HF propagation in a wideband ionospheric fluctuating reflection channel: Physically based software simulator of the channel

Authors


Abstract

[1] A wideband HF simulator has been constructed on the basis of a detailed physical model of propagation which can generate a time realization of the HF wideband channel for any HF carrier frequency, bandwidth, transmitter receiver path and background, and stochastic (irregularity) ionosphere models. To accomplish this, a comprehensive solution has been obtained on the basis of the complex phase method (Rytov's method) to the problem of HF wave propagation for the most general case of a three-dimensional (3-D) inhomogeneous ionosphere with time-varying electron density fluctuations. A simulation is presented for a 1000 km path for which E and low- and high-angle F mode paths exist. The time-varying field owing to each of these paths is summed at the receiving location, enabling the calculation of the scattering function and also the time realization of the received signal shown as a function of both fast and slow time.

1. Introduction

[2] With the advent of digital HF broadcasting (e.g., DRM) and communications via the ionosphere, significantly higher data rates have become possible. However, the channel has not yet been well characterized for wideband (>8 kHz) digital signals. The ionospheric radio propagation channel is very complex and the ultimate success of new digital ionospheric radio systems will depend on a good understanding of important parameters of the channel such as Doppler shift, Doppler spread and multipath dispersion. The time variation of these parameters is also important, particularly the faster variations due to fluctuating ionospheric irregularities. Further complications arise from the geographical variations of the channel parameters with differences between equatorial, mid and high latitudes being particularly marked. To this must be added diurnal, seasonal, solar cycle and geomagnetic storm time variations. Because of the great variation of the ionosphere with different times and locations, it is also more difficult to adequately test out new HF communication systems. To cover all possible conditions, even for one fixed path, requires many trials to be performed. For a system that it is desired to deploy globally or for varying link distances and path locations, the necessary trials generally become prohibitively costly and time-consuming. Thus there is a need for a wideband simulator able to characterize the ionosphere response for any conditions, transmitter and receiver locations, transmission frequencies and bandwidths and taking into account not only the background ionosphere but all the fluctuating electron density irregularities. This should be able not only to generate realistic values of Doppler spread and delay spread for different paths, but also produce a time series output representative of the effect of the medium on the transmitted signal.

[3] Further, since based purely on physical models and parameters, the simulator will enable the correspondence between the characteristics of the received field and the physical parameters of the model to be investigated. This permits fine-tuning of the model by comparison of received field and predicted output for a variety of conditions as well as providing a way of estimating the physical parameters from the characteristics of the received field. The theoretical basis of such a simulator, as outlined above, is described in section 2, the necessary steps and equations to construct it are explained in section 3 and the production of random time series employing it and some preliminary results are given in section 4.

2. Theoretical Basis for the Wideband HF Simulator

[4] The problem of HF propagation in the ionosphere is one of the classical issues in the theory of radio wave propagation in near-Earth space. When treating HF propagation in the real ionosphere, it should be considered that the medium of propagation is a 3-D smoothly inhomogeneous (in terms of wavelengths of the HF band) anisotropic dispersive background medium, which is additionally disturbed by local deterministic and random inhomogeneities of the ionospheric electron density over a wide range of scales. As far as propagation in the background smoothly inhomogeneous medium is concerned, this problem can be considered to be accomplished as the methods to construct the high-frequency asymptotic solutions to this sort of problem are fairly well known. These are the classical geometrical optics approximation [Kravtsov and Orlov, 1980] or appropriate integral representations of the wave field in terms of geometrical optics type component waves known as the interference integral [Orlov, 1972], or oscillatory integral [Arnold, 1982] (see also classical works on high-frequency asymptotic solutions in mathematical physics [Ludwig, 1966; Maslov, 1965; Kravtsov, 1968]).

