A radio wave scattering algorithm and irregularity model for scintillation predictions

Authors


Abstract

[1] An algorithm for calculations of phase and amplitude scintillation of satellite signals in the equatorial region will be described in detail. The algorithm will be developed by initially transforming the discrete version of the Huygens-Fresnel integral into a convolution involving a series of coefficients with decreasing amplitudes. Next, the Fourier transform of the corresponding series of coefficients is stored for all the frequencies of interest, and the fast Fourier transform algorithm is used to evaluate discrete convolutions. Two phase screen models will be described. The first assumes that the phase fluctuations of the wave front emerging from the bottom of the irregularity layer are proportional to electron density fluctuations directly obtained from satellite in situ measurements. The second assumes that the same phase fluctuations can be obtained from their power spectral densities and phase spectra, represented by analytical functions with parameters provided by physics-based or morphological models of ionospheric irregularities. The propagation algorithm will be applied to both phase screen models, assuming five frequencies in the high-VHF to low-SHF band (suffering strong to weak scattering), to display its potential in the prediction of phase and amplitude scintillation of satellite signals.

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 25 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.

2. Slant Propagation Through an Irregularity Layer

[6] The slant propagation of a plane wave through a bidimensional irregularity layer of thickness L and bottom height h can be represented by the wave equation

display math

In the above equation, k is the free space wave number and n is the refractive index of the medium. The horizontal x axis is aligned with the bottom of the layer, and the z axis points downward. Without any loss of generality the field U(x, z) can be represented by

display math

where θ is the zenith angle of the wave vector. Substituting the right-hand side of equation (2) for U into equation (1) and assuming that the scale size of the vertical fluctuations of the complex amplitude u is large in comparison with (2k cos θ)−1, one gets

display math

Under the above approximation the second derivative of u with respect to z becomes much smaller than the third term in equation (3) and can be discarded. This approximation is reasonable for sufficiently high frequencies and small zenith angles. It is seen that the slant propagation of the plane wave is now represented by a parabolic equation, indicating that backscattering has been neglected.

[7] The Fourier split-step algorithm [Kuttler and Dockery, 1991] can be used to numerically solve equation (3). This algorithm assumes that u(x, −L) = 1, divides the layer into thin horizontal slabs, and substitutes the Fourier representation

display math

into equation (3) to obtain

display math

Equation (5) yields

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Calculating the inverse Fourier transform of the above equation, it follows that

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In the above equation, δz is the step size. This equation should be repeatedly applied to produce the field from the top to the bottom of the layer. It should then be applied once more with δz = h and n = 1 to produce the field on the ground. It should be observed that each step in the solution of the parabolic equation by the Fourier split-step algorithm begins with the application of the Huygens-Fresnel diffraction integral within curly brackets [Goodman, 1968] to the field at the top of each slab. A phase shift and a displacement are then performed to produce the field at the bottom of the same slab. Note that phase shift and the displacement are not necessary when n = 1 and θ = 0°, respectively.

[8] Equation (7) is only an exact solution to the parabolic equation (3) when (n2 − 1) is constant. There are errors associated with the above solution when the refractive index is position-dependent [Knepp, 1983], which is the case of interest. Adapting the derivation by Kuttler and Dockery [1991] to slant propagation of plane waves through an irregularity layer in the ionosphere, one obtains the following conditions for negligible errors in each step:

display math

In their application of equation (7) to propagation in the troposphere (with θ = 0°) the first upper bound in the right-hand side of inequality (8) was equal to 20 km. Kuttler and Dockery claimed that their numerical experiments using the Fourier split-step algorithm revealed that the ratio within the absolute value bars in the second upper bound is greater than 1, except in the vicinities of nulls in the field. Additionally, they claimed that the dimensions and the number of these vicinities increase with the frequency. As a result, the maximum acceptable value for δz should decrease with the frequency. Kuttler and Dockery [1991] then reminded their readers that the total error in the calculations depended on the accumulation of the errors from each step. No analytical treatment of error accumulation is available. However, they observed that step sizes of several hundred wavelengths were adequate for the convergence of the solution in their application and presented an excellent agreement between corresponding results from the Fourier split-step algorithm and from a waveguide mode propagation model.

