=Paper=
{{Paper
|id=Vol-2843/paper25
|storemode=property
|title=Theoretical studies of the Earth's ionosphere inhomoge neities influence on the propagation of HF radio waves (paper)
|pdfUrl=https://ceur-ws.org/Vol-2843/paper025.pdf
|volume=Vol-2843
|authors=Andrew Kryukovsky,Dmitry Lukin,Eugene Palkin,Eugene Ipatov,Dmitry Rastyagaev
}}
==Theoretical studies of the Earth's ionosphere inhomoge neities influence on the propagation of HF radio waves (paper)==
https://ceur-ws.org/Vol-2843/paper025.pdf
Theoretical studies of the Earth's ionosphere inhomoge-
neities influence on the propagation of HF radio waves*
Andrew S. Kryukovsky1, Dmitry S. Lukin1, Eugene A. Palkin1, Eugene B. Ipatov2 and
Dmitry V. Rastyagaev1
1
Russian New University, 22, Radio Street, Moscow, 105005, Russian Federation
2
Moscow Institute of Physics and Technology, 9, Institutsky per., Dolgoprudny, 141700, Rus-
sian Federation
rdv@rosnou.ru
Abstract. In the article algorithms for studying the characteristics of frequency-
modulated (FM) radio signals reflected from the ionosphere containing local
plasma inhomogeneities is studied. Mathematical simulation of vertical sound-
ing ionograms for various inhomogeneities both in the case of an ordinary wave
as well as in the case of an extraordinary wave is carried out by the method of
Hamilton-Lukin bi-characteristics. An algorithm for mathematical modeling of
LFM HF radio signals propagation in a magnetically active ionospheric plasma
is considered. The algorithm is based on a multipath description of the received
HF signal, the amplitude-phase characteristics of the components of which are
determined on the basis of the extended bi-characteristic system. The proposed
method is focused on predicting the polarization characteristics taking into ac-
count the geometry of the inclined trajectories of the probing rays. On the basis
of the bi-characteristics method, an algorithm recovering the effective fre-
quency of electron collisions in the ionosphere, using data on signal attenuation
during vertical sounding of the ionospheric plasma with a continuous fre-
quency-modulated decameter signal has been developed. The results of numeri-
cal simulation are presented. The work is of an overview nature. The study was
supported by a grant from the Russian Science Foundation (project No. 20-12-
00299).
Keywords: ionosphere, irregularities, ionograms, ordinary and extraordinary
waves, vertical sounding, bi-characteristic system, effective frequency of elec-
tron collisions, polarization characteristics, Stokes parameters.
1 Introduction
Diagnostics and monitoring of the ionosphere, permanent monitoring of extreme phe-
nomena in the atmosphere are the urgent tasks due to the significant influence of the
state of the ionosphere on the operation of radio systems for various purposes: radio
communication, navigation (positioning), radar, etc. The efficiency of the systems of
short-wave radio communication and radio navigation depends on an adequate de-
*
Copyright © 2021 for this paper by its authors. Use permitted under Creative Commons License Attribu-
tion 4.0 International (CC BY 4.0).
scription of the propagation medium taking into account all sufficient processes in the
ionosphere.
The most promising method for solving this problem is mathematical modeling of
the propagation of radio waves in a disturbed ionospheric plasma, connected directly
to the operational data of oblique and vertical sounding. Vertical sounding (VS)
ionosondes are one of the most effective diagnostic tools for the ionosphere with a
long history of development [1].
The ray-tracing method is one of the efficient mathematical modeling methods for
the radio waves propagation problems. But when studying the propagation of HF
radio waves by ray methods, the problem of describing fields in the caustics arises.
The relevance of the study of caustic structures is determined by their special role
with respect to ray structures, since caustics, which are the envelopes of ray families,
divide the physical space into regions with different propagation patterns [2]. We also
note that the field in the vicinity of the caustic increases substantially. So the diffrac-
tion-ray approach to the description of HF waves propagation in the ionosphere is
used, and it gives the possibilities to describe as regular ray field structures, so as
diffraction field structures in the caustics regions.
Here we shall consider the wave field structure in the vicinity of the caustic near
the Earth’s surface without taking into account the radio waves reflected from the
surface, but taking into account the absorption and divergence of the radio signal in
the ionospheric anisotropic plasma. We shall consider the polarization characteristics
of HF signals at the receiving point in the diffraction-ray approach.
