radar and scattering parameters through falling hydrometeors with axisymmetric shapes

Radar and scattering parameters through fallinghydrometeors with axisymmetric shapes

Alessandro Battaglia, Franco Prodi, and Orazio Sturniolo

Falling raindrops and other hydrometeors have, in general, nonspherical shapes and mean cantingangles that are due to aerodynamic and gravitational forces. We use the T matrix and the quantumtheory of angular momentum to compute extinction matrices, scattering and absorption cross sections,backscattering matrices, and, from these quantities, radar parameters. A monodisperse population ofrain with axially symmetric distribution over orientations where the axis of symmetry is the localdirection of air flow about the raindrops is considered. Oblate spheroids with axial ratios that dependon size and appropriate series of Chebyshev polynomials were assumed for definition of the shapes ofraindrops. Computations were performed at common microwave frequencies for several temperatures,incidence angles, and degrees of particle wobble about a preferred orientation. Results reveal theimportance of the role of orientation distribution and particle size and the shape of radar parameters inthese computations for a region of moderate-sized raindrops. © 2001 Optical Society of America

OCIS codes: 290.0290, 290.1350, 290.2200.

1. Introduction

Computation of extinction and scattering of micro-wave radiation from hydrometeors is widely em-ployed in atmospheric sciences for the retrieval ofpassive and active instruments. For these purposes,in most practical cases, hydrometeors can be modeledas axisymmetric particles such as spheroids, cylin-ders, and series of Chebyshev polynomials, so theWaterman T-matrix computation method is particu-larly effective. Despite its great ability to computescattering properties with large size parameters, ex-treme geometries, or both for low dielectric constantsand negligible loss ~see, e.g., Refs. 1 and 2!, in micro-wave studies of raindrops, this method has the draw-back of not converging ~with our computationalresources! for equivolume radii req greater than 2.5mm with the actual equilibrium shape ~that is, byexpanding the drop shape in a series of Chebyshevpolynomials!. Eventually, gravitation and the airstream tend to define the preferential orientation for

A. Battaglia [email protected]! and F. Prodi are with the Depart-ment of Physics, University of Ferrara, Via Paradiso 12, 44100Ferrara, Italy. O. Sturniolo [email protected]! is withthe Institute of Atmospheric and Oceanic Studies, National Re-search Council, Via Gobetti 101, 40129 Bologna, Italy.

Received 7 June 2000; revised manuscript received 16 January2001.

0003-6935y01y183092-09$15.00y0© 2001 Optical Society of America

3092 APPLIED OPTICS y Vol. 40, No. 18 y 20 June 2001

hydrometeors in the atmosphere, generating an axi-symmetric orientation distribution about the direc-tion of alignment.

The T-matrix approach together with vector spher-ical function quantum theory has the advantage ofanalytical expressions for computing scattering pa-rameters averaged over an arbitrary quadraticallyintegrable orientation distribution function.3 Refer-ence 3 updates the method developed by Khlebtsov4

and by Mishchenko and Travis1 for randomly ori-ented and by Mishchenko5 for axially oriented non-spherical particles; whereas Khlebtsov4 analyticallysolved the scattering problem, Mishchenko5 imple-mented a code for forward scattering only, becausehis major interest is in radiative transfer theory.The latter solution is better for this purpose, becauseit involves only averages of the T matrix. Par-amonov’s research has the drawback that the formu-las for computing the average of TT* elements, whichis necessary for computing the scattering matrix, in-volve highly nested summations, so their efficientnumerical implementation is problematic.

In this paper we implement numerically the for-malism that was presented in Ref. 3 for scatteringcross sections and backscattering ~and hence for ra-dar parameters! of hydrometeors composed of axi-symmetric particles by using an axisymmetricprobability-density function as an orientation distri-bution about the direction of alignment. Startingfrom the T matrix,2 of Mishchenko et al.,2 we imple-mented a procedure for analytically computing T,

†

iqc

ca~t

fim

o

fi

w

l

i

t W W

fdt

3

tm

TT*, and TT averaged over orientations by exploit-ng the symmetries of the problem. From theseuantities all other parameters of interest can beomputed.

