Modelling propagation effects in the use of S-band polarisation-diversity radars R. McGuinness D.H.O. Bebbington A.R. Holt Indexing terms: Radar and radionavigation, Radio-wave propagation (microwave) Abstract: The effects of propagation through pre- cipitation on polarisation diversity radars at S-band are modelled. It is shown that in propaga- tion through rain, although linearly polarised waves need not be significantly affected, the effect on circularly polarised returns is important. It is shown that a gate-by-gate correction process is not a satisfactory method of removing the propa- gation effects. However, it is shown that a derived linear parameter ZDR can be obtained from the circular parameters which is less affected by pro- pagation. The modelling is extended to targets which comprise rain and hail. 1 Introduction Since its early days, radar has been used to measure reflectivity patterns in storms. Reflectivity alone, however, is not sufficient to provide information on drop- size distribution, precipitation phase etc. Various attempts have therefore been made to obtain more detailed information on storms. These have included dual-wavelength systems, Doppler systems and polarisa- tion diversity. Precipitation particles, whether rain or hail, are generally nonspherical [1-2]. Consequently, not only will a radar signal be depolarised when reflected by precipitation, but also different incident polarisations will give rise to different reflected characteristics. The recent advances in the use of polarisation diver- sity have developed from studies initiated in the late 1960s at the National Research Council, Ottawa, Canada. Two polarisation-diversity systems were designed and built, and both used circular polarisation. The first, at 16.5 GHz, was used for research at Ottawa [3-5] and the second, at 2.88 GHz, has been in oper- ational use since then with the Alberta Research Council [6, 7]. The mode of operation was to send one circularly polarised wave, and to measure four return signal param- eters — the powers in the main and orthogonal channel, and the complex correlation. The principles have been discussed by McCormick and Hendry [8], who define the circular depolarisation ratio (CDR) (the ratio of the power in the orthogonal channel to that in the main Paper 5487H (Ell), first received as a single paper 8th December 1986, and in revised form as two papers 14th April 1987 Dr. McGuinness and Dr. Holt are with the Department of Mathe- matics, and Dr. Bebbington is with the Department of Electronic Systems Engineering, University of Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, United Kingdom channel), and parameters ORTT and ALD. More recent- ly, the Ottawa group have commenced studies at 9.6 GHz [9]. The S-band system in Alberta has had the par- ticular aim of distinguishing rain from hail. However, quite early in its use, it was observed that returns from the further sections of storms could be affected by passage through the precipitation lying closer to the radar [10]. Although the mode of operation in Alberta is always to send the same polarisation (left-hand), switched polarisation transmission has been developed in Ottawa [11]. The use of linear polarisation was proposed by Seliga and Bringi [12]. The method they suggested was to send alternate vertical and horizontal polarisations, and in each case to measure the copolar return power. The ratio of the reflectivities for horizontal and vertical polarisa- tions they called the differential reflectivity (ZDR). The method has been used both by Seliga and Bringi [13] and by the group at the Rutherford Appleton Laboratory [14, 15]. If a 2-parameter model is assumed for the drop-size distribution, information on drop-size distribution can be derived throughout the storm. As, unlike the CDR system, the measurements are independent of the phase of the return signals, and as, in the S-band, differential attenuation is negligible, propagation has normally no significant affects on ZDR, provided that the linear pol- arisation is aligned with the principal axes of the rain- drops. If there is appreciable tilt, the propagation effects can not be discounted [16]. McCormick [17] demonstrated that it is possible to obtain values of ZDR from data obtained using circular polarisation. This was confirmed by model calculations by Holt [18] who also performed a sensitivity analysis of various model assumptions across a wide frequency range. In this paper, we discuss the interpretation of data from rain, and rain and hail mixtures. By means of calcu- lations using a rain and hail storm model we demon- strate in Sections 2 and 3 the effects of propagation at S-band, and show that the interpretation of CDR data can be very difficult because of corruption by propaga- tion effects. We also demonstrate how the method of McCormick [17], used in obtaining ZDR from circular polarisation, improves the interpretation of the CDR data significantly. 2 Model calculations for rain A signal transmitted by a radar will undergo depolar- isation and power loss by absorption and scatter along a IEE PROCEEDINGS, Vol. 134, Pt. H, No. 5, OCTOBER 1987 423
Transcript
Modelling propagation effects in the use of S-bandpolarisation-diversity radars
R. McGuinnessD.H.O. BebbingtonA.R. Holt
Indexing terms: Radar and radionavigation, Radio-wave propagation (microwave)
Abstract: The effects of propagation through pre-cipitation on polarisation diversity radars atS-band are modelled. It is shown that in propaga-tion through rain, although linearly polarisedwaves need not be significantly affected, the effecton circularly polarised returns is important. It isshown that a gate-by-gate correction process isnot a satisfactory method of removing the propa-gation effects. However, it is shown that a derivedlinear parameter ZDR can be obtained from thecircular parameters which is less affected by pro-pagation. The modelling is extended to targetswhich comprise rain and hail.