2 Extended bi-characteristic system for describing the
parameters of a short-wave signal in a magnetically active
ionospheric plasma
Based on diffraction-ray approach the description of HF waves propagation in the
ionosphere reposes on a Hamiltonian system of ordinary differential equations (ODE)
for the spatial coordinates of the rays r ( x, y , z ) , the component of the wave vectors
k ( k x , k y , k z ) and on the concomitant system of ODE for the partial derivatives that
definer the geometric divergence of radiation fluxes in the coordinate-momentum
space (in the space coordinates and wave vectors) [3; 4]:
1
dri c 2
ki
dt 2 ki 2
(1)
1
dki
i = 1, 2, 3,
dt 2 ri 2
dr 2 2 2
2c k
dt k
dk 2 2
dt r
(2)
dr 2 2 2
2c k
dt k
dk 2 2
dt r
With initial conditions for spatial coordinates in the report system associated with
the radiation source, and components of wave vectors:
k1 (0) k x (0) 0 cos cos ,
c
k2 (0) k y (0) 0 sin cos ,
c (3)
k3 (0) k z (0) 0 sin ,
c
r1 (0 ) x (0 ) x0 , r2 (0 ) y ( 0 ) y 0 , r3 (0 ) z ( 0 ) z 0 ,
And corresponding initial conditions for partial derivatives with respect to the pa-
rameters of the ray family:
k x (0) 0 sin cos , k x (0) 0 cos sin ,
c c
k y (0) 0 sin sin , k y (0) 0 cos cos ,
c c (4)
k z (0) 0 cos , k z (0) 0.,
c
r (0) 0, r (0) 0
The initial conditions (3-4) are set at the location of the radiation source on the
Earth's surface, assuming that the influence of the ionosphere can be neglected at this
level. In formulas (1-4), is the circular frequency of radiation, с is the speed of
light, t is the parameter along the ray path and r , k , is the effective permittivity
of the wave propagation medium, which takes into account the properties of inhomo-
geneous magnetically active ionospheric plasma for two normal components of an
electromagnetic wave (the "ordinary" and "extraordinary" wave components with
circular polarization) [1,5].
Within the framework of approximation involved, if the deflecting influence of ab-
sorption is neglected, which is acceptable for the ionospheric plasma, then the expres-
sion for the effective permittivity has the form [1; 5]:
2v(1 v)
1 (5)
2(1 v) u sin u 2 sin 4 4u (1 v)2 cos 2
2
The following notations are introduced in expression (5):
p 4 e2 N (r )
2
v (6)
me 2
– ratio of the square of the plasma frequency to the square of the working fre-
quency:
2
e2 H 2
u H 2 2 0 2 (7)
me c
– ratio of the square of the gyrofrequency to the square of the operating frequency.
In formulas (6) and (7), me is the mass of the electron, e is the charge of the elec-
tron, H0 is the value of the magnetic field of the Earth, and the function N (r ) is the
electron concentration at a fixed point in space.
In addition to the functions u and v, the angle between the Earth's magnetic field
strength H 0 ( H 0 x , H 0 y , H 0 z ) and wave vector k enters into formula (5) too:
H 0 x H 0 cos cos , H 0 y H0 cos sin , H 0 z H 0 sin (8)
The orientation of the magnetic field is given by the angles γ and φ. When calculat-
ing, you only need to know cos 2 :
( Н 0 x kx Н 0 y k y Н 0 z kz )2
cos2 2 (9)
H 02 k
The sign "+" in formula (5) corresponds to an ordinary wave, and the sign "–" cor-
responds to an extraordinary wave (o-wave and an x-wave, respectively). To construct
ray paths, we use the bi-characteristic system method, described above (see also [6–
8]).
3 Ray trajectories calculation
As an example of the ray trajectory calculation algorithm we consider the following
problem. Figure 1 shows the dependence of the electron concentration on height, and
Figure 2 shows the dependence of the electron collision frequency [9; 17]. It is as-
sumed that a monochromatic signal is emitted with an operating frequency
f = 3.3 MHz.
To calculate the field in the vicinity of the caustic, a high-latitude night ionospheric
plasma model corresponding to March, 80 North latitude and 30 Eastern longitude
was used. The angle 83 , the angle 90 , H0 = 0,551 Oe. We assume that the
radiation source is point-wise and is located on the surface of the earth at the coordi-
nate origin. Radiation of an electromagnetic wave occurs in the plane (x, z).
Fig. 1. Dependence of the electron concentration on height.
Fig. 2. Dependence of the electron frequency of collisions on height.