2. T-Matrix Method

The T-matrix method is based on the idea that, be-ause of the linearity of Maxwell’s equations, there islinear relation between the expansion coefficients

we use the expansion in vector spherical functionshat was used by Mishchenko et al.2! of the incident

and the scattered fields. Matrix T gives these coef-ficients. In case of axisymmetric particles, in thenatural reference frame the T matrix is characterizedby the well-known symmetry

Tm,n,m9,n9i, j 5 dm,m9 Tm,n,m,n9

i, j ; Tm,n,n9i, j , (1)

Tm,n,m,n9i, j 5 ~21!i1j T2m,n2m,n9

i, j . (2)

Before resuming the analytical procedure for obtain-ing an orientational average, we define some usefulframes of reference.

A. Frames of Reference

The physical model of hydrometeor canting togetherwith the usual assumption of rotationally symmetricshape and the geometry of the radar dictate the use offour right-handed Cartesian coordinate systems:

• The natural frame of reference of particle Axed to a scattering particle is chosen such that the Tatrix is simple ~zA axis along the axis of symmetry,

xA and yA axes arbitrary because of rotational sym-metry!.

• The hydrometeor frame of reference H, withrientation fixed in space and the zH axis directed as

the force ~the gravitational field or the electrostaticeld! that causes alignment. xH and yH are arbi-

trary if there is rotational symmetry about this axis.• The electromagnetic wave frame of reference W,

the zW axis in the direction of propagation of theave, the xW ~or vertical component! axis in the plane

defined by the propagation direction, and the localvertical axis ~zL!. The yW axis ~horizontal compo-nent! is chosen perpendicular to the vertical compo-nent such that xW 3 yW 5 zW. The component of theelectric field in the OzLzW plane is the vertical com-ponent.

• The laboratory frame of reference L, with ori-entation fixed in space, zL axis directed in the verticalocal direction, xL axis such that the electromagnetic

wave is propagating in the OZX plane, with elevationangle x and yL [ yW.

In the most common case the orientation directionis just the vertical direction, so L and H coincide. Byusing an approach similar to that of Holt6 and Stur-niolo et al.7 ~but for the hydrometeor’s symmetry axisnstead of the particle’s axis; see Fig. 1!, it is possible

o compute the direction ~usym, fsym! 5 ~c, a! of theaxis of symmetry of the hydrometeor, OZH, in W as aunction of its direction ~usym

L , fsymL ! in L and the

irection of elevation angle x @which gives the direc-ion of radiation, ni

L 5 ~2cos x, 0, sin x!# by

cos c 5 cos usymL sin x 2 sin usym

L

3 cos x cos fsymL , (3)

cos ac sin c 5 cos usymL cos x 1 sin usym

L

3 sin x cos fsymL ,

sin ac sin c 5 sin usymL sin fsym

L , (4)

where usymL is the true canting angle with respect to

the local vertical plane and ac is the canting angle inthe polarization plane, i.e., the angle between thevertical polarization axis and the projection of thedirection of alignment onto the polarization plane.

Using the relationships described above, we com-pletely define all the variables in the problem oncethe direction of incidence ~ui, fi! and the direction ofsymmetry of the hydrometeor ~usym, fsym! are knownin the L frame. Note that when H [ L ~i.e., usym 50! we simply have c 5 py2 2 x 5 ui and ac 5 0.

In the T-matrix context, transformation betweentwo frames is accomplished by a Euler transforma-tion ~see Ref. 2!. The relevant Euler rotation anglesfor the problem under examination are ~a, b, g!H3L 5~p, usym

L , p 2 fsymL !, transforming H in L, and ~a, b,

g!H3W 5 ~p, c, p 2 ac!, transforming H in W.

3. Probability-Density Function over Orientations

With respect to reference H, we define a probability-density function over orientations HP~a, b, g!H3Athat is square integrable in the closed domain @0, 2p#

@0, p# 3 @0, 2p#, which can be expanded in a seriesof Wigner ~D and d! functions as follows:

HP~a, b, g!H3A 5 (n50

`

(m52n

n

(m952n

n 2n 1 18p2

Hpm,m9n

3 Dm,m9n ~a, b, g!H3A. (5)

As the orientation is typically axisymmetric, we useas an orientation function P~a, b, g! 5 Hp~b!y4p2 ~b ishe angle formed by the direction of the axis of sym-etry and the z direction of H!, with corresponding

expansion in Legendre polynomials Pn:

Hp~b! 5 (n50

` 2n 1 18p2

Hpn D0,0n ~b!

