CC BY
61
Vol. 22, No. 03, 2019
OvO Aviation High Technologies
UDC 621.396.96
DOI: 10.26467/2079-0619-2019-22-3-57-66
RADIO-WAVES REFLECTION AT REMOTE SENSING OF
UNDERLYING COVERS
L.R LIG THART1, A.I. KOZLOV2, A.I. LOGVIN2, I.V. AVTIN2
1Delft University of Technology, Delft, Netherlands 2Moscow State Technical University of Civil Aviation, Moscow, Russia
Monitoring technologies are rapidly developing at present and allow to extract and use non-coordinate information about objects. Non-coordinate information is the information about the type and properties of an object under study. Remote sensing is the main method of solving monitoring problems where special positioning belongs to the radar methods, based on space-time processing of signals and, in particular, on methods of radio polarimetry. It is necessary to have information about the surface in order to solve the monitoring task. The slightest changes in the electrical and physical properties of such areas as salinity, humidity, soil composition, etc. will lead to a change in the basic electrodynamics of the surface, notably its complex dielectric permittivity. The article demonstrates the precise solutions to the problems of radio-waves reflection from a layered surface with various laws of changes of the complex permittivity 8 in depth. Media with exponential and quadratic laws of variation 8 for arbitrary angles of incidence of the radio wave on the surface are considered. Precise decision is obtained for layered media with the law of change in the complex permittivity the polynomial and linear characteristics. A similar problem for the parabolic layer is considered separately. The detailed analysis of radio waves reflection from the medium with a matching layer is carried out. The nature of the electromagnetic field inside the transition layer is studied in detail. The article is illustrated by the graphs showing the dependences of an electromagnetic wave reflection coefficient on the layered medium with linear and exponential laws of variation of the complex dielectric constant over depth.
Key words: polarization, scattering matrix, underlying surface, complex dielectric permittivity, scanning, remote sensing.
1. INTRODUCTION
In order to solve the remote sensing issues, it is very important to know polarization patterns of radio waves reflected from underlying surfaces. We initially need the information on elements of the scattering matrix of illuminated areas on the surface. It is clear that variations of electrical-and-physical properties (salinity, moisture, soil composition, etc.) of such areas will cause variations in the main electrodynamic characteristics of the surface, notably its complex permittivity 8. Complex permittivity variation results in variation of reflecting characteristics of the underlying surface (i.e. characteristics of its elements in the scattering matrix).
2. GENERAL RELATIONS
In radiolocation, the scanning of underlying surfaces is carried out from top to bottom (from the aircraft board, satellite, stationary tower, etc.). Thus, the dimensions of an illuminated area of the underlying surface are determined by the height (H) from which scanning is carried out, the scanning angle (a), the antenna beam width [2, 4, 9, 15] (in two mutually perpendicular planes Aa and Ap Fig. 1).
The wave radiated by the antenna induces (on the surface D) currents which are the sources of Ha scattered field [13, 14]. In general, the energy is scattered omnidirectionally and, in particular, in the direction of the antenna.
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We shall consider a medium which fills the half-space z < 0. Electric-and-physical properties of this medium depend only on the distance to the medium surface.
Suppose that a plane electromagnetic wave is incident on a boundary (medium surface) at the angle. The electric vector of this wave is perpendicular to the plane of incidence (horizontal polarization). The field in the upper half-space will be included into the incident and reflected waves [2]. The electric vector of the reflected wave should meet the differential equation: AE (x, z) + k2s(z) E (x, z) = 0. This equation after substitution of E (x, z) = g (z) exp {ikx sin 9} will be reduced to the following differential equation:
g:z + к2 [e(z )-sin2 0] g (z ) = 0.
(1)
It is possible to derive the formula for the reflection coefficient on the horizontal polarization by using standard conditions of the continuity of tangential components of the vectors E and H :
RHP (0) = [ikg (0 ) cos 0 + g ; ( 0 )] / [ikg ( 0 ) cos 0- g'z ( 0 )]. (2)
The magnetic vector for a vertical polarization may be represented by the following form:
H (x, z) = n (z) v (z) exp (-ikx sin 0), (3)
where n (z) = ^/s(z), the function v (z) is the solution for the differential equation:
ii
[ n (z ) v(z )] zz + k 2 [n2 (z )-sin2 0] n (z ) v (z ) = 0.