Figures 3 and 4 [17] show the ray structure of the radio signal in the (x, z) plane for
an o-wave and an x-wave. The angle of the ray exit varies from 0° to 90°. As a back-
ground, as in Figure 1 and 2, the electron concentration of the ionosphere is shown.
At an altitude of about 300 km, the maximum of the F layer is clearly visible, and at
an altitude of 115 km the maximum of the E layer too. Rays corresponding to the
propagation of an o-wave are reflected from the E and F layers with a small exit angle
and return to the ground, and rays with large exit angles pass through the ionosphere.
Fig. 3. Ray structures in the (x, z) plane, o-wave.
Fig. 4. Ray structures in the (x, z) plane, x-wave.
Figure 3 shows that the family of rays forms a complex caustic structure containing
three caustic points. In accordance with the classification of wave catastrophe, this is
a catastrophe of A3 [7; 10]. The lower caustic cusp is caused by layer E, and the two
upper caustic cusps are determined by ionospheric layers forming the main maximum.
In the case of an x-wave, all rays are reflected from the ionosphere and returned to
the Earth. Part of the rays forms the upper caustic, and part of the rays with small exit
angles form a caustic cusp, due to layer E. It can be seen that three lower branches of
caustics, forming the A3 singularities, descend to the earth at distances of 333.755 km,
575 km and 750 km [17]. The paper considers the structure of the wave field in
the vicinity of the first caustic (yellow rays) without taking into account the surface
wave, the effect of which is negligible in this conditions. According to Figure 4 there
is an x-wave field that modulates the caustic field of an ordinary wave in this region
too.
4 Wave field calculation
Let an isotropic radiation source create an electric field E0 at a distance r0. Then:
30W
E0 V / m . (10)
r0
In formula (10), W is the radiation power, and r0 is the distance to the emitter. In
this work, it was assumed that W = 1 kW and r0 = 1 m.
Then the wave field at the receiving point is formed as a result of coherent or inco-
herent addition of the partial fields of individual beams:
N
j 1
E (rrec , t ) A j ( j , j , t ) exp i ( j , j , t ) t
rrec
(11)
Here N is the number of rays coming to the observation point; j , j – the angular
coordinates of the output of the ray with the number j; falling at the location of the
receiver, ( j , j , t ) - the phase path, A j ( j , j , t ) – the vector of the electric
(magnetic) field of the wave corresponding to the j-th ray. In formula (11), the slow
dependence of the field on time t, which occurs in the presence of non-stationary in-
homogeneities in the ionosphere, is distinguished. It is also assumed that the reception
of ionospheric reflected radiation occurs on the Earth's surface, that is, the influence
of magnetically active plasma at the receiving point can be neglected.
As fallows from the ray structure analysis, the field of an o-wave to the right of the
caustic is defined as the sum of the contributions of two rays:
Ego A1o exp i o1 / 2 A20 exp i 20 (12)
The first ray has already touched the caustic, and the second is not yet. The radia-
tion field of an x-wave in this region is single-ray:
Egx A1x exp i x1 / 2 (13)
This ray has already touched the upper caustic. In formulas (12) – (13), the ampli-
tude coefficients can be represented as:
J0
A j E0 exp[ j ] (14)
Jj
In expression (14), J j is the Jacobian of the divergence calculated at the observa-
tion point using the extended bi-characteristic system, J 0 is the Jacobian of the diver-
gence calculated at a distance r0 from the source, j is the absorption determined by
the electron collision frequency, and j is the phase calculated like absorption, along
the ray path [8]. In our notation, the first ray is the ray that has already touched the
caustic.
Since the Jacobian J j on the caustic is equal to zero due to ray focusing, solution
(14) becomes infinite. Therefore, the field on the caustic and in its vicinity is deter-
mined using uniform asymptotic through the Airy function and its derivative (see, for
example, [7,11]):
dAi ( )
Ec exp(i ) l1 Ai ( ) i l2 (15)
d
In expression (15):
Ai ( ) exp i ( 3 ) d (16)
Is the Airy function, is the phase of the traveling wave, and is the argument of
the Airy function, which in the light region (where two rays intersect) are defined as:
1 3
1 2 , 4/3 1 2
2/3
(17)
2 2
The coefficients of the asymptotic expansion (15) l1 and l2 in a first approximation
have the form:
1 3 1
l1 A1 A2 4 3 , l2 A1 A2 4 (18)
2 2 3
The problem of the field determination on the caustic and in its vicinity is due to
the need to solve the “shooting” problem, that is, calculating with high accuracy at
one point the phases and amplitudes of two rays that came along different but near
trajectories. For the lower branches of the caustic, this task is especially time-
consuming, since the caustic and the rays in its vicinity are quasi-parallel.