5 (n50

` 2n 1 12

Hpn Pn~cos b!. (6)

It is important to transform the probability-densityfunction over the orientation of the particles from

20 June 2001 y Vol. 40, No. 18 y APPLIED OPTICS 3093

o

wepdH

dmi

cTr

1F

3

reference H to reference L. Using the addition the-rem for Wigner D functions results in

LP~a, b, g!L3A 5 (n50

`

(m52n

n

(m952n

n 2n 1 18p2

Lpm,m9n

3 Dm,m9n ~a, b, g!L3A,

Lpm,m9n ; F (

m52n

nHpm,m9

n Dm,m9n ~a, b, g!H3LG ,

(7)

hich expresses the expansion coefficients of the ori-ntation distribution in the L frame. A similar ex-ression is true for a general S frame, so, when theistribution is axisymmetric with respect to reference, Eq. ~7! becomes

Spm,m9n 5 dm9,0

Hpn D0,mn ~a, b, g!H3S

5 dm9,0Hpn d0,m

n ~b! exp~2img! ; Spmn . (8)

Note that

Sp2mn 5 ~21!m exp~2img! Spm

n 5 ~21!m @Spmn #*.

Examples of possible orientation functions ~ran-om orientation, perfect orientation, Gaussian align-ent! with the corresponding expansion coefficients

n the hydrometeor are given by Mischchenko.5

Fig. 1. Scattering geometr


4. Scattering Properties

Computing scattering properties needs the evaluationof averages over orientations of various combinationsof T-matrix elements and of expansion coefficients ofincident plane waves @see Eqs. ~39! and ~40! of Par-amonov3!. We briefly summarize the first issue.

A. Orientation Average of Elements of a T-MatrixCombination

To determine various scattering properties we com-puted analytically the average over orientation ~indi-ated by angle brackets! for several combinations of-matrix elements in the most convenient frame ofeference.

. Average of T-Matrix Elementsor a generic reference frame S,

^Tm,n,m9,n9i, j &S 5 (

m152M

m15M

~21!m91m1 (l5un2n9u

n1n9Spm2m9,0

l

3 Cn,m,n9,2m9l,m2m9 Cn,m1,2n9,2m1

l,0 Tm1,n,n9i, j ~ A!, (9)

where M 5 min~n, n9!. Note that the symmetry re-lation 5 that is present in

^Tm,n,n9i, j &S 5 ~21!i1j ^T2m,n,n9

i, j &S (10)

is satisfied for S 5 H.

otation is defined in text.
y. N

Iti

2. Average of the TT* Matrix ElementBy using the symmetry of the T matrix in reference A,we arrive at the average of the TT* matrix element:

t

i

The scattering problem can be solved in anyframe, but it is clear that, when the H reference isused as the principal frame, the expansion coeffi-cients of the orientation function are clear; simpli-fied expressions can be obtained in the W framealso, by use of the simplicity of the coefficients of theincident wave.

B. Cross Sections

The extinction and scattering cross sections for a sin-gle particle are

ssca 51k2 (

mn@u pm,nu2 1 uqm,nu2#, (12)

sext 5 21k2 ReH(

m,n@a*m,npm,n 1 b*m,nqm,n#J . (13)

n an ensemble of particles with an arbitrary orien-ation distribution function it is enough to substitutento Eqs. ~12! and ~13! the average of T ~^T&! and of

T*T~^T*T&!, as calculated in Eqs. ~9!–~11!. The al-bedo and the absorption cross section are easily com-puted from Eqs. ~12! and ~13!. We obtained thesequantities by using both H and W as the principalframes as a cross check.

C. Extinction Matrix

The determination of an extinction matrix has greatimportance both for radiative transfer applicationsand for the computation of some radar parameters.Computation of the extinction is simple with H as the

KL~n! 5 3sext~uH! cos 2Vspol~u

cos 2Vspol~uH! sext~uH!sin 2Vspol~uH! 0

0 sin 2Vscpol~u

principal frame because in this case extinction matrixelements angularly depend only on the polar angle.Following Mishchenko,5 in a linear polarization rep-

resentation the extinction matrix in the H frame,KH~u!, has a block-diagonal form: all nonzero termsare linear functions of the average T matrix, as in Eq.~9!, with S 5 H. For radiative transfer purposes, theextinction matrix needs to be computed in the Lframe. This is simply done with the transformationof Stokes parameters under rotation, so

where uH 5 c and V is the angle of the rotation abouthe direction of propagation of the plane through ~ni,

z! and through ~ni, Rsym! ~see Fig. 1!. The first planes individuated by the y axis; the second, by the vector

Sv~1!v~2!v~3!