[5] To treat the problem of the effects of local inhomogeneities of the ionosphere (including the effects of random inhomogeneities) on HF propagation, a solution to the scattering problem for the case of a 3-D inhomogeneous, dispersive and, strictly, anisotropic background medium with local inhomogeneities must be constructed. If the spatial scales of the local inhomogeneities are greater than the appropriate main Fresnel zones size, the scattering problem can still be solved in the geometrical optics approximation. However, as the ionospheric turbulence has a wide spatial spectrum characterized by an inverse power law, a reasonable fraction of the random inhomogeneities has spatial scales less than the appropriate Fresnel zone size. This means that the contribution of diffraction should be properly accounted for when treating the scattering problem. This together comprises a very complicated problem for which a comprehensive solution should be given for a variety of realistic models of the ionosphere and geometry of propagation. This explains why different empirical models have been developed [Watterson, 1981; Vogler and Hoffmeyer, 1993; Mastrangelo et al., 1997; Sudworth, 1999; see also Proakis, 1983], which are widely employed [Angling et al., 1998; Messer, 1999; Nieto and Ely, 1999] to characterize the HF fluctuating channel of propagation. By contrast to the empirical approach, we present here a rigorous treatment of the HF propagation in a 3-D inhomogeneous medium disturbed by fluctuations of the electron density of the ionosphere.

[6] Concerning the way to properly account for the wave polarization, when dealing with propagation in a smoothly inhomogeneous isotropic medium without fluctuations and considering power (quadratic) characteristics of the field, the wave polarization does not affect the result. However, when considering the scattering by local random inhomogeneities, then, for a completely rigorously treatment, the vector character of the scattered field should be taken into account. However, there is a physical reason to remain within the framework of the scalar approximation. It is well known that the differential scattering cross section of the same inhomogeneity is not the same for the scalar and vector field scattering, but the difference almost vanishes in the case of the scattering by large-scale inhomogeneities, in other words, in the case of forward scattering. The complex phase method we have employed just describes this case. All the inhomogeneities we consider are large scale in terms of the wavelength. Thus we consider that it is a reasonable basis to consider the problem in the scalar approximation, at least, to the zero-order approximation.

[7] The theoretical consideration of the problem of HF propagation in the disturbed ionosphere can be split into two parts. The first part is the HF propagation in the background 3-D smoothly inhomogeneous medium (section 2.1) and the second is the description of the effects of scattering of the HF field by local random inhomogeneities of the ionosphere (section 2.2). We will also present the description of the software simulator of the fluctuating channel of propagation, developed on the basis of rigorous treatment of the appropriate equations governing the propagation.

2.1. Propagation in the Background Ionosphere

[8] As mentioned above, the description of the HF propagation in the 3-D smoothly inhomogeneous medium is the simplest part of the problem. Characteristic scales of the background ionosphere in all the directions are sufficiently large to allow the geometrical optics approximation to be employed to describe the HF field. Appropriate codes for calculation of ray paths and ray pencil divergences are available to enable the construction of simulated oblique sounding ionograms. These can then be employed, using the appropriate model of the background ionosphere, to determine possible paths (modes) connecting transmitter and receiver locations. Quantities such as divergences along the paths of propagation are also used when determining the scattering of the field by local random inhomogeneities along each actual mode of propagation between a transmitter and receiver.

2.2. Scattering of HF Field by Random Ionospheric Inhomogeneities

[9] This is the most complicated part of the propagation problem. The simultaneous presence of several scales of ionospheric density variations is very demanding when treating the scattering problem. Generally speaking, the solution should be obtained for the scattering problem for the case of a 3-D inhomogeneous dispersive background medium, accounting also for the contribution of diffraction effects in the scattering by local random inhomogeneities.

[10] The case of weak or moderate fluctuations of the amplitude of the field can be treated in the framework of perturbation theories. Among them the complex phase method (or the generalized Rytov's approximation) handles the scattering problem in the most comprehensive form as it can also account for diffraction by local random inhomogeneities and partly accounts for multiple scattering effects. Additionally, it enables construction of the appropriate two-position, two-frequency correlation and coherence functions of the random field for the condition of a strongly inhomogeneous and dispersive medium; a condition which is fully pertinent to the ionosphere. These functions are the core quantities when modeling the fluctuating channel of propagation both in terms of the statistical moments of the field propagated through the channel and random time sequences of the field. The method limitation is determined by the range of validity of the complex phase method, which can be roughly stated as that the variance of the fluctuations of the log-amplitude (level) of the field cannot be large. This is a well known limitation of the Rytov's approximation (or the complex phase approximation, which is its extension). In addition, for our application, puts certain limitations on the variance of the electron density fluctuations. The codes are arranged in a way that this is controlled for any given path and conditions of propagation. In turn, this means that there is no one particular universal limit for the fractional electron density fluctuations. However, for a typical one-hop path of propagation for a link distance of the order of 1000 km, it results in a limit of the order of 1% for the r.m.s. of the fractional electron density fluctuations. The same criteria needs to be applied for high and low latitudes as for midlatitude paths, but in the former cases the possible occurrence of strong scintillations can lead to a break down of the theory's validity.

