Oddelek za fiziko

Seminar IIb – 4th year, Old University Program

Hydrometeor classification with dual polarization weather radars

Author: Primož Ribarič

Mentor: doc. dr. Gregor Skok

Co-mentor: mag. Anton Zgonc, ARSO

Ljubljana, April 2016

AbstractMeasurements from rain gauges give very important data for the operational meteorologist, but they validate only precipitation in a very small area around measurement locations. For large areas without rain gauges conventional observations from satellite or radar can be used. Single polarization radar systems can only obtain rainrate and cannot distinguish between different types of precipitation. On the other hand, polarimetric quantities from dual-polarized radars can be used for precipitation classification, as they depend on hydrometeors' geometrical and dielectrical properties. In the seminar I would like to present one of widely used methods for classification of hydrometeor types – the fuzzy logic scheme. In the last part of the seminar I will present some results and images made by dual-polarized system (obtained by radar on Pasja ravan in Slovenia.)

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1. INTRODUCTION

Weather radars went through three generations during their development [1]. In 50's of 20th century the first operational classical weather radars were invented, which measured only radar reflectivity. Ground clutter, which is the biggest problem in radar measurements, was filtered out with static (clutter) maps [2]. In the 80's Doppler weather radars were coming in use. There sophisticated Doppler filters are used for elimination of ground clutter echoes. The best systems can totally separate precipitation echoes from ground clutter echoes, except in intense orographic areas. Form approx. 2005 onwards Doppler radars with dual-polarization began to be put into operational use. They enable better quantitative assessment of precipitation and hydrometeor classification within precipitation clouds.

In December 2013 the brand new C-band dual-polarization weather radar was installed on Pasja ravan in Polhograjski Dolomiti Mountains (west of Ljubljana) [3].Operational measurements with the radar on Pasja ravan take place only in a horizontal polarization like at the weather radar on Lisca which was renewed in August 2013.

Testing of measurements in dual polarization mode started at the beginning of this year which among others enables hydrometeor-classification in precipitation clouds [4].Hydrometeor classifications results from weather radar Pasja ravan are not very reliable yet, because so called Zdr calibration – differential reflectivity calibration – is not yet reliable due to minor problems with some microwave components.

2. HOW DOES RADAR WORK

Figure 1 presents a classical pulse radar in a simplified way:

Fig. 1: Schematic illustration of a classical pulse radar [5]

Each radar consists from at least four main components [6]:

• The transmitter, which creates the energy pulses. There are four main types of transmitters - klystron, magnetron (as in case of Pasja ravan and Lisca), TWT (traveling wave tube) and SST (solid state transmitter).

• The transmit/receive switch (also called duplexer) that switches the antenna system between the transmission and reception mode. This feature means that only one antenna is needed. It was invented nearly at the same time as magnetron, in 1940.

• The antenna that send pulses out into atmosphere and during the reception mode receives reflected echoes. Pulse waveform consists of repetitive short-duration pulses usually of duration of few microseconds.

• The final chain is a receiver, which detects, amplifies and transforms the received signals into video format. The received and heavily processed signals are then either displayed on a display system or storedto computer files.

The antenna (see fig. 2b) can be a parabolic reflector, planar arrays or electronically steered phased arrays. For

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operational weather radars parabolic dish is applied because array systems are extremely expensive and thus limited to a military or aviation systems [7]. The feedhorn is the extended end of the flared waveguide and provides increased directivity if the aperture of the flare is large and the horn is long (the angle of cone is low). Inmost radars, the feedhorn is covered with a window of polystyrene fiberglass to prevent moisture and dirt from entering the open end of the waveguide [8].

The above system is only capable of measuring reflectivities, for velocity observation an additional Doppler module is needed using considering Doppler effect.

3. CLASSICAL SINGLE POLARIZATION RADAR

Single polarization weather radar transmits rectangular EM (electromagnetic) pulses within narrow beam of typically 1° width into the atmosphere and measures reflected echoes from weather targets, i. e. hydrometeors – the precipitation particles [2]. Horizontal polarization is used because the reflectivities from larger oblate drops are higher in horizontal polarization. (see fig. 3)

Fig. 2: A typical appearance of the key components of the radar. a) The transmitter and receiver units in the radar cabinet inside the building of the radar center, b) the transmitting/receiving antenna system on to of the radar tower [9].