[9] In the application of the Fourier split-step algorithm to analyze the slant propagation of plane waves through an irregularity layer in the ionosphere, the upper bounds in the right-hand side of inequality (8) can be rewritten in the forms

display math

where N is the electron density and LN is its gradient scale size, if upper VHF or even higher frequencies are assumed. The minimum value of LN has been estimated, on the basis of a reconstruction of the high-resolution electron density in situ data displayed by McClure et al. [1977], to be of the order of 200 m. Therefore the first upper bound for the step size in the present application is consistent with the value obtained by Kuttler and Dockery [1991]. Considering the arguments presented in the previous paragraph in relation to the second bound, one could expect that step sizes of several hundred wavelengths would also be adequate for the convergence of the solution in the present application. However, it is important to observe the convergence of the solution as the step size is reduced to determine a value of δz that is adequate to the present application.

[10] It has already been mentioned that the maximum acceptable value for δz should decrease with the frequency. This will increase the processing time necessary to perform the calculations. Additionally, the second term in the convolution integral within curly brackets in equation (7) oscillates increasingly faster with |xx′|, and this behavior is intensified as the frequency of the plane wave also increases. Therefore, in the evaluation of the convolution integral, a threshold in |xx′| will be reached beyond which these oscillations become so fast that u(x′, z) can be considered a constant within each cycle of the second term. Beyond this threshold, which characterizes an effective integrating interval, the contribution of the integrand becomes negligible and can be discarded. Note that the length of the effective integrating interval decreases (indicating that propagation becomes increasingly localized) as the frequency increases, approaching the optical regime.

[11] The straightforward numerical evaluation of the convolution integral should only be performed over the effective integrating interval; that is, it should be associated with truncation. Otherwise, the fast oscillations in the second term of the convolution integral will not be adequately sampled (assuming a uniform sampling), and errors will rapidly accumulate. Simply decreasing the sampling interval will only delay the beginning of the error accumulation process, in addition to unnecessarily increasing the processing time spent in the calculations. However, note that all the sampled elements of the second term of the convolution integral in equation (7) have unit amplitudes. Therefore it is not easy to determine the length of the effective integrating interval in the x domain.

[12] In section 3 the discrete version of the integral will be written as the convolution between a series of sampled values of the plane wave at a fixed height and a series of coefficients with decreasing amplitudes. The use of these coefficients provides an alternative approach that is capable of circumventing the difficulty described in the previous paragraph.

3. Description of the Scattering Algorithm

[13] As mentioned in section 2, the term within curly brackets in equation (7), denoted v(x, z + δz), results from the convolution of two functions. Therefore it has a discrete representation that can be written in the form

display math

In the above equation, m and q play roles analogous to those of x and x′ in equation (7), respectively. Equation (10) allows an efficient calculation of the signal vm(z + δz) through an “a priori” calculation of the coefficients Cq and the storage of the Fourier transform of the corresponding series for each frequency of interest, combined with use of the FFT algorithm to evaluate discrete convolutions.

[14] To derive an expression for the coefficients Cq displaying decreasing absolute values with the index q, the horizontal variables x and x′ will be initially normalized by the horizontal sampling interval δx. That is, the changes of variables x = mδx and x′ = s δx will be introduced into equation (7), and the result will be written as a summation of integrals over unit s-intervals as follows:

display math

where zf = (2λ δz)1/2 is the Fresnel scale size and λ is the wavelength of the transmitted signal.

