## 1. Introduction to Theory

[2] Satellite radio tomography is known as an efficient method for mapping the structure of ionosphere [*Kunitsyn and Tereschenko*, 2003; *Pryse*, 2003; *Leitinger*, 1999]. This approach is based on the analysis of radio waves transmitted from flying satellite, then scattered by ionospheric irregularities, and measured by ground-based receivers. As the radio wave passes through the ionosphere, phase and amplitude of the signal change. Phase fluctuations are mostly due to large-scale irregularities with scale sizes much higher than the Fresnel radius. Spatial distribution of such irregularities can be reconstructed by means of ray radio tomography. The measured phase φ is proportional to the integral of electron density *N* along the ray *s* from a satellite to a receiver

[3] Phase measurements at a set of intersecting rays provide a linear system of integral equations like (1). Solving the system for *N* by means of tomographic inversion of the data yields the spatial distribution of electron density within the ionospheric volume under study.

[4] Scattering from small-scale irregularities with scale sizes of the order of, or less than, the Fresnel radius mostly contributes to amplitude fluctuations of radio waves. These irregularities may fill ionospheric volumes of various shapes and sizes so that the scattered field depends on a large set of scatterers and is therefore chaotic. Each separate irregularity cannot be investigated individually, and a statistical consideration is preferable. In this approach the ionosphere is regarded as a randomly inhomogeneous medium, and a relationship between statistical characteristics of the ionosphere and statistical parameters of the scattered radio wave is sought for.

[5] Consider a VHF radio wave scattered by electron density fluctuations of ionospheric plasma. Let inhomogeneous ionosphere be a plane layer, *zL* and *zU* being its lower and upper boundaries, respectively. Incident spherical wave is detected by a ground-based receiver. The *z* axis joins the source and the receiver and points in the direction of radio wave propagation. Coordinate is transverse and perpendicular to *z* axis. The receiver is located at (*z*_{0}, _{0}). Field (, *t*) at a point (*z*, ) within fluctuating ionosphere satisfies the equation

where conductivity is a tensorial quantity and depends on various parameters of the medium denoted symbolically as ρ_{i}, *i* = 0, 1, 2,… In equation (2), *t* is time and *c* is the light velocity. In a fluctuating medium, each parameter can be represented as a sum of its mean value and deviation from the mean:

[6] Angular brackets denote statistical averaging. If fluctuations are small, one can expand in a Taylor series and retain the first-order terms only:

[7] Assuming 〈ρ_{i}〉 and δρ_{i} as time independent, the solution to equation (2) can be represented as a monochromatic wave

where ω is frequency.

[8] Assuming “cold” plasma approximation, conductivity tensor at frequencies not exceeding the hyrofrequency can be expressed as , where is a unity tensor. Then

where 〈*N*〉 is average electron concentration, δ*N* electron density fluctuations, *e* electron charge, and *m* electron mass. Bearing in mind equations (3)–(5) and taking into account that in the absence of exterior charges *div*ε = 0 where ε is dielectric permittivity, one obtains equation (2) in the form

where *n* = is a refractive index, is squared plasma frequency, and is wave number.

[9] In case of VHF waves propagation through the ionosphere the depolarization term grad(− grad ε) can be neglected [*Rytov et al.*, 1987]. Besides, the frequency of incident VHF wave is much higher than the plasma frequency; therefore one can assume *n* ≈ 1. Taking into account where *r*_{e} = 2.82 · 10^{−15} m^{−3} is classical electron radius, we obtain equation (6) in the form

[10] Vector equation (7) splits into three scalar equations for field components *E*_{x}, *E*_{y}, and *E*_{z},

which derives a relationship between amplitude of scattered field and the parameters of small-scale ionospheric irregularities. We shall use Rytov's approximation valid for comparatively weak scintillation. Solution to equation (9) can be present in the form

Substituting equation (9) into equation (8), expanding the solution of the obtained equation into a series of small parameter δ*N* and grouping terms of the same order; we obtain a system

[11] The first of these equations is a zero approximation that describes wave propagation through the medium containing no fluctuations. Subsequent equations describe wave interaction with fluctuations in the propagation medium. Assuming fluctuations to be weak, it is sufficient to consider only the first-order approximation and to solve an equation

With the solution in the form

equation (8) appears as

Solution to this equation is

Then

The solution to the equation of zero-order approximation is a spherical wave

where *A*_{0} is amplitude of the wave. Taking into account equation (16), one obtains an expression for Ψ_{1}:

The field in the first-order Rytov's approximation can generally be described as

However, field can be written as = *A*()exp (*iS*()), where *A* is amplitude and *S* is phase, whence it follows that

[12] Here χ denotes logarithmic relative amplitude. Equation for the variance of log relative amplitude σ_{χ}^{2} can be easily obtained from equation (18). Let ′(*z*′, ′) and ″(*z*″, ″) be current points of integration. Using paraxial approximation one arrives at

Converting to difference and summary coordinates = ′ − ″, and integrating over the summary coordinate , one obtains

where *K*(, *z*) = is correlation function of electron density fluctuations that is related to spectral density Φ_{N} by Fourier transform [*Ishimaru*, 1978]:

where κ_{z} and are wave vector components along and perpendicular to z axis, respectively. Substitution of equation (21) into equation (20) and integration of the obtained expression yields finally

where λ is wavelength, *R*_{F} = [λ*z*′ · (*z*′ − *z*_{0})/*z*_{0}]^{1/2} is the Fresnel radius, and ≡ .

[13] Assuming that the same mechanism of irregularities generation operates all over the ionospheric region under study, spectral density can be written as

where σ_{N}^{2}(z) is a variance of electron density fluctuations that depends only on absolute value of *z*, and Φ_{0} (, *z*) is normalized spectral density depending on the direction of . Applying to equation (22) the Kirchhoff formula linking the field in the fluctuating medium with the field outside, one obtains the equation for the variance of logarithmic relative amplitude at the receiving point:

The double integral in equation (24) depends on the spectral shape only and not on the variance of electron density fluctuations.

[14] For further calculations the spectral shape should be specified. The power law spectrum is believed the most suitable model of ionospheric irregularities. High-latitude irregularities are known to be strongly anisotropic [*Aarons*, 1982]; therefore we shall use a model of 3-D anisotropic spectrum of irregularities elongated in the direction of geomagnetic field and in some field-perpendicular direction [*Fremow and Secan*, 1984].

[15] Here α is relative elongation of irregularities in the direction of geomagnetic field vector , and β is relative elongation of irregularities in a plane perpendicular to the geomagnetic field (so as the irregularity axes relate as α: β: 1), Γ gamma function. κ_{∥} is wave vector component parallel to the geomagnetic field, κ_{x⟂} is a component in the direction of cross-field elongation of the irregularities, and κ_{y⟂} is a component perpendicular to κ_{∥} and κ_{y⟂}. Coordinate system used in equation (25) is portrayed in Figure 1 by thick lines. Since at small κ the factor sin^{2}κ^{2} in equation (24) is of the order of κ^{4}, the contribution of large-scale irregularities with sizes exceeding the Fresnel radius is insignificant. Thus one can make use of the smallness of the Fresnel radius to outer scale ratio, and take the spectral shape as follows:

Finally, after integration of equation (24), we obtain for 0 < *p <* 4 a formula for the variance of logarithmic relative amplitude of the wave scattered by 3-D anisotropic ionospheric irregularities

where

Here ψ is orientation angle of cross-field anisotropy, θ(*z*) an angle between the direction of wave propagation and the geomagnetic field vector, and *F* hypergeometric function. Coordinate system is oriented so that the geomagnetic field vector is contained in *yOz* plane as shown in Figure 1 (thin lines). Unit vector in the figure indicates the direction of cross-field elongation of irregularities.

[16] There is some specificity in application of the above equation to the analysis of experimental data. In a real experiment it is impossible to measure the field in the absence of irregularities. However, it turns out that within Rytov's approach an averaged measured amplitude *A*_{0,e} can be successfully used instead of idealized quantity *A*_{0}. Let us show that an experimental estimate of the variance of logarithmic relative amplitude is identical to its theoretical value. Denote experimental logarithmic relative amplitude χ_{e} = ln(*A*/*A*_{0e}), where *A* is the measured amplitude and *A*_{0e} = 〈*A*〉 is experimental approximation of hypothetical amplitude in the medium containing no irregularities.

[17] Averaged amplitude 〈*A*〉 is linked with the amplitude of undisturbed field (in the absence of irregularities) *A*_{0} by a relation [*Rytov et al.*, 1987]

Making use of equation (28), we obtain

where χ = ln(*A*/*A*_{0}) is theoretical logarithmic relative amplitude. Averaging equation (29) and taking into account that 〈χ〉 = −σ_{χ}^{2} [*Rytov et al.*, 1987] gives

Finally, allowing for equality 〈χ^{2}〉 = σ_{χ}^{2} + 〈χ〉^{2}, we arrive at