[11] For the case of strong scintillation, such methods as Markov's parabolic equations for the statistical moments of the random field [Ishimaru, 1978; Rytov et al., 1978] and the path integral technique [Dashen, 1979; Flatte, 1983] should be mentioned, which permit description of the effects of strong scintillation in some cases. Many problems of wave propagation in random media have been considered in the scope of these methods and we cannot here provide a complete bibliography. However, it is our current conviction that neither Markov's approximation, nor the path integral technique is yet capable of handling the problem of constructing spaced position and frequency coherency in the ionosphere-type medium, i.e., for the essentially inhomogeneous and dispersive background medium with local random inhomogeneities embedded. This led us to consider it best to confine the present treatment of the problem of scattering of the HF waves by local random ionospheric inhomogeneities within the framework of the complex phase method, at the same time taking account of the constraints of this method and its range of its validity as discussed above. Toward the end of the paper (in section 4.2) a numerical example is given for an ionosphere including the effect of the geomagnetic field. There is additional complexity for this case introduced by the anisotropy of the medium of propagation. In this paper we just present the theory for the isotropic case as we consider that the complexity of the anisotropic case requires special consideration. We intend to give a full description of this in a subsequent paper.

3. Complex Phase Method: General Case of 3-D Inhomogeneous Background Medium

[12] The complex phase method is the extension of the classic Rytov's approximation [Rytov et al., 1978], dated back to 40 s, to the case of the point source field and the inhomogeneous background medium. The first extension of the method was performed by Zernov [1980], who considered the HF field in a stratified ionosphere, disturbed by local inhomogeneities. The extended Rytov approximation was further employed in a series of papers [Gherm and Zernov, 1995, 1998; Gherm et al., 1997, 2001a] to construct and study the statistical moments of the random HF field in the plane-stratified ionosphere disturbed by fluctuations of the electron density.

[13] Obviously, the following extension of the method must include the general case of a 3-D inhomogeneous medium. In particular, this is necessary when characterizing the HF fluctuating ionospheric channel of propagation, which is horizontally inhomogeneous (i.e., containing horizontal gradients of electron density). The appropriate generalization has been recently performed by Gherm et al. [2001b] in a paper written and issued in Russian. Here we will briefly reproduce the milestones of this extension.

[14] In the present consideration the scalar equation

equation image

widely used to describe HF propagation, is employed, where k is the wave number in vacuum, ɛ0(r) is the dielectric permittivity of the background medium and ɛ(r) is the dielectric permittivity of local inhomogeneities. r is the point of observation, r′ the variable of integration and is the position of the source of the field (the transmitter). Quantity A characterizes, in some sense, the power of a source. In order to account for the time dependence of the electron density fluctuations function ɛ(r) is also allowed to be a function of the slow time in the quasi-stationary approximation.

[15] Depending on the given model of the background medium ɛ0(r), the undisturbed (incident) field E0(r), which satisfies equation (1) with ɛ(r) = 0, may have a multipath structure, i.e., several paths of propagation may occur, which connect the transmitter and receiver. The field propagating along each of m paths can be well described in the geometrical optics approximation, so that the full undisturbed field is represented by the sum of the geometrical optics type fields as follows:

equation image

[16] Acceptance of the representation given by (2) for the undisturbed field implies limitation to the case when the observation points are far from any caustic (far from the skip distance, if the transmitter and receiver are located on the Earth's surface). This implies that the main Fresnel volumes for different paths of propagation do not overlap. In the same fashion the Green's function for the undisturbed equation (1) is also represented in a form similar to equation (2) by the sum of geometrical optics contributions

equation image

providing r, r′ are not near any caustic.

[17] To account for the effects of local random inhomogeneities of the ionosphere on every geometrical optics component E0mGO of the undisturbed field, its own complex phase ψm is introduced for each component, so that the full field disturbed by local ionospheric inhomogeneities is given as follows:

equation image

According to the complex phase method each ψm is represented by the perturbation series in powers of the disturbances ɛ(r), and the technique of the method permits solutions to the appropriate equations for different orders of ψm in the following invariant form [Zernov, 1980]:

equation image
equation image

We have presented here only the disturbed complex phases of the first and second orders, which are employed in the following treatment.