[15] It will then be assumed that u(s, z) is a piecewise linear function of the first variable, with breakpoints at its integer values. That is,

display math

Note that the samples un and un+1 are obtained at the altitude z. To simplify notation, this dependence has been omitted in expression (12). Substituting its right-hand side for u(s, z) in equation (11), it follows that

display math

where

display math
display math
display math

[16] Substituting the right-hand side of equation (15) for Imn2 in equation (13), collecting terms that multiply the same sample un in this equation (or, equivalently, changing indices n +1 → n in the second summation), one gets

display math

A final change of indices n = mq yields

display math

It should be observed that the above equation has exactly the same format as equation (10) and that its term within curly brackets should be identified with the coefficient Cq.

[17] The right-hand side of equation (14) can be written in terms of Fresnel integrals [Abramowitz and Stegun, 1972]

display math

On the other hand, a straightforward integration yields

display math

Substituting the right-hand sides of expressions (19) and (20) for Iq1 and Jq2, respectively, into the term within curly brackets in expression (18), with appropriate consideration for their indices, one gets

display math

where

display math

Computationally, it is convenient to express the Fresnel integrals in expression (22) in terms of the auxiliary functions fq) and gq) [Abramowitz and Stegun, 1972]. Therefore

display math
display math

where

display math

The coefficients cq(β) can be calculated using both the small-argument and asymptotic expansions for f(x) and g(x) provided by Abramowitz and Stegun [1972]. Expression (24) indicates that the coefficients Cq can be obtained iteratively. More specifically, since the values cq−1(β) and cq(β) are available from previous iterations, only the value of cq+1(β), calculated using expression (25), is necessary to obtain Cq. Note that in addition to this prescription (that is, in addition to c1(β)), c0(β) should also be calculated to produce C0 at the first iteration.

[18] From expressions (21) and (22), as well as from the properties of the Fresnel integrals (both are odd functions of their arguments), it can be shown that Cq = Cq. Furthermore, the dominant terms in the asymptotic expansions of the auxiliary functions fq) and gq) are such that [Abramowitz and Stegun, 1972]

display math

and

display math

Therefore the dominant term in the asymptotic expansion of the coefficient cq(β) is

display math

It is seen that the absolute value of cq(β) decreases asymptotically with the second power of the index q and the third power of the parameter β. Expression (24) shows that Cq is the second central difference of cq(β). Therefore it should decrease asymptotically at least as fast as the right-hand side of expression (28), as desired.

4. Phase Screen Models

[19] The scattering algorithm will be applied to describe the slant propagation through an irregularity layer, represented by two different phase screen models. These phase screen models characterize an irregularity layer by its bottom height h (also the height of phase screen), the thickness L and the average electron density No and neglect amplitude fluctuations in the wave front emerging from the bottom of the irregularity layer.

[20] The first phase screen model assumes that the phase fluctuations φ(x) of the wave front emerging from the bottom of the irregularity layer are proportional to electron density fluctuations δN(x) resulting from satellite in situ measurements such as those reported in the literature [McClure et al., 1977; Basu et al., 1983]. As an example of the first model, Figure 7 of McClure et al. [1977], displaying electron density fluctuations detected by the AE-C satellite was enlarged and digitized. Next, it was sampled at a constant interval to simulate a sampling distance of in situ measurements equal to 40 m. Finally, it was low-pass filtered to remove digitization-induced noise.

