CC BY
7PHYSICS AND MATHEMATICS
INDICATRIXS OF MILLIMETER AND TERAHERTZ WAVE ANGULAR SCATTERING
BY WATER DROPS AND RAIN
Malyshenko Yu.I., Roenko A.N.,
Senior researchers, PhD, Kostina V.L.
Research assistant, A. Ya. Usikov Institute for Radiophysics and Electronics National Academy of Sciences of Ukraine, Kharkov
ABSTRACT
Detailed information about millimeter and terahertz wave scattering diagrams in rains is needed to develop millimeter and terahertz range radio engineering systems. It is rather difficult to measure scattering diagrams, the easier way is to calculate them. The prepared tables of scattering diagrams can be used for microwave range, but there are no such data for the terahertz one. This paper presents the results of scattering diagram calculation of millimeter and terahertz waves both by separate raindrops and rains. The calculations were executed using the exact formulas for all interval of raindrop sizes appeared in precipitations. On the basis of obtained results the database of scattering diagrams was developed. It gives a wide opportunity to calculate scattering diagrams at various combination of initial data, i.e. radiation wavelength, raindrop size distribution function and rain intensity value. The results obtained may be used to estimate the radiation spatial distribution in millimeter and terahertz ranges.
Keywords: scattering diagram, millimeter and terahertz ranges, drops size distributions, indicatrix database.
Introduction.
As is known, rains cause attenuating and masking interferences mainly for millimeter transmission lines and radars. To describe this rain effect, specific attenuation (y, dB/km) and specific backscattering coefficients (n, m-1) for the following two directions mostly are used:
1) for transmission lines - in the radio waves propagation direction (6=0°)
2) for radars - in the inverse direction (6=180°).
Remote sensing technologies deal with mediums
containing scattering particles. Theoretical study of such mediums doesn't restricted anybody by two mentioned above directions only [1, 2], it considers more detailed angular scattering coefficients, usually called as scattering indicatrixs. For example, they are used in hydrology to predict possible floods and determine accumulated water storage in snow covers during winter season by onboard remote sensing tools [3]. This method is based on theoretical model of angular scattering of electromagnetic radiation by snow crystals. Indicatrixs of radiation scattering by vegetation elements (steams, leaves, grain) are also used to predict future harvests (it's necessary for grain crops) in agrol-ogy while tracing accumulating speed of green biomass during vegetation season by means of remote sensing [3, 4].
From time to time ice reconnaissance requires finding ice floes, which can be a reliable base for future drift-ice research units among a lot of ones identically covered with snow. An old ice floes are suitable for this purpose only. They are the ones that can be found using radio physical onboard radar or radiometric tools. It appeared that knowledge of angular characteristics of increased microwave emission by bubbles in the thickness of sweetened old ice floes helped here.
As was shown in the examples above, angular characteristics of radiation scattering by particles of studied medium in all directions play significant role in the theory and practice of remote sensing. They are also important when exploring rain rate and its features in metrology tasks. In this case we refer to angular characteristics of scattering by rain drops that are usually called as scattering indicatrixs.
They are widely used in optical and microwave ranges (including millimeter one). Their values are calculated and published in the table form [7-10]. Let's remind that microwave tools (onboard radars and radiometers) complete the tools of optical range in solving rain remote sensing tasks as they can detect the places of rainfalls through the clouds. Up to now scattering indicatrixs have been absent in intermediate area of frequencies between two mentioned ranges only, in the terahertz one.
At some point it is due to more difficulties in their calculation at these frequencies comparing to microwave and optical ones. In the first case their calculation gets easier because the radiation wavelength is higher than raindrop sizes in microwave range and it is enough to use simple formulas containing only some of the additive components of Mie rows or even one additive component in most cases (Rayleigh approximation [12]). Reverse ratio between raindrop sizes and wavelength that allows using simple (asymptotic) formulas can be seen in optics. Wavelength can be compared with the sizes of raindrops in intermediate and terahertz ranges. Due to this fact, well-known Mie resonances can be seen in these wave ranges and that is why rains greatly effect on radiation of these ranges [11]. This leads to the fact that the largest amount of components should be considered exactly in infinite Mie rows [14].
In this paper scattering indicatrixs of raindrops are calculated for terahertz range. It appeared that they had a number of useful features for remote sensing technologies.
1. Techniques to calculate the values of scattering indicatrixs.
Formulas to calculate the values of scattering indicatrixs can be found in [12-14], in particular. They
are based on the Mie solutions of vector wave equation in the problem of electromagnetic radiation scattering on spherical particles.
