## 1. Introduction

[2] The phase-screen theory has been frequently used to study the fluctuations of a radio signal due to its propagation through an irregularity layer in the ionosphere. This theory assumes that the medium is equivalent to a diffracting screen with random phase fluctuations that are proportional to the irregularities in the total electron content. The theory further assumes that the phase fluctuations are frozen in the uniform background and move with a fixed velocity. As the wave propagates in free space beyond the screen, fluctuations in amplitude begin to develop. This approach was used by many authors to derive different moments of the amplitude and the phase of the received signal [*Bowhill*, 1961; *Mercier*, 1962; *Briggs and Parkin*, 1963; *Budden*, 1965; *Salpeter*, 1967; *Tatarskii*, 1971; *Yeh and Liu*, 1982]. These analytical studies were performed both in the thin-screen (small RMS phase fluctuation) and in the thick-screen (large RMS phase fluctuation) regimes, assuming appropriate statistics for the phase fluctuations of the emerging wave front.

[3] An alternative to the above analytical studies is the application of the Huygens-Fresnel diffraction theory [*Goodman*, 1968] to an irregular ionosphere represented by single or multiple phase screens to numerically characterize the signals received on the ground. This approach repeatedly applies the known field at a certain screen into a convolution integral to determine the field at the next screen. It can be used when scattering is relatively strong and can also be applied to a diffracting screen with phase fluctuations characterized from in situ measurements or by models assuming arbitrary shapes for their power spectral densities [*Knepp*, 1983; *Rino and Owen*, 1984]. The most general version of this approach is equivalent to the numerical solution of the parabolic equation that describes wave propagation when partial reflection can be neglected [*Wagen and Yeh*, 1989; *Martin*, 1993]. This equation has been solved directly, using classical techniques such as the Crank-Nicholson algorithm [*Wernik et al.*, 1980]. It can also be solved by the Fourier split-step algorithm [*Kuttler and Dockery*, 1991].

[4] In addition to the known field the Huygens-Fresnel convolution integral involves a unit-amplitude term that oscillates increasingly faster. Therefore a threshold will be eventually reached beyond which the field can be considered as a constant within each cycle of the second term. The numerical evaluation of the integral should only be performed up to this threshold. Otherwise, the fast oscillations in the second term will not be adequately sampled, and errors will rapidly accumulate. However, since the second term of the convolution integral has unit amplitude, it is not so easy to determine this threshold. One of the main objectives of this paper is to show that the discrete version of the Huygens-Fresnel integral can be written as the convolution between a series of sampled values of the field and a series of coefficients with decreasing amplitudes. The use of these coefficients provides an alternative approach that is capable of circumventing the described difficulty. Additionally, an efficient propagation algorithm can be designed by a priori calculating these coefficients and storing the Fourier transform of the corresponding series for all the frequencies of interest, combined with use of the fast Fourier transform (FFT) algorithm to evaluate discrete convolutions.

[5] Sections 2–5 will also describe models for the phase fluctuations in a diffracting screen and will apply the designed algorithm to the prediction of phase and amplitude scintillation of satellite signals in the equatorial ionosphere. The phase screen models and the propagation algorithm can be easily integrated with physics-based or morphological models of ionospheric irregularities to forecast equatorial scintillation.