Department of Physics

Seminar Ia - first year, Second cycle

Electromagnetic Attenuation due to theLiquid Water Content in the Atmosphere

Maks Kolman

supervised byDoc. Dr. Daniel Svensek

Dr. Gregor Kosec

December, 2015

Abstract

In this seminar we tackle an influence of Liquid Water Content (LWC), i.e. the mass of the waterin the atmosphere, on the attenuation of a communication signal between a ground antenna andcommunication satellite. Three main attenuation models that relate the LWC and attenuationare presented. Besides models also the analysis of correlation between the measurements ofLWC by 5.6 GHz weather radars and attenuation of communication between Ljubljana StationSatProSi 1 and communication satellite ASTRA 3B on the 20 GHz band is considered.

Contents

1 Introduction 2

2 Electromagnetic attenuation 32.1 Rain rate, LWC and DSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Lord Rayleigh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Mie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Marshall-Palmer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Attenuation in clouds and fog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Measurements 73.1 Measurements of signal attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Measurements of rainfall rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Correlation between rainfall rate and signal attenuation . . . . . . . . . . . . . . 9

4 Conclusion 11

1 Introduction

The increasing demands for higher communication capabilities between terrestrial and/or earth-satellite repeaters requires employment of frequency bands above 10 GHz [1]. Moving to suchfrequencies the wavelength of electromagnetic radiation (EMR) becomes comparable to the sizeof water droplets in the atmosphere. Consequently, EMR attenuation due to the scattering onthe droplets becomes significant and ultimately dominant factor in the communications quality[2]. During its propagation, the EMR waves encounter different water structures, where it canbe absorbed or scattered, causing attenuation. In general, water in all three states is present inthe atmosphere, i.e. liquid in form of rain, clouds and fog, solid in form of snow and ice crystals,and water vapour, which makes the air humid. Regardless the state, it causes considerableattenuation that has to be considered in designing of the communication strategy [2]. Therefore,in order to effectively introduce high frequency communications into the operative regimes, anadequate knowledge about atmospheric effects on the attenuation has to be elaborated. In thisseminar we focus only on the Liquid Water Content (LWC).

Naturally, the task begins with measurements. We use measurement of both involved quan-tities. The LWC measurements are provided by two weather radars, one situated in Lisca and,a newer one, in Pasja Ravan, both operated by Slovenian Environment Agency [3]. For theattenuation we use in-house measurements of signal strength between Ljubljana Station Sat-ProSi 1 and communication satellite ASTRA 3B [4]. The main purpose of this seminar is toinvestigate correlation between precipitation measured in 3D with meteorological radar and theattenuation.

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2 Electromagnetic attenuation

Many empirical relations featuring non whole exponents will appear in following discussions.For the sake of readability we will adopt subscript notation to denote unit-less quantities. Fora given quantity X we define its dimensionless quantity measured in yz as

X[yz] =X

1 yz.

Electromagnetic attenuation describes energy losses during wave propagation through a medium

Attenuation[dB] = 10 log10Input intensity

Output intensity.

Specific attenuation α is often used. It describes attenuation per travelled distance and isusually measured in dB/km [2, 5]. In this seminar we deal with the attenuation due to theLWC, which is basically myriad of droplets in the atmosphere. There are several approaches toconsider electromagnetic propagation through the atmosphere filled with water droplets [2,5,6].In the following discussion we will present some of them.

Specific attenuation for almost all frequencies under 100 GHz is under 1 dB/km with a spiketo 10 dB/km around 60 GHz. For snow these numbers range from a couple ten to hundreddB/km depending on snowfall intensity.

Basically, raindrops can be considered as dielectric blobs of water that polarize in the presenceof an electric field. When introduced to an oscillating electric field, such as electromagneticwaves, a droplet of water acts as an antenna and re-radiates the received energy in arbitrarydirection causing a net loss of energy flux towards the receiver. Some part of energy canalso be absorbed by the raindrop, which results in heating. Absorption is the main cause ofenergy loss when dealing with raindrops large compared to the wavelength, whereas scatteringis predominant with raindrops smaller than the wavelength [2]. In our case, as shown later inthe subsection 2.1, the raindrops will always be smaller than their wavelength, and thereforewe will only dive into the scattering side of the attenuation.