The common-known radar equation relates the parameters of transmitted pulses and measured echoes from ensemble of scatterers distributed within the radar pulse volume at a distance r [1]. It is proportional to volume density of back-scattering cross-sections, called reflectivity (ηb):

ηb≡1V ∑σb →

V → 0∫

0

∞

σb(D)N (D)dD (1)

Here N(D) denotes the number density of hydrometeors (n) per unit diameter:

n≡∂N 1

∂VN (D)≡

∂n∂ D

(2)

The radar equation for any type of scatterers is as follows

Pr (r )=( c τ2 )[ P tG 0

2 λ2

(4 π)3 ] [ πθ1ϕ1

8 ln2 ] ηb

r2(3)

In the radar equation the backscattering cross-section plays the essential role and it depends not only on the distribution of the shape and size but also the type of hydrometeors present within the pulse volume.If the rain drop is spherical with diameters much smaller then wavelength of incident wave, then back-scatter cross section can be approximated using Rayleigh scattering theory and is written as sixth-order dependence of drop diameter D:

σb=(π5

λ4 )|Kw|2 D6 [m²] (4)

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(a) (b)

The scattering cross section of a target is a fictional area such if this area scattered the incident power isotropically by one angle (Rayleigh theory), the power received by the radar would be the same as that from target (neglecting attenuation on beam path between target and radar – r).

The factor Kw (for liquid water) is known as the Clausius-Mossotti factor and is related to dielectric constant of liquid water at the radar frequency:

Kw=m2 – 1m2+2

, (5)

where m is the complex index of refractionm=n+iκ (6)

Fig. 3 a) Wind tunnel measurement of drop shapes, showing the oblateness dependence on drop sizeb) Axis ratio vs. equivalent spherical diameter from a fit to the numerical model of Beard and Chuang for equilibrium drop shapes. De is equivolume spherical diameter given by π/6 De³ = 4π/3 a²b, where a and b are semi-minor and semi-major axes, respectively. Included are the results from an artificial rain experiment, using 2-D video disdrometer [10].

The factor Kw is therefore a complex quantity[2]. In the region of weather frequency bands (X,C,S), its squared absolute value for water is nearly constant having a value of 0.91-0.93 while ice spheres have values of 0.18.As back-scattering cross-sections are proportional to the 6th power of hydrometeor diameters, the reflectivity factor Z is used in radar meteorology instead of reflectivity:

Z≡1V ∑V

D6

→V →0

∫0

∞

D6N (D)dD (7)

It is the volume density of 6th power of hydrometeor diameters in the resolution volume (related to back-scatter cross section of hydrometeors). It is usually shortened to just 'reflectivity'. As hydrometeors are often non-spherical, the volume-equivalent diameter De is assumed in equations if not stated otherwise. Reflectivity is measured in dBZ unit which is defined as

dB Z≡10 logZ

1mm6/m3

(8)

If we apply the back-scattering cross section equation for water droplets and use reflectivity Z, the radar equation for droplets within a radar volume at distance r is then:

Pr (r )=( c τ2 )[ Pt G0

2

λ2(4π)3 ][

πθ1ϕ1

8 ln2 ] π5|Kw|

2

r2Ze(r ) (9)

All three parentheses in the equation represent radar physical-technical properties while the rest of the equation depends on the propertied of observed scatterers. c is the speed of light, Pt the transmit power, G0 is the radar antenna gain, τ represents pulse duration (pulse length), λ is the radar transmitted wavelength, θ1 and φ1 are antenna 3-dB (half-width) beam-widths in the azimuth and elevation planes. The above equations is valid for water droplets. Since we don't know anything about the aggregate state of scatterers, water droplets are usually assumed. The reflectivity is therefore termed as the equivalent reflectivity Ze.

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(a) (b)

4. DUAL-POLARIZATION MEASUREMENTS OF PRECIPITATION CLOUDS

Modern polarimetric radars receive H and V components of the back-scattered signals via two receivers simultaneously while the transmission in both polarizations also runs simultaneously [12]. This is called the STAR mode (Simultaneous Transmission And Reception). The transmitted power is split equally to two halfs (i. e. reduced by 3 dB) into two channels by a power splitter and then applied to two ports of dual polarization feed horn. The sensitivity in any of receiving channels is also reduced by one half (3 dB). The STAR mode is fully compatible with Doppler processing, clutter filtering and range-velocity methods used in single-polarized radars.