Another algorithm is inherently close to the method of interpolating local asymp-
totics [14] and is alternative to the local approach [15]. The angle of exit of the ray
forming the caustic was initially calculated. Knowing this angle, the ray family in the
vicinity of the caustic was divided into two subfamilies: a subfamily of rays that
touched the caustic and did not touch it. Relative to each ray, the point of its intersec-
tion with the earth's surface and all the necessary radiation parameters at this point
were determined. Then for each ray subfamily, the interpolation formulas for the
phases and amplitude coefficients were constructed using the least squares method.
Then the parameters of two intersecting rays were found at each point, and the radia-
tion and caustic fields were calculated using the formulas given above.
Figure 5 [17] shows the ray amplitudes in the vicinity of the caustic in the ray ap-
proximation. The solid line shows the amplitudes of the rays of an o-wave that have
not yet touched the caustics, and the dotted line – that have already touched. The posi-
tion of the caustic is marked by a vertical red line. The dashed line shows the ampli-
tudes of the x-wave.
In Figure 6 the amplitude of the field of an o-wave in the vicinity of the caustic is
shown. Uniform asymptotics calculated by formulas using the Airy function and its
derivative (15) are shown in black and blue colors [17].
The black color shows the calculations made taking into account the absorption,
and the blue line shows the calculations without taking into account the absorption.
The red dot on the horizontal axis shows the position of the caustic. The maximum
value is shifted to the region of light relative to the position of the caustic as expected.
On the caustic, the field amplitude is close in value to the average values of the field
amplitude in the light region. In addition, this figure shows a comparison of the field
amplitude on the caustic calculated using the uniform asymptotics (15) and the ray
approximation (12). The green color indicates the geometrical-optical (GO) approxi-
mation without taking into account absorption, and the purple color shows the ap-
proximation taking into account absorption. It should be noted that the GO approxi-
mation in the region of light very well coincides with uniform asymptotics up to the
slopes of the main maximum.
Fig. 5. The amplitudes of the rays in the vicinity of the caustic without taking into account
absorption (green) and taking into account absorption (purple).
It follows from the figure that in order to estimate the maximum value of the field
amplitude in the vicinity of the caustic using the ray approximation, it is enough to
determine where the curve forms the inflection point before going to infinity.
Fig. 6. The amplitude of the electric field intensity modulus versus the distance along the
horizontal axis; o-wave.
Figure 7 shows the amplitudes of the field of an o-wave, taking into account the
contribution of the x-wave in the vicinity of the caustic in the caustic, but excluding
the absorption. We can see that the x-wave makes a significant contribution and
strongly modulates the caustic structure of the o-wave [17].
Fig. 7. The amplitude of the electric field strength modulus versus the distance along the
horizontal axis without taking into account absorption, but taking into account the contri-
bution of the x-wave.
5 Algorithm for calculating the polarization characteristics of
HF signals at the receiving point
To calculate the polarization characteristics of HF signals at the receiving point we
use the formula (11), where the total field is the result of coherent or incoherent addi-
tion of the partial fields of individual beams. It is also assumed that the diffraction
effects due to the caustic regions are not presented at the receiving point.
In (11) it is necessary to determine coherently summing components and non-
coherently summing ones, the fact taking into account that different components have
different degrees of coherence, since they correspond to different trajectories of wave
propagation through spatially separated regions of the inhomogeneous ionosphere. In
this case, to determine the polarization characteristics, we use the Stokes parameter
system [16]:
N
S0 Ax2 j ( j , j , t ) Ay2 j ( j , j , t )
rrec
j 1
(19)
S1 I x , j I y , S2 I 45, j I 45, j , S3 I О , j I Н , j
rrec rrec rrec
In formula (19) I x , j , I y , j , I 45, j , I 45, j , I О , j , I Н , j are respectively the intensity of
the linear components of the field strength vector in the specified directions
x, y, 450 , 450 , as well as the "ordinary" and "extraordinary" components of the
waves measured for the coherent part of the total field. S0 is the total intensity of the
wave field at the receiver. In (19) the local coordinate system associated with the
selected direction signal (orientation of main lobe of receiving antenna diagram) is
used with respect to which the components x, y of vectors Aj ( j , j , t ) so as the
"mixed" direction of linear polarization oriented at angles 450 are determined. The
number j is determined by all the coherent components of the signal. In particular, in
the model numerical calculations constructed for the corresponding models of the
effective permittivity of ionospheric plasma, the components of the "ordinary" and
"extraordinary" waves were used to calculate the coherent part of the Stokes parame-
ters, providing they correspond to ray trajectories close in the exit direction. For sim-
plicity, in the model calculations we assume that the corresponding components of the
"ordinary" and "extraordinary" waves are completely coherent, while the modes cor-
responding to rays reflected from different ionospheric layers or local inhomogenei-
ties are not coherent. It is also necessary to note that the partial depolarization of "or-
dinary" and "extraordinary" waves exists, that occurs when the direction of arrival of
a transverse electromagnetic wave does not coincide with the main direction of recep-
tion in polarization measurements. In addition, if there is a significant (for the re-
cording equipment used) difference in the signal arrival times for different modes, it
is possible to determine the polarization characteristics for individual groups of re-
corded signals that correspond to different propagation paths in the ionosphere.