[21] The thick solid curve in the upper panel of Figure 1 shows the normalized (zero-mean, unit standard deviation) phase fluctuation function φ(x) resulting from this model. The middle panel of Figure 1 shows the corresponding maximum-entropy method (MEM) power spectrum (thick solid curve). This curve displays an outer scale size Lo ≈ 25.0 km and a power law spectral index p = 1.86 for spatial frequencies between fo = 1/Lo ≈ 0.04 km−1 and a breakpoint located at the spatial frequency fb = 1/Lb ≈ 2.50 km−1. For spatial frequencies higher than fb a power law spectral index q = 3.00 is observed. These parameters are consistent with the ones obtained by Basu et al. [1983], except for the spatial frequency fb, which surpasses their highest value. On the other hand, rocket in situ electron density data [Hysell et al., 1994; Hysell, 2000] from altitudes above 280 km have shown double-slope spectra with spectral indices between 1.7 and 2.5 for large-scale sizes and between 4.5 and 5.0 for small-scale sizes. The scale size of the spectral breakpoint has been found in the vicinity of 100 m (spatial frequency fb = 10 km−1). These results seem to indicate that spread F irregularities sampled in the east-west direction by satellites display power spectra with larger breakpoint scale sizes and smaller slopes in the small-scale size regime than those obtained from sampling the same irregularities in the vertical direction by rockets.

Figure 1.

(top) Normalized phase fluctuations φ(x) resulting from the first model (thick curve, in situ data reproduced from McClure et al. [1977]) and the second model (thin curve), assuming fo = 0.04 km−1, fb = 2.50 km−1, p = 1.86, and q = 3.00). (middle) Corresponding MEM power spectral densities (same curve code). (bottom) Corresponding phases of the FFT components after the original ±π discontinuities have been eliminated, and the resulting curve has been artificially broken into sections, to allow the phase curve to be displayed within the selected vertical scale (same curve code).

[22] The lower panel of Figure 1 shows the phase ψ(fx) of the FFT components of φ(x) as a function of the spatial frequency fx (thick curve). Initially, the original ±π discontinuities have been eliminated to produce a continuous curve that fits (in the least squares sense) the function

display math

with RMS error equal to 7.80 rad. Next, the resulting curve has been artificially broken into sections, to allow all the phase curve to be displayed within the selected vertical scale.

[23] The second phase screen model assumes that the power spectral density function Sφ(fx) of the phase fluctuations φ(x) is represented by

display math
display math
display math

The second phase screen model initially generates Φ(fx), a realization of the desired phase fluctuation φ(x) in the Fourier transform domain, using

display math

That is, the square root of the right-hand side of expression (30)(30c) is sampled at equally spaced spatial frequencies, and each sample is multiplied by a phase factor characterized by a prescribed random variable ψ(fx) [Franke and Liu, 1983]. The inverse Fourier transform of expression (31) then provides a realization of the random phase fluctuations φ(x) at the bottom of the irregularity layer. Fougere [1985] used a more involved approach for the generation of φ(x).

[24] The example using the second model assumes that Lo = 1/fo = 25.0 km, Lb = 1/fb = 0.40 km, p = 1.86, and q = 3.00, which are the parameters of the first model. It also assumes that the samples of ψ(fx) are mutually independent random variables uniformly distributed between 0 and 2π. Additionally, attempts have been made to use the second model to produce realizations of φ(x) that display phase distributions of their FFT components closely matching that of the first model. The thin solid curves in the three panels of Figure 1 display the corresponding results.

[25] It should be observed that the normalized phase screen curve created from the in situ data is characteristically different from the results provided by the second model. The thick curve in the upper panel of Figure 1 displays, particularly in the first 15 km of data, asymmetric structures with steep edges. Many rocket [Kelley et al., 1976; Morse et al., 1977; Hysell et al., 1994] and satellite data [Dyson et al., 1974; Basu et al., 1983] also show similar features. These were accepted as an indication that steep edges resulting from the nonlinear steepening of individual waves by plasma instability processes could dominate the power spectrum of the electron density fluctuations and produce the observed spectral shape [Costa and Kelley, 1978; Wernik et al., 1980]. This process would exhibit some degree of self-similar scale invariance and of coherence in the phase distribution of the Fourier components of the electron density fluctuations [Costa and Kelley, 1978; Hysell et al., 1994]. On the other hand, the asymmetric structures with steep edges are not present in the thin curve of the upper panel, which is more representative of a turbulence-like process, also resulting in essentially the same power spectrum for the wave amplitudes, but with uniformly distributed random phase.