In dual polarization radars, a slightly different approach is used to derive radar equation. First, we rewrite the definition for the scattering cross-section using the fact that power of EM signal is proportional to E2:

σ≡4 π r2|E r

Ei|2

=4 π r2|S|2 (10)

We introduced here the backscattering coefficient S which is the ratio between the square incident (Ei) and square reflected field (Er) [7]. The coefficient is complex (phasor) because it contains also phase change of the reflected waves.

The main feature of that type of dual-polarization radars is transmitting and receiving signals in both polarizations (H-horizontal and V-vertical). Incident and reflected electric field strength E – written in polarization plane – at the scatterer are related with 2×2 scattering matrix S, also called the Jones' matrix:

S=[Shh Shv

Svh S vv ] → [Ehr

Evr ]=S[Eh

i

Evi ] (11) , (12)

The first subscript means H for horizontal or V for vertical polarization of the received signal while second subscript means H or V polarized signal on a incident electrical field at the scatterer (Ei). Elements Sij relate the two components of the incident electric field to back scattered electric field. Their units are in meters.

Comparing with the preceding equation for backscattering coefficient (eq. 10), we realize that the backscattering matrix can be expressed with backscattering cross sections:

S=1

√4π [√σhh √σhv

√σvh √σvv ] (13)

Each element describes the scattering processes where H and V polarized waves interact with the scatterer. For simplicity we omitted the phase terms ( e iϕ ) in the matrix elements [4].

Non-diagonal coefficients present depolarization rate of incident EM wave of one polarization into co-polar (orthogonal) reflected wave component of EM field on particles. For understanding the meaning of each matrix component let's take a look at examples.

As we have seen in single polarization case for spherical rain drops using Rayleigh scattering theory, back-scattercross-section is written as a sixth-order dependence of drop diameter D but can be defined also as [1]:

σb(D)=4 π|Ssphere|2

⇒ Ssphere(D)=π

2|K w|D3

2λ2 [m ] (14)

Now, we turn back to polarimetric observations. If all drops in observation volume are spherically shaped, then scattering matrix is diagonal, polarization of reflected E will not change and both diagonal elements are equal [1]:

Shh=Svv=Ssphere(D) Shv=Svh=0 (15)

In the case of oblate spheroidal droplets, which are a satisfactory approximation of actual droplets with

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erythrocite-like oblate shapes, with zero canting angle (vertical tilt angle of the axis of symmetry), the matrix is still diagonal, only both coefficients differ. It turns out that this is valid for the real droplets as well. That means that scatterers with vertical axis of symmetry do not depolarize incident EM waves, i. e. do not reflect EM waves with some co-polar component:

Shh≠Svv , Shv=Svh=0 ⇒ S=[Shh 00 Svv ] (16)

But if the canting angle is non-zero, we have non-spherical droplets with tilted vertical axis of symmetry, then thenon-diagonal elements are also non zero and equal. This is the so-called principle of reciprocity [1].

Shh≠Svv Shv=Svh≠0 (17)

Non-zero canting angle occurs when wind shear is present inside precipitation clouds. Hydrometeors do not fall exactly vertically in such cases. Typical canting angles are of order of magnitude 10°, with dispersion between 2°~14° [11].

The theory behind these coefficients for oblate spheroidal droplets far too complex for this lecture. It is explained in detail in [18]. I will try to make a simplified summary of it. In Rayleigh approximation, droplets are small enough to see the incident (oscillating) electrical field Ei as constant in space throughout their volumes. The induced dipoles within droplets produce additional internal field Eint which oscillates synchronously with the Ei . If droplets are perfectly spherical in shape, the Eint equals P/3ε0 ,where P is polarization, i. e. volume density of electric dipoles. Polarization is proportional to the volume density of scatterers n and polarizability α – ability of matter to produce induced electrical dipoles. Polarization is also proportional to the ambient electrical field whichis a sum of the incident field and the internal field:

P=nα(Ei+P /3ϵ0) (18)We know that electrical displacement field D is defined as electrical field E displaced due to polarization P:

D=ϵϵ0 E=ϵ0 E+P (19)From this equation the polarization can be expressed with familiar formula

P=(ϵ−1)ϵ0 E (20)Comparing with (18), we realize that polarizability α and permittivity ε are related via Lorenz-Lorentz formula

nα=3ϵ0(ϵ−1)/ (ϵ+2) (21)If we proceed further, we arrive to the backscattering cross-section for spherical droplets (4).