For example, figure 8 shows the results of modeling the frequency dependence of
the main parameters of the multipath structure of the recorded signal, which allow us
to construct the frequency dependence of the Stokes parameters [20]. Figure 8 shows
the frequency dependence of the wave arrival angles j relative to the local horizon at
the receiving point at a distance of 350 km from the source and the intensity of differ-
ent modes (normalized by the field intensity at a distance of 1 m from the source).
The data obtained allow us to determine the mode structure of the signal at differ-
ent frequencies, which is necessary for polarization characteristics forecasting. The
calculation was performed only in the geometrical-optical approximation, since the
diffraction effects in the focal regions have significantly smaller scales of frequency
variations.
Fig. 8. Frequency dependence of the arrival angles and the intensity for different modes of
the received signal (two branches of the graph respond to the "ordinary" (NH = -1) and
"extraordinary " (NH = +1) components of the wave).
The graph in figure 9 shows the results of numerical prediction of the frequency
dependence of the normalized Stokes parameter S3 S3 / S0 of the probing signal that
characterizes the ratio of right-and left-polarized components in the total field at the
receiving point [20]. When performing calculations, it was assumed that for each
frequency value there is one dominant pair of coherent signal modes that have a
maximum intensity of waves.
In the diffraction zones corresponding to the regions of spatial focusing of individ-
ual modes, a jump-like change in the mode structure occurs, which can be easily
traced from the graph of figure 3. In these regions, there are small-scale oscillations of
a diffraction nature. They are not shown in the graph.
Fig. 9. Frequency dependence of the normalized Stokes parameter S3 S3 / S0 of the re-
ceived signal.
6 Diagnostics of the effective frequency of electronic collisions
in the ionosphere based on analysis of the amplitude
characteristics of continuous LFM radio signals
Also, we consider an algorithm for determining the effective collision frequency
based on the analysis of the amplitude characteristics of linear frequency modulated
(LFM) signals [18]. When the chirp ionosonde is operating, the signal delay and am-
plitude are usually recorded depending on the radiation frequency (f). In this paper,
the case of vertical sounding is considered. It is assumed that the electron density
profile N(z) as a function of the height z is known, that is, it has already been recon-
structed from the dependence of the signal delay on the radiation frequency. As for
the effective collision frequency, it should be reconstructed from the attenuation of
the amplitude (A) of the probing signal, which can be represented in the form (11).
Knowing the amplitude A, E0 and the divergence D, we can proceed to determining
the effective collision frequency in accordance with formula (11). Absorption along
the path is defined as [3]:
t
2 0
2 dt (20)
Formula (20) includes an approximate expression for the imaginary part of the di-
electric constant of the medium having the form [3,8]:
2 Z 1 Z 2 , Z e
1
(21)
If we assume that the effective collision frequency at heights of the order of 80 km
does not exceed 106 s-1, then in (18) the denominator can be neglected, and it can be
assumed that 2 Z . Then, taking into account (21), equation on the energetic
characteristics can be rewritten as:
t
Vc m A 4 e 2
2 e
N dt L L ln Vc (22)
0
D E0 me
In formula (22), L is a function of the operating frequency and is determined at the
receiving point. tm is a function of frequency too. This is the time it takes for the sig-
nal to travel from the source of the reflection point. As for e and N, they are func-
tions of the height z, which in turn are functions of the group time t, the function z(t)
is calculated along the trajectory and also depends on the frequency. Equation (22) is
essentially Volterra integral equation of the second kind. One of the methods for solv-
ing it is the iteration method [18]. Let's introduce the notation: G = e N. We will
assume that up to a certain frequency f0, for which the time tm0,
tm 0
G dt 0
0
(23)
We divide the frequency interval (f0, flev) into n parts (f0, f1,…, fj,…, fn = flev), on
each of which (with number j) we will assume the function G to be constant (G = Gj).
