The characteristics of random ionospheric inhomogeneities are described, including their dependency on the geophysical parameters. They are used in a propagation model to estimate the propagation errors and the fades of the transmitted signals. Margins to be included in a budget link are subsequently determined. Features of our propagation model, Global Ionospheric Propagation Model (GIM), are presented with typical results for VHF and L band links.
 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.
 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.
 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].
 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  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.
 This PE technique has been used by Kiang and Liu  and Wagen and Yeh  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  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.
 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.
 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.
 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.
2. Inhomogeneity Characteristics
 In order to calculate the statistical properties of the transmitted signals through the ionosphere we need to specify the following parameters: (1) the spectral density of their electron density fluctuations, (2) their correlation distance, (3) the altitude to which they develop, and (4) their velocity and direction of displacement.
2.1. Spectral Density
 Scintillations as a random process are most completely characterized by temporal or spatial power spectra, from which one can pass to spectra that give rise to irregularity scintillations through the use of the phase screen theory. Since logarithmic spectra of scintillations often exhibit a linear variation with the logarithm of frequency, their most important parameter is the inclination index of the spectrum p.
 Temporal power spectra of amplitude scintillations have been estimated from measurements. On most occasions the logarithmic spectrum shows a linear variation when plotted against the logarithm of frequency. The cutoff frequency is related to the correlation length of these inhomogeneities.
 The value of p depends on the specific conditions of development of the turbulence, in particular the instability process involved, the latitude, the altitude, … As observed by measurements, in particular with satellite ETS-2 [Afraimovich et al., 1994], the slope value is usually between −2 and −4.
2.2. Correlation Length
 The correlation length is of primary importance in the characterization of the inhomogeneities. With respect to these inhomogeneities the medium may be assimilated to a number of scatterers randomly distributed in space. One important parameter to consider is the size of the first Fresnel zone at the altitude under examination. As an example, for a transmitted signal at frequency 137 MHz and at height 300 km the size of the first Fresnel zone is km. Inhomogeneities whose size is lower or comparable to this dimension will create scintillation phenomena.
 As for the power spectrum index value, the coherence length of inhomogeneities varies with local specific conditions. To perform a propagation calculation, the medium is considered as statistically homogeneous. To do this, we assign to each particular region of the ionosphere (altitude, latitude) typical characteristics of the inhomogeneities in the region under examination. The mean value of the correlation length deduced from measurements is about 800 m at the F region altitude [Afraimovich et al., 1994].
 Drift of the irregularities causes a Doppler shift of the diffraction pattern. Both the direction and the modulus of velocity are important and are taken into account in the characterization of the statistical properties of the medium.
2.3. Height of Irregularities
 The height at which the instabilities develop may be obtained from the diffraction pattern of the transmitted signals. This last is related to the location of the first Fresnel zone. The corresponding frequency is obtained from the spectral density spectrum of the irregularities. This frequency being measured, the altitude where the irregularities develop is easily obtained. Histograms of measurements show a peak value at altitudes between 300 and 500 km, consequently at the F region altitude [Afraimovich et al., 1994; McDougall, 1981].
2.4. Dependency of Inhomogeneities on Latitude
 Scintillations are most severe and prevalent in and north of the auroral zone and near the geomagnetic equator [Aarons, 1982]. The equatorial region extends approximately from −20° to 20°, and auroral regions extend from 55° to 90°. These boundaries change with the time of the day, the season of the year, the sunspot number, and the magnetic activity.
2.4.1. Equatorial scintillations
 Scintillation is predominantly a nighttime phenomenon in the equatorial region occurring for more than 40% of the year during the 2000–0200 local time period [Basu et al., 1976]. It also shows a strong seasonal dependence with a pronounced minimum at the southern solstice and relatively high scintillation activity at the northern solstice. Equatorial scintillations also show a tendency to occur more often during years in which the sunspot number is high. The RMS amplitude of electron density irregularities is equal to 20% in the most severe cases.
 Two regimes may be identified. For values of the scintillation index (S4) below approximately 0.5 the RMS values of phase and intensity fluctuations seem to be linearly correlated and approximately equal. For greater values of S4 there is no obvious correlation, and measured values are greater for intensity than for phase [Doherty et al., 2000].
 If we considered the case of GPS L2 scintillations, the typical value of S4 at equatorial regions is 0.3. Its occurrence is related to the season and the solar activity. It may reach a value of 0.5 with an occurrence 10% below and a value of 0.8 or even 1 in a few cases.