But we pay attention to the nonspherical droplets now. If a droplet is nonspherical but still symmetrical to the incident Ei (zero canting angle), the contribution of induced dipoles to the Eint differs from P/3ε0 for a geometricalfactor which is greater than 1 for both horizontal axes if droplet is oblate – more precisely rotationally symmetrical along the Ei axis. Polarizability is not a scalar (isotropic) quantity anymore, but a diagonal tensor α:

α=[α 0 00 α 00 0 αz

] (22)

If a droplet is also asymmetrical to the incident Ei (nonzero canting angle), the Eint is not parallel to Ei any more and the reflected Er is depolarized, i. e. contains the co-polar component and also the 3rd component which cannotbe measured by radar. The polarizability tensor contains also nondiagonal elements. For oblate spheroids with a given canting angle, analytical expressions are derived in [1]. An extensive procedure is given in [1] to relate the elements of α with the backscattering matrix (11) or (13).

5. REQUIREMENTS FOR RELIABLE DUAL-POLARIZATION MEASUREMENTS' PERFORMANCE

Dual polarization radars require additional technical criteria for reliable measurements [12]. Antenna main H- andV-beams should be aligned as much as possible. The squint angle should be less than 0.1°. The same holds for thebeamwitdh difference. This requires careful positioning of the feed horn in the antenna focus. Antenna has a

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radiation pattern, i. e. angular distribution of power density. The energy is not confined completely into a narrow conical beam – the mainlobe, some of it falls outside the mainlobe into sidelobes. In modern systems the sum total of power in sidelobes can reach some 10 promiles of that transmitted within the mainlobe. Because the scatterers are distributed also within the sidelobes of radiation pattern, receiver can get sometimes significant energy from those directions, especially from mountains and intense storm cells with big hail stones. The antennaand receiver should be well cross-pol isolated inside mainlobe area and also over sidelobes, better than -20 dB. The Pasja ravan radar has a very good cross-pol isolation of -35 dB.

Co-polar correlation function (RhoHV) defines the level how the magnitude and phase of the measurements between the horizontal and vertical channels match together. It is a value normalized between 0 (total de-correlation) and 1 (perfect correlation). Imperfect beam or worse isolation between channels can decrease the value. These systematic errors can strongy affect the quality of H and V magnitude and phase measurements and also other polarimetric observables. I will describe that observables in the next section. The higher the maximum RhoHV measurable by radar is the better the quality of the antenna. In case of light rain – consisting of small nearly spherical droplets, research groups recommend the highest RhoHV values that are above level 0.98, but for valuable observation of rainrate RhoHV levels above 0.99 should be reached. The test measurements of the Pasja ravan radar often reach maximum RhoHV value up to 0.998, which proves the qualityof the radar.

Benefits of Dual Polarimetric Weather radars [11]:• provides significant improvement over conventional radar when measuring actual rainfall and quality

comparisons of data, but we should be aware of systematical errors!• enables hydrometeor classification• improves detection of storm cells and especially hail storm cells• enables much better calculation of the melting-layer height and thus the 0°C – isotherm surface

6. DUAL-POLARIZATION OBSERVABLES 6.1. REFLECTIVITIES

As we said earlier, the back-scattering cross-sections are proportional to the 6th power of hydrometeor diameters [1]. The reflectivity is termed as the equivalent reflectivity Ze because we assumed that scatterers are water drops and we didn't know anything about the aggregate state of hydrometeors. Combining radar equation (eq. 9) and back-scattering cross-section for water drops (eq. 4) the measured reflectivity can be defined as:

Ze=λ4

π5|K w|

2∫σb(D)N (D)dD [mm6/m3

] , (23)

and that can be applied for any type of scatterers with single or dual-polarization type of radar.