D 5 F sin x sin usym sin fsym

2cos x cos usym 2 sin x sin usym cos fsym

cos x sin usym sin fsym

G ,

(15)

such that

cos V 5v~2!

~@v~1!#2 1 @v~2!#2 1 @v~3!#2!1y2 .

As a cross check we computed the extinction matrixon a circular basis also ~details can be found in Ref. 3!by computing averages of the T matrix directly in S 5L. By so doing, with the usual matrices of passagefrom CP to LP representation, we verified the consis-tency of our procedure.

sin 2Vspol~uH! 00 2sin 2Vscpol~uH!

sext~uH! cos 2Vscpol~uH!2cos 2Vscpol~uH! sext~uH!

4 , (14)

^Tm,n,m9,n9p,q @Tm,n,m9,n9

p,q #*&S 5 (m1,m152M

M

~21!m12m1m12m9 Tm1,n,n9p,q ~ A! @Tm1,n,n9

p,q ~ A!#*

3 (n15un2nu

n1n

Cn,m,n,2mn1,m2m Cn,m1,n,2m1

n1,m12m1 (n25un92n9u

n91n9

Cn9,m9,n9,2m9n2,m92m9 Cn9m1,n9,2m1

n2,m12m1

3 (n35un12n2u

n11n2Spm2m1m92m9,0

n3 Cn1,m2m,n2,m92m9n3,m2m1m92m9 Cn1,m12m1,n2, m12m1

n30

5 ~21!m1m9 (n1

Cn,m,n,2mn1,m2m (

n2

Cn9,m9,n9,2m9n2,m92m9 (

n3

Spm2m1m92m9,0n3 Cn1,m2m,n2,m92m9

n3,m2m1m92m9

3 (m1,m1

~21!m11m1 Tm1,n,n9p,q @Tm1,n,n9

p,q #* Cn,m1,n,2m1

n1m12m1 Cn9,m1,n9,2m1

n2,m12m1 Cn1,m12m1,n2,m12m1

n3,0 . (11)

H!

H!


tP

ma

F

3

D. Scattering Matrix

On a circular ~C! or a linear ~L! basis the scatteringmatrix that relates the Stokes vector to the scatteringplane is defined by

IsC,L~ns! 5

1r2 ZC,L~ni; ns!Ii

C,L~ni!, (16)

with ni along z. Introducing some linear combina-ion of the T matrix and following the notation ofaramonov,3 we arrive at the orientationally aver-

aged scattering matrix elements at a scattering angle~us, fs!:

^Cp,qC*p,q& 5 (n,n9,n,n9,m,m

~ 2 1!q1q1m1m

4

3 tn,n9t*n,n9dp,mm ~us!dp,m

m ~us!exp@i~m 2 m!fs#

3 ^Tm,n,q,n9p,q @Tm,n,q,n9

p,q #*&W,

Tp,q 5 T1,1 1 qT1,2 1 pT 2,1 1 pqT 2,2,

tn,n9 ; ~i!n92n@~2n 1 1!~2n9 1 1!#1y2, (17)

where p, q, p, q 5 1, 21.

E. Backscattering Matrix

Using the values of the Wigner functions at backscat-tering transforms Eq. ~17! into

^Cp,qCp,q&uus5p 5~21!p1p1q1q

4 (n,n9,n,n951

nmax

~21!n1ntn,n9t*n,n9

3 exp@2i~ p 2 p!fs#

3 ^T2p,n,q,n9p,q @T2p,n,q,n9

p,q #*&W.

The fs dependence takes into account the transfor-ation of Stokes parameters on a circular basis when

n angle fs is rotated ~in the clockwise direction withrespect to the direction of propagation!. Clearly, theintrinsic properties of backscattering matrices ~Ref.8, p. 15 and Chap. 3! and single-particle and popula-tion symmetries are fully exploited for computationonly of the independent parameters in a backscatter-ing matrix.

