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Based on the geometrical-optics approximation, we propose a method for calculation of statistical moments of the radio-wave phase in the case of total internal reflection from a randomly inhomogeneous ionosphere with a monotonic height profile of regular dielectric permittivity. To take into account the radio-wave scattering at the reflection point in a correct way, we perform analytical transformation of the eikonal equation solution derived in a first approximation of the perturbation method.Allowance for the features of the phase fluctuations of a radio wave in the case of total internal reflection from the ionosphere is significant for solutiion of the problems of prediction of radio-path characteristics and diagnostics of ionospheric irregularities [1]. Profound analysis of the reflection region on phase fluctuations was performed in [2][3][4][5], where the problems of scattering of radio waves vertically and obliquely incident on the linear layer of the ionosphere with random dielectric-permittivity irregularities given by the Gaussian correlation function were studied on the basis of the geometrical-optics approximation and the perturbation method. The result was an important conclusion that the ionospheric irregularities located in the vicinity of the reflection point of a radio wave have the dominant effect on its phase fluctuations.Solving the problems of remote sounding of the ionosphere under the actual geophysical conditions [6] requires a method for calculation of statistical moments of the radio-wave phase in a multiscale randomly inhomogeneous medium where the profile of the regular dielectric permittivity ε 0 (z) is different from the linear one. In the present paper, we consider the possibility of calculation of the phase moments of the reflected radio wave in the case of its normal incidence on the turbulent ionospheric layer with the monotonic dependence ε 0 (z).Let the wave from the source located at the point z = −z p be incident vertically upward on the ionospheric layer with random irregularities of the dielectric permittivity. Represent the dielectric permittivityε(x, y, z) of the medium as the sumwhere the functionε 1 (x, y, z) characterizes the ionosphere turbulence.Assuming that the fluctuation component of the medium satisfies the condition |ε 1 (x, y, z)| ε 0 (z),for calculation of the phase fluctuationΦ 1 of the wave we will use the solution of the eikonal equation in a *

Characteristic U-shaped traces (cusps) on ionograms have been identified as off-angle echoes from sloping electron density contours caused by the presence of traveling ionospheric disturbances (TIDs). Temporal evolution of the cusps is associated with horizontal drift of the disturbances. A potential for reducing calculation time in numerical synthesis of vertical ionograms is under discussion. Since numerical ray tracing is computationally intensive, we have developed simplified formulation for this study. The suggested model of compound parabolic layer allows us to analytically calculate ray paths. Changes in the shape of the ionogram cusp caused by varying TID characteristics are examined.

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