SCINTILLATION EFFECTS AND THE SPATIAL POWER SPECTRUM OF SCATTERED RADIO WAVES IN THE IONOSPHERIC F REGION

Differential equation for two-dimensional spectral function of the phase fluctuation is derived using the modify smooth perturbation method. Second order statistical moments of the phase fluctuations are calculated taking into account polarization coefficients of both ordinary and extraordinary waves in the turbulent collision magnetized plasma and the diffraction effects. Analytical and numerical investigations in the ionospheric F region are based on the anisotropic Gaussian and power law spectral functions of electron density fluctuations including both the field-aligned anisotropy and field-perpendicular anisotropy of the plasma irregularities. Scintillation effects in this region are investigated for the smallscale ionospheric irregularities. The large-scale background plasma structures are responsible for the double-humped shape in the spatial power spectrum taking into account diffraction effects. Numerical calculations are based on the experimental data of the navigation satellites.


INTRODUCTION
At the present time the features of light propagation in random media have been rather well studied [1]. Many articles and reviews are related to the statistical characteristics of scattered radiation and observations in the ionosphere [2][3][4][5]. The analysis of the statistical properties of small-amplitude electromagnetic waves that have passed through a plane turbulent plasma slab is very important in many practical applications associated with both natural and laboratory plasmas [6].
Radio signals from communication satellites experience the scintillation phenomenon when received on the ground. The scintillation is caused by the electron density irregularities in the ionosphere. Diffraction effects are importand for small scale plasma irregularities. Interference between propagated radio wave and scattered radiation in the magnetized turbulent ionospheric plasma lead to the scintillation effects at the observation points.
Investigation of statistical characteristics of scattered electromagnetic waves in the ionosphere is of great practical importance. Measurements of the statistical characteristics of scattered electromagnetic waves by satellite, ground-based radar systems or meteorological-ionospheric stations gives the information about ionospheric plasma irregularities. Statistical characteristics of the spatial power spectrum (SPS) (broadening and displacement of its maximum) in the collision magnetized plasma was considered in [7,8] applying the complex geometrical optics approximation and the smooth perturbation methods. Scintillation effects of scattered ordinary and extraordinary waves in the ionospheric plasma for both power-law and anisotropic Gaussian correlation functions of electron density fluctuations have been investigated in [9]. The "Double-Humped Effect" for both above mentioned spectra was investigated analytically and numerically in [9][10][11] using the modify smooth perturbation method.
The problem is formulated in section 2, where equation for the two-dimensional spectral function of the phase fluctuation is derived and second order statistical moments are presented for arbitrary correlation function of electron density fluctuations. In section 3 analytical and numerical results are presented based on experimental data. Peculiarities