3.1. Geometrical Optics Field

[18] It is convenient to specify the representations (5) and (6) in ray-centered variables (s, q1, q2), where the reference ray is a given m th curvilinear path connecting the communicating points in the 3-D inhomogeneous background medium, so that every path gives rise to its own ray-centered coordinate system. (From now on we omit subscript m referring to the m th path of propagation.) In these coordinates variable s is measured along the reference ray in the direction from the source to the receiver, and q1 and q2 lie in the plane perpendicular to the reference ray at each point. For this coordinate system, Lamé coefficients are as follows:

equation image

Here n2(s, q1, q2) = ɛ0(s, q1, q2) and n(s, 0, 0) = [ɛ0(s, 0, 0)]equation image. In the following we denote n(s, 0, 0) = n0(s). In the introduced ray-centered variables we take the coordinates (0, 0, 0) for the transmitter and (s0, 0, 0) for the receiver. The variable of integration r′ in the integrals in (5) and (6) is now given by (s, q1, q2).

[19] To construct E0mGO and GmGO in equations (5) and (6) in the form of the geometrical optics approximation for type such as A0 exp (ikϕ), the appropriate eikonal equation

equation image

for the phase function ϕ and the main transport equation for the amplitude A0,

equation image

must be solved for each path of propagation. The solutions of equations (8) and (9) locally nearby the reference ray are sought for in the form of a series in the transverse variables q1 and q2 as follows:

equation image
equation image

The representation (10) means that the finite curvature of the front of the undisturbed (incident) field is accounted for to the accuracy of the main terms, which are given by the full quadratic form in the square brackets. The linear terms vanish here because the medium is isotropic so that the wave front must be orthogonal to the wave direction.

[20] Performing necessary expansions for n2 and hs in a series in the transverse plane to the reference ray variables and equating to zero coefficients at different powers of q1, q2 yields for the amplitude A00(s)

equation image

where the functions b11(s), b22(s), b12(s) satisfy a set of differential equations of Riccati type, which may be conveniently written in the matrix form as follows:

equation image

[21] In the last equation,

equation image
equation image

When considering the equations in (13), the solution should reduce to the spherical wave near a source in the small-angle approximation (assuming that the source is in vacuum) as follows:

equation image

This is also the recipe as to how to properly choose the constant and the limits of integration in (12). Then (10)–(12) finally yield the following expression for the undisturbed (incident) field

equation image

where b11(s), b12(s), b22(s) are properly chosen solutions of equations (13)–(15). Formally this should be considered in the limit when the small quantity r0 tends to zero. This equation reduces to the spherical wave (16), when n0(s) = 1. Small finite values of r0 are employed when numerical solutions of equations are realized to properly specify equation (13)–(15).

[22] In the same manner, the representation for the Green's function G(r, r′) with r = (s0, 0, 0) and r′ = (s, q1, q2) may be written

equation image

Here the variable s1 is measured along the same reference ray, but in the direction from the receiver to the transmitter. Making use of this variable, the elements of the matrix equation imageg = {bikg}, bikg = 1, 2, i, k = 1, 2 satisfy the same set of equations (13)–(15) as the matrix equation image. If the substitution s = s0s1 is performed under the sign of integration, equation (18) becomes

equation image

When written by means of variable s, matrix equation imageg satisfies the set of equations

equation image

which differs from the set of equations (13)(15) only by the sign at the first derivative.

3.2. First-Order Complex Phase

[23] Finally, putting together all the necessary representations gives the following equation for the first-order complex phase from equation (6):

equation image

To derive the last equation the relationship E0(s0r0, 0, 0) ≈ E0(s0, 0, 0) has been used.

[24] To further transform equation (21) some necessary relationships for the new matrix

equation image

should be derived. Subtracting equation (20) from (13) and performing simple transformations yields

equation image

where

equation image

Then, the integral expression, which is the first term in the exponential in equation (21) is just:

equation image

This allows us to finally write the quantity from equation (21) in the following form:

equation image

To obtain this expression, the relationship

equation image

was used. Matrix equation image+, involved in calculations according to (26), is given by equation (24), where, in turn, elements of matrixes equation image and equation imageg satisfy sets of differential equations (13) –(15) and (20) respectively. When and where it is necessary to have the representation for the second-order complex phase ψ2 the quantity k2 ɛ in equation (26) should be replaced by (∇ψ1)2.