Normalized scattering indicatrixs are determined by the following expression [12]
œ
(o, p) =
F (e, p)
2n n
(1)
{ dpjF(0, p) SinOdO
where F(0, p) is an angular distribution of intensity, scattered by the particle with radius a, complex
dielectric refractivity m = yfs = nx — jn2 and wave parameter X = ka, k = 2ft/A, A is a wavelength, 0 is a scattering angle in relation to direction of wave propagation. According to the Mie theory F(0, p) can be represented in the form [13]
F(0, p) = i (0) Sin2 (p) + i (0) Cos2 (p).
(2)
Here
i (0)
i (°) =
„ 2n +1 / , x
> —7-r(a n + b T )
_/_.-|\\n n n n /
n=1 n(n + 1) 2n +1
(3)
>
(b
n + a t
n=i n(n +1)
where a and b are the expansion coefficients of fields scattered by spherical harmonics in Mie task; ft
n n r J r ' n
and Tn are angular coefficients, which are functions from Cos(0) only. Equations for an, bn, ftn and Tn
are described in details in [14].
Formula (1) can be presented in the following form:
2
2
œ(0,p) = i^Sinp iOC°s2(p)' (4)
scat
where CxRt (m,x) = 2j i^n +1)an\ + \bn \) is scattering cross section.
k n=i ^ '
As opposed to the scattering indicatrixs of separate raindrops, equation (4) needs to be completed with raindrops size distribution function N(d) Ad to calculate scattering indicatrixs of polydisperse medium (rain).
One of the well-known equations, for example, Marshall-Palmer, Laws-Parsons, Gamma or Lognormal distribution can be used for the microwave range. In the terahertz range, where small drops should be considered, in our opinion, it is possible to use distributions of Best and Marshall-Palmer in first approximation only, but it is better to use distribution, proposed by the authors of [15]. So, considering specific raindrop size distribution N(d),
we get the following equation to calculate scattering indicatrix of polydisperse medium (rain).
S (i (0) Sin2 (q>) + i (0)Cos2 (p)) N (d,R)Ad
i£rain = dmin---, (5)
dmax
^k2CscatN (d,R)Ad
where R is rain intensity, d is drop diameter.
As follows from the formula (4), the data about complex properties of water deflection
m = yfs = n — ik for all wavelengths and temperatures are required for these calculations.
As is generally known, the values of permittivity can be obtained by means of well-known Debye equations [16] with sufficient accuracy for microwave and millimeter wave ranges. But at higher frequencies of submillimeter wave range the mechanism of orienta-tional polarization of water molecules is changed step-by-step by faster one of resonant polarization, related to vibrations of separate fragments of water molecule. To describe the form of spectrum resonant line, nearest to submillimeter wave range, the Frohlich's model [16] was applied in [17]. Equation accounting both of above mentioned mechanisms, existed simultaneously in submillimeter wave range, was proposed in this work also. It is suitable to calculate water permittivity in wide frequency (0,03-3 THz) and temperature (0°-70°C) ranges.
Convenient nomogram for quick recalculation of the water dielectric permittivity values from one temperature value to another was introduced in this calculation model. As follows from this nomogram, temperature dependence is much weaker (about twice weaker) in terahertz range than in microwave one. And that is why it is not required to apply the same temperature step while calculating scattering indicatrixs. Once more similar calculation model of Elisson [18], covering wide frequency and temperature ranges, was published. This model was based on Lorentz form of resonance lines. Almost the same results in submillimeter wave range were obtained after comparing calculated values
of water complex dielectric permittivity of both models [17, 18]. It is worth mentioning that it is better to use spherical model of raindrops in this range and upgraded algorithm [19], which sufficient level of accuracy should be used when calculating using Mie formulas.
2. Scattering indicatrixs of separate drops.
Scattering indicatrixs of separate drop with 1.0 mm diameter in polarization plane with ty = 0° on a few wavelengths are shown as an example in Fig.1. Indicatrixs in orthogonal plane (ty = 90°) are similar and not presented hereinafter to reduce the publication volume.
Let's point out that indicatrixs of separate raindrops do not have any significant application in actual use, but their calculation is the necessary stage of transfer to indicatrixs of polydisperse mediums, which are widely used in practice. At the same time analysis of indicatrixs of separate drops is interesting for understanding of some physical features of electromagnetic wave scattering by hydrometeors.