THEORETICAL CONSIDERATIONS
The electric field in the magnetized ionospheric plasma with anisotropic electron density irregularities satisfies the wave equation: with the components of the second order permittivity tensor [12]: where: plasma is a random function of the spatial coordinates: . First term is a regular (unperturbed) component; the second one describes turbulence: rr . Scintillations are caused due to electron density fluctuations. At high frequency the effect of ions can be neglected. The geomagnetic field leads to the birefringence and anisotropy of scattered electromagnetic waves.
Polarization coefficients in the collision magnetized plasma are [11] expressed via magneto-ionospheric parameters: where:  We introduce the wave field as [10,13]: Taking into account inequalities characterizing the smooth perturbation method [1,14] in the first approximation we have [10,11,13]: where: ( , ) xy kk  ae is the wave number vector perpendicular to the Z axis, 10 () where: where: I S S N 2 3 4 7 -3487 V o l u m e 1 3 N u m b e r 1 J o u r n a l o f A d v a n c e s i n P h y s i c s 4596 | P a g e F e b r u a r y 2017 w w w . c i r w o r l d . c o m  and x  are distances between observation points spaced apart in the principle and perpendicular planes, respectively. Phase fluctuations at different observation points are not independent and they correlate. The asterisk indicates the complex conjugate, the angular brackets indicate the statistical average.
The fluctuations in the ionosphere can be characterized by the phase structure function, which is the variance of the phase difference between two probes as a function of their separation [1,6,14] and can be calculated using equation (7): The angle-of-arrivals (AOA) in the principle and perpendicular planes are [1,14]: The small fluctuations in the AOA of the received signal can be used to obtain some information about the ionosphere, where  is the angle between the direction of the uniform drift velocity 0 V and vector η . This velocity of the irregularities assumed to be frozen in a direction transverse to the signal path. In this case function is anisotropic due to the presence of the wind direction even at isotropic correlation function of phase fluctuations. Calculation of the second order moment (10) allows to estimate geometrical parameters of the anisotropic plasma irregularities and the drift velocity with which the plasma irregularities drift across the signal path without changing their characteristics.
The standard relationship for weak scattering between the scintillation level 4 S and the 2D phase spectrum describing 2D diffraction pattern on the ground is [16]: where: is the Fresnel wavenumber. The double integral in the wave number space not depends on the intensity fluctuations and depends on the shape of the fluctuation spectrum. The sinusoidal term is responsible for oscillations in the scintillation spectrum. The value of the scintillation index depends on geometry, frequency and the ionization irregularity structure. Phase scintillations are usually observed as a phase difference between the antennas of an interferometer system. The spatial autocorrelation function of the diffraction pattern could be measured with a suitable two-dimensional array of sensors.
Transverse correlation function of a scattered field containing variances of the phase fluctuations in the first and second approximations has the following form [17,18]: where 2 0 E is the intensity of an incident radiation.
SPS of a scattered field in case of an incident plane wave ( , ) W k k  is easily calculated by Fourier transform of the transversal correlation function of a scattered field [19]: I S S N 2 3 4 7 -3487 V o l u m e 1 3 N u m b e r 1 J o u r n a l o f A d v a n c e s i n P h y s i c s 4597 | P a g e F e b r u a r y 2017

NUMERICAL RESULTS
The incident electromagnetic wave has the frequency of 40 MHz ( 0 0.84 k  1 m  ). Plasma parameters at the altitude of 300 km are: The first Fresnel's radius and the Fresnel wavenumber are equal to 1.5 km and 2.4 An RH-560 rocket flight was conducted from Sriharikota rocket range (SHAR), India ( 0 to study electron density irregularities during spread F. It was found that the irregularities were present continuously between 150 and 257 km. The most intense irregularities occurred in three patches at 165-178 km, 210-257 km and 290-330 km [20,21]. Studing the equatorial spread F irregularities using RH-560 rocket instrumented with Langmuir probes launched from SHAR it was established [20,21] that the relationship between the spectral index, p and the mean integrated spectral power (in 20 m to 200 m scale size range) could be represented by a Gaussian function.
The anisotropic 3D Gaussian autocorrelation function describing narrow-band process has the following form [7]: where: For small-scale ionospheric irregularities having characteristic linear scale ~1 km the ratio of the diffusion coefficients along and transverse directions with respect to the geomagnetic field on the altitude 300 km is 2  minutes. We should also to point out that lifetime of ionospheric irregularities is determined by turbulence varying substantially along latitude but not by diffusion.
Measurements of satellite's signal parameters passing through the ionospheric slab and measurements aboard of satellite show that in the ionospheric F-region irregularities have power-law spectrum with different spatial scales and is defined as [7]: where k  and || k indicate the perpendicular to the magnetic field and field aligned wave numbers respectively. In this formula  is the gamma function, p is the power index. Although generation mechanisms of irregularities in the magnetized turbulent ionospheric plasma is substantially different, we will use both 3D Gaussian and power law spectra.
We consider both large (characteristic linear scales about ten hundred km) and small scale (dimensions several hundred meters) irregularities. Anisotropy degree of small scale irregularities varies from 1.1 up to 5.5. Slope angle of elongated irregularities with respect to the geomagnetic lines of forces varies in the interval 00 8 28  . Mean anisotropy factor of large scale irregularities is equal to 2.8, maximum is 6 [22]. An RH-560 rocket flight was conducted to study electron density irregularities during spread F caused due to the gradient drift instability. It was found [21] that the average spectral index of transitional (10-100 m) scale size range for 160-178 km, 210-250 km 250-280 km and 290-330 km regions is -3.99, -4.33, -3.36 and -3.17, respectively. The irregularities in this scale are presented in plenty in 220-250 km and 290-330 km regions with an average spectral index of -4.32.The average spectral index of intermediate (100 m-2 km) scale irregularities in 160-170 km was found to be -3.78; in 220-250 km and 290-320 km regions can be presented by 3.4 0.5  [21].
Substituting equation (14) into (7), we obtain: where: This correlation function is valid for all levels of scintillations.

