Correction off propagation effects in S-band circularpolarisation-diversity radars

D.H.O. BebbingtonR. McGuinnessA.R. Holt

Indexing terms: Radar and radionavigation, Radio-wave propagation (microwave)

Abstract: A new method for correcting propaga-tion effects in circular polarisation diversity radarsat S-band is described. By considering the trans-formation of the coherency matrix under the influ-ence of propagation, it is shown that correctionscan be derived from the correlation data of indi-vidual range gates. This avoids the use of any iter-ative gate-by-gate procedure previously shown tobe unstable. Assuming the mean orientation of theprecipitation particles is known to within 1-2°, itis found that essentially all the polarisation infor-mation can be recovered with an acceptably smallerror. Application of the correction procedure tostorm data verifies that use of the circular polari-sation technique at S-band can now be extendedto heavy precipitation regions.

1 Introduction

In the preceding paper in this issue [1], we discussed theimpact at S-band of precipitation induced propagationeffects on the polarisation diversity radar technique asapplied to the study of rain and hail. In particular, it haslong been recognised [2] that circularly polarised systemsare highly sensitive to such effects, and in Reference 1 weshowed by model calculations, that, even with S-band,severe corruption of the circular depolarisation ratio(CDR) may occur on paths through heavy precipitation.

About a decade ago, Seliga and Bringi [3] proposedthe use of a linear H-V dual polarisation technique. Thishas the distinct advantage of being relatively immune topropagation effects. The cost of this alternative choice ofpolarisation basis is that fewer independent parameterscan be measured. If only we could eliminate, or greatlyreduce, the propagation effects, a circularly polarisedsystem with cross-correlation facilities could supply fourparameters (ZE, CDR, ORTT, ALD) [1], instead of thetwo (Z, ZDR) available with a dual linear system. Thefundamental reason for this is due to the nature of theanisotropy of rain-filled media: a linear polarised wavealigned or orthogonal to the mean canting angle of theraindrops can change in magnitude and phase, butremains in the same pure polarisation state. In contrast, a

Paper 5488H (Ell), first received as a single paper 8th December 1986and in revised form as two papers 14th April 1987Dr. Bebbington is with the Department of Electronic Systems Engineer-ing, and Dr. McGuinness and Dr. Holt are with the Department ofMathematics, University of Essex, Wivenhoe Park, Colchester, EssexCO4 3SQ, United Kingdom

IEE PROCEEDINGS, Vol. 134, Pt. H, No. 5, OCTOBER 1987

circularly polarised wave not only experiences theseeffects, but in addition is depolarised, generally to a par-tially polarised elliptical state. While this makes the cir-cularly polarised system useful as a tool for investigatingprecipitation targets (because of the information deriv-able from the depolarisation), the same properties areresponsible for changing the polarisation states on theoutward and return paths.

In this paper, we shall show that the situation withS-band circular polarisation can be saved, albeit at theexpense of one measured parameter, ALD, whichdepends on canting angles. In essence, our methoddepends on being able to distinguish between the respec-tive contributions of the propagation and backscatter tothe depolarisation of the return signal. The vehicle for theanalysis of the problem is the (matrix) transformation ofthe coherency matrix [4] which contains all the polarisa-tion information of the partially coherent electromagneticfield. Accordingly, we call this approach the coherencytransformation method (CTM). Both the process of pro-pagation and scattering, and the correction method mayreadily be interpreted through a new polarisation param-eterisation, which was first introduced in Reference 1.

Finally, we show in Section 4 some applications of theCTM to circular polarisation data obtained in Alberta,Canada.

2 Propagation correction by coherencytransformation

By means of a geometric interpretation of the transform-ations of polarisation state occurring due to propagationthrough the anisotropic rainy medium, it can be seenhow even small propagation effects can give rise tomarked changes in the apparent CDR and correlationparameters. The problem, once suitably posed is easilysolved in the S-band case when the following simplifyingassumptions are made:

(a) propagation is well described by differential phasebetween vertical and horizontal polarisations with negli-gible mean and differential attenuation

(b) the backscattering components introduce no differ-ential phase between linear polarisations

(c) the degree of polarisation of the backscattered fieldis near unity.