[25] Formula (26) is the final result, which extends the classic Rytov's method to the case of the point source in an arbitrary 3-D inhomogeneous medium. It permits different limiting cases. In particular, when the background medium is homogeneous it yields the known result for the complex phase of a spherical wave in a homogeneous background medium, disturbed by a local inhomogeneity ɛ(r) [Tatarskii, 1971; Ishimaru, 1978]. In this case b11 = b22 = s−1 = x−1, b12 = 0; b11g = b22g = (s0s)−1 = (x0x)−1, b12g = 0, and equation (26) yields:

equation image

Another limiting case for the general representation (26) is when the quantities (b11 + b11g), (b22 + b22g), (b12 + b12g) are large compared to the transversal characteristic scales of the inhomogeneities ɛ along all the path of integration in variable s. Then integration in q1 and q2 in equation (26) may be performed explicitly employing the steepest descent method to produce, to the first approximation, the first-order correction of the geometrical optics approximation as follows:

equation image

This is the case of local inhomogeneities with large spatial scales compared to the main Fresnel zone size along the path of propagation.

[26] To utilize the generalized complex phase given by (26), a special numerical code has been produced to solve the matrix equations (13) and (20), This was combined with a general ray-tracing code for the 3-D inhomogeneous background medium.

4. Simulation of Random Time Series and Statistical Moments of the HF Field

[27] When characterizing the ionospheric fluctuating HF reflection channel of propagation, both the random time series and statistical moments of the pulsed signal propagated through the channel are of interest.

4.1. Random Time Series

[28] A random realization of a pulsed signal propagated through the fluctuating ionosphere can be represented as the following Fourier integral in the frequency domain

equation image

Here P(ω) is the spectrum of a launched pulse, E0mGO represents the transfer function for a given m th path in the undisturbed channel, i.e., the functions from equation (2). Once the model of the 3-D background ionosphere is given, the quantities E0mGO are calculated employing the appropriate ray-tracing code, which also permits calculation of the ray tube divergence. A random phasor Rm (r, ω, T) is introduced in (30) to account for the effects of fluctuations of the electron density of the ionosphere. Variable t is the flight time of a pulse and T denotes slow time dependence of fluctuations, which can be treated in the quasi-stationary approximation. The background channel is assumed to be stationary that is time-independent. A corollary of this is the absence of slow time dependence in the transfer functions of the background channel in (30). The summation in (30) is performed over all paths of propagation from a transmitter to a receiver.

[29] According to the complex phase method, described above, random phasors Rm (r, ω, T) are represented utilizing complex phases as follows:

equation image

where the first and second-order complex phases ψm in powers of the disturbances of the dielectric permittivity are given by equations (5) and (6) and specified in ray-centered variables by equation (26). Complex phases

equation image

are random functions with the real part χm representing log-amplitude fluctuations and Sm giving the fluctuations of the phase of the field.

[30] To produce the time series of a pulsed signal given by the Fourier integral (30) for a point of observation r, two real random functions χm and Sm must be generated in the two-dimensional domain (ω, T) for a given value of r. This demands knowledge of the probability density functions for χm and Sm, as well as their autocorrelation and cross-correlation functions. In the scope of the complex phase method, these functions are given as follows:

equation image
equation image
equation image

All these functions can be found making use of the two main autocorrelation functions of the first-order complex phase Bψ1 = 〈ψ11, T11*(ω2, T2)〉 and Bψ2 = 〈ψ11, T112, T2)〉. Their explicit expressions will be presented below.

[31] As far as the probability density functions for the random functions χm1 and Sm1 are concerned, equation (26) shows that χm1 and Sm1 are represented by linear integrals over many random inhomogeneities. This guarantees, according to the central limit theorem, that both the random functions χm1 and Sm1 are normally distributed. When averaging in (33)–(35) it is also implied that the electron density fluctuations along different paths of propagation are not correlated. This is in a reasonable agreement with the requirement that the main Fresnel volumes of the neighboring rays do not overlap.