1 Introduction
Since its early days, radar has been used to measurereflectivity patterns in storms. Reflectivity alone,however, is not sufficient to provide information on drop-size distribution, precipitation phase etc. Variousattempts have therefore been made to obtain moredetailed information on storms. These have includeddual-wavelength systems, Doppler systems and polarisa-tion diversity. Precipitation particles, whether rain orhail, are generally nonspherical [1-2]. Consequently, notonly will a radar signal be depolarised when reflected byprecipitation, but also different incident polarisations willgive rise to different reflected characteristics.
The recent advances in the use of polarisation diver-sity have developed from studies initiated in the late1960s at the National Research Council, Ottawa,Canada. Two polarisation-diversity systems weredesigned and built, and both used circular polarisation.The first, at 16.5 GHz, was used for research at Ottawa[3-5] and the second, at 2.88 GHz, has been in oper-ational use since then with the Alberta Research Council[6, 7]. The mode of operation was to send one circularlypolarised wave, and to measure four return signal param-eters — the powers in the main and orthogonal channel,and the complex correlation. The principles have beendiscussed by McCormick and Hendry [8], who define thecircular depolarisation ratio (CDR) (the ratio of thepower in the orthogonal channel to that in the main
Paper 5487H (Ell), first received as a single paper 8th December 1986,and in revised form as two papers 14th April 1987Dr. McGuinness and Dr. Holt are with the Department of Mathe-matics, and Dr. Bebbington is with the Department of ElectronicSystems Engineering, University of Essex, Wivenhoe Park, Colchester,Essex CO4 3SQ, United Kingdom
channel), and parameters ORTT and ALD. More recent-ly, the Ottawa group have commenced studies at 9.6GHz [9]. The S-band system in Alberta has had the par-ticular aim of distinguishing rain from hail. However,quite early in its use, it was observed that returns fromthe further sections of storms could be affected bypassage through the precipitation lying closer to theradar [10]. Although the mode of operation in Alberta isalways to send the same polarisation (left-hand), switchedpolarisation transmission has been developed in Ottawa[11].
The use of linear polarisation was proposed by Seligaand Bringi [12]. The method they suggested was to sendalternate vertical and horizontal polarisations, and ineach case to measure the copolar return power. The ratioof the reflectivities for horizontal and vertical polarisa-tions they called the differential reflectivity (ZDR). Themethod has been used both by Seliga and Bringi [13]and by the group at the Rutherford Appleton Laboratory[14, 15].
If a 2-parameter model is assumed for the drop-sizedistribution, information on drop-size distribution can bederived throughout the storm. As, unlike the CDRsystem, the measurements are independent of the phaseof the return signals, and as, in the S-band, differentialattenuation is negligible, propagation has normally nosignificant affects on ZDR, provided that the linear pol-arisation is aligned with the principal axes of the rain-drops. If there is appreciable tilt, the propagation effectscan not be discounted [16].
McCormick [17] demonstrated that it is possible toobtain values of ZDR from data obtained using circularpolarisation. This was confirmed by model calculationsby Holt [18] who also performed a sensitivity analysis ofvarious model assumptions across a wide frequencyrange.
In this paper, we discuss the interpretation of datafrom rain, and rain and hail mixtures. By means of calcu-lations using a rain and hail storm model we demon-strate in Sections 2 and 3 the effects of propagation atS-band, and show that the interpretation of CDR datacan be very difficult because of corruption by propaga-tion effects. We also demonstrate how the method ofMcCormick [17], used in obtaining ZDR from circularpolarisation, improves the interpretation of the CDRdata significantly.