It is possible to derive the formula for the reflection coefficient by using standard conditions of the continuity of tangential components of the vectors E and H :
Rp -[ gZ (0 ) + ikg (0 )e( 0 ) cos 0] / [ gZ ( 0 )-ikg ( 0 )e( 0 ) cos 0].
(4)
3. EXPONENTIAL LAYER
The complex permittivity varies according to the exponential law s (z) = P • exp {2a • z} = ( + ip2 ) exp {2a • z}; Im a = 0. The function g(z) will be the solution for the equation (1):
g (z ) = H fi-kVß. exp {a-z}j,
(5)
where is the Hankel function of the second kind. The order of this function is £ =—sin 9. In case we use the formula for the Hankel function derivative we get:
k, • a
rhp -
<|k#y,0-#Hg!Îk#ï / iHfii#y'0+,/pHg>ik,/p
a
a
a
a
(6)
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4. QUADRATIC LAYER
The complex permittivity varies according to the quadratic law: s(z) = (az + p)2. For such a
medium it is impossible to express the solution (1) with the use of the known functions. The following solution may be derived with small a, :
* (z ) =
1
4 (az + ß)2
- sin2 0
exp j -ik j^(az + ß)2 - sin2 zdz I.
(7)
5. VERTICAL SCANNING
Vertical scanning is quite often used when solving remote sensing issues. In this case there is no difference between vertical and horizontal polarizations and the scattering matrix becomes the identity matrix. However, the reflected-wave power depends to a great extent on the behavior of dielectric properties of the surfaces under study [7, 12, 14, 15, 16, 17, 18]. Let us consider how such reflection for several laws of the complex permittivity changes with depth.
5.1. Polynomial layer
The law of the complex permittivity change is described by relation: s(z ) = (az + bf. The formula will be the solution of (2) which satisfies the condition of infinity [15]:
g (z) = Vaz + b • J 1
m+2
m+2
2 — ! 4 (az + Ь)
m+2 2
m + 21 a
(8)
5.2. Linear layer
The law of the complex permittivity change is described by the relation: s(z) = az + b . It results in the relation [2, 18]:
R =
H,?f2—V ifbH -2», f2—
3a
3a
W2—V i4i • h-2> f
3a J
3a
(9)
Analysis of this expression with arbitrary complex values a and b results in awkward, hardly soluble expressions. Let us consider only extreme cases of small and large |a| and also the relations between IrI and |a| for several typical and practically important cases.
With small |a| (when the complex permittivity modulus slowly varies in depth), we derive:
R =
1 - b0'5 +
ib~h5 -(0,104 + 0,146b0'5 )
1+b0'5 +
ib-1'5 - (0,104 - 0,146b0'5 )
(10)
When a sharp boundary is absent (i.e. b = 1), we have: R = 0,004|a|2 A2. In this case there is almost no reflection.
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Another extreme case is when the complex permittivity increases with a high speed, i.e. |a| is large. In this case we derive:
f2kb4b^1/3
R --1 +1,58
3a
(l - i • 1,73). This formula shows that the reflection coefficient is close
to 1 for large values of |al. With a further increase of |al and 1 it tends to 1.
The above mentioned behavior of R remains mainly for other kinds of the polynomial dependence. Fig. 2. shows the dependence of the reflection coefficient | R |2 on the parameter |a| for a linear layer.
5.3. Parabolic layer
Let us introduce the following notation: s (z) = (az + b)2 = [(a1 + ia2) z + (b1 + b2)]2. In this case:
R -
H0225 (w) - ibH-2)75 (w)] / [H0225 (w) + ¿bH-^ (w)
(11)
where w -
kb2 2a
When a l « b, i.e. p >> 1, we have
R -[(1 - b)kw + i(0,0094 + 0,156b)] / [(1 + b)kw + i(0,0094 - 0,156b)].
(12)
When a sharp boundary is absent (b = 1): R = 0,016| a| X2. In another extreme case when 5,1(1 - i)
«1, we have: R = -1 + —^— 7.
via x
Том 22, № 03, 2019_Научный Вестник МГТУ ГА
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5.4. Matching layer
In a number of extreme cases a thin intermediate layer is formed on the medium-atmosphere boundary. In this layer, the complex permittivity smoothly varies from 1 (atmosphere) to its value in the depth of the medium and the curve which represents this dependence has no kinks. The presence of such a layer in a number of cases results in a substantial decrease of the reflection coefficient due to the decrease in reflections that take place on the boundary.