[32] If, additionally, the hypothesis of the “frozen drift” of random inhomogeneities in the ionosphere is adopted, slow time T is expressed through the position of the inhomogeneity structures, so that actually appropriate two-frequency, two-position autocorrelation and cross-correlation functions must be constructed. We have studied in detail these type of functions in the scope of the complex phase method for the case of a plane-layered background medium [Gherm et al., 1997; Gherm and Zernov, 1998]. Having the representation (26), which extends the complex phase method to the case of a fully 3-D inhomogeneous medium, appropriate statistical moments of the complex phase can be constructed for an arbitrary 3-D inhomogeneous background medium. In particular, the abovementioned correlation functions Bψ1 and Bψ2, which permit expressing the correlations (33)–(35), are of the following form

equation image
equation image

Here k = ω/c and Bɛ(s; 0, κn, κτ) is the three-dimensional spatial spectrum of the electron density fluctuations with zero value of the spectral variable, Fourier-conjugated to the difference variable along the path. It is also a function of the central variable along the reference ray. The spectral variables κn and κτ are Fourier-conjugated to the spatial variables q1 and q2 lying in the plane perpendicular to the reference ray at each point. The quantities Δn and Δτ are the components of the vector of distance between the rays corresponding to the frequencies ω1 and ω2, which also depend on s. Additionally, the hypothesis of the “frozen drift” of random inhomogeneities is utilized, so that vn and vτ are the components of the frozen drift velocity also depending on the point along the reference ray, and T = T1T2 is the difference in slow time. The central slow time T+ is not involved in equations (36) and (37), because of the assumption of the statistical homogeneity of the fluctuations. The coefficients Dn, Dτ, and Dnτ are the elements of the matrix equation image = (equation image+)−1, which is the inverse of the matrix equation image+(24). These also depend on the variable s.

[33] In the numerical calculations, a turbulence model of the ionospheric fluctuations is considered having an anisotropic inverse power law spatial spectrum of the form

equation image

Here CN2 is a known normalization coefficient. Ktg = 2πltg−1, where ltg is the outer scale of the turbulence along the geomagnetic field, and Ktr = 2πltr−1, where ltr is the outer scale of the turbulence across the magnetic field. Function ɛ0(s) is the distribution of the dielectric permittivity of the background ionosphere along the reference ray in the 3-D inhomogeneous background ionosphere and σN2 (s) is the distribution of the variance of the relative fluctuations of the electron density of the ionosphere along the reference ray in the 3-D inhomogeneous ionosphere. As a result, functions (36) and (37) are, with a very high degree of generality, valid for arbitrary three-dimensional models of the background ionosphere and fluctuations of the ionospheric electron density.

[34] All the abovementioned results permit to uniquely produce random series of functions χm and Sm in the domain (ω, T), if, additionally, the cross correlation (35) is also properly accounted for. To generate the time series, spectra of the correlation functions of χm and Sm (power spectra) are calculated in the domain (τ, Ω), where τ is Fourier-conjugated to ω and Ω is Fourier-conjugated to T correspondingly. Complex valued Fourier spectra of random realizations of χm and Sm are assumed to have their absolute values equal to the square roots of the appropriate calculated power spectra and arguments uniformly distributed in the interval 0–2π. A correct cross correlation of the χm and Sm realizations is then provided by the proper choice of two basic sequences of random numbers having their cross-correlation coefficient defined by the mutual correlation of χm and Sm [see, e.g., Devroye, 1986]. In turn, these permit generation of random values of the phasor Rm (r, ω, T ) in the same domain, and finally to generate the random series of a signal that has propagated through the fluctuating ionosphere employing the appropriate methods of numerical calculation of the integrals in equation (30).

[35] Below we shall present some results of a simulation obtained using the developed technique and simulator. All the results have been calculated for a single-hop path of length 1000 km oriented to the west from St. Petersburg, Russia. The IRI model for July at 0700 LT was chosen for the transmitter site at St. Petersburg and for the receiver site 1000 km to the west of St. Petersburg. For this path, horizontal gradients of the electron density resulted in a difference of 0.5 MHz in foF2 between the transmitter and receiver. The carrier frequency was 8.1 MHz.

[36] The fluctuations of the ionospheric electron density were characterized by the inverse power law anisotropic spatial spectrum with the spectral index of 3.7, the scale of random inhomogeneities across the geomagnetic field of 3 km and the aspect ratio of 5. The variance of relative fluctuations of the electron density was assumed to be uniform along the path of propagation and equal to 3 × 10−6. The hypothesis of frozen drift of the random inhomogeneities was utilized with the same horizontal longitudinal and latitudinal velocity of 0.5 km/s. The bandwidth of the rectangular transmitted pulse was 20 kHz.

