Scattering from Hydrometeors:Scattering from Hydrometeors:Clouds, Snow, RainClouds, Snow, Rain

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Clouds, Snow, RainClouds, Snow, RainMicrowave Remote Sensing INEL 6069

Sandra Cruz PolProfessor, Dept. of Electrical & Computer Engineering,

UPRM, Mayagüez, PR

Outline: Clouds & RainOutline: Clouds & Rain

1. Single sphere ( Mie vs. Rayleigh )2. Sphere of rain, snow, & ice ( Hydrometeors )

Find their εεεεc, nc, σσσσb

3. Many spheres together : Clouds, Rain, Snow

2

a. Drop size distributionb. Volume Extinction= Scattering+ Absorptionc. Volume Backscattering

4. Radar Equation for Meteorology5. TB Brightness by Clouds & Rain

Clouds Types on our AtmosphereClouds Types on our Atmosphere

3

40

50

60

70

hexagonalplatesbullet rosettes

%

Cirrus Clouds Composition

4

0

10

20

30

Ice Crystals

dendrites

others

%

EM interaction with EM interaction with Single Spherical ParticlesSingle Spherical Particles

� Absorption– Cross-Section, Qa =Pa /Siiii

– Efficiency, ξξξξa= Qa /ππππr2

� Scattered

Si

5

� Scattered – Power, Ps

– Cross-section , Qs =Ps /Siiii

– Efficiency, ξξξξs= Qs /ππππr2

� Total power removed by sphere from the incident EM wave, ξξξξe = ξξξξs+ ξξξξa

� Backscatter , Ss(ππππ) = Siiiiσσσσb/4ππππR2

Mie Scattering: Mie Scattering: general solution to EM general solution to EM scattered, absorbed by dielectric spherescattered, absorbed by dielectric sphere..

� Uses 2 parameters (Mie parameters)– Size wrt. λλλλ : 2

λπχ r=

6

– Speed ratio on both media:

bλ

bn

nn p=

Mie SolutionMie Solution

� Mie solution

)|||)(|12(2

),( 2

1

22 m

mms bamn ∑

∞

=

++=χ

χξ

7

� Where am & bm are the Mie coefficients given by eqs 5.62 to 5.70 in the textbook.

}Re{)12(2

),(1

2

1

mm

ma

m

bamn ∑∞

=

=

++=χ

χξ

χ

Mie coefficientsMie coefficients

1

1}Re{}Re{

WWm

n

A

WWm

n

A

a

mmm

mmm

m

−

+

−

+=

−

−

χ

χ

coλ

πrr ελπχ 2

2

p

==

8"'

1

1

cossin

}Re{}Re{

jnnn

jWwhere

WWm

nA

WWm

nA

b

o

mmm

mmm

m

−=

+=

−

+

−

+=

−

−

χχ

χ

χ

oc

cb

cp

b k

j

n

nn

)( p αβε

ε

ε −====

NonNon--absorbing absorbing sphere or dropsphere or drop((n”=n”= 0 for 0 for a a perfect dielectricperfect dielectric, , which is awhich is anonnon--absorbingabsorbingsphere)sphere)

9

oook

k

jjnnn

call

εµω

αβ

=

−=−=o

)("'

Re

χ =.06

Rayleigh region |nχ|<<1

Conducting (absorbing) sphereConducting (absorbing) sphere

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χ =2.4

Plots of Mie Plots of Mie ξξeeversus versus χχ

Four Cases of sphere in air :

n=1.29 (lossless non-absorbing sphere)

n=1.29-j0.47 (low loss sphere)

n=1.28-j1.37 (lossy dielectric sphere)

n= perfectly conducting metal sphere∞

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� As n’’ increases, so does the absorption (ξa), and less is the oscillatory behavior.