2.4.2. High-latitude scintillations
 In contrast to equatorial fluctuations the polar fluctuations exhibit more phase than intensity fluctuations. The scintillation index is usually quite low. It seldom exceeds 0.2, and, same as before, the probability of occurrence is very low in summer and, in any case, below the values obtained at equatorial regions. The situation is the reverse for phase fluctuations. They may exist all year. The values are quite high and seem, moreover, to be related to the magnetic activity. This has been observed, in particular, in the case of the 14–15 July 2000 magnetic storm.
3. Propagation Model
 The first calculation is to determine the line of sight. This is done solving the Haselgrove equations by a ray-tracing technique. The curvature of the main propagation axis, due to the variation of the electron density inside the medium, is obtained by the model. Other mean errors are subsequently obtained. Inputs of this first-order model are the values of the electron density inside the medium. They are obtained by the NeQuick model from the knowledge of the geophysical parameters. The problem solved is a three-dimensional (3-D) problem. Consequently, horizontal gradients of the index are considered.
 The scintillations are then calculated using the MPS technique, solving the PE for a random incident electric field along the line of sight as main propagation axis. The inputs required for this second-order model are the following: the slope of the spectral density of fluctuations, the average size of the inhomogeneities, the value of the ratio of the RMS electron density to its mean value, and the drift velocity of the inhomogeneites. The first three data are required for phase synthesis. The drift velocity allows one to perform Doppler spectrum calculation. In GIM these data may be defined by the user. However, typical values are suggested.
 The fluctuating medium is divided into successive layers, within which the statistical properties of the medium are specific. The measurements available make it possible to estimate the probability of occurrence of inhomogeneities as a function of latitude, altitude, and time, as well as their dimensions. Consequently, we may associate statistical characteristics with each one of the different layers.
 For a given layer the transmitted phase spectral density of the signal is
where q0 = 2π/L0 and L0 in meters is the average size of the inhomogeneities.
 The autocorrelation function obtained by inverse Fourier transform can be calculated analytically. For propagation problems through the medium, we look at phase fluctuations in the transmitted signal in a plane perpendicular to the propagation axis. These phase fluctuations are directly related to the electron density through
where L (in meters) is the thickness of the medium, Cs is the medium's structure constant, and σΦ is the phase standard deviation (in radians). Moreover, the signal's phase is a centered Gaussian random variable.
 The phase fluctuations in the medium are synthesized using a technique borrowed from numerical filtering methods. The spectral density of the phase at the output of the medium is equal to the product of the Fourier transform of a centered Gaussian random variable and the square root of the spectral density of the signal that we want to synthesize. The resulting random variable meets the required conditions. The corresponding signal is obtained as the inverse Fourier transform of this product.
 Propagation inside the medium is calculated using the phase screen theory, by alternating scattering and propagation calculations (split-step technique). The related equation is the PE, and the corresponding technique applies, regardless of the level of ionization, within a screen and, in particular, to the propagation outside the inhomogeneity layers. The PE is solved in the Fourier domain. It can be shown that the corresponding integral used to calculate the scattered field is equivalent to the Kirchhoff integral. Cartesian coordinates are maintained for resolution. However, the main propagation axis reproduces the bending of the rays, which is usually not significant as regards this effect on signal fluctuations, for frequencies under consideration.
 An example of the intensity of a transmitted signal (in decibels) after propagation through the ionosphere is presented in Figures 1 and 2 both for a VHF (137 MHz) and for a GPS (1.57 GHz) link. These two cases correspond to the same geophysical conditions and identical locations of transmitter and receiver. The only difference is the operating frequency considered. In both cases the propagation path intercepts the equatorial region. The satellite altitude has been chosen to be 22,000 km. This infers the timescale of the graphics, which is related to the apparent satellite and medium velocities. The extent of the timescale depends on the screen size, this size being chosen such that both the plane wave approximation condition and requirements for the use of the fast Fourier transform (FFT) algorithm are fulfilled.
3.2. Probability of Transmitted Signal
 The variance of the intensity (from which we deduce the scintillation index S4) is calculated numerically using the results of the intensity of the transmitted field. Results presented in Figures 3 and 4 show the probability of transmitted signal for the two previous cases. It fits to a Rayleigh process in the first case, of high fluctuation (S4 = 0.76) (Figure 3), and to a Nakagami process in the second case, of low fluctuation (S4 = 0.25) (Figure 4).
3.3. Channel Availability
 The cumulative probabilities and, consequently, the margins are easily obtained. The average duration of fades versus their level is also calculated by the model. The fade levels obtained for the same two cases as previously (VHF band and GPS frequency) are presented in Figures 5 and 6. The percentage of time applies to a nighttime fluctuation event.