As discussed in Reference 1, there is theoretical [5] andexperimental evidence [6] that mean canting angles arenear vertical; this and the fact that at S-band for a rainrate of 10 mm/hr, A0 ~ 0.05° per km, and differentialattenuation ~0.002 dB/km justifies assumption (a).Numerical modelling in Section 3 of Reference 1 indi-cated a differential phase in backscatter which is typically

431

less than 0.5°, and a degree of polarisation greater than0.99 in rain, and greater than 0.95 in hail.

The most suitable approach for elucidating the trans-formation of polarisation with backscattering and propa-gation is via the coherency matrix [4]

E\E2(1)

which is Hermitian, Ex and E2 being the orthogonal fieldcomponents. The angle brackets here denote time averag-ing. Quite generally J can be parameterised [4] by*

/ = l2

(2)

where the real (observable) parameters constitute theStokes vector. Such representations, which are widelyused in optics [4] and radar [7], lead to the useful dia-grammatic representation in the projection of the triplecomponent % = (£lt £2, £3) on the Poincare sphere. Thisvector vanishes for an unpolarised field, and so containsall the information concerning the polarised components,and £0 represents the total power.

To illustrate the process of radar scatter, let us firstconsider the case of a single scatterer, in the absence ofpropagation effects. Suppose the incident polarisation tobe RHC corresponding to ^ = (0, 0, — 1) on the sphere inFig. 1 (scaled to unit radius: absolute magnitude need

LHCg

RHC

Fig. 1 Derived coordinates displayed on the Poincare sphere

not concern us at this stage — we normalise to the totalpower). After scattering, the resultant state will be some-where else on the sphere. For spherical scatterers, thebackscatter is again circular, but of the opposite hand (i.e.LHC), here represented by ^ = (0, 0, 1). For spheroidalscatterers, however, this symmetry is broken, and wemight represent the point by polar coordinates (6, a), witha = 0 the azimuth containing the ^ axis. The CDR isobtained by resolving the components with respect to thetwo circular states. In general, these are £0 ± £3, so thatby resolving components in Fig. 2,

* The convention for time dependence used here is the complex factore~ia", with the wave vector aligned with the z axis.

CDR = 10 log10

= 10 log101 - p cos 2/?1 + p cos 2/? (3)

It is worth considering at this stage, what typical valuesare encountered in meteorological radars observinghydrometeors. For example, at 2.88 GHz CDR ^—13 dB for all oblate spheroidal raindrops [8]. Thisnecessitates /? < 12.5°. For rainrates of 10 mm/hr,CDR ~ —21 dB typically, which requires jS small and pnear unity.

Fig. 2 Resolution of orthogonal channels after circular depolarisation

We now can begin to consider what occurs in the pre-sence of propagation effects. We have to determine howthe incident polarisation transforms, how the scatteredstate is affected by this change, and finally, how the pol-arisation state alters on the return path. First we considerthe propagation effects. It is well known that (in theabsence of wind shear), raindrops are strongly orientedwith symmetry axis near vertical, and at S-band the pro-pagation effect is dominated by differential phase propa-gation between horizontal and vertical polarisations. Inthis idealised approximation, the coherence matrix willundergo a unitary transformation of the form [9]

/ ' = UJU* (4)

where * denotes Hermitian transpose, and, for a differen-tial phase 2<p between horizontal and vertical polarisa-tions,

00

(5)

Note particularly that </> does not have to increase uni-formly along the path. The general effect of any unitarytransformation acting on / causes the £ vector on thePoincare sphere to rotate about the axis connecting theeigenpolarisations [9, 10], in this case the ^ axis (Fig. 3).