[26] The realizations provided by the models should then be adjusted for the proper average fluctuation amplitude. They are normalized to create a zero-mean, unit standard deviation series of values, which are then multiplied by [Costa and Kelley, 1977; Rino, 1979]

display math

In the above expression, re = 2.8179 × 10−15 m is the classical electron radius, λ is the wavelength, and θ is the zenith angle of the incident rays. Further, Lo, L, and No have been previously defined, and 〈δN2〉 is the variance of the (zero-mean) electron density fluctuations. That is, the last term in expression (32) is the fractional RMS electron density fluctuation. Naturally, the parameters associated with the first model are defined by the data set in this case and, to make the overall model self-consistent, should be used in expression (32). Finally, the geometric factor G combines the effects of the anisotropy in the three-dimensional power spectral density of the electron density fluctuations with the orientation of the propagation direction with respect to the geomagnetic field [Rino, 1979].

5. Amplitude and Phase Scintillation of the Signal Received on the Ground

[27] Figure 2 displays the variations with the index q and the frequency of the absolute values of the coefficients Cq (decibels) defined in equations (10) and (23) to (25), assuming h = 400 km, δx = 40 m, θ = 30°, and f = 136 MHz, 360 MHz, 800 MHz, 1500 MHz, 4000 MHz. These curves can be compared with their correspondents in Figure 3, obtained with the same parameters but for vertical propagation (θ = 0°). This comparison shows that the number of coefficients necessary to simulate propagation using expression (10) increases with the zenith angle. Based on the observed behavior of the curves in Figures 2 and 3, it is relatively easy to design a truncation criterion to limit the number of coefficients to be used in a particular analysis. For example, only the coefficients within the lobes with peaks above (C0 − 40) dB have been displayed in these figures and used in the calculations. Figures 2 and 3 also illustrate how fast the number of coefficients necessary to simulate propagation decreases with frequency, that is, how fast the local character regime of propagation is reached as the frequency increases.

Figure 2.

Absolute value of the propagation coefficients Cq for h = 400 km, δx = 40 m, θ = 30°, and (a) f = 136 MHz, 360 MHz, 800 MHz or (b) f = 1500 MHz, 4000 MHz.

Figure 3.

Absolute value of the propagation coefficients Cq for h = 400 km, δx = 40 m, θ = 0°, and (a) f = 136 MHz, 360 MHz, 800 MHz or (b) f = 1500 MHz, 4000 MHz.

[28] Figure 4 displays the amplitude and the phase of the received signals at 360 MHz, as well their respective MEM power spectral densities (PSDs), for the two phase screen models of Figure 1. These calculations have assumed No = 2.5 × 1011 el/m3, L =100 km, h = 400 km, δx = 40 m, θ = 0°, G = 1, and fractional RMS electron density fluctuation of 5%. As before, results from the first and the second models are displayed with thick and thin curves, respectively. Additionally, a constant value (two amplitude units) has been added to the second amplitude signal, for display purposes only. It is interesting to observe that, differently from their originating phase screen models, the two amplitude signals at 360 MHz display the same features. That is, the amplitude fluctuations in the first received signal are as homogeneously distributed in space as the ones from the second signal. This feature results from the nonlocal character of propagation at 360 MHz. Also, the observed large-scale fluctuations of the phase screen models have been filtered from both amplitude signals. On the other hand, it can be seen that the phase signals preserve the features of their originating phase screen models.

Figure 4.

Amplitude and phase of the received signals at 360 MHz, as well the respective MEM power spectral densities (PSDs), for the two phase screen models of Figure 1, assuming No = 2.5 × 1011 el/m3, L = 100 km, h = 400 km, δx = 40 m, θ = 0°, G = 1, and fractional RMS electron density fluctuation of 5%. Thick and thin curves have been used to display the results from the first and the second models, respectively. Two amplitude units have been added to the second amplitude signal, for display purposes only.