Let us discuss the reflection of the following structure: a region of the space z is filled with a medium with the complex permittivity 8k . A "matching" layer is located within h. The complex permittivity 8s of this layer varies according to the law: 8s = + 1 8 cos n h • This relation shows
that when z = 0 the complex permittivity is 8s = 1, when z = h - 8 = 8s and 8^ (0) = 8^ (h) = 0, i.e.
the matching over the derivative takes place on boundaries of the layer. The wave equation for the matching layer is reduced by means of substitution of an independent variable into the Mathieu equation.
Fig. 3. shows the results of calculation of the reflection coefficient with different values of the complex permittivity
\S\2
Fig. 3. Dependence of s(z ) upon the thickness of an intermediate layer - h.
In an intermediate layer s(z) changes from s = 1 up to s = 65-40/ according to the linear (1), harmonical (2) and exponent (3) laws. X = 3cm
This figure shows that the reflection coefficient varies from |(l - Vs ) / (l + Vs )| to zero. The
figure demonstrates the comparison of the similar curves for the matching layer with a linear dependence (without the matching over the derivative). The figure reveals that a decrease in the reflection coefficient for the linear layer is faster than for the matching "cosinusoidal" layer. For a thick layer
f\ \
(h^œ) R = 0,002 |l - 8k\2 - ; for a thin layer (h^0) R =
h
20,4 h
j
l-8k
5.5. Intermediate layer
The law of the complex permittivity change, determined by (3), corresponds to an intermediate layer. In this case the formulae in 3 remain valid. For a medium with small a we have:
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R = ■
1 - + 0,125 -- (l + 3 )
_-0_
1 + ^e"ib/2 + 0,125 -L (l - 3^е"г5/2У
(13)
where -0 = к
M
a
,-ib/2
Ô is the angle of losses. This formula shows that the reflection coefficient is
determined first and foremost by the complex permittivity on the boundary. In particular, for a medium
with a very small permittivity (sweet water) we derive: R =
1 -#i
1 +
M
1 -
0,14aÀ
-1
Л
. In another case
when a is large we derive: R « -1 + 2N0 (p0 )/ N0 (p0 +2i«Jfie'5/2 / rcp0 ) . Fig. 4 shows a number of dependencies of the reflection coefficient upon a with different values [5, 6, 8, 17] of P and 5.
Fig. 4. Dependence R|2 upon |a| cm 1 for an exponent medium.
1. e(z) = exp(1 + i0,1az); 2. e(z) = (1 + i) exp(az); 3. e(z) = (3,5 + 2i) exp(az)
It is important to mention that, regardless of the law of variation of the complex dielectric constant from depth, there is almost a linear dependence of the reflection coefficient on the determining parameter.
CONCLUSION
The paper deals with the problem of reflection of electromagnetic waves from electrically layered inhomogeneous media. Most natural formations belong to the class of such media. The corresponding problems of reflection of radio waves always arise during remote sensing of natural objects under observation. The purpose of the relevant monitoring is to remotely determine the physical characteristics of the underlying surfaces (humidity, hardness, temperature, etc.). The listed properties are determined by the electrophysical characteristics of the surface: dielectric permittivity and conductivity, which, as it is known, are combined into a complex permittivity s.
Layered structures in which the laws of variation of the complex permittivity in terms of depth were exponential, polynomial, parabolic and linear were considered as models of remote sensing objects. The field inside each of the matching and transition layer was considered separately.
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Knowledge of the reflection coefficients dependence in such media opens the way to unambiguous definition of the complex dielectric permittivity, and, consequently, of the physical characteristics of the probed surfaces.
Unfortunately, it is impossible to express the value of the complex dielectric constant explicitly as the function of the reflection coefficient and the viewing angle by means of the obtained strict dependences of the reflection coefficient on the parameters of the laws of the change in the complex dielectric constant. However, this problem is solved quite easily by numerical methods, which, in the final analysis, make it possible to determine the desired physical characteristics.
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13. Biard, R.U. and Mak, Lejn T.U. (2015). Malye bespilotnye letatelnye apparaty: teoriya i praktika [Small unmanned aerial vehicles: theory and practice]. Per. s angl. Moscow: Tekhnosfera, 120 p. (in Russian)
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INFORMATION ABOUT THE AUTHORS
Leo P. Ligthart, Doctor of Philosophy, Honorary Doctor of Moscow State Technical University of Civil Aviation, Delft University of Technology, International Research Centre for Telecommunications-Transmission and Radar, Delft, The Netherlands, l.p.ligthart@its.tudelft.nl.