We now turn to the scattering process, first consider-ing a single scatterer. If we consider a scatterer alignedwith its symmetry axis in the vertical plane about thewave normal, it has the form

S =0

»22

432 IEE PROCEEDINGS, Vol. 134, Pt. H, No. 5, OCTOBER 1987

which may be represented as

(6)

Fig. 3 Differential phase propagation, 5 being the phase differencebetween H and V

When we consider the effect on an incident polarisa-tion state represented by /,

/ ' = SJ&

we obtain, on expansion,

\4e" °, /«. + «

JWi)«2"

« 0 "

+ i{3

• ti)e-

(J)

\(e'(7)

The effect of this is to alter (proportionally) only the Sicomponent. That is, the transformation shifts the latitudefrom the plane Si = 0. (Fig. 4.)

Fig. 4 Backscatter in the case of a real diagonal scattering matrix

In the case of many scatterers, with random orienta-tion, but zero canting angle, for an incident polarisationwith Si = 0, the Si shift is again found to be independent

of the S2: S3 ratio. In summary, for a transmitted circularpolarisation, the % vector is rotated about the Si axis,shifted in latitude from Si = 0 by an amount independentof the propagation differential phase, and finally rotatedagain about Si- Essentially, the effects can be separatedunambiguously, because the eigenpolarisations for propa-gation are the same as for backscatter, but in the onecase there is a differential phase, and in the other a differ-ential amplitude. The problem reduces^ to elementaryspherical trigonometry. In Fig. 5 the arc S3 S of length 2ft

——,

fa20

'

Fig. 5 Comparison of received polarisation in the propagation-freecase (£) and with differential phase 28 each way (£') as projection of thePoincare sphere parallel to LHC, RHC axis

along with the degree of polarisation p characterises thedepolarisation. Because of the invariant properties of thebackscattering process, under differential phase propaga-tion, the image S' of an incident polarisation on the S3 S2great circle must lie on the small circle (of constantcolatitude with respect to S\) containing S- Rotationabout, and shifts parallel to, S\ obviously commute, sothat the shift and phase rotation (4<5 for a 2-way trip) canbe calculated independently. The arc S3 <f of length 2(1'now represents the depolarisation with respect to theideal circular polarisation, and 2a is the measured correl-ation phase. Applying the cosine rule,

7T 7T

cos 2/?' = cos - sin 2/? + sin - cos 2/? cos 2<f>

= cos 2(3 cos 0

whereas the sine rule gives

sin 20' cos 2ft

sin 2<f) sin 2a'

(8)

(9)

Hence, knowing a' and ft', the measured correlationparameters, the two trigonometric equations above,(which assume the true a is zero) can be taken together toeliminate the unknown propagation 'round trip' phase (f)to give

cos 2ft = 2ft' + sin2 2ft' sin2 2a' (10)

Quantitatively, two points are immediately apparent. Forsmall ft, the change in measured correlation phase 2a' isgreatly magnified in comparison to the propagationphase shift </>. Even a few degrees in 0 such as may occur

IEE PROCEEDINGS, Vol. 134, Pt. H, No. 5, OCTOBER 1987 433

over several kilometres of moderate rain makes a signifi-cant jump in a. Secondly, the arc /?' may then be signifi-cantly extended. As the CDR is a very sensitive functionof P for p close to unity, this accounts for the observeddegradation in the CDR due to propagation effects.

3 Model calculation using the coherencetransformation method

In Reference 1 it was shown how rain along a radar pathaffects the radar observables. However, in Section 2 itwas seen that it is reasonable to assume that if the truemean canting angle of the drops is zero, all deviations inthe radar observable arg (W) can be attributed to thedifferential phase at S-band. Thus arg (W) can then beapplied to correct the three remaining circular radarobservables Wu W2 and \W\. This method is onlyneeded for (and applicable to) the circular polarisationradar method as the ZH, ZDR radar does not measurethe phase of the returning signal. This Section examinesthe method's sensitivity to the assumption that the truemean canting angle is zero and also examines the effect ofquantisation errors.