REFLECTIVITIES IN DUAL-POLAR VERSION

Dual-polarization radars receive both reflected echo components simultaneously[1]. Backscattering cross-sections which contribute to both reflectivities are σb

hh ,σbvv and can be expressed in terms of scattering

coefficients in matrix S (for one scatter and crowd of them at the distance r from the radar), that we learned also earlier, by:

Z h , v=λ4

π5|K w|

2 ∫0

D max

σbhh , vv

(D)∂ n∂D

dD [mm6/m3

] σbhh , vv

(D)=4 π|Shh ,vv(D)|2 (24)

Here we take into account that droplet equivalent diameters does not exceed Dmax≈5mm . If they do, they quickly spontaneously break-up due to high turbulent drag forces aroud them.

6.2. DIFFERENTIAL REFLECTIVITY Zdr

If radar operates in dual-pol mode, then ratio between signal magnitudes (H vs. V channel) is called differential reflectivity Zdr [1].

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Zdr=10 log10( Zh

Z v)=10 log10( ∫0

Dmax

|Shh(D)|2N (D)dD

∫0

Dmax

|Svv(D)|2N (D)dD ) , [dB] (25)

These observations can detect difference between H and V reflectivity factors on ensemble of scatterers distributed within the radar pulse volume at a distance r, that can estimate drop shapes. (see fig. 4)Zdr values for rain is typically over 1 up to 5 dB, smaller drops are more spherical-like, larger are more flattened, Zdr of dry snow flakes is around 0 dB, wet and melted snow have higher values, hail between -1 dB and 1 dB (larger values for melting hail). (See fig. 5) Zdr measurements are heavily susceptible to relative miscalibration between the H and V channel. For reliable hydrometeor classification, Zdr should be relatively calibrated up to 0.1dB. This is not trivially achieved and requires more precise maintenance and calibration of the radar systems comparing with the traditional single-polarization systems [11]. This is not yet achieved with the Pasja ravan radar, because there are minor issues with microwave components to be solved until this summer season[4].

Fig. 4. Differential reflectivity with different orientations of droplets [11].

Fig. 5. Summary of typical Zdr values of raindrops of various sizes and hail. The black arrow on the hail particle represent the tumbling motions as it falls in a thunderstorms [13]

6.3. DIFFERENTIAL PHASE Φdp AND SPECIFIC DIFF. PHASE Kdp

Each EM wave (thus H and V polarized) has its own amplitude and phase [1]. This waves are beingattenuated while propagating through air mass that contains scatterers. The phase lag of waves is also the result of the same effect. The oblateness of drops or deviations in shape of hydrometeors from spherical form in horizontal direction causes higher attenuation and phase lag of horizontally polarized wave relative to vertical polarized wave.

The differential propagation phase Φdp is defined as a difference or delay between horizontal and vertical phase shift (see fig. 6) and is sensitive to whole water mass (density) on a path between radar and target.Φdp [°] is differential phase shift upon back scatter deduced from the differential phase shift along the propagationpath. δ is the phase difference in H and V polarizations caused by backscattering or 'back scatter differential

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(a) (b)

phase'. It becomes significant in the Mie region when radar wavelength becomescomparable to the size of scatterers. It occurs more frequently in X-band radars, and in case of hail stonesalso in C-band radars. It can be caused also by badly outfiltered ground clutter or by birds. δ occurs as local perturbation in monotonically increasing Φdp . It should be filtered out from the Φdp range profile before further applications.

Fig. 6. – shows conceptual phasor diagram which explains Φdp. ΦHH and ΦVV are cumulative differential phase shift for total round trip between radar and resolution volume: Φdp=(ΦHH – ΦVV) – δ [14];

The specific differential phase Kdp is the radial gradient of difference between propagation constants for H and V polarized EM waves [2]. In homogeneous medium (in resolution volume with the same scattering properties) is obtained from Φdp shifts at two range locations:

Kdp=12

ddr

Φdp=Φdp(r2)−Φdp(r1)

2(r2−r1)(26)

In homogeneous rainfall, the differential propagation phase is known to be monotonically increasing function of arange by a fixed amount per unit distance along range and Kdp [°/km] is constant. So in this case Kdp represents liquid water mass at specific locations between two ranges (r1 and r2).If forward-scattering is bigger for H than V waves, then drops are flattened in vertical direction and Kdp is above zero [11]. In inverse case Kdp value is below zero and scatterers are flattened in horizontal direction. Typical Kdp values are near 0 °/km for dry hail, between -1 and +0.5 °/km for snow, and 0 to 5 °/km for rain (larger Kdp indicates larger drops and/or increased drop concentration). (see fig. 7)