2 Model calculations for rain
A signal transmitted by a radar will undergo depolar-isation and power loss by absorption and scatter along a
IEE PROCEEDINGS, Vol. 134, Pt. H, No. 5, OCTOBER 1987 423
rain filled path up to the point of interest. After back-scatter the signal undergoes similar propagation effectsalong the return path to the radar. To model the effectsof propagation through rain on polarisation-diversityradar a model, consisting of a uniformly rain filled path10 km long is examined for several rainrates up to80 mm/hr. Although such cells do not exist in real rainsituations the model reveals important information onthe limitations of the radar system and the conditionswhere reliable data can be collected.
2.1 Drop canting angleRaindrop canting angles are believed to be small andnarrowly distributed about a mean of zero. McCormickand Hendry [3] found that from 1212 measurementsproducing rain on the ground the mean canting anglewas narrowly distributed about an average of 0.48° witha standard deviation of 1.77°. Beard and Jameson [19]developed a theoretical model which predicted cantingangles narrowly distributed about a mean close to zero.Holt [18] compared the effects of a canting angle dis-tribution and random azimuthal distribution with that ofaligning all the drops at a fixed canting angle and azi-muthal orientation. The difference between the threecases of (i) a fairly broad distribution (up to 10°), (ii) anarrow canting angle distribution with a mode of 2°, and(iii) all aligned drops with 2° canting angle were not sig-nificant. In this analysis therefore the all-aligned dropmodel is used with a mean canting angle of 2°.
2.2 TheoryWe consider the example of a radar sensing a region ofrain as shown in Fig. 1.
Fig. 1 Range-gated radar sensing a raincell
The scattering of a plane electromagnetic wave, ofwavevector k0 (^(cm"1) = 2n/X) and polarisation vectore0, by a dielectric particle of complex refractive index isdetermined by the scattering amplitude vector f(9, (f>)where the field at a large distance r from the scatterer is
where a time dependence of exp (— icot) is assumed. Forscattering into the final polarisation state ef, the scat-tering amplitude is
fof{d,4>)=f{d,(f))ef (2)
For the example considered both forward and backwardscattering mechanisms are involved. The forward copolaramplitudes are denoted as/K K, /H H where the subscriptsV and H denote vertical and horizontal polarisation,respectively. The backward scattering of the incidentwave
E — Eve EHeH (3)
is described using the formalism of McCormick andHendry [8] as
rEyi\_EHJSCAT
12TEVI exp(ifcor)(4)
If, in Fig. 1, gate n is being sensed, then prior to reflectionthe EM wave will have passed through (n — 1) gates con-taining rain. For the purpose of these model studies it isassumed that each of these gates can be characterised byan exponential dropsize distribution (DSD) of the form
= iVoexp(-3.67D/Do) (5)
where N(D)dD is the number of drops whose equivolumediameter lies in the interval D, D + dD. Do is the medianvolume drop diameter. (The broad features should beindependent of the DSD model used). Forward scatterhas traditionally been described in terms of the equi-volume drop radius a such that
n(a) da = N(D) dD (6)
Forward propagation through a given range gate oflength L (cm) is described by means of the propagationconstants Kv H [20]:
[Jo
fvv, HH(a)n( (7)
If the medium consists of aligned drops such that theelectromagnetic wave is incident in a principal plane, thefield in the forward direction is given by the matrixproduct
INC
where
A = diag (dlt d2)
and
d1<2 =
(8)
(9)
KVHL) (10)
In practice raindrops will not all lie in the principalplane and will be canted from the vertical. Assuming thatthe drops are all aligned with a mean apparent dropcanting angle of a, as seen by the incident wave, theforward field is
(11)
where
„ , fcos a —sin a~||_sin a cos a j
Combining the up path propagation with the back-scattered field and the down path propagation, andassuming that gate n is being sensed, the received (R) andtransmitted (T) fields are related by
(12)
where
Mn = D
M22 = D
i S u cos2
2 S n sin2
a
a
+
+Dl
DlS22
S22
sin
cos
424
(13)
M12 = M21 = -{D\S22 - DfSn) sin a cos aj
IEE PROCEEDINGS, Vol. 134, Pt. H, No. 5, OCTOBER 1987
and
£>1.2= E M U (14)
where d{<2 are the elements for gate;, and Sxl and S12
are functions of raindrop diameter. For linearly polarisedradars, the vertical, horizontal and differential reflec-tivities are
ZV,ZH = \C\2 N(D)\Mlu22(D)\2dDJo
ZDR = 10 log10 (ZH/ZV)
and
(15)
(16)
where n0 is the refractive index and I the wavelength incm.