3.1 Effect of the mean canting angleIn Reference 1 the effects of rain along the path of ZE,CDR and ZDRC were described. At 80 mm/hr after 10 kmZE has decreased by 1.9 dB, CDR increased by 9.8 dBand ZDRC decreased by 0.9 dB. Correcting for propaga-tion using the coherence transformation method is sensi-tive to the true mean canting angle along the path. If theangle is zero, the method can correct for propagationvery well. At 80 mm/hr and after 10 km ZE has beencorrected to within 0.8 dB, CDR to within 0.2 dB andZDRC to within 0.1 dB. However, if the true canting angleis 2° these three quantities are then correct to within 0.8,1.4 and 0.5 dB, respectively. If the true canting angle is 5°,they are correct to within 0.9, 4.4 and 1.4 dB, respectively.Therefore the calculation of CDR and ZDRC is sensitiveto the true mean canting angle along the path. To showwhat effects this can have on rainrate predictions Figs.6-8 present the percentage error in the radar estimate of

2.5 7.5 10

range, km

Fig. 6 Percentage error in the radar estimate of rainrate along a 10km path with a 20 mm/hr rainrate, applying the coherence transformationmethod.

(ZE, CDR)(ZE, ZDRC)

the rainrate for the three uniform rain paths in Reference1 (which assumes a canting angle of 2°) with the coher-ence transformation method applied. Both (ZE, CDR)and {ZE, ZDRC) estimates are similar providing predic-tions within 15% up to 40 mm/hr and within 40% at 80mm/hr. However, if the true canting angle is 5°, the errorsincrease to 50% at 40 mm/hr and to greater than 100%at 80 mm/hr. Therefore the assumption of a meancanting angle of between 0° and 2° is important.

3.2 Effects of quantisation errorQuantisation error and measurement uncertainty willincrease the errors when estimating path rainrate. ZE

10

2.5 5.0

range, km

7.5 10.0

Fig. 7 Percentage error in the radar estimate of rainrate along a 10km path with a 40 mm/hr rainrate, applying the coherence transformationmethod

(ZE, CDR)(ZE, ZDRC)

30

20

10

2.5 5.0range,km

7.5 10.0

Fig. 8 Percentage error in the radar estimate of rainrate along a 10km path with a 80 mm/hr rainrate, applying the coherence transformationmethod

(2E CDR)'(ZE, ZDRC)

and CDR are assumed to have quantisation levels of0.5 dB. It is assumed that the quantisation level in ORTTis 5% and in arg (W) it is 1°. Figs. 9 and 10 present the

434 1EE PROCEEDINGS, Vol. 134, Pt. H, No. 5, OCTOBER 1987

percentage error in the radar estimate of the rainrate foruniform rain paths of 40 and 80 mm/hr. They can becompared with Figs. 7 and 8 to show the effects of quan-tisation. The effects of quantisation on the {ZE, CDR)

20

15

10

2.5 50range,km

7.5 10.0

Fig. 9 Percentage error in the radar estimate of rainrate along a 10km path with a 40 mm/hr rainrate, applying the coherence transformationmethod and quantisation imposed

(ZE, CDR)(ZE, ZDR1)

estimate are greater than on the {ZE, ZDRC) estimate.The {ZE, ZDRC) estimate provides reasonably good rain-rates with a maximum error of 40% at 80 mm/hr. Ifarg {W) is quantised in 5° steps, the error increases butstill remains reasonable.

This analysis shows that if the mean degree of dropcant along the path remains below approximately 2°, therainrate estimates given by {ZE, ZDRC) are reasonableeven in very heavy rain, even when quantisation is intro-duced. If the mean degree of drop canting angle increases,the errors increase significantly.

4 Real rain data analysis

We now present linear and circular polarisation-diversityS-band radar data in heavy precipitation situations. Thedata were provided by the Rutherford Appleton Labor-atory (RAL) linear polarisation-diversity radar at Chil-bolton, UK, which uses a 25 m antenna and a frequencyof 3.0765 GHz, and the Alberta Research Council (ARC)circular polarisation-diversity radar at Penhold, Canada,using a 6.7 m antenna and a frequency of 2.88 GHz.