The most ambitious way to mathematically describe above model is to create an exact physicalmodel described by set of Partial Differential Equations and solve it on a droplet level. However,for practical computation simplified or statistical models are used.

The rain is the most important mechanism of the attenuation of EMR in frequencies inorder of tens of GHz (wavelength λ ∼ cm) range [2]. Leading reasons for attenuation at otherfrequencies can be seen in 1.

Figure 1: Transmission specter in the atmosphere for different wavelengths. Each drop in trans-mission is marked with the substance that causes the attenuation. [7]

It is typically characterized by rain rate R[mm/h], which is also relatively easy to measure.The attenuation is also gravely dependent on the size of the droplets, which relates, not onlyon the rain rate, but also on the type of the rain, e.g. a storm and a shower might have thesame rain rate, but a different drop size distribution (DSD).

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2.1 Rain rate, LWC and DSD

Before we proceed to attenuation modeling, a basic models to relate LWC and DSD to rainfallrate are discussed.

A simplified DSD model, where the type of rainfall is not considered, is presented in [8]

N[mm−1m−3](D) = 8.3 · 103 exp(−4.1R−0.21[mm/h]D[mm]), (1)

where D stands for drop diameter and N(D) describes number of droplets of size D to D+ dDin a unit of volume. The DSD was also determined experimentally for different rain rates [5].The experimental data is presented in figure 2, where we can see that the typical diameter ofdrops is in range of mm.

Figure 2: DSD measured in Czech Republic (one year measurement, rain rate R is the parameterof particular curves) [5]. Lines are values of a DSD model described in (1).

The physical background of the exponential distribution is non-trivial and is studied in [9].They discovered that larger drops are unstable because they are deformed by high air resistance.This causes fragmentation into smaller droplets, which in turn dominate in the distribution. Itis also worth noting that the typical time between collisions is much longer than the time scalein which the droplet decays into stable fragments. The disparity confirms that explanationusing a model with non-interactive drops is sufficient.

The LWC can also be extracted from rain rate. To get LWC we would also need the terminalvelocity of droplets, that we can extract from their sizes. And as previously mentioned their sizeor rather DSD is also a function of rain rate so we can resort to an empirical relation, definedas [10]

LWC [kg/m3] = 6.7 · 10−5R0.846[mm/h] (2)

Note that typical values of LWC for rain are between 3-8g/m3.LWC is a great measure for the amount of water in the atmosphere as it can be used in rain,

fog and clouds alike. Electromagnetic phenomena on water droplets are always dependent onthe quantity of water present, which is why LWC is extensively used as the key parameter inthis field.

2.2 Lord Rayleigh

The very first model for atmospheric scattering was introduced by lord Rayleigh [11]. BothRayleigh scattering and Mie scattering outlined in 2.3 describe scattering on a single sphere.We will consider scattering in rain as independent scattering events on individual droplets. Withthis we neglect the scattered EMR from nearby droplets. This is so-called multiple scattering

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and is only important when droplets are extremely closely packed compared to light wavelengthe.g. visible light in fog.

For small dielectric spheres (e.g. droplets), where πD � λ, we can consider the electric fieldof the incident wave as a spatial constant only oscillating in time as

E(r, t) = E(t) = E0e−iωt . (3)

Sphere polarization is dependent on the electric field as

P = (n2 − 1)ε0E(t) (4)

where n is the refractive index of the scatterer. This results in uniformly polarized but oscillatingsphere of water. In turn the sphere radiates as a dipole. Finally, a solution of above problemleads to an expression for beam intensity I0 of unpolarized EMR.

I = I0

(n2 − 1

n2 + 2

)2(2π

λ

)4(D2

)6 1 + cos2 θ

2r2, (5)

where I is scattered intensity at the distance r from the droplet at the angle θ from the incidentwave.

