## 1. Introduction

[2] Ionospheric scintillations are especially important in equatorial and polar regions. They appear after sunset and may last a few hours. They are related in particular to the solar activity and the season. The main factors whose dependency has been established are indicated hereafter.

[3] Only the random fluctuations of the medium are considered here. The traveling disturbances which act as long-term fluctuations with respect to the timescales considered do not contribute to the scintillations. These last are created by random fluctuations of the medium's index, these fluctuations being related to inhomogeneities inside the medium. These inhomogeneities in the ionosphere are the result of several mechanisms. The fluctuation of the electric field is one of them. Local related inhomogeneities evolve and can develop. Also, highly energetic particles and precipitation can create such inhomogeneities. This can be the case in particular in polar regions.

[4] The irregularities develop under the effect of instability mechanisms (*E* × *B* gradient drift, streaming instabilities (Kelvin-Helmholtz), Rayleigh-Taylor, etc.). Characteristic dimensions and different growth rates correspond to each one of the existing processes. The overall problem is very complex and difficult to reproduce theoretically on a large scale [*Béniguel*, 1996].

[5] The inhomogeneities create a number of modifications of transmitted signals, among them phase and intensity fluctuations, fluctuations of the arrival angle, dispersivity, Doppler spectrum, …. This problem has been studied extensively in the past. *Yeh and Liu* [1982] have published a review paper on this topic including both theoretical approaches and experimental results. Methods presented include the Rytov approximation in the case of weak fluctuations, the phase screen theory, and the parabolic equation (PE) method. Statistical results on the fluctuation spectrum and on the probability of transmitted signals are given. The multiple-phase-screen (MPS) technique, which is the object of this paper, consists in a resolution of the PE for a medium divided into successive layers, each of them acting as a phase screen.

[6] This PE technique has been used by *Kiang and Liu* [1985] and *Wagen and Yeh* [1986] for HF propagation. Kiang and Liu considered a medium with a linearized index, allowing a simplified resolution of the PE. This was applied to a vertical incidence. This work has been extended by Wagen and Yeh for the case of oblique incidence. *Knepp* [1983] considered the case of transionospheric propagation. Assuming a quadratic approximation for the phase structure function, he derived an analytical solution for the mutual coherence function (MCF) in the case of a statistically homogeneous medium.

[7] The model which is described in this paper is composed of two models: The first one provides the mean errors. It is based on a solution of the ray equations. The medium's index is calculated from an accurate value of the electron density inside the medium, which is provided by the NeQuick model developed by University of Graz (Austria) and Institute of Theoretical Physics, Abdus Salam University, Trieste, Italy [*Radicella and Leitinger*, 2001; *Hochegger et al.*, 2000]. The second one is based on the MPS technique. It provides the signal scintillations.

[8] Inputs of the model consist in the geophysical parameters, the inhomogeneity data, and the operating data (carrier frequency, etc.). The inhomogeneity data are deduced from measurements. This is detailed in section 2.

[9] Global Ionospheric Propagation Model (GIM) provides the statistical characteristics of the transmitted signals, in particular the scintillation index, the fade durations, and the cumulative probability of the signal, allowing one, consequently, to determine the margins to be included in a budget link. Maps of the scintillation index S4 are easily obtainable.