[29] To successfully perform the convolution indicated by equation (10), one should have

display math

In this inequality, Nscr is the number of samples characterizing the phase screen, NCq is the number of propagation coefficients, and NRx is the number of desired samples in the received signal, assuming a common value of δx for the three series. The second phase screen model allows one to generate a phase screen with as many samples as necessary to meet inequality (33). However, the number of samples provided by in situ data to the first model may be limited, as in the present case. This limitation is particularly critical at lower frequencies, as observed in Figures 2 and 3. To apply the first model and still satisfy inequality (33), a sufficiently long series of samples has been generated from the available in situ data observed in Figure 1 by consecutive specular reflections around the last sample. This artifact may have affected the results from the first model at 360 MHz and be responsible for the observed differences between the PSDs of the two signals.

[30] Figure 5 displays the amplitude and phase of the received signals at 1500 MHz, as well their respective MEM PSDs, for the same phase screen models. A constant value of −0.5 has been added to the first amplitude signal, for display purposes only. It is observed in Figures 1 and 5 that the amplitude fluctuations in the first received signal are collocated with the steep edges in the first 15 km of the in situ data. This feature results from the local character of propagation at 1500 MHz. The same feature cannot be observed in the second amplitude signal, because of the homogeneous distribution of irregularities generated by the corresponding phase screen model. Also, the observed large-scale fluctuations of the phase screen models have been filtered from both amplitude signals even more deeply. On the other hand, it can be seen that the phase signals again preserve the features of their originating phase screen models.

Figure 5.

Amplitude and phase of the received signals at 1500 MHz, as well as the respective MEM power spectral densities (PSDs), for the two phase screen models of Figure 1, assuming No = 2.5 × 1011 el/m3, L = 100 km, h = 400 km, δx = 40 m, θ = 0°, G = 1, and fractional RMS electron density fluctuation of 5%. Thick and thin curves have been used to display the results from the first and the second models, respectively. A constant value (−0.5) has been added to the first amplitude signal, for display purposes only.

[31] Note from Figures 2 and 3 that the limitation imposed by inequality (33) can be more easily met by the first phase screen model at 1500 MHz. Consequently, the PSDs of the two amplitude and phase signals display a good agreement.

[32] The width of the power spectrum is usually characterized by the largest spatial frequency at which the PSD is 3 dB below its maximum value. The 50% decorrelation distance is the inverse of this spatial frequency. Comparing the power spectra of the second amplitude signal at 360 MHz and 1500 MHz, one finds that the former is wider by a factor of 2.5. Therefore the 50% decorrelation distance at 1500 MHz is longer than the one at 360 MHz by the same factor.

[33] Table 1 shows, for each phase screen model, the variation with frequency of the scintillation index S4 and the RMS phase deviation ϕo (radians) resulting from the input data listed above in this section. They correspond to essentially saturated scintillation at the two lower frequencies. On the other hand, both parameters decrease approximately as f−1 for the three upper frequencies. For each phase screen model, two columns with results from S4 calculations have been presented, using all (25 km) or only the first 15 km (corresponding to the presence of steep edges) of the amplitude signal. The corresponding results from the two 25-km calculations are reasonably consistent, which seems to indicate that the different characteristics of the phase screen models do not severely affect S4. However, the difference between corresponding values from the two 15-km calculations for the three upper frequencies clearly shows the important effects of steep edges observed in the first phase screen model on gigahertz scintillation [Wernik et al., 1980].