The backscattering cross sections can be expressedin terms of backscattering matrix elements definedby Eq. ~16! as

sv,v 5 2p@Z11L 1 Z12

L 1 Z21L 1 Z22

L #, (18)

sh,h 5 2p@Z11L 2 Z12

L 2 Z21L 1 Z22

L #, (19)

sv,h 5 sh,v 5 2p@Z11L 2 Z22

L #. (20)

The analytical method for computing extinctionand the backscattering matrix was compared with anumerical method ~that is, a method in which the Tmatrix is averaged numerically over orientations!and with the benchmark results of Mishchenko.5The agreement is perfect. Time-consuming consid-erations yield the conclusion that, for a backscatter-ing matrix, numerical integration over orientationangles is preferable to analytical integration in all


cases except that for randomly oriented particles.For the extinction matrix, however, an analytical pro-cedure is always faster.

F. Radar Parameters

For a general extinction matrix in which all lengthsare expressed in meters and n0 is the total number ofparticles per cubic meter, we can define

• Specific attenuation at vertical and horizontalpolarization:

Av 5 4.343 3 103 n0svext @dB km21#, (21)

Ah 5 4.343 3 103n0shext @dB km21#. (22)

• Specific differential phase:

KDP 5 103~180yp!n0 KL~3, 4! @deg km21#. (23)

or a general backscattering matrix we can define

• Differential reflectivity, the ratio between thefraction of horizontally polarized backscattering andthat of vertically polarized backscattering:

ZDR 5 10 log10

ZHH

ZVV

5 10 log10

Z11L 2 Z12

L 2 Z21L 1 Z22

L

Z11L 1 Z12

L 1 Z21L 1 Z22

L . (24)

• Linear depolarization ratio, defined as the ratioof the power backscattered at vertical or horizontalpolarization to the power backscattered at horizontalor vertical polarization for a horizontally or a verti-cally polarized incident field, respectively:

LDRH 5 10 log10

ZVH

ZHH

5 10 log10

Z11L 2 Z12

L 1 Z21L 2 Z22

L

Z11L 2 Z12

L 2 Z21L 1 Z22

L Uback

, (25)

LDRV 5 10 log10

ZHV

ZVV

5 10 log10

Z11L 1 Z12

L 2 Z21L 2 Z22

L

Z11L 1 Z12

L 1 Z21L 1 Z22

L Uback

, (26)

LDRT 5 10 log10

ZHV 1 ZVH

ZVV 1 ZHH5 10 log10

Z11L 2 Z22

L

Z11L 1 Z22

L Uback

.

(27)

• Circular depolarization ratio, defined as the ra-tio of the power backscattered at left-hand circularpolarization to the power backscattered at right-hand

da

t

c

1f8

saG

polarization for a left-hand circularly polarized inci-dent field:

CDR 5 10 log10

ZLL

ZRL

5 10 log10

Z11L 2 2Z14

L 1 Z44L

Z11L 2 Z44

L . (28)

5. Computational Results

A. Rain Modeling

We approximated raindrops as oblate spheroids withan aspect ratio following Andsager et al.9:

G

ro1a

and using a series of Chebyshev polynomials withcoefficients given by Chuang and Beard.10 For theseparticles the T-matrix method converges to 4.5-mm

iameter ~in fact, raindrops in the millimeter regimere deep inside the resonance region!, whereas for

spheroids convergence is obtained for a whole rangeof physical diameters ~D # 9 mm!.

As illustrative examples of rain modeling we com-pute results for various scattering parameters of amonodisperse population of raindrops with differentorientation distributions: at 19.4 GHz with refrac-tive indices m 5 5.3364 1 i2.9744 at 0 °C and m 56.74841 1 i2.76453 at 20 °C, at 37 GHz with refrac-tive indices m 5 3.9305 1 i2.4295 at 0 °C and m 55.1083 1 i2.8087 at 20 °C, and at 85.5 GHz withrefractive indices m 5 2.84950 1 i1.51920 at 0 °Cand m 5 3.48450 1 i2.08120 at 20 °C. This analysisis similar to that studied by Sturniolo et al.,7 but, inhat research, Wang particles were used.