For us a more interesting quantity is the scattering cross section σ. It is an effective area thatquantifies the intrinsic likelihood of a scattering event when an incident beam strikes a targetobject. The scattering cross section is equivalent to the cross section of a hard object, if, andonly if, the probabilities of hitting them with a ray are the same. For Rayleigh scattering it isequal to

σ =2π5

3

D6

λ4

(n2 − 1

n2 + 2

)2

. (6)

Specific attenuation is proportional to the sum of all total cross sections of droplets in a unitof volume. We can write this as an integral [5]

α = 4.343× 103∫ ∞0

σ(D)N(D)dD , (7)

where α is specific attenuation in units dB/km.However, due to its simplifications this model accurately describes scattering on raindrops

only for frequencies up to 5 GHz, and is therefore not useful as our high frequency attenuationmodel.

2.3 Mie

A more general scattering model was developed by Mie in 1908 [2]. Derivation of Mie scatteringstarts with finding a complete set of solutions for the wave equation

∇2ψ + k2ψ = 0 (8)

in empty space with the exception of a sphere in the origin. The solutions are written inspherical coordinates as

ψen(r, θ, φ) = cosφP 1n(cos θ)zn(kr), (9)

ψon(r, θ, φ) = sinφP 1n(cos θ)zn(kr), (10)

where e and o stand for even and odd functions of φ respectively, P 1n is associated Legendre

functions of first kind, first order and degree n, and zn stands for spherical Bessel function of

the first kind jn inside the sphere and h(1)n , spherical Hankel function of first kind, outside of

the sphere.Incident field Ei is linearly polarized plane wave propagating in z direction. Because of the

sphere we get an additional electric field, which we will call scattered field Es. The overallelectric field outside of the sphere is then

E = Ei + Es (11)

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By expanding it into an infinite series over the solutions described in (9) and (10), and ap-plying boundary conditions between the sphere and the surroundings (continuity of electricdisplacement field’s normal component and electric field’s tangential component) we get thesolution [12] [

E⊥E‖

]=

exp(−ikr + ikz)

ikr

[S1(θ) 0

0 S2(θ)

] [E0⊥E0‖

], (12)

where E0⊥, E0‖ are perpendicular and parallel components of incident radiation to the scatteringplane and k is its wave vector. The distance from the scatterer is denoted by r and θ is thescattering angle at which we observe scattered radiation E⊥ and E‖.

The amplitude functions S1(θ) and S2(θ) contain infinite series

S1(θ) =∞∑n=1

2n+ 1

n(n+ 1)(anπn(cos θ) + bnτn(cos θ)) (13)

S2(θ) =∞∑n=1

2n+ 1

n(n+ 1)(anτn(cos θ) + bnπn(cos θ)) (14)

where πn and τn are angular functions defined with corresponding associated Legendre polyno-mials

π(cos θ) =1

sin θP1n(cos θ) , τ(cos θ) =

d

dθP1n(cos θ) . (15)

Mie coefficients an and bn are dependent on relative permittivity εr of the scattering sphere, inour case water, and its diameter D. These coefficients are known and are available on-line1. Inour case (λ ∼ cm) and all drop sizes it is enough to end the sum around n = 10 [2].

We can graphically represent Mie scattering and how it compares with Rayleigh scatteringas shown on figure 3. We see that opposed to the Rayleigh scattering we get asymmetricalscattering pattern.

Figure 3: Comparison between Rayleigh scattering and Mie scattering. Length of the line rep-resents intensity of scattered EMR in the given direction [11].

We will not show it here, but total scattering cross section can be derived as [2]

σ =λ2

2π

∞∑n=1

(2n+ 1)(|an|2 + |bn|2

). (16)

We could again use (7) to get a more accurate approximation of attenuation.