Let us calculate the values of tmj as a function of frequency for each fj. Then
12 22 t t
G1 L1 , G2 L2 G1 m1 m 0
tm1 tm 0 tm 2 tm1 tm 2 tm1
(24)
2j j 1
tm,i tm,i 1
Gj Lj Gi , j2
tm, j tm, j 1 i 1 tm, j tm, j 1
Knowing the values of Gj and assuming that this is the product of the effective col-
lision frequency of electrons and the electron concentration at the reflection point, it is
easy to find the dependence of the effective collision frequency on height.
Below there is an example of the implementation of this algorithm using model
calculations. In Figure 10 the dependence of the electron concentration on the height,
and in Figure 11 effective collision frequency are shown [18].
Fig. 10. Dependence of electron concentration on height.
Fig. 11. Dependence of the logarithm of the collision frequency on the height.
It can be seen that a two-layer model has been selected (Figure 10). This structure
of the ionosphere inevitably leads to the loss of data at some frequencies during verti-
cal sounding (see also Figures 12-15). In Figure 11, the thin red line shows the ap-
proximation of the effective collision frequency by the hyperbolic dependence:
lg e a b z , a 0.217841, b 487.771 (25)
which is typical for heights below the maximum of the F2 layer. The purpose of
the simulation is to calculate the field amplitude at different frequencies at the receiv-
ing point from the known data, using the above formulas, and then, using the tech-
nique outlined above and formulas (24), (25) restore the effective collision frequency
and compare model and calculated values.
Figure 12 shows the dependence of the signal reflection height on the frequency,
calculated from the initial data [18]. In the region of 3 MHz, a gap is seen due to the
interlayer valley. In Figure 13 the dependence of the time of arrival of the ray at the
point of reflection on the frequency is shown [18]. A sharp increase in the curves in
the region of the interlayer valley and the maximum of the F2 layer inevitably leads to
instability of the iterative process when the effective collision frequency is restored.
Fig. 12. Reflection height versus frequency.
Fig. 13. Frequency dependence of the arrival time of the ray at the reflection point.
By eliminating the frequency f, a graph of the reflection height versus delay can be
plotted (Figure 14) [18]. Finally, Figure 15 shows the dependence of the function L on
frequency f [18].
Fig. 14. Reflection height versus delay.
Fig. 15. Logarithm of the amplitude function of frequency.
The simulation results are shown in Figures 16 and 17 [18]. The thick violet line
shows the model dependence of the effective collision frequency on the height, and
the thin green line shows the calculated values. It is seen that the model and calcu-
lated values for the lower ionosphere coincide with acceptable accuracy. This area is
especially significant, since the main absorption of the radio wave is formed here.
In Figure 17, an extrapolation model is constructed from the calculated data (thin
blue line). Obviously, at high frequencies, it needs to be corrected using chirp data.
Fig. 16. Comparison of the dependence on the height of the model and calculated collision
frequency.
Fig. 17. Approximation of the effective collision frequency.
7 Conclusion
An algorithm is developed for studying the characteristics of frequency-modulated
radio signals reflected from the ionosphere, which contains local plasma irregularities
with low and high electron density [19]. Mathematical modeling of vertical sounding
ionograms for various inhomogeneities both in the case of an ordinary wave and in
the case of an extraordinary wave has been performed using a system of bicharacteris-
tic equations.
In the material presented an algorithm for determining the polarization characteris-
tics of the HF signal is given for the numerical simulating experiments on ionospheric
inhomogeneities radiosounding. Since from the point of view of this approach the
polarization characteristics are determined by the amplitude-phase relations of the
interfering components, these characteristics are presented to sensitive instruments for
recording and investigation local ionospheric inhomogeneities of various nature. Car-
rying out a large-scale model experiments of this kind for various models of inho-
mogeneities will allow us to identify typical frequency patterns for identifying fea-
tures observed in experiments.
Thus, using the bicharacteristics method, an algorithm for recovering the effective
frequency of collisions of electrons in the ionosphere was developed using data on
signal attenuation during vertical sounding of the ionospheric plasma with a continu-
ous frequency-modulated decameter signal. A numerical experiment has been carried
out.
The study was supported by a grant from the Russian Science Foundation (project
No. 20-12-00299).
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