� Optical limit (r >>λ) is ξe =2.� Crossover for

– Hi conducting sphere at χ =2.4

– Weakly conducting sphere is at χ =.06

n= perfectly conducting metal sphere∞

Rayleigh Approximation |Rayleigh Approximation |nnχχ|<<1|<<1

� Scattering efficiency

� Extinction efficiency

...||3

8 24 += Ks χξ

12

� Extinction efficiency

� where K is the dielectric factor

...||3

8}Im{4 24 ++−= KKe χχξ

2

1

2

12

2

+−=

+−=

c

c

n

nK

εε

Absorption efficiency in Rayleigh Absorption efficiency in Rayleigh regionregion

esea K ξχξξξ ≅−=−= }Im{4

13

esea K ξχξξξ ≅−=−= }Im{4

i.e. scattering can be neglected in Rayleigh region(small particles with respect to wavelength)|nχ|<<1

Scattering from HydrometeorsScattering from Hydrometeors

Rayleigh Scattering Mie Scattering

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λ >> particle size λ comparable to particle size--when rain or ice crystals are present.

Single Particle CrossSingle Particle Cross--sections vs.sections vs.χχ

� Scattering cross section

� Absorption cross section

][m ||3

2 2262

KQs χπλ=

For small drops, almost no scattering, i.e. no bouncing from drop since it’s so small.

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� Absorption cross section

In the Rayleigh region (nχ<<1) =>Qa is larger, so much more of the signal is absorbed than scattered. Therefore

][m }Im{ 232

KQa χπλ=

as ξξ <<16

RayleighRayleigh--MieMie--GeometricOpticsGeometricOptics� Along with absorption, scattering is a major cause of the

attenuation of radiation by the atmosphere for visible. � Scattering varies as a function of the ratio of the particle

diameter to the wavelength (d/λ) of the radiation.� When this ratio is less than about one-tenth (d/λ<1/10),

Rayleigh scattering occurs in which the scattering

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Rayleigh scattering occurs in which the scattering coefficient varies inversely as the fourth power of the wavelength.

� At larger values of the ratio of particle diameter to wavelength, the scattering varies in a complex fashion described by the Mie theory;

� at a ratio of the order of 10 (d/λ>10), the laws of geometric optics begin to apply.

Mie Scattering Mie Scattering (d/(d/λλ≈≈11), ),

� Mie theory : A complete mathematical-physical theory of the scattering of electromagnetic radiation by spherical particles, developed by G. Mie in 1908.

� In contrast to Rayleigh scattering, the Mie theory embraces all possible ratios of diameter to wavelength. The Mie theory is very

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possible ratios of diameter to wavelength. The Mie theory is very important in meteorological optics, where diameter-to-wavelength ratios of the order of unity and larger are characteristic of many problems regarding haze and cloud scattering.

� When d/λ 1 neither Rayleigh or Geometric Optics Theory applies. Need to use Mie.

� Scattering of radar energy by raindrops constitutes another significant application of the Mie theory.

≈

Backscattering CrossBackscattering Cross--sectionsection� From Mie solution, the backscattered

field by a spherical particle is

Observe that

( )2

2

12

))(12(11

),(r

bamn bm

mm

mb π

σχ

χξ =−+−= ∑∞

=

19

Observe that� perfect dielectric(nonabsorbent) sphere exhibits large oscillations for χ>1.� Hi absorbing and perfect conducting spheres show regularly damped oscillations.

Backscattering from metal sphereBackscattering from metal sphere

5.0nfor

||4 24

<=

χχξ Kb

� Rayleigh Region defined as

where,

20� For conducting sphere (| n|= ) 49 χξ =b∞

=Kwhere,

Scattering by HydrometeorsScattering by Hydrometeors

Hydrometeors (water particles)� In the case of water , the index of

refraction is a function of T & f. (fig 5.16)

� @T=20C − GHz 1 @ 25.9 j

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� For ice.� For snow, it’s a mixture of both above.

