation for the two-frequency, two-position coherence function in the case of an arbitrary structure function of the refractive index of fluctuations and an inhomogeneous stratified background medium. The technique employs quasi-classic representation in terms of complex trajectories. It has no constraints pertinent to the initial conditions in the form of an incident plane wave, and is also valid beginning with the zero distance from a boundary surface and zero spaced frequency. In the case of the quadratic structure function it produces the known exact solution of the problem [*Sreenivasiah et al.*, 1976] in automatic fashion.