6.4. CROSS-POLAR CORRELATION COEFFICIENT ρhv

The correlation coefficient ρhv between H and V input at zero lag (simultaneously polarimetric radar systems) depends on the shape, oscillation and canting angle distribution of hydrometeors [1]. It is defined as the correlation between H and V polarized return signals (samples of signals at certain pulse volume atthe distance r from the radar):

ρhv≡⟨ Svv Shh

∗⟩

√[ ⟨|Shh|2⟩⟨|Svv|

2⟩ ]

(27)

The complex conjugate value denotes asterisk *. Beside the oscillations of large drops and the phase difference inbackscattering signals, ρhv is influenced by particle mixture and irregular particle shapes (e.g. hail, graupel, rain/hail mixture etc.). That amplifies the importance of ρhv for identifying different precipitation types.

This correlation value is a measure of how homogeneous are the targets in the sample volume. It is a way of discriminating meteorological from non-meteorological echoes, such as bio-scatterers (insects, birds) and radio-frequency interferences (WiFi at 5 GHz).Little differences from pulse to pulse between H and V pulses are caused from dry snow and rain [11]. The correlation values can reach high over 0.97, if hydrometeors are not uniform (mixture of different precipitation particles) lower values are expected (0.8 to 0.97). If differences are very big then it is a high probability, that echoes come from non-meteorological objects and values can be below 0.8.

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Fig. 7. Specific differential phase with different orientations and concentrations of droplets [11].

6.5. LINEAR DEPOLARIZATION RATIO LDR

If the scatterers have their axis of symmetry tilted from the vertical, the incident energy will be slightly depolarized during the scattering process [1]. Part of the energy from one polarization could be resolved in the direction of the other polarization. The rate of LDR depends on numerous factors – mostly by hydrometeor size, the axis ratio, the degree of canting and radar beam elevation. That will also cause in non-zero values for the magnitudes of Shv and Svh in scattering matrix S.

For an ensemble of scatterers, the cross-polar reflectivity is calculated using analogous relation to co-polar reflectivity Zhv and cross-polar back-scatter cross section σhv .

Z hv=λ4

π5|K w|

2 ∫0

D max

σbhv(D)N (D)dD [mm6

/m3] σb

hv(D)=4π|Shv(D)|

2(Zvh=Zhv) (28)

We have assumed also the principle of reciprocity for co-polar reflectivities.

The LDRs above are defined by the ratios between cross-polar reflectivity and co-polar reflectivityand given by:

LDRhv=10 log10( Zhv

Zh) : LDRvh=10 log10(Zvh

Z v) (29)

In most rain cases it is hard to detect the cross-polar component because of limitations due to antenna cross-polar performance. Melting snowflakes on the other hand often cause sufficient LDR to be detected easier. Radar on Pasja ravan does not measure LDR operationally and thus LDR is not used in the fuzzy logic algorithms for hydrometeor classification [4].

7. FUZZY LOGIC SCHEME

The values of dual-polarization observables that delineate the different hydrometeor types are not sharply definedin reality or can even simply overlap [1]. Therefore, a simple classical logical approach, that is using pre-defined sharp boundaries is inaccurate as it can be seen in the next figure (fig. 8). That approach can only obtain few dominant expected hydrometeor types (rain, hail, wet snow) and the classification could only be made then with logic trees with a lot of if classes. That is the main reason why newest classification methods use fuzzy logic to obtain hydrometeor types (classes) .

It seems that very sharp boundaries are also not natural in 'real' life. I.e.: Let's say that temperatures below 15°C presents cold conditions, temperatures above 15°C warm conditions. That would mean that it is cold if T=14.9°C, but warm if T=15.1°C. But that is nonsense! Only 0.2°C higher temperature doesn't mean 'much' warmer condition compared to 0.2°C lower temperature. It is practical to define transient boundary in field of temperature, where cold conditions are crossing smoothly to warm weather conditions (i.e from 10 - 20°C). In this wider boundary temperature area the condition-probability value (is warm or not) slowly rises from 0 (not warm) to 1 ('totally' warm).