[37] In the first step, the oblique sounding ionogram was constructed for the chosen model of the background ionosphere, which indicated possible high- and low-angle F and E mode paths of propagation. In Figure 1, the random walk as a function of T is shown for the phasor corresponding to the E mode propagation path for a fixed frequency component ω, whereas Figure 2 demonstrates the same for the phasor of the high-angle F mode path. Clearly, the spread of possible random values of the phasor on the plot of Figure 2 is significantly wider due to the higher density of the background ionosphere at the altitude of the F layer, leading to the higher values of the absolute fluctuations of the electron density.

Figure 1.

Random walk of the phasor Rm (r, ω, T) for the E mode. The fluctuations of the field are weak.

Figure 2.

Random walk of the phasor Rm (r, ω, T) for the high-angle F mode. This shows stronger fluctuations of the field.

[38] In a similar way, phasors for all possible paths of propagation, connecting transmitter and receiver for the given conditions (the model of the background ionosphere and geometry), are produced. Then, calculating numerically the quantity U(r, t, T) according to the integral (30) random time sequences of a pulsed signal propagated through the fluctuating ionosphere are generated for different moments of slow time provided that the spectrum P(ω) of the transmitted signal is specified. In Figure 3 the results of generating the random sequences for a transmitted rectangular pulse are presented, as a function of the flight (fast) time, for different moments of slow time.

Figure 3.

Realization of the received signal plotted in slow time and fast time variables.

4.2. Scattering Functions

[39] The scattering function of a pulsed signal is introduced [Proakis, 1983; Vogler and Hoffmeyer, 1993; Mastrangelo et al., 1997; Gherm et al., 2001a] as the appropriate Fourier transform of the autocorrelation function of the random channel impulse response on the difference slow time variable. Utilizing (30), the autocorrelation function of a pulsed signal on slow time T can be written as follows:

equation image

Here the spatial variable r was suppressed and the following relationships have been introduced:

equation image
equation image

Equation (40) shows explicitly the amplitude and phase of the field E0mGO, which are calculated for each possible mode of propagation, defined by the model of the background medium. Relationship (41) is the definition of the two-frequency two-time correlation function of the random phasor Rm (ω, T).

[40] It is convenient to work with the central and difference variables in the frequency and slow time domains

equation image
equation image

Utilizing new variables in (39) and performing Fourier transformation on difference slow time T, the following equation for the scattering function (which is sometimes termed the wideband scattering function) is obtained:

equation image

Here the summation is performed over all paths of propagation from the source to the receiver. The scattering function of the channel S(t, T+, ωd) depends on the Doppler variable ωd, (Fourier-conjugated to T), the group delay t and, generally, on the slow time T+. The latter dependence vanishes when the random ionospheric fluctuations are assumed to be statistically stationary. Group delay time tgm+) is given by the equation

equation image

and is calculated for each mode of propagation.

[41] In the framework of the complex phase method, the frequency and time correlation functions of the random phasor ΨRm in the integral (44) are expressed through the statistical moments of the complex phase [Gherm and Zernov, 1998; Gherm et al., 2001a] as follows:

equation image
equation image

The function Ψψ denotes the autocorrelation function of the complex phase, with the main term being obtained employing the complex phase method within the relationship

equation image

The functions (46)–(48) have been studied in detail in the work of Gherm and Zernov [1998] for the case of plane-layered background medium. The scattering function (44) has been studied in the work of Gherm et al. [2001a], also for the layered background medium. The extension of the complex phase method obtained above for the case of a 3-D inhomogeneous background medium (equation (26)) naturally permits the extension of the technique of calculation of the scattering functions to the general case of a 3-D background. The statistical moments of the complex phases involved in equations (44) and (46)–(48) are given through the representations (36) and (37). They have all been derived analytically for the general case of an arbitrary 3-D inhomogeneous background medium. The derivation is based on the general equation (26) for a random realization of the complex phase. The calculations are then performed numerically for a given model of the background ionosphere. In Figure 4, the scattering function is presented in the form of a contour plot, calculated according to the described technique as the appropriate statistical moment of the signal.

Figure 4.

Scattering function calculated theoretically as a statistical moment of the field.