Anatoly I. Kozlov, Doctor of Physical and Mathematical Sciences, Professor of Air Transport Radio Electronic Equipment Maintenance Chair, Moscow State Technical University of Civil Aviation, Moscow, Russia, vilandes@yandex.ru.
Alexander I. Logvin, Doctor of Technical Sciences, Professor of Air Traffic Control Chair, Moscow State Technical University of Civil Aviation, Moscow, Russia, vilandes@yandex.ru.
Igor V. Avtin, Post-graduate Student of Air Transport Radio Electronic Equipment Maintenance Chair, Moscow State Technical University of Civil Aviation, Moscow, Russia, igarrykavt@gmail.com.
ОТРАЖЕНИЕ РАДИОВОЛН ПРИ ДИСТАНЦИОННОМ ЗОНДИРОВАНИИ
ПОДСТИЛАЮЩИХ ПОКРОВОВ
12 2 2 Л.П. Лигхарт , А.И. Козлов , А.И. Логвин , И.В. Автин
1 Делфтский технологический университет, г. Делфт, Нидерланды
2 Московский государственный технический университет гражданской авиации,
г. Москва, Россия
В настоящее время активно развиваются технологии мониторинга, которые позволяют извлекать и использовать некоординатную информацию об объектах. Некоординатная информация - это информация о типе и свойствах исследуемого объекта. Основным методом решения задач мониторинга является дистанционное зондирование, где особое место принадлежит радиолокационным методам, опирающимся на пространственно-временную обработку сигналов, и в частности на методы радиополяриметрии. Для решения задачи мониторинга необходимо иметь информацию о поверхности. Поскольку даже небольшие изменения электрических и физических свойств (солености,
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влажности, состава почвы и т. д.) таких зон приведут к изменению основной электрофизической характеристики поверхности - комплексной диэлектрической проницаемости. В статье приводятся строгие решения задач отражения радиоволн от слоистой поверхности с различными законами изменения комплексной диэлектрической проницаемости - е по глубине. Рассматриваются среды с экспоненциальным и квадратичным законами изменения е при произвольных углах падения радиоволны на поверхность е. Для слоистых сред с законами изменения комплексной диэлектрической проницаемости, носящими полиномиальный и линейный характер, строгое решение получено для случая вертикального визирования. Отдельно рассматривается аналогичная задача для параболического слоя. Проводится детальный анализ отражения радиоволн от среды с согласующим слоем. Подробно исследуется характер электромагнитного поля внутри переходного слоя. Статья иллюстрируется графическими зависимостями коэффициентов отражения электромагнитной волны от слоистой среды с линейным и экспоненциальным законами изменения комплексной диэлектрической проницаемости по глубине.
Ключевые слова: поляризация, матрица рассеяния, подстилающая поверхность, комплексная диэлектрическая проницаемость, сканирование, дистанционное зондирование.
СПИСОК ЛИТЕРАТУРЫ
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17. Kozlov A.I., Ligthart L.P., Logvin A.I. Modeling and verification of earth-based radar objects. Vol. 7. Requirements to accuracy and reliability of the equipment of determination of the objects parameters and signal characteristics. Moscow - Delft, 1998. 112 с.
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СВЕДЕНИЯ ОБ АВТОРАХ
Лигхарт Лео Петрус, доктор философских наук, почетный доктор МГТУ ГА, Делфт-ский технологический университет, Международный исследовательский центр телекоммуникаций и связи и радиолокации, Делфт, Нидерланды, l.p.ligthart@its.tudelft.nl.
Козлов Анатолий Иванович, доктор физико-математических наук, профессор кафедры технической эксплуатации радиоэлектронного оборудования воздушного транспорта, Московский государственный технический университет гражданской авиации, Москва, Россия, vilandes@yandex.ru.
Логвин Александр Иванович, доктор технических наук, профессор кафедры Управления воздушным движением, Московский государственный технический университет гражданской авиации, Москва, Россия, vilandes@yandex.ru.
Автин Игорь Викторович, аспирант кафедры технической эксплуатации радиоэлектронного оборудования воздушного транспорта, Московский государственный технический университет гражданской авиации, Москва, Россия, igarrykavt@gmail.com.
Поступила в редакцию 23.07.2018 Received 23.07.2018
Принята в печать 21.05.2019 Accepted for publication 21.05.2019