Fig. 2. Difference between Zh of perfectly horizontally orientedpheroids and Z of equivolume spheres as a function of elevationngle and of particle size ~radius of equivolume sphere! at 85.5Hz and T 5 0 °C.

B. Scattering Properties and Radar Parameters

As we increase the typical resonance patterns withridges and valleys are observed in all scattering prop-erties with increasing oscillations ~even if with de-reasing depth!.

With regard to Zh, great departures ~as much as 18,2, and 8 dB at 19.4, 37, and 85.5 GHz, respectively!rom equivolume spheres are found ~e.g., for Fig. 2,5.5 GHz!: As expected, Zh for spheroids overesti-

mates ~underestimates! Z for spheres at the nadir~side! view. A spheroid approximation seems towork well at side incidence, however, for both tem-peratures at the three frequencies the Chebyshevmodel never differs from the spheroid model by more

than 0.15, 0.5, and 0.3 dB, respectively. At verticalincidence, differences start becoming noticeable at 37GHz @e.g., an oblate spheroid model with req 5 2.2mm has a shh value that is 60% ~2 dB! greater thanthat of the equilibrium shaped model# and at 85.5GHz @e.g., an oblate spheroid model with req 5 1.6mm has a shh value that is 40% ~1.5 dB! greater thanthat of the equilibrium shaped model#; see Fig. 3.

enerally, spheroid approximation Zh overestimates~underestimates! the Chebyshev Zh at the nadir~side! view. At 19 GHz, differences never exceed0.15 dB for every size and incident angle. Note that,as is well known, the temperature dependence on therefractive index has a great effect on reflectivities ~for

Fig. 3. Zh at vertical incidence of raindrops at 20 °C modeled asperfectly oriented ~PO; diamonds! and Gaussian oriented ~withsu 5 10°; hexagons! Chebyshev particles and as PO ~continuouscurve! and Gaussian oriented ~with su 5 10°; dashed curve! sphe-oids of raindrops at 0 °C modeled as PO ~asterisks! and Gaussianriented ~with su 5 10°; squares! Chebyshev particles, and as su 50° ~dotted curve! spheroids. PO spheroids at 0 °C are assumeds zero level.

ab

5 H1.0 D # 0.1 cm1y~1.0048 1 0.0057 D 2 2.628 D2 1 3.682 D3 2 1.677 D4! 0.1 # D # 0.9 cm (29)


3rfsrf~wa

smcp

u2

a1scqad

0is

co0

a

3

example, in Fig. 3, the departure of the continuouscurve from zero is due to differences in temperatureonly!.

ZDR exhibits fluctuations near the side view thatare fewer in number but stronger in depth as thefrequency decreases: at 19 GHz it becomes negativefor very large drops ~req . 3.2 mm; Fig. 4!, whereas at7 and 85.5 GHz negative values are found, even ateq 5 1.7 and req 5 1.25 mm, respectively. Thereforeor polydispersion of raindrops, this parameter hasmall values at the two higher frequencies. Sphe-oids give a good approximation of Chebyshev shapesor this parameter, especially at 19 and at 85.5 GHzdifferences never greater than 0.05 and 0.25 dB!,hereas at the intermediate frequency, departures ofs much as 0.65 dB are found ~Fig. 5!.LDRV signals for spheroids are always low; circular

depolarization ratio values are bigger than LDR onesand are influenced less by orientation distribution,

Fig. 4. ZDR for horizontally PO spheroids as a function of eleva-tion angle and of particle size ~radius of equivolume sphere! at 19.4GHz and T 5 0 °C.

Fig. 5. Departures from ZDR at side incidence for PO spheroids at°C ~zero level!. Continuous curve, PO spheroids at 20 °C; aster-

sks and diamonds, PO Chebyshev particles at 0° and 20 °C, re-pectively. Frequency, 37 GHz.