  
These statistical characteristics gives the information of plasma irregularities at high frequency radio waves propagation in the ionospheric F region.  For relatively small irregularities diffraction effects are important. In such cases the interference between the direct ray from the transmitter and the scattered ray from the irregularity can result in a rapid fading of the received signal and hence scintillation. At investigation of the ionosphere by radiophysical methods parameters of inhomogeneous structure mainly are determined from the analyses of the diffraction pattern on the ground. Structure of the diffraction pattern substantially depends as on the ratio of the scale of irregularities and the Fresnel radius as well as on the phase fluctuations in the inhomogeneous slab. Observation of the scintillation of radio waves is connected with the investigation of the diffraction pattern on the ground level. Ionospheric scintillation depends on the 2D spectral correlation function of electron concentration fluctuations. The two asymptotic simplifications 1   is associated with a significant filtering, nonfully developed diffraction pattern, whereas 1   is associated with a fully developed scintillation. Shaded area corresponds to a transition region between these two regions.
The relation between the phase fluctuations and scintillation indices is of interest. Substituting equation (16)   10    , corresponding to the altitudes 40-400 km. Analyses show that the collision between plasma particles has no substantial influence on the scintillation level in the high-latitude ionospheric F region.   In high latitude ionosphere the second order statistical moment of the phase fluctuations is:

CONCLUSION
Second order statistical moments of scattered radio wave in the turbulent collision magnetized plasma are investigated for both anisotropic Gaussian and power-low correlation functions of electron density fluctuations using the modify smooth perturbation method taking into account polarization coefficients of the ordinary and extraordinary HF wave modes and the  Correlation function of the phase fluctuations fast decreases for both wave modes, while oscillations are observed at big anisotropy factors of elongated irregularities. The behavior of the phase structure function of the ordinary and extraordinary waves is different in the principle (location of an external magnetic field) plane while the same in the perpendicular plane. Anisotropy, diffraction effects and collision between plasma particles have substantial influence on the phase structure function of the extraordinary wave in the principle plane. This statistical characteristic allows to calculate AOAs. Varing anisotropy factor the AOA in the principle plane less than in the perpendicular one. They are determined by geometry of ionospheric plasma irregularities which are necessary for the solution of the inverse problems.
Scintillation effects are investigated numerically using anisotropic Gaussian correlation function at field-aligned and field-perpendicular small scale anisotropic irregularities of electron concentration fluctuations. It was shown that scintillation level of scattered radio wave depends on the spectrum of electron density irregularities and substantially decreases increasing frequency of an incident wave from 3 MHz up to 40 MHz scintillation level. Splashes caused by the strong phase fluctuations are revealed in the normalized scintillation level at 40 MHz frequency. The study of scintillations enable to obtain some information about the nature of the irregularities. Measuring the intensity scintillation index and the variance of phase it might be able to deduce parameters of the ionospheric irregularities producing scintillation. Doublehumped shape has revealed in the SPS for large scale plasma irregularities. Decreasing both anisotropy factor and collision frequency the depth of a curve increases at fixed collision frequency.
The obtain results might be useful for remote sensing purposes and for communication-channel modeling.