Separate raindrops scattering indicatrixs are close to Rayleigh ones which remind of eight in the plane of polarization and circle in an ortogonal plane in centimeter and decimeter ranges, where radiation wavelengths exceed the raindrop sizes substantially (in this case wave parameter X = 2.HÜ. / X is much less than unit). Scattering intensities of radiation falling on a drop back and forth are approximately the same at these conditions. Scattering in a backward semi-sphere even exceeds scattering in a front one, so-called Thomson effect shows itself, in some cases.
Fig. 1. Scattering indicatrixs of separate drop with 1,0 mm diameter at 0,1; 0,5; 1,0; 3,0 and 8,3 mm wavelengths
(curves 1-5 correspondingly).
It is partially seen at the wavelength X = 8 mm in Fig. 1. Generally radiation of these ranges depends
on rains moderately. Another situation occurs in millimeter wave range: its radiation wavelengths almost
fully coincide with raindrop sizes, and this causes strong resonant interaction. That's why millimeter wave range together with adjacent submillimeter one appeared to be the most vulnerable to the weakening and dispersive rain expose. Significant changes of scattering indicatrix types happen here as well. The shorter submillimeter wavelength, the strongly prolate narrow
front lobe appears and grows quickly in the scattering indicatrixs, they start to look like the directional pattern of large mirror aerials (Fig. 2). This lobe is surrounded by considerably weaker sidelobes. Their positions in an angular plane for different drop sizes do not coincide, which results in overlapping of dips between them and smoothing of polydisperse media indicatrixs.
Fig. 2. Scattering indicatrix in polar coordinate system.
The higher wave parameter of separate drop, the greater front lobe size of scattering indicatrix. Its value can be estimated easily by K.Shifrin limiting formula for drop indicatrixs in forward direction [20] at wave parameter X > 20
œ (0=0°) =
X
X
4 k 2C
8^
where scattering cross section Csnaf takes on its
2
^ ïfï _ I
limiting value C cat =
1 +
m —1
m +1
m = n — jn2 is rain water complex refractive index
n ,2 + n2 +1
[17] and % = ■ 1 2
(nx + 1)2 + n22
Estimation of scattering indicatrix front lobe amplitude for the waves of middle part of submillimeter range for a drop with 2,86 mm diameter (thus wave parameter is X = 20) conducted according to this formula gives a value of 28, and exact calculation gives a value of 31,2. The values of front lobes of scattering indicatrixs are equal to 70 and 280 for drops with 1 mm and 2 mm diameters in short-wave part of submillimeter range (at 3 THz frequency) and an exact calculation gives values of 76 and 290 correspondingly. Thus, Shifrin's formula provides evaluation precision of percents' units. It is useful for the program debug of scattering indicatrixs calculation, as it allows to get the values of indicatrixs in control points quickly.
As it was mentioned before, lateral radiation in polarization plane is absent in the Rayleigh scattering in-dicatrixs. This property, but in a weaker type, keeps being saved in drop indicatrixs in the whole millimeter wave range. The deepest scattering minimums near lateral angles of 80-100° (Fig. 1) for X = 8,3; 3 and 1 mm are presented exactly on these indicatrixs. But, this feature of single drop indicatrixs gradually disappears with further frequency increase (to middle part of submillimeter wave range and higher).
3. Scattering indicatrixs of polydisperse medium.
Considering radars and communication lines operation in rains it is necessary to take into account that radio waves emitted by antennas light up a lots of drops with a wide range of sizes simultaneously. As mentioned already, in this case, positions of main and side lobes of indicatrixs of separate drops of different sizes do not coincide, which leads to overlapping of dips between them and smoothing out the scattering indicatrix of polydisperse medium. That's why more smoothed resulting scattering indicatrixs of rains can be seen in practice. Values of such indicatrixs for millimeter and terahertz ranges at temperature of 25°C and for several values of rain intensity R from 1,25 to 100 mm/h and for all known and widely used raindrops size distributions have been calculated.
Both classic distributions of Best, Marshall-Palmer, Laws-Parsons, Gamma and Lognormal one, and the original distribution suggested by the authors of this work specifically for terahertz range [15] are among the last ones. The feature of the original distribution suggested here is that it allows taking into account the smallest raindrops (with 0,05-0,6 mm diameters) when calculating. These raindrops were not considered in classic distributions before, but are quite essential for terahertz range. Practical presence of modern disdrometers in meteorological measurements confirmed the fact of their (these raindrops) existence in the rains in many geographical areas of the Earth, which are distant from equatorial belt.