 The dependency of the margins on the scintillation index is presented in Figure 7 for confidence levels equal to 0.9, 0.95, and 0.99. The fade margins increase with S4 for the Nakagami process and decrease with S4 for a Rayleigh process. Curves in Figure 7 combine the two probability dependencies. In the low-fluctuation regime (for values of S4 typically below 0.5) the Nakagami law applies (solid curves). In the high-fluctuation regime (values of S4 > 0.5) the Rayleigh law applies (dashed curves).
3.5. Autocorrelation Function
 The autocorrelation function of the transmitted field can be calculated from the transmitted signal intensity. It allows one to determine the space and time coherence lengths and the Doppler spread spectrum by a Fourier transform with respect to the time.
 An analytical solution of the MCF is, in the general case (spherical wave, arbitrary phase structure function, …), very difficult to obtain. Many papers have been published on this topic. Each solution assumes a simplified hypothesis in order to obtain a tractable solution. The simplest approximation is the HF approximation, which may be cast into the form
where DΦ (in square radians) is the phase structure function. The first term stands for frequency dependency, and the second term stands for space dependency. Numerical experiments show that this expression provides a result very close to the numerical result even in the VHF band. Results of the MCF and of the Doppler spectrum presented in Figures 8 and 9 correspond to the VHF case (S4 = 0.76). The arrival angle fluctuations are simply deduced from the phase structure function, which may be obtained from this autocorrelation function.
3.6. Dependency of S4 on the Free Space Propagation Distance/Anisotropy of the Medium
 Results presented in Figure 10 show the dependency of S4 on the free space propagation distance noted z, varying the RMS phase from low- to high-fluctuation conditions. Generally speaking, the scintillation index increases with distance for low values up to an asymptotic value equal to 1 in the case of strong scintillations.
 The analysis technique implemented in GIM solves a 2-D problem. The first dimension is the main propagation axis. The second dimension is the transverse direction with respect to the propagation axis. In order to estimate the influence of the anisotropy of the medium due to the fact that inhomogeneities may be aligned with the terrestrial magnetic field, it is necessary to solve a 3-D problem considering two directions in the transverse plane.
 In this case, variable q2 in the expression of the spectral density of electronic density fluctuations in the medium (equation (3)), is replaced by Aqx2 + Bqy2, which includes two directions qx and qy and where the ratio B/A represents the anisotropy index.
 The numerical synthesis technique is identical. However, this time, 2-D Fourier transforms are involved. Comparison results are presented in Figure 11 for both cases of isotropic and anisotropic media. In the first case, the B/A ratio is equal to 1, and the medium is isotropic. In the second case, this ratio is equal to 10. The simulation results help quantify the influence that the medium's anisotropy has on the scintillation index of the transmitted signal.
 As observed in Figure 11 the discrepancies may reach approximately 30% in the first regime when S4 increases with the free space propagation distance. Both curves merge in the asymptotic regime, which is to be expected.
 A typical application of GIM is the calculation of S4 and phase fluctuation maps. Two examples are presented. In the first one (Figure 12) an S4 map has been plotted. The coordinates are subsatellite points. The total electron content (TEC) map is plotted correspondingly (Figure 13). As the model is based on a background ionosphere related to the mean value of the electron density provided by the NeQuick model, it reproduces the equatorial anomaly in the scintillation process.
 The second example is a scintillation result obtained from a ground station depending on the elevation angle. The ground station is located on the equator at longitude 0° for this example (Figure 14). The F10.7 flux number is equal to 133 in this case. Peak values correspond, as assumed, to low elevation angles.
 Most of the scintillation characteristics can be obtained with GIM. They include the RMS values of scintillations, the time series synthesis of transmitted signal phase and intensity, the mutual coherence function, the average duration of fades, the Doppler spectrum, and the probability density function.
 No hypothesis is made with respect to the level of fluctuations. The model allows one to consider any conditions: low or high fluctuation. Values of S4 greater than 0.5, corresponding to high fluctuation, may be reached in the VHF band both at equatorial and polar latitudes and in a few occasions at GPS frequency. Moreover, the GIM numerical model allows the estimation of the statistical characteristics of the transmitted signal. A fading margin calculation is subsequently performed. Most of the computation is based on FFT technique, allowing one to obtain scintillation characteristics for a specific link in a short time. In the case of weak fluctuations some of the calculations are also performed in a simpler way as described in Wide-Band Scintillation Model (WBMOD) [Secan et al., 1995; Rino, 1979]. When applicable, these results are calculated for comparison.
 Results shown in this paper have been obtained for a VHF (137 MHz) and an L band (1.57 GHz) link. Extensive comparisons with measurements for GPS links are conducted at the moment. So far, reasonable agreement has been obtained.