For circularly polarised radars W2 and Wx describe themain and orthogonal powers:
D 2 S U +D22S22\
2N(D)dD
Wl^i\C j\D21S11-D 2
2S22\2N(D)dD
(18)
(19)
The complex correlation between the two channels isdefined as
)1S11 — D2S22)(D1Sli + D2S22)
x exp (2i<x)N(D) dD (20)
Four meteorological parameters are defined as the mainchannel power (ZE) and the circular depolarisation ratio(CDR), the magnitude of particle alignment (ORTT) andthe mean canting angle (ALD),
ZE = 10 log10 (W2)
CDR = 10 log10 (WJW2)
ORTT = \W\l{WlW2)112
ALD = i(arg (W) - n)
(21)
(22)
(23)
(24)
McCormick [17] showed that the combination of Wu W2
and W should give a very good estimate of ZDRassuming that the mean apparent canting angles aresmall. To a good approximation,
ZDRC = 10 log10 ((1 - /i)/(l + ii))
where
fi = 2 Re (W/W2)/(l + 10(CD*/10))
(25)
(26)
Holt [18] showed this approximation to be valid at2.88 GHz and 9.6 GHz in the absence of propagationeffects.
2.3 Effect of rain on radar observablesThe results presented assume the radar is range gatedwith a 1 km range gate and is examining a 10 km longuniform rain path. The uniform rain cell model assumes adropsize distribution given by eqn. 5 and uses drop fallspeeds given by Gunn and Kinzer [21]. The rainrate(mm/hr) is given as
x (9.65 - 10.3 exp (-0.6/))) dD (27)
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Assuming that there are no propagation effects and thatthe drop fall speeds are accurate, R depends only on No
and Do. The No, Do parameters are obtained using either(ZH, ZDR), (ZE, CDR) or (ZE, ZDRC). A particular rain-rate can be the result of many different combinations of(No, Do). Combinations of (ZH, ZDR) or (ZE, CDR) havebeen chosen for rainrates up to 80 mm/hr which aretypical of those found in real data. Table 1 presents theeffects of propagation on radar observables in the5-band. Circular polarisation parameters (ZE, CDR, Re(W/W2), ZDRC) and linear polarisation parameters (ZH,
ZDR) are presented for uniform rainrates of 20, 40 and80 mm/hr. The observables should remain constant alongthe 10 km path at a value equal to their value at 1 km;deviations are the result of propagation effects.
The most significantly affected parameter is the CDR.The effects are fairly small at 20 mm/hr (1 dB over 10 km)but increase at 40 mm/hr (nearly 4 dB over 10 km) and at80 mm/hr they are particularly severe. After 4 km theCDR has increased by 2.5 dB and after 10 km it increasesby nearly 10 dB to -4 .7 dB.
The effect of propagation on the main channel powerZE or ZH is not significant although it is always greaterfor ZE. The effect of propagation on the ZDR is onlydetectable at 80 mm/hr and even after 10 km it hasdecreased by only 0.3 dB from its true value. The effect ofpropagation on the circular polarisation parameter ZDRC
(derived from Re (W/W2)) although it is more significantthan on the ZDR, represents a significant improvementon the CDR. At 80 mm/hr the effect is almost linear at0.1 dB/km. It must be remembered, however, that ZDRC
is dependent on the mean canting angle (taken as 2° inthis analysis). If the mean canting angle increases, ZDRC
can be significantly affected. The effect of drop tem-perature and elevation angle (up to 10°) are not signifi-cant.