2.4 Marshall-Palmer

Another possible approach towards computing attenuation is to use statistical models. A pop-ular model is presented in [8], where attenuation is related only to the rain rate. The model isreferred to as a Marshal-Palmer relation, and it is stated as

α[dB/km] ∼ aR b[mm/h] . (17)

1http://omlc.org/software/mie/

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Coefficients a and b are determined empirically by gathering data and finding the best fit tothe model. In general, coefficients depend on the incident wave frequency and polarization, andambient temperature. Some example values for different frequencies [5] are presented in belowtable.

f[GHz] 10 12 15 20 25 30

a 0.0094 0.0177 0.0350 0.0722 0.1191 0.1789

b 1.273 1.211 1.143 1.083 1.044 1.007

The Marshal-Palmer relation is derived from the Rayleigh scattering model through relation(7). In the derivation we assume a more general DSD similar to the one described in (1) butadding an extra power factor to account for smaller droplets. It is

N(D) = ADµ exp(BR−δD), (18)

where A, B, µ and δ are coefficient independent of D and R and will later become coefficientsin the Marshall-Palmer equation.

The only relevant piece of data in the scattering cross section for us is the sixth powerdependence to D as seen in (6). The integral can then be written as

α = A · C∞∫0

D6Dµ exp(BR−δD)(BR−δD→x)

= AC B−µ−7Rδ(µ+7)

∞∫0

xµ+6 exp(−x). (19)

The integral that is left is equal to the gamma function Γ(µ+ 7) and with that we get the formwe were looking for. By defining

a = AC B−µ−7Γ(µ+ 7)

andb = δ(µ+ 7)

we get the exact form of the Marshall-Palmer relation.

2.5 Attenuation in clouds and fog

Water in clouds or fog, mostly small droplets with diameter smaller than 0.01 cm, also influencesthe attenuation, however with much lower amplitude than rain [2]. Typical values of LWC inclouds are from 0.03 g/m3 up to 3 g/m3. Again we can use one of empirical relations to assessthe attenuation, for example attenuation dependency on the LWC formulated as

α[dB/km] = KC LWC [g/m3] , (20)

where KC is the attenuation constants dependant on wave frequency and ambient temperature[2]. The same model can also be used for attenuation in fog. In addition, a model dependingon wavelength λ, temperature T0 and fog density ρ

α[dB/km] =

(−1.347 + 0.0372λ[mm] +

18

λ[mm]− 0.022T0[K]

)ρ[g/m3] (21)

is also used [13].

3 Measurements

In previous sections we dealt with the models relating the attenuation and LWC. In this sectionwe focus on measurements of those two quantities. We deal with two expressively different typesof measurements. While the attenuation measurements are relatively well posed, the LWC ismuch harder to determine. Although, the pluviographs provide accurate measurements, thespatial resolution of such measurements limits theirs useability for our purposes. For examplein Slovenia we have only 27 operative pluviographs providing real-time data about the rainrate [3]. Besides, the pluviographs provide only ground rain rate. Therefore, a meteorologicalradars seems a much better choice.

In following discussions we tackle some of the related topics.

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3.1 Measurements of signal attenuation

Jozef Stefan Institute (JSI) and European Space Agency (ESA) cooperated in SatProSi-Alphaproject that included measuring attenuation of the communication link between on groundantennas and a satellite. The satellite utilized is ASTRA 3B, a geostationary communicationsatellite located on the 23.5◦E longitude over the equator. The in-house receiver is a 1.2 mparabolic antenna positioned on the top of the JSI main building (figure 4) and has gain ofabout 47 dBi. The energy loss every 0.15 seconds, resulting in over 500000 daily data points, ismeasured from 1. 10. 2011.

Figure 4: Antenna SatProSi 1 on the roof of JSI.

3.2 Measurements of rainfall rate

In this section we focus on measurements of LWC with meteorological radars. The basic ideabehind such approach is to measure EMR that reflects from water droplets, i.e. the radiationthat scatters directly back at the radar. However, with this method we might have hard timedistinguishing between precipitation and cloud, but because of 100 times smaller droplet sizesin clouds their scattering goes almost undetected (as suggested by the 6th power dependencein equation (5)).

Radar reflectivity factor Z is formally defined as the sum of sixth powers of drop diametersover all droplets per unit of volume. or as an integral

Z =

∞∫0

N(D)D6dD . (22)

It is usually measured in units mm6m−3. When conducting measurements a so-called EquivalentReflectivity Factor

Ze =ηλ4

0.93π5(23)

is used, where η means reflectivity, λ is radar wavelength and 0.93 stands for dielectric factorof water. As the name suggests both are equivalent for large wavelengths compared to the dropsizes. This is because it is based on Rayleigh scattering model and we can quickly see how bycomparing above equations to (6) and (7).