Indicatrixs of radiation scattering by rain with the intensity of R = 50 mm/h and drop size distribution described in [15] are shown as an example in Fig. 3. Scattering indicatrixs of rain of the same intensity and Laws-Parsons drop size distribution from [8] is given for comparison. It is evident that the main energy of submillimeter waves, which lights up the rain volume, leaves it directly forward as a narrow bunch (Fig. 2.)
Let's point out that this phenomena significantly corrects solving the beam energy transfer equations, reducing the role of multiple scattering for this range and also cross noise level in communication lines during rains. Besides that, it improves the work of radars here as compared to a microwave range, where rain masks air targets better because of the Thomson effect. The usage of radiation with circular polarization suppresses
this interference (clutter) only partially in that range. manage to find results of similar calculations of rain Comparison of the achieved results with well-known scattering indicatrixs for submillimeter wave range. data showed quite good correspondence. We did not
N «• % t \\ \ • :
[ ,.•- OS •
• : V- \: ',s > ) ) * v\ ■ X. t i. : 4 ■r • *" ' rt".
' N \ . \ 1
i 1 ■ 1 i 1 i 1 i 1 i 1 ■ ( i 1 i
0 20 40 60 80 100 120 140 160 9°
Fig. 3. Scattering indicatrixs of rain with 50 mm/h intensity and drop size distribution from [16] for vertical polarization at wavelengths of 0.1;0.5;1.0;3.0 and 8.3 mm (curves 1-5 correspondingly); 6 - at wavelength of
1.0 mm for Laws-Parsons distribution [8].
4. Scattering indicatrixs database.
Software package, allowing calculation of scattering indicatrices at any wavelength for millimeter and submillimeter wave ranges both for separate drops at the whole interval of their diameter changes from 0.05-7.0 mm, and for the rain, also for all types of drop size distributions, which are widely used in microwave and millimeter ranges nowadays has been created while performing this work. Distribution of the increased size spectrum of the drops from [15] has been used for submillimeter range.
Scattering indicatrixs of both types (separate drops and polydisperse medium) for a certain set of wavelengths with 2 = 8,3; 5,0; 3,19; 3,0; 2,14; 1,0; 0,83; 0,5; 0,3 and 0,1 mm have been calculated to demonstrate the abilities of this software package. Achieved results gave a chance to create the database of wave scattering indicatrixs of millimeter and terahertz ranges by drops and rain itself.
Database provides efficient access to all data arrays, smart data control, data selection as per the request function and data processing. This software package is easy and convenient, doesn't require any specific programming skills or certain experience in data processing.
Conclusions.
Calculations of indicatrixs of wave scattering by raindrops for millimeter and terahertz ranges have been executed. Calculations have been performed according
to exact formulas for the whole range of raindrop sizes, that can be spotted in rainfalls. Wave scattering indicatrixs of microwave, millimeter and terahertz ranges by polidesperse medium (rain) with different types of drop size distributions and different values of rain intensity have been calculated. Calculation results can be used for theoretical problem solving and practical application in radar and wireless communication. Achieved results are presented as the database of scattering indica-trixs to increase the efficiency of their application.
References
1. Tsang L. Theory of microwave remote sensing / Tsang L., Kong J.A., Shin R.T. -N.Y.: J. Wiley and sons, 1985. - 613 p.
2. Ulaby F.T. Microwave remote sensing - active and passive / Ulaby F.T., Moore A.K., Fung A.K. - Ar-tech House, Dedham, MA, Vol. 1-3 - P. 1982-1985.
3. Wilson L.L. Mapping snow water equivalent by combinating a spatially distributed snow hydrology model with passive microwave remote sensing of environment / Wilson L.L., Tsang L., Hwang J.N., Chen C.N. - IEEE Tr.on GRS-37, 1999, No.3 - P.690-701.
4. Marliani F. Simulating coherent backscattering from crops during the grooving circle / Marliani F., Palosia S., Pampaloni P., Kong J.A. - IEEE Tr.on GRS-40, 2002, No.1 - P.162-177.
5. Ferrazdi P. Multifrequency emission of wheat: modeling and application / Ferrazdi P., Wigneron J.P.,
Guerriero L., Chanzy A. - IEEE Tr.on GRS-38, 2000, No.6 - P.2598-2607.
6. Gloersen P. Microwave signature of first year and multiyear sea ice / Gloersen P., Nordberg W., Schmugge T.J., Wilheit T.T. - J. of Geophysical research, Vol.78, No.18 - P. 3564-3572.