Table 1. Variation of S4 and ϕo With Frequency
Frequency, MHzFirst Phase Screen ModelSecond Phase Screen Model
S4ϕo, radS4ϕo, rad
25 km15 km25 km15 km
1360.970.9423.190.990.9819.13
3600.820.889.401.000.989.42
8000.150.200.660.160.150.66
15000.090.130.350.080.070.35
40000.020.030.090.020.020.09

6. Conclusion

[34] In this paper, the Huygens-Fresnel convolution integral has been explored to describe the slant propagation of a plane wave through a bidimensional irregularity layer. Instead of the straightforward application of the integral, one of the main contributions has been to show that its discrete version can be transformed into a convolution between a series of sampled values of the field and a series of propagation coefficients with decreasing amplitudes. These coefficients have been expressed in terms of the auxiliary functions involved in the definition of the well-known Fresnel integrals. An efficient propagation algorithm has been developed 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 FFT algorithm to evaluate discrete convolutions. It has been observed that the number of necessary propagation coefficients decreases (indicating that propagation becomes increasingly localized) as the frequency increases, approaching the optical regime.

[35] Next, two different phase screen models have been described. The first model assumes that the phase fluctuations of the wave front emerging from the bottom of the irregularity layer are proportional to electron density fluctuations directly obtained from satellite in situ measurements. The second model assumes that the same phase fluctuations at the bottom of the layer can be obtained from their power spectral densities and phase spectra, represented by analytical functions with parameters provided by physics-based or morphological models of ionospheric irregularities. In principle, the two models should provide equivalent results. Indeed, good agreement has been observed between their PSDs, and reasonable agreement has been observed between their phase spectra. However, it has been observed that the phase fluctuations created from the in situ data are characteristically different from the results provided by the second model. The in situ data display asymmetric structures with steep edges. These have been interpreted as an indication that nonlinear steepening of individual waves by plasma instability processes could dominate the power spectrum of the electron density fluctuations and produce the observed spectral shape. This process would exhibit some degree of self-similar scale invariance and of coherence in the phase distribution of the Fourier components of the electron density fluctuations [Costa and Kelley, 1978; Hysell et al., 1994]. The available models have not implemented these features, which would probably explain the observed difference between their characteristics and those of in situ data. In particular, the asymmetric structures with steep edges are not present in the results from the second model, which are more representative of a turbulence-like process, also resulting in essentially the same PSD, but with a noncoherent random phase.

[36] Finally, the propagation algorithm has been applied to both phase screen models, assuming frequencies in the high-VHF to low-SHF band. It has been observed that, differently from their originating phase screen models, the amplitude signals from the two models for frequencies in the low-UHF band and below display the same features. That is, the amplitude fluctuations in the first received signal are as homogeneously distributed in space as the ones from the second signal. This feature results from the nonlocal character of propagation in this relatively low frequency band. Also, the observed large-scale fluctuations of the phase screen models have been filtered from both amplitude signals. On the other hand, it has been observed that the amplitude fluctuations become collocated with outstanding small-scale size structures (such as the steep edges) that could be present in the phase screen model for frequencies in the L band and above. This feature results from the local character of propagation in this high-frequency band. Also, the observed large-scale fluctuations of the phase screen models have been filtered from both amplitude signals even more deeply. It has been seen that the phase signals preserve the features of their originating phase screen models throughout the total frequency band used in the exercise. Several parameters of the received signals have been obtained. In particular, it has been shown that the 50% decorrelation distance at 1500 MHz is longer than the one at 360 MHz by a factor of 2.5. For the assumed parameters of the ionosphere the obtained values for the scintillation index S4 and the RMS phase deviation ϕo at 136 MHz and 360 MHz correspond to essentially saturated scintillation. For the three upper frequencies (f ≥ 800 MHz) both parameters decrease approximately as f−1.

Acknowledgments

[37] Work at CETUC-PUC/Rio has been performed under subcontract 966-1 (of AF contract F19628-97-C-0094) between Boston College and PUC/Rio. The work at Air Force Research Laboratory was supported by AFOSR Task 2311AS. The authors thank the two reviewers for carefully reading the manuscript and for the helpful comments and discussions, which significantly improved this paper.

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