The meteorological radars at Pasja Ravan (Figure 5b) and Lisca (Figure 5a) emit short (1 µs)electromagnetic pulses with the frequency of 5.62 GHz and measure strength of the reflectionfrom different points in their path.

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(a) Lisca (b) Pasja Ravan (c) Reflectivity factor image from radar in Lisca

Figure 5: Images of both Slovenian radars and an example of an image recorded by radar inLisca. [3]

Reflectivity factor and rainfall rate are then related through Marshall-Palmer relation as

Z[mm6m−3] = aRb[mmh−1] , (24)

where Z[mm6m−3] is reflectivity factor measured in mm6m−3 and R[mmh−1] is rainfall rate mea-

sured in mm h−1. In general, empirical coefficients a and b vary with location and/or season,however, are independent of rainfall R. Most widely used values are a = 200 and b = 1.6 [6, 8].Meteorologists rather use dimensionless logarithmic scale and define

dBZ = 10 log10Z

Z0= 10 log10 Z[mm6m−3] , (25)

where Z0 is reflectivity factor equivalent to one droplet of diameter 1 mm per cubic meter.Radars collect roughly 650000 spatial data points per radar per every scan. Example of suchmeasurement can be found in Figure 5c, which shows reflectivity factor reading on May 12015 at 13:50 UTC from Lisca radar. There are three different projections on the image, thecenter image showing latitude-longitude, latitude-height on the right and lastly longitude-heightprojection is shown on the top of the image. Every projection is divided into square bins witheach bin presenting maximum reflectivity factor of all measurements that fall within its borders.

In addition to reflectivity radars also measure the radial velocity of the reflecting particles bymeasuring the Doppler shift of the received EMR. In this seminar only the data collected fromthe radar in Lisca, which records reflectivity every 10 minutes, will be considered.

We can get a sense of radar accuracy from the graph on figure 6. First, we can see a greatscattering of data around our theoretic model. This is mostly because both rainfall rate andreflectivity are functions of DSD and we cannot have an exact function connecting the two. Forexample a storm and an afternoon shower might have the same rain rate but a different DSD,resulting in different attenuation.

3.3 Correlation between rainfall rate and signal attenuation

In the next step we want to correlate rainfall rate and attenuation measurements. The satelliteASTRA 3B is in a geostationary orbit with radius 42 164 km, above the equator at 23.5◦E. Toproperly aim the antenna at the satellite we have to elevate it for 36.3◦. All weather activityis strictly under the altitude of 15 km [14]. That means our electromagnetic ray will rise aboveall rain and clouds within the radius of 20 km.

In first order approximation we look at correlation between maximal measured reflectivitywithin 100 closest data points and measured attenuation. Using equations (17) and (24) we can

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Figure 6: Z-R relationship. Blue scatter dots are measurements, black dots and line describethe theoretical model for a = 300, b = 1.5 and magenta line for a = 200 and b = 1.6. [5]

construct a α-Z relation

α[dB/km] = a

(Z[mm6m−3]

a

) bb

. (26)

With antenna we measure total attenuation A on the path to the satellite, but so far we onlydeveloped models for specific attenuation α. We will approximate that all attenuation happenson the path of length L with constant specific attenuation. We can then write

A = Lα . (27)

To get a linear relation we will use a logarithm on the equation (26). Combined with equation(27) we get

log10A[dB] = log10 L[km]a−b

blog10 a+

b

blog10 Z[mm6m−3] . (28)

We will group all additive constants together into one constant C, and use the definition of dBZfrom equation (25) to get

log10A[dB] =b

10 bdBz + C . (29)

Since log10A and dBZ should have a linear connection we can see resemblance if we plotthem on the same graph with differently scaled y axes. On figure 7 we plotted two weeks worthof data for both sets of data. There is clearly a correlation between them.