2.4 Estimating path rainrateIf the radar is being used to estimate the rainrate alongthe path it is sensing, it will require (No, Do) to calculateR using eqn. 27. If it is a linearly polarised radar (ZH,ZDR) is used to calculate (No, Do), whereas if it is circu-larly polarised (ZE, CDR) or (ZE, ZDRC) is used. If theeffects of propagation are small, the radar estimated rain-rate using the radar observables for the three uniformrain paths in Table 1 should remain constant for all 10range gates. Figs. 2-4 present the percentage error in theradar estimate of the rainrate for the three uniform rainpaths given in Table 1 and for estimates using (ZH,ZDR), (ZE, CDR) and (ZE, ZDRC). The linear radar esti-mates (ZH, ZDR) remain within 10% up to 80 mm/hrrepresenting a very good estimator over all types of rain.The circular radar estimate (ZE, CDR) is significantlyaffected by propagation even in relatively light rain of20 mm/hr. After 2 km within a cell of 80 mm/hr the CDRmeasurement is so corrupted as to be incapable of calcu-lating a rainrate. The second circular radar estimate (ZE,ZDRC) is a significant improvement on the (ZE, CDR)estimate but does not perform as well as the (ZH, ZDR)estimates. Up to 40 mm/hr over 10 km, the (ZE, ZDRC)estimate remains within 15% of its true value and it canstill provide reasonably sensible estimates at 80 mm/hr.
2.5 Correcting for propagation effectsTable 1 has shown that in moderate to heavy rain, theeffects of propagation can be significant. One method ofattempting to obtain the correct (No, Do) for the nthrange gate out from the radar is to use an iterative gate-
425
by-gate correction algorithm in which (No, Do) are calcu-lated for the first gate with rain. They are then used tocalculate the propagation effects on the returns from gate
10r
Si -10o
-20
-302.5 5.0 7.5
range, km10.0
Fig. 2 Percentage error in radar estimate of rainrate along a 10 kmpath with 20 min/hr rainrate
(ZE, CDR)(ZE,ZDRC)(ZH, ZDR)
2 0 r
-20
-60
-802.5 5.0
range, km7.5 10.0
Fig. 3 Percentage error in radar estimate of rainrate along a 10 kmpath with 40 mm/hr rainrateSec Fig. 2 for key
2, and thus (No, Do) are calculated for this gate. This pro-cedure is repeated radially outwards through the storm.
The stability of this method depends upon such factorsas the rain along the path, the true dropsize distribution,
the measurement error and quantisation error of theradar observables.
Radar observables such as ZH, ZDR, ZE, CDR areusually measured with quantisation levels of 0.1, 0.25 or
A0r
30
20
en2 10
-10
-202.5 5.0
range, km7.5 10.0
Fig. 4 Percentage error in radar estimate of rainrate along a 10 kmpath with 80 mm/hr rainrateSee Fig. 2 for key
0.5 dB and hence a measurement uncertainty of at leastthe same order of magnitude. In this analysis a quantisa-tion level of 0.5 dB is imposed on ZH, ZE, CDR, a quan-tisation level of 0.25 dB on ZDR, and finally 0.0125 onRe (W/W2).
Figs. 5-7 present the rainrate percentage error apply-ing the gate-to-gate correction procedure (with the above
1 0 r
-10
-202.5 5.0
range, km7.5 10.0
Fig. 5 Percentage error in radar estimate of rainrate along a 10 kmpath with 20 mm/hr rainrate applying the gate-by-gate correction pro-cedure and quantisation imposedSee Fig. 2 for key
quantisation levels imposed). The linear radar estimates(ZH, ZDR) remain within 20% up to 80 mm/hr showingreasonably good stability. The circular radar estimate(ZE, CDR) is much less stable and at 80 mm/hr it termin-ates after 7 km as the CDR increases to —13 dB forwhich no real Do exists. One of the major reasons whythe corrected (ZE, CDR) estimates are less stable than the
426 1EE PROCEEDINGS, Vol. 134, Pt. H, No. 5, OCTOBER 1987
Table 1: Effect of propagation on radar observables sensedthrough a 10 km uniform raincell for rainrates of 20, 40 and80 mm/hr
Frequency = 2.88 GHz, t = 10°C, elevation = 5°, canting angle = 2°
corrected (ZH, ZDR) estimates is because when Do islarge (heavy rain) a significant change in Do causes littlechange in CDR, but causes a much larger proportionate
3 0 r
20
10
-10
-20
80
60
20
-202.5 5.0
range,km7.5 100 2.5 5.0 7.5
range, km10.0
Fig. 6 Percentage error in radar estimate of rainrate along a 10 kmpath with 40 mm/hr rainrate applying the gate-to-gate correction pro-cedure and quantisation imposedSee Fig. 2 for key
Fig. 7 Percentage error in radar estimate of rainrate along a 10 kmpath with 80 mm/hr rainrate applying the gate-by-gate correction pro-cedure and quantisation imposedSee Fig. 2 for key
change in ZDR. The second circular radar estimate (ZE,ZDRC) provides a significant improvement on the {ZE,CDR) estimate remaining within 10% up to 40 mm/hrwith improved stability.