Fig. 11 presents RAL data along a 15 km radial linefrom the radar for a strong convective precipitation cellon 3rd August 1982. The {ZH, ZDR) measurements cor-respond to 300 m range gates and each {ZH, ZDR) pairare spaced 1.5 km apart. Very good correlation betweenZH and ZDR exists along the intense cell. No significantdegradation in ZDR is detectable throughout the entire

storm. Fig. 11 shows the ZDR remains well correlated toZH even at the far edge of the storm.

Figs. 12-14 present ARC radar data for three differentconvective precipitation cells. The {ZE, CDR) measure-ments are made with 1.05 km range gates. Figs. 12a, 13a

10.0

Fig. 10 Percentage error in the radar estimate of rainrate along a 10km path with a 80 mm/hr rainrate, applying the coherence transformationmethod and quantisation imposed

(ZE, CDR)(ZE, ZDRC)

50

30

\

35 40range,km

Fig. 11 (ZH, ZDR) radar data collected by the Chilbolton radar on3rd August 1982 at 1824 (BST), elevation 0.5°

ZHZDR

and 14a show the radar measured ZE and CDR alongwith the CDR corrected, by applying the coherence trans-formation method. Figs. 12b, 136 and 146 show the radarmeasured ZE along with ZDRC, which is derived usingthe coherence transformation method.

Fig. 12 presents a strong precipitation cell on 24thJuly 1983 along a 15 km radial line from the radar. Theground rainfall rate was recorded at a point 47 km fromthe radar and was estimated to be approximately90 mm/hr at this time. The effects of propagation on theradar measured CDR are obvious as it increases from

IEE PROCEEDINGS, Vol. 134, Pt. H, No. 5, OCTOBER 1987 435

— 20 dB to —5 dB over the 15 km path. If propagation isnot considered, rainrate predictions are not possible asthe measured CDR increases above —12 dB for which noDo value exists. The model calculations in Reference 1describe a very similar behaviour in the CDR at 80mm/hr to the measured CDR in Fig. 12a. The correctedCDR, however, gives a much better correlation with ZEthroughout the entire path. This is also true of the ZDRC

calculations presented in Fig. 12b which clearly indicatesthe validity of the method.

Fig. 13 presents a precipitation cell mixture of rain andmelting hail on 5th July 1984 along a 40 km radial linefrom the radar. Ground hail and rain was reportedbetween 22 and 26 km. Hail of 5 mm diameter and arainrate of approximately 100 mm/hr were recorded. Theregion of high reflectivity at 30 km has obvious effects on

0

-5

CD

cc"-10Qu

-15

-20

-25

-

-

N"CD

T5

M

-

-

60

50

40

30

20

1042.5 47.5 50 52.5

range, km

.- 2

60

50-

40

j /

AA' \

/ W

/ /

/ \

\

3042.5 45.0 47.5 50.0 52.5

range,kmb

Fig. 12 Radar data collected by the Penhold radar on 24th July 1983at 2041 {MDT), elevation 0.6°

a (ZE, CDR)ZEradar measured CDRcoherence transformation CDR

b (ZE, ZDR*)ZEcoherence transformation ZDRC

436

the measured CDR which can be seen, in Fig. 13a, todegrade rapidly in this region rising from —20 dB to- 5 dB in less than 10 km. As ZE begins to fall at 50 kmthe CDR remains almost constant at approximately —3dB. An RHI scan at this time revealed ice which hadmelted almost completely into rain when it reached theground. The corrected CDR values show a significantimprovement compared with the measured CDR withvalues typically 15 dB lower at the far edge of the precipi-tation cell. The ZDRC values given in Fig. 13b again showvery good correlation along the entire 40 km path. This

60

Fig. 13 Radar data collected by the Penhold radar on 5th July 1984at 1840 {MDT), elevation 1.1°a (ZE, CDR)

ZEradar measured CDRcoherence transformation CDR

b (ZE, ZDR<)ZEcoherence transformation ZDRC

1EE PROCEEDINGS, Vol. 134, Pt. H, No. 5, OCTOBER 1987

shows that the method can be applied not just for shortprecipitation paths but also for much longer paths.