7. Denman H. Angular scattering functions for water spheres / Denman H. Keller W., Pangonis W.J. -Weyne State University Press, Detroit, 1966 - 280 p.
8. Shifrin K.S. The scattering indecatrixes of centimeter radiation by water drops / Shifrin K.S., Chernyak M.M. // Proc. Glavnoj Geophys. observatory. - 1967. - No. 203. - P. 123 - 136.
9. Setzer D.E. Anisotropic scattering due to rain at radio-relay frequencies / Setzer D.E. // Bell System Technical J. - 1971. - 50, No. 3. - P. 861-868.
10. Guschina I.Ya. Angular distributions of scattering millimeter waves by rain / Guschina I.Ya., Ma-linkin V.G., Sokolov A.V., Sukhonin E.V. - Preprint IRE RAS, Moskov, No.19/322, 1981 - 31 p.
11. Malyshenko Yu.I. Terahertz radio waves specific attenuation due to rain with small raindrops / Malyshenko Yu.I., Roenko A.N. // J. Atmospheric Electricity. - 2014. - 34, No. 1. - P. 9-19.
12. Van de Hulst H.C. Light scattering by small particles / Van de Hulst H.C.; J. Wiley N.Y., 1957.
13. Stratton J.A. Electromagnetic Theory / Strat-ton J.A. - N.Y: McGraw Hill book Co., 1941.
14. Deirmendjian D. Electromagnetic Scattering on Spherical Polydispersials / Deirmendjian D.; - N.Y., 1969.
15. Malyshenko Yu.I. Taking into account of small droplets in rain drop size distribution function for terahertz wave range/ Malyshenko Yu.I., Roenko A.N. // Radiophys. and Electron. IRE NASU, Vol. 14, No.3, P.323-330, 2009.
16. Fröhlich H. Theory of Dielectrics / Fröhlich H.; - Clarendon Press, Oxford, England, 1958.
17. Malyshenko Yu.I. Water permittivity model for millimeter and terahertz wave ranges / Malyshenko Yu.I., Kostina V.L., Roenko A.N. // Ukr. Phys. J., Vol.52, No.2, P.155-164, 2007.
18. Ellison W.J. Permittivity of pure water at standard atmospheric pressure over the frequency range 0-25 THz and temperature range 0-100 °C / Ellison W.J. // J. Phys. Chem. Ref. Data. - 2007. - 36, No. 1. - P. 1-18.
19. Wiscombe W.J. Improved Mie scattering algorithms / Wiscombe W.J. // Appl. Opt. - 1980. - 19, No. 9. - P. 1505-1509.
20. Shifrin K.S. Light Scattreing in Muddy Medium / Shifrin K.S. - M.: Gostechizdat, 1951. - 288 p.
К РАЗРАБОТКЕ ЭКСПЕРТНЫХ СИСТЕМ ИССЛЕДОВАНИЯ СЛОЖНЫХ ОБЪЕКТОВ
Гаджиев Ф.Г.
Азербайджанский Государственный Университет Нефти и Промышленности, Баку, доцент
Гасымов Г.Г.
Азербайджанский Государственный Университет Нефти и Промышленности, Баку, доцент
Керимов В.А.
Азербайджанский Государственный Университет Нефти и Промышленности, Баку, доцент
THE DEVELOPMENT OF EXPERT SYSTEMS FOR THE STUDY OF COMPLEX
OBJECTS
Hadjiyev F.H.,
Azerbaijan State Oil and Industry University, Baku, assosiative professor
Gasimov G. G.,
Azerbaijan State Oil and Industry University, Baku, assosiative professor
Karimov V.A.
Azerbaijan State Oil and Industry University, Baku, assosiative professor
АННОТАЦИЯ
В статье рассматривается проблема построения и развития экспертных систем, ориентированных на исследование сложных объектов. Обосновывается применение лингвистического подхода, эффективность которого обеспечивается как экспоненциальными функциями принадлежности,так и логическими аппроксимационными процедурами.
ABSTRACT
The article deals with the problem of constructing and developing expert systems focused on the study of complex objects. A linguistic approach is sought, the effectiveness of which is provided by both exponential membership functions and logical approximation procedures.
Ключевые слова: экспертные системы, знания, функции принадлежности, терм-множество, универсальное множество
Keywords: expert systems, knowledge, membership functions, term-set, universal set
1. Постановка проблемы. К настоящему времени в концепции экспертных систем установились