3 Modelling or rain-hail mixtures
3.1 Extension to hail within rainCircular polarisation radar characteristics of rain, hailand rain-hail mixture in the S-band were previously con-
IEE PROCEEDINGS, Vol. 134, Pt. H, No. 5, OCTOBER 1987 427
sidered by Torlaschi et al. [22]. They established anumber of salient features, particularly the dominance ofreflectivity of hail over that of rain at comparable rates ofprecipitation, and the marked tendency to lower degreesof orientation when hail is present. But their model was apurely additive one, taking no account of multiplicativeeffects due to propagation. Realistically, one must con-sider not only the backscattering characteristics of rain-hail mixtures, but the effects of propagation betweenradar and target and back through intervening precipi-tation. The computational model discussed belowincludes the effect of rain on the path. Although haileffects could also have been included, it was found thatthey could be neglected in comparison to rain for all rea-sonable combinations, and the form of the hail-dependent propagation effects in the S-band is notdifferent in any qualitative sense. As propagation effectson circular polarisation measurements are known to besignificant, as has been discussed in Section 2, for purerain, effects for rain-hail mixtures should not be unex-pected. Indeed, this was strongly suspected after lookingat S-band data in which rain-hail mixtures were knownto be present. What was initially surprising to us was theappearance of sudden large steps in the correlation phaseat points where hail was encountered. This behaviourwould not be expected in an additive model because W isclose to real for both rain and hail. We were thereforevery interested to see whether this phenomenon could beaccounted for in a model which included propagationeffects.
3.2 The modelThe computational model was developed as an extensionof the rain modelling described in Section 2. The propa-gation matrix terms are essentially the same, and thepower backscattering terms, both co- and crosspolar areadditive. The principal differences arise in taking inte-grals over particle size and orientation distributions. Forsize distribution, the Cheng-English [23] distribution,
(28)N = 115A363
was adopted. Hailstones were taken to be oblate spher-oids of axial ratio 0.8 [2]. In this study, we consideredonly dry hailstones (homogeneous ice), which, as they arerigid bodies, would not be expected to be able to stabilisetheir alignment to the extent that liquid drops or liquidcoated particles would. A low degree of alignment wouldbe expected, and some thought was given to the descrip-tion of the orientation. As there is not an establishedempirical or theoretical model, it was decided to allowany distribution rotationally symmetric about an arbi-trary mean canting axis. Using the approximation ofHolt and Shepherd [24] for the backscattering ampli-tudes in terms of canting angles, McGuinness et al. [25]have shown that the required orientation integrals can beexpressed in terms of the second and fourth moments ofthe orientation distribution with respect to its axis ofsymmetry.
3.3 ResultsThe number of possible rain-hail configurations leaves alarge parameter space to be explored. The behaviour ofthe computer model was found to give good agreementwith the results of Torlaschi et al. [22], for hail, and rain-hail mixtures when considered without propagationeffects, and with earlier model calculations obtained by
us for rain only cases. One of the first considerations formodelling was whether behaviour could be producedqualitatively similar to that seen in Alberta storm datawhere hail was believed to be present. A very strikingfeature of these examples was a very large and sharp stepin the phase of W.