especially near the side view ~see Figs. 6 and 7!. Inparticular, for all frequencies, LDRV is always lessthan 220 dB for Gaussian-oriented particles with

u 5 3°, except for a narrow size interval ~3.7–4.2m! near the side view at 19.4 GHz ~see Fig. 6, lower

urve!. However, when there is a greater degree ofarticle wobble ~such as su 5 10°!, the LDRV signal

remains greater than 230 dB for almost every angleand size at the two lowest frequencies ~see Fig. 6,

pper curve!, whereas it always remains less than26 dB at 85.5 GHz.For these depolarization parameters, a spheroid

pproximation works well only at a side view. At9.37 GHz, the differences between Chebyshev andpheroid approximations gradually increase with in-reasing size and elevation angle; for the highest fre-uency, however, even particles with req 5 2.5 mmre well inside the resonant regime, so the shapeetails are important and strong differences in depo-

Fig. 6. LDRV for Gaussian-oriented spheroids ~su 5 10°, topurve; su 5 3°, bottom curve! as a function of elevation angle andf particle size ~radius of equivolume sphere! at 19.4 GHz and T 5°C.

Fig. 7. Circular depolarization ratio ~CDR! for spheroids ~Gauss-ian oriented with su 5 10°, shaded surface; PO, lighter surface! as

function of elevation angle and of particle size ~radius of equiv-olume sphere! at 85.5 GHz and T 5 0 °C.

fbap

og

larization parameters are found between the two ap-proximations, even with small particles and atintermediate elevation angles ~see, e.g., Fig. 8 for thecircular depolarization ratio at 85.5 GHz!. There-ore, if the resonant regime is effective, the interplayoth of orientation distribution and of particle sizend shape will crucially influence the depolarizationarameters.With respect to propagation parameters, the spher-

id approximation is precise in the region of conver-ence. KDP becomes negative with increasing

particle size, and this phenomenon is more pro-nounced for shorter wavelengths. Therefore nega-tive values of this parameter are expected forpopulations that correspond to a high rainfall rate,especially at the two upper frequencies, as is shown,for example, in Fig. 9, where the behavior of KDP as afunction of elevation angle and rainfall rate for apopulation of Marshall–Palmer raindrops approxi-mated as spheroids is plotted.

Fig. 8. Difference between circular depolarization ratio ~CDR!computed with Chebyshev and spheroid approximations as a func-tion of elevation angle and of particle size ~radius of equivolumesphere! at 85.5 GHz and T 5 0 °C.

Fig. 9. KDP in degrees per kilometer for PO spheroids at 37 GHzas a function of elevation angle and rainfall rate ~RR! for a popu-lation of Marshall–Palmer spheroidal raindrops at T 5 0 °C.

6. Conclusions and Applications

An analytical procedure for averaging T-matrix ele-ments over an axisymmetric probability-density ori-entation distribution has been revisited andnumerically implemented, with particular attentionpaid to extinction and backscattering matrices.Some preliminary results are shown for rain; otherswill follow in subsequent papers for ice and mixedclouds, always modeled with spheroids.

The method described in this paper can be usefulfor the most-recent radiative transfer codes that areable to take into account preferential orientation ofhydrometeors ~see, e.g., Refs. 11 and 12! by comput-ing the extinction matrix and the dependence ofscattering parameters on incident angle. More spe-cifically, the computation of radar parameters can bedirectly compared with experimental outputs of cloudradars ~see, e.g., Refs. 13 and 14! in which naturalprecipitation was observed or of scatterometers ~see,e.g., Refs. 15 and 16! in which synthetic hydromete-ors were observed.

We acknowledge the use of the T-matrix code de-veloped by Michael Mishchenko. We used the codein the computation of the T matrix for the singlescatterer.

References1. M. I. Mishchenko and L. D. Travis, “Light scattering by

polidispersions of randomly oriented spheroids with sizescomparable to wavelengths of observation,” Appl. Opt. 33,7206–7225 ~1994!.

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putations of polarized microwave radiation,” presented at the

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using the 33 GHz and 95 GHz polarimetric cloud profiling

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13th International Conference on Clouds and Precipitation,Reno, Area, Nev., 14–18 August 2000.

3. A. Tokay and K. V. Beard, “The influence of raindrops oscilla-tions on dual-polarization radar measurements of precipita-tion,” Preprints, 26th International Conference on RadarMeteorology, Norman, OK, 24–28 May 1993 ~American Mete-orological Society, Boston, Mass., 1993!, pp. 100–102.

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radar system ~CPRS!,” IEEE Trans. Geosci. Rem. Sens. 35,256–270 ~1997!.

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radar and scattering parameters through falling hydrometeors with axisymmetric shapes

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