A more precise approach is to take into account also spatial dependence of specific attenuationα, consequently total attenuation is computed as

A =

∫path

α(r(s))ds . (30)

Since we have a full 3D spatial LWC information, the above integral could be numericallyevaluated. First, the path from JSI main building to ASTRA 3B satellite would be discretizedwith N nodes. In each computational node a LWC would be approximated from nS closestdata points from radar measurements (Figure 8). However, this is already beyond the scope ofthis seminar.

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Figure 7: Logarithm with base 10 of attenuation measurements and maximum reflectivity abovethe antenna.

Figure 8: A conceptual diagram of LWC integration. Blue dots stands for measured radardata points, green dots represent nodes of approximation on the antenna-satellite link(green line). Magenta circle shows the nearby region of the selected (orange) node.

4 Conclusion

In this seminar we dealt with the attenuation of communication link working on 20 GHz, wherethe LWC determines the quality of the link. The first goal of this seminar is to shed somelight on the models describing the relation between LWC and attenuation. The most knownmodels are presented, i.e. Rayleigh model, Mie model and Marshal-Palmer model for rain, andin addition also model for attenuation of signal due to the LWC in clouds and fog.

Besides models we also dealt with related measurements. The LWC is measured with a helpfrom meteorological radars and attenuation with in-house receiver linked with communicationsatellite ASTRA 3B. The basic correlation between those two measurements is presented fortwo weeks in November 2014.

Future work will be focused on implementation of better correlation models. First, we will

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integrate LWC along the communication path. In second step we will compare theoreticalmodels and measured correlations. Further improvements might also include using SlovenianEnvironment Agency’s weather simulator ALADIN to predict attenuation for the upcomingdays.

References

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[2] Milda Tamosiunaite, Stasys Tamosiunas, Mindaugas Zilinskas, and Milda Tamosiuniene.Atmospheric attenuation due to humidity. Electromagnetic Waves, pages 157–172, June2011.

[3] Slovenian Environment Agency. (Agencija Republike Slovenije za Okolje).

[4] Andrej Vilhar, Gorazd Kandus, Arsim Kelmendi, Urban Kuhar, Andrej Hrovat, andMichael Schonhuber. Three-site Ka-band diversity experiment performed in Slovenia andAustria. In Antennas and Propagation (EuCAP), 2015 9th European Conference on, pages1–5. IEEE, 2015.

[5] Ondrej Fiser. The role of dsd and radio wave scattering in rain attenuation. Geoscienceand Remote Sensing New Achievements, 2010.

[6] Remko Uijlenhoet. Raindrop size distributions and radar reflectivity–rain raterelationshipsfor radar hydrology. Hydrology and Earth System Sciences, pages 615–627, August 2001.

[7] Sr Nicholas M. Short. Electromagnetic spectrum: Distribution of radiant energies.http://fas.org/irp/imint/docs/rst/Intro/Part2 4.html. Accessed: 2016-1-2.

[8] J.S. Marshall and W.McK. Palmer. The distribution of raindrops with size. Journal ofmeteorology, pages 165,166, August 1948.

[9] Emmanuel Villermaux and Benjamin Bossa. Single-drop fragmentation determines sizedistribution of raindrops. Nature Physics, pages 697 – 702, July 2009.

[10] P. Pytlak, P. Musilek, E. Lozowski, and J. Toth. Modelling precipitation cooling of overheadconductors. Electric Power Systems Research, 81(12):2147–2154, 2011.

[11] Blue sky and rayleigh scattering. http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html. Accessed: 2015-12-5.

[12] W. A. de Rooij and C. C. A. H. van der Stap. Expansion of Mie scattering matrices ingeneralized spherical functions. Astronomy and Astrophysics, 131:237–248, February 1984.

[13] E. E. Altshuler. A simple expression for estimating attenuation by fog at millimeter wave-lengths. Antennas and Propagation, IEEE Transactions on, 32(7):757–758, July 1984.

[14] J. Rakovec and T. Vrhovec. Osnove Meteorologije. Drustvo Matematikov Fizikov in As-tronomov Slovenije, 2000.

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