Fig. 8 shows an example of a fairly complicated con-figuration of rain and hail. Superimposed on widespread
-10r
10 15range, km
Fig. 8 Modelled response to rain and mixed rain/hail along a path2£CDRORTT
uniform rain is an intense rain cell, and in addition thereare two hail cells, one in light rain, and the other inheavy. Fig. 9 also shows the behaviour of a new derived
Q 50< 40S30- 20
100
| 30
a,'20a.^ 10o
20 255 10 15range, km
Fig. 9 Derived parameters for the same configuration as in Fig. 8
P is defined in eqn. 29
parameter (3, which, together with the correlation phasea = arg W, describes the polar co-ordinates of the Poin-care sphere mapping of the polarisation state. This is a
428 IEE PROCEEDINGS, Vol. 134, Pt. H, No. 5, OCTOBER 1987
convenient descriptor, because we can form a 2-parameter equivalent representation of the cancellationand orientation parameters via the relations (see theAppendix)
CDR = log10
ORTT =
1 - p cos 2/?
1 + p cos 2jS
p sin 2jS
7(1 - p2 cos2
(29)
where p is the degree of polarisation of the return [26].The significance of this transformation becomes apparentwhen one considers that for S-band propagation, thedegree of polarisation is almost invariant: in as much asthe propagation matrix is nearly unitary, p depends onlyon the trace and determinant of the coherency matrix,which are unitary invariants. With this new param-eterisation, then, we may expect that the propagationeffects on CDR and ORTT are almost wholly expressedin the behaviour of /?. In the following paper in this issue[27], we address this problem by relating a and /? to thetransformations of the coherency matrix. In this paperattention is confined to the behaviour of the a, fi param-eterisation of the model. First, consider the correlationphase, a. In the absence of propagation effects, a wassmall throughout, always less than 1°. The curve in Fig. 9is therefore almost wholly attributable to propagationeffects. Note that in light rain it increases gradually andsteadily, but rises sharply in the presence of hail. After thefirst hail cell, it decreases again, but does not whollyrecover. At the start of the second hail cell, a very strongstep occurs. The hail is encountered this time in quiteheavy rain. The other parameter, p\ appears to be anincreasing function of rain-rate when there is no hailpresent, but is reduced to a very low value whenever hailis present. In the hail regions, the /? value does not seemto be strongly dependent on hail or rain rate.
4 Conclusions
At S-band linear polarisation radar does not suffer sig-nificant propagation effects except when sensing throughheavy rain over long paths. The resultant two parameterdropsize distribution obtained from (ZH, ZDR) gives rea-sonable estimates of the liquid water content and rain-rate. When correcting for propagation, instability isintroduced with quantisation of the radar parameters.However, if the assumed dropsize distribution is correct,the estimates remain within 20% of their true value forrainrates up to 80 mm/hr over a 10 km path.
At S-band, circular polarisation radar suffers frompropagation effects at lighter rainrates. The instabilitydue to quantisation in the correction method for (ZE,CDR) is much more significant. However the model cal-culations reveal that if the real part of (W/W2) can beobtained from the complex correlation channel very goodestimates of the rainrate can be obtained up to 40 mm/hrover a 10 km path.
The rain-hail model revealed that hail embedded in arain cell was distinguished using a circular polarisationradar by both a sudden large step in the correlationphase ALD and a drop in the orientation ORTT.
5 Acknowledgments
This work was supported in part by the Alberta ResearchCouncil, to whom grateful thanks are due for permission
to publish, under contract ARC: UOE 100.00.85. We alsoacknowledge many helpful conversations with Dr. R.G.Humphries, Mr. A Hendry and Dr. G.C. McCormick.