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Fig.8: Hydrometeor classification decision boundaries in the (a) Zdr-Zh, (b) Kdp-Zh, (c) LDR-Zh and (d) ρco – Zh planes for S-band, from Straka and Zrnic (1993) [16]

In fuzzy logic all possible values are matter of a degree [15]. The limit values that determines set of elements intoclasses are not sharply (in form of rectangle) but »fuzzy« bounded. In classical concept, the only question is, if anelement is contained or is not in a set, a fuzzy set is defined as set where its elements have different degrees of membership other than full or zero membership.

Four main steps in fuzzy-logic schemes [1]: (i) selection of the input vector of observables may consist of all possible observables that can contribute to determine the final result – selection of hydrometeor (HM) class (ii) definition of the membership functions (MF) is the most important step in fuzzification. Those functions quantify the conditional probability if echo(s) with certain values of input observables belongs toa particular class. The decision for a shape of membership function depends on subjective aspects and semi-empirical knowledge of physics of a certain problem. In our case trapezoidal(beta) functions are used as MF. Figure (9) shows MFs for 'warm' and 'cold' season in case of prototype WSR-88D radar in Oklahoma. Input observables are Zh , Zdr and ρhv.(iii) The next step is to weigh the membership functions according to the step of confidence of the measurement and must be also normalized. The same routine is used for all input observables. (iv) the aggregation method & defuzzification The range of the aggregation (Q) function must also lie between 0 and 1. In our case Q value is calculated by operation of a summation and multiplication of weighted MFs for each class (rain, hail, wet snow, etc.) separatelyand so the Q function consists of all subset of MFs. The last step is defuzzification (sharpening), because only one final result is allowed as the output (the crisp value). For a set of input observation values the fuzzy logic scheme produces only one hydrometeor class as the output result. That can be made by finding class with max(Q) value for certain(all) echo(es) at the certain(all) location(s) r from radar.

Fig. 9. a) MF for three classes in **'warm season': big drops (BD), moderate rain (MR) and rain/hail mixture (HA) shown as a function of Zh , Zdr and ρhv; b) MF for four classes in ***'cold season': dry snow(DS), wet snow (WS), stratiform rain (SR) and convective rain (CR) shown as a function of Zh , Zdr and ρhv from Schuur et al (2003) [17]

**Warm season scheme is used mainly for the climate conditions where mainly thunderstorms or tropical cyclones are expected.***Cold season scheme is appropriate for mid-latitude frontal systems, where large areas of stratiform precipitation with low level bright-bands and moderate convecting systems develop

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(a) (b)

8. CONCLUSIONS

While single-polarization radar transmit and measure reflectivity in one polarization only, the dual-polarization radar sends pulses and receives echoes from scatterers distributed on the path of the radar-beam through two portsof dual polarization feed horn which is located in the radar antenna. The power of incoming reflected signals are connected with transmited power by the weather radar equation. The ratio between the square incident and squarereflected electric field is called back-scattering coefficient S and is related to backscattering cross-section. In caseof the dual-polarization measurement the S is represented as scattering matrix (2×2) which describes interaction of scatterer(s) with the incident and reflected energy. The form of the matrix depends on aggregate state of the particle, size, flatteness, and its canting angle (vertical axes of symmetry). There from a huge set of polarimetric observables is calculated. Since the dual-poalrization 'era' is still in the beginning phase we cannot experience whole benefits from this approach. Big importance for reliable measurements belongs to well-callibrated radar system (hardware, antenna, feed horn) and quality of the isolation at the input/output H and V channels. We can reveal 'lost' precipitation intensity by using the specific differential phase Kdp in case where the signal is attenuated on the way to target region by densier rainfall. Smart selection of the membership functions and arithmetic operation between them in fuzzy logic can vastly improve the hydrometeor classification.

9. LITERATURE

[1] V. N. Bringi et. al., Dual-Polarization Weather Radar Handbook, Gematronik. 2005[2] R. J. Doviak and D. S. Zrnić, Doppler radar and weather observations, 2nd edition, Academic Press Inc.,

San Diego, 1993[3] ARSO websites: http://www.arso.gov.si, searched on 19. 4.2016[4] personal contact with the radar-meteorologist mag. Anton Zgonc (ARSO)[5] Australian Bureau of Meteorology, Radar components,

http://www.bom.gov.au/australia/radar/images/radcomp.gif, searched on 15.4.2016[6] How Radar Works, Australian Bureau of Meteorology.

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