An operational method for calculating fluctuations of the Doppler frequency shift of a radio signal reflected from a randomly inhomogeneous ionosphere is proposed. The method is based on a numerical and analytical solutions of stochastic ray equations. The integral expressions are obtained for the average and root-mean-square deviations of the frequency of the radio signal along the oblique sounding path in the approximation of the perturbation method. The motion of chaotic ionospheric irregularities is taken into account in the framework of the hypothesis of the frozen turbulence transfer. The integral expressions for statistical moments are reduced to a system of ordinary differential equations of the first order and are solved numerically together with the ray equations in the regular ionosphere. This can significantly reduce the computer time spent calculating the Doppler frequency shift of the radio signal in a randomly inhomogeneous ionosphere. The results of mathematical modeling of frequency fluctuations of a decameter radio signal on a single-hop track in various geophysical conditions are presented.
An operational method is proposed for calculating the refraction of decameter radio waves in a randomly-inhomogeneous upper atmosphere. The method is based on the numerical-analytical solution of stochastic equations of geometric optics. An integral expression is obtained for the dispersion of the refraction angle of a radio wave on the atmospheric path using the approximation of the perturbation method. For a quick calculation of the statistical moment of the refraction angle, the integral expression is reduced to an ordinary first-order differential equation. The joint numerical solution of the unperturbed ray equations and the equations for the statistical moment allows doing an operational estimate of beam width of radio waves arriving at the observation point. The results of numerical calculations of the standard deviations of the refraction angles of radio waves on paths of various lengths are presented.
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