[42] Finally, there is also another possible method of obtaining the scattering function, namely from the random time series as represented in the plot shown in Figure 3. This is a numerical processing of the simulated time series of a signal analogous to what is really done to real experimental data. The result is presented in Figure 5 in the form of a contour plot. In both Figures 4 and 5, the adjacent contours are separated by 5 dB and range from 0 to −30 dB. Strictly speaking, these plotted values in Figure 5 are not statistical moments, but a sort of realization of the scattering function, obtained after averaging over a finite number of realizations of the received signal. If the period of this averaging is increased, then the number of random realizations will also increase and the resulting plot will converge to the true scattering function, which is the rigorous statistical moment presented in Figure 4.

Figure 5.

Scattering function retrieved from the field realization shown in Figure 3.

[43] As far as the effects due to the magnetic field of the Earth are concerned, we have confined this consideration to the isotropic refractive index case. However, the appropriate extension of the theory (of the complex phase method) has also been developed to describe the effects of ordinary and extraordinary modes, so that the magnetoionic splitting can also be accounted for. It was not really practical to give the detailed description of the anisotropic version including the theory and simulator in the framework of a single paper. An additional paper is planned to be devoted to this subject. Some results for the anisotropic case have been recently reported in the work of Gherm et al. [2003]. To conclude this paper, in Figure 6 the anisotropic case is briefly presented for conditions analogous to the isotropic case in Figure 5. Again, the retrieved scattering function is represented in the form of a contour plot. It is clearly seen that the high-angle F mode (the uppermost local maximum) is split into o components and e components, whereas low-angle F mode and E mode are not resolved into o components and e components.

Figure 6.

Scattering function retrieved from the field realizations for the case when the Earth's magnetic field is taken into account.

5. Practical Use of the Simulator

[44] The inputs required for the simulator are the geographic location of the transmitter and receiver, the nature of the transmitted signal and the characteristics of the background and stochastic ionosphere (time-varying irregularities) components. The background ionosphere can be fully 3-D such as being represented as a fit to the IRI model over a given latitude and longitude range. Alternatively, it can be specified in terms of the parameters of a number of Chapman, parabolic or quasi-parabolic layers which can include linear latitudinal and/or longitudinal gradients of electron density and/or height of the electron density maximum. Slow time variation such a layer movement or TIDs can also be incorporated and will result in Doppler shift whereas the time-varying irregularities result in Doppler spread. The stochastic component of the ionosphere is specified in terms of the variance of the fractional electron density, the exponent of the inverse power law spatial spectrum, the outer scale of the irregularities along and transverse to the geomagnetic field direction and the direction and speed of the irregularities in three dimensions. The E and H field patterns of the transmitting and receiving antennas can be taken into account when determining the strength of the transmission at different azimuths and elevations and when summing the E fields of the different modes at the receiver. The initial azimuth and elevation angles of the signal for each multipath component are determined by the homing-in program and so are known. This information can also be obtained from the ray-tracing program for the end of the ray path at the receiver location. Either vertical or horizontal antennas can be used for the link.

6. Conclusions

[45] The general description of HF propagation in the ionosphere with 3-D inhomogeneous background and local random inhomogeneities embedded presented above comprises the physical basis for producing a software simulator for the wideband ionospheric fluctuating reflection HF channel. The simulator is capable of producing both random time sequences of a pulsed signal propagated through the fluctuating ionosphere and its statistical moments, e.g., scattering functions. The programs are arranged in the way that any given 3-D model of the background ionosphere can be utilized and multimode propagation can be included for any geometry of propagation. The software simulator utilizes the inverse power law spatial spectrum of fluctuations of the electron density of the ionosphere with given spectral index and different spatial scales of inhomogeneities along and across the magnetic field. Fluctuations are assumed to be statistically homogeneous in time (stationary). The simulator is capable of producing results for signals with bandwidths up to 0.5 MHz. A noise model, described by Lemmon and Behm [1991], has also been added. Bulk plasma motion of the background ionosphere can also be included, giving a Doppler shift in addition to the Doppler spread resulting from diffraction by the moving irregularities.

[46] This propagation model and simulator, since based purely on physical models and parameters, also enables the correspondence between characteristics of the received field and the physical parameters of the model to be investigated. This permits fine-tuning of the model by comparison of received field and predicted output for a variety of conditions as well as providing a way of estimating the physical parameters from the characteristics of the received field.

Acknowledgments

[47] NNZ and VEG thank the EPSRC (UK) for financial support as visiting fellows under grant VF GR/R37517/01.

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