6 References
1 PRUPPACHER, H.R., and PITTER, R.L.: 'A semi-empirical deter-mination of the shape of cloud and rain drops', J. Atmos. Sci., 1971,28, pp. 86-94
2 BARGE, B.L., and ISAAC, G.A.: 'The shape of Alberta hailstones',J. Rech. Atmos., 1973, 7, pp. 11-20
3 McCORMICK, G.C, and HENDRY, A.: 'Polarisation propertiesof transmission through precipitation over a communication link',ibid., 1974, 8, pp. 175-187
4 McCORMICK, G.C, and HENDRY, A.: 'Polarisation relatedparameters for rain: measurements obtained by radar', Radio Sci.,1976,11, pp. 731-740
5 McCORMICK, G.C, and HENDRY, A.: 'Techniques for the deter-mination of the polarisation properties of precipitation', ibid., 1979,14, pp. 1027-1040
6 BARGE, B.L.: 'Polarisation measurements of precipitation back-scatter in Alberta', J. Res. Atmps., 1974, 8, pp. 163-173
7 BARGE, B.L., and HUMPHRIES, R.G.: 'Identification of rain andhail with polarisation and dual-wavelength of radar'. Proceedings of19th Radar Meteorology Conference, Miami, FL, USA, 1980,pp. 507-516
8 McCORMICK, G.C, and HENDRY, A.: 'Principles for the radardetermination of the polarisation properties of precipitation', RadioSci., 1975,10, pp. 421-434
9 HENDRY, A., and ANTAR, Y.M.M.: 'Precipitation particle identi-fication with centimeter wavelength dual-polarization radars', ibid.,1984,19, pp. 115-122
10 HUMPHRIES, R.G.: 'Depolarization effects at 3 GHz due to pre-cipitation'. PhD Thesis, McGill University, Canada, 1974
11 HENDRY, A., ANTAR, Y.M.M, ALLAN, L.E, and COOK, PR.:'Application of waveguide switching in dual-channel polarizationdiversity radar, and preliminary results'. Proceedings of 21st Con-ference on Radar Meteorology, Edmonton, Alberta, Canada, 1983,(Boston-AMS) pp. 352-357
12 SELIGA, T.A, and BRINGI, V.N.: 'Potential use of radar differen-tial reflectivity measurements at orthogonal polarisations for mea-suring precipitation', J. Appl. Meteorol., 1976,15, pp. 69-76
13 BRINGI, V.N, CHANDRASEKHAR, V, SELIGA, T.A, SMITH,P.L, and HARI, T.: 'Analysis of differential reflectivity (ZDR) radarmeasurements during the co-operative convective precipitationexperiment'. Proceedings of 21st Conference on Radar meteorology,Edmonton, Alberta, Canada, 1983, pp. 494-499
15 GODDARD, J.W.F, and CHERRY, S.M.: 'The ability of dual-polarisation radar (copolar linear) to predict rainfall rate and micro-wave attenuation', Radio Sci., 1984,19, pp. 201-208
17 McCORMICK, G.C: 'Relationship of differential reflectivity tocorrelation in dual-polarisation radar', Electron. Lett., 1979, 15, (10),pp. 265-266
18 HOLT, A.R.: 'Some factors affecting the remote sensing of rain bypolarisation diversity radar in the 3- to 35-GHz frequency range',Radio Sci., 1984,19, pp. 1399-1412
19 BEARD, K.V, and JAMESON, A.R.: 'Raindrop canting', J. Atmos.Sci., 1983,40, pp. 448-454
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26 BORN, M., and WOLF, E.: 'Principles of optics' (Pergamon Press,Oxford, 19.75, 5th edn.)
27 BEBBINGTON, D.H.O., McGUINNESS, R., and HOLT, A.R.:'Correction of propagation effects in S-band circular polarisationdiversity radars', IEE Proc. H, Microwaves, Antennas & Propag.,1987,134, (5), pp. 431-437
7 Appendix: Relationship between representationof polarisation
We take as the fundamental representation the coherencematrix J, which, in linear polarisation, is related to theStokes parameters £ by
trace (J) f 1 -(30)
We may transform this to a circular polarisation basis bythe unitary transformation
/ = UJU1
where
f 1 + i '-'Giving
trace (J)
(31)
(32)
(33)
On transforming the Stokes' vector from cartesian topolar co-ordinates as
£1 = p sin 2/? cos a
£2 = p sin p sin 2a (34)
£3 = p cos 20
we find
trace (J)
L -P sii
— p cos 20 — p sin 2/te 2
2 I - p s i n 20e2ia 1 + p cos 201Pi(35)
Now, comparing this with McCormick's circular repre-sentation,
W, W*
W W2(36)
we can make the following identifications:
trace (J) = trace (f) = Wl + W2
Wx \-p cos 20
1 + p cos 20 ~
(37)
ORTT- -» V(l-p2cos20)
= i(7t - arg (WO) = - a
Finally, we note from the identity (unitary invariant)
p =4 det
(trace (J))2
that p represents ihe degree of polarisation [26].
(38)
430 IEE PROCEEDINGS, Vol. 134, Pt. H, No. 5, OCTOBER 1987