Fig. 14 presents a strong precipitation cell on 12thJuly 1983 along a 15 km radial line from the radar. Hail-stones exceeding 40 mm diameter and rainrates of

-5

S-10<rao

-15

-20

60

50

30

20

-25L 10

IKS

V-'-'

—̂ /V

100 106.25 112.5range.km

a

A

3

2

C D 1

T3a'aM

0

-1

-2

-

-

mT3M

-

-

60

50

40

30

20

10100 106.25

range , kmb

112.5

Fig. 14 Radar data collected by the Penhold radar on 12th July 1983at 1628 (MDT), elevation 0.3°a (ZE, CDR)

ZEradar measured CDRcoherence transformation CDR

b (ZE, ZDRC)ZEcoherence transformation ZDRC

approximately 50 mm/hr were recorded on the groundover the 15 km path. The region of high reflectivity at thefront of the storm does not cause rapid degradation inthe measured CDR as shown in Fig. 14a. This is con-firmed by the relatively small difference between the mea-sured CDR and the correct CDR. Fig. 146 reveals thatZDRC is low through the storm. Negative ZDR values arerecorded at approximately 105 km when the reflectivity isgreater than 65 dBZ. The combination of high Z, low ornegative ZDRC is indicative of large ice particles whichare either spherical or tumbling.

In the three examples described above it can be seenthat ZDRC calculation is more useful than the correctedCDR. The combination of ZE and ZDRC has proved tobe a very good discriminator of precipitation type.

5 Conclusions

We have shown in this paper that it is possible to inter-pret 4-channel CDR radar data at S-band so as toobtain 3-channel data which is free from propagation-corruption. The most useful channels to obtain wouldseem to be reflectivity, ZDR and ORTT. By means ofexample we have shown that in regions of high reflec-tivity where hail has been known to be present, it is pos-sible to obtain either very low values of ZDR, or else veryhigh values. The latter we believe to be melting hail.

The method used here has relied on the fact that in theS-band differential attenuation is negligible. In a futurepaper we shall discuss the situation in the C- and X-bands.

6 Acknowledgments

This work was supported in part by the Alberta ResearchCouncil under contract ARC: UOE 100.00.85, to whomgrateful thanks are due for permission to publish. Wealso acknowledge many helpful conversations with Dr.R.G. Humphries, Mr. A. Hendry and Dr. G.C. McCor-mick.

Our thanks are also due to the Rutherford AppletonLaboratory for the use of their data.

7 References

1 McGUINNESS, R., BEBBINGTON, D.H.O., and HOLT, A.R.:'Modelling of propagation effects in S-band circular polarisation-diversity radars', 1EE Proc. H., Microwaves, Antennas & Propag.,1987,134, (5), pp. 423^30

2 HUMPHRIES, R.G.: PhD Thesis, McGill University, Canada, 19743 SELIGA, A., and BRINGI, V.N.: 'Potential use of radar differential

reflectivity measurements at orthogonal polarisations for measuringprecipitation', J. Appl. MeteoroL, 1976,15, pp. 69-76

4 BORN, M., and WOLF, E.: 'Principles of optics' (Pergamon Press,Oxford, 1975, 5th edn.)

5 BEARD, K.V., and JAMESON, A.R.: 'Raindrop canting', J. Atmos.Sci., 1983, 40, pp. 448-454

6 McCORMICK, G.C., and HENDRY, A.: 'Polarisation propertiesof transmission through precipitation over a communication link', J.Rech. Atmos., 1974, 8, pp. 175-187

7 CLOUDE, S.R.: 'Target decomposition theorems in radar scat-tering', Electron. Lett., 1985, 21, pp. 22-24

8 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

9 GOODMAN, J.W.: 'Statistical optics' (Wiley & Sons, New York,1984), Chap. 4

10 MARATHAY, A.S.: 'Operator formalism in the theory of partialpolarization', J. Opt. Soc. Am., 1965,55, pp. 969-981

IEE PROCEEDINGS, Vol. 134, Pt. H, No. 5, OCTOBER 1987 437