−−−

=−=GHz 300 @ 47.4.2

GHz 30 @ 5.22.4

GHz 1 @ 25.9

'''

j

j

j

jnnnw

78.1' =in

Liquid water refractivity, n’Liquid water refractivity, n’

22

Sphere pol signatureSphere pol signature

Co-pol

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Cross-pol

Sizes for cloud and rain dropsSizes for cloud and rain drops

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SnowflakesSnowflakes

� Snow is mixture of ice crystals and air

� The relative permittivity of dry snow

0=aρ 3g/cm3.005.0 ≤≤ sρ3g/cm 916.0=iρ

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� The relative permittivity of dry snow

� The Kds factor for dry snow

−−=−

''

'

'

'

2

1

3

1

dsi

ds

i

s

ds

ds

εεε

ρρ

εε

5.01.1 ≈≅

i

i

ds

ds KK

ρρ

2

1

+−=

i

iiK

εε

24

652

4

652 ||

4

D ||

D i

ods

osbbs KKr

λπ

λππξσ ≈==

Volume ScatteringVolume Scattering

� Two assumptions:– particles randomly distributed in volume--

incoherent scattering theory.

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incoherent scattering theory.– Concentration is small-- ignore shadowing.

� Volume Scattering coefficient is the total scattering cross section per unit volume.

rdrQrp ss ∫= )()(κ [Np/m]rdrrp bb ∫= )()( σκ222 / / / rrQrQ bbaass πσξπξπξ ===

DdDDN bb ∫== )()( σκη

Total number of drops per unit volumeTotal number of drops per unit volume

DdDNrdrpNv ∫∫ == )()( in units of mm-3

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oDDo

c

eNDN

earrp/

/

)(

)(−

−

=

= γαα

Volume ScatteringVolume Scattering

� It’s also expressed as

λ∫∞3

[Np/m]

χπ

λπξλπχ ddrrQr osso 2

and / , /2 2 ===Using...

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� or in dB/km units,

χχξχχπλκ ∫

∞

=0

,,2

2

3

,, )()(8

dp beso

bes

[dB/km]

[Np/m]

DdDDN bbdB

∫∞

×=Κ=0

3 )()(1034.4 ση

[s,e,bstand for scattering, extinction and backscattering.]

For Rayleigh approximationFor Rayleigh approximation

� Substitute eqs. 71, 74 and 79 into definitions of the cross sectional areas of a scatterer.

265

2 ||D 2

KrQππξ ==

29

24

652

322

24

2

||D

)Im(D

||3

wbb

waa

wss

Kr

KrQ

KrQ

λππξσ

λππξ

λπξ

==

−==

==

D=2r =diameter

Noise in Stratus cloud imageNoise in Stratus cloud image--scanning Kscanning Kaa--band radarband radar

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Volume extinction from cloudsVolume extinction from clouds

� Total attenuation is due to gases,cloud, and rain

� cloud volume extinction is (eq.5.98)epcega κκκκ ++=

∫∫ −== dDDKdDQ 32

}Im{πκ

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� Liquid Water Content LWC or mv )

� water density = 106 g/m3

∫∫ −== dDDKdDQ wo

ace3}Im{

λκ

∫∫ == dDDdrrm wv363

610

3

4 ππρ

=wρ

Relation with Cloud water contentRelation with Cloud water content

� This means extinction increases with cloud water content.

mκκ =

32

where

and wavelength is in cm.

][ )Im(6

434. 3111 mgdBkmK

o

−−−=λπκ

vce m1κκ =

Raindrops symmetryRaindrops symmetry

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Volume backscattering from CloudsVolume backscattering from Clouds

� Many applications require the modeling of the radar return.

� For a single drop65π

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� For many drops (cloud)

24

652 ||

D wbb Kr

λππξσ ==

ZK

dDKdDDN

w

wbvc

24

5

624

5

||

N(D)D||

)(

λπη

λπσση

=

==== ∫∫

Reflectivity Factor, ZReflectivity Factor, Z

� Is defined as

so that∫= dDDNZ )(D 6 ZK wo

vc2

4

5

||

λπσ =

35

� and sometimes expressed in dBZ to cover a wider dynamic range of weather conditions.

� Z is also used for rain and ice measurements.

ZdBZ log10=

Reflectivity in other references…Reflectivity in other references…

24

512 ||

10 ZKw

oλπη −=

36

36

1-

/mmmin expressed is

and cmin is where

Z

η

Reflectivity & Reflectivity FactorReflectivity & Reflectivity FactorR

efle

ctiv

ity, η

[cm

-1]

dB

Z fo

r 1

g/m

3

η Z (in dB)

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Ref

lect

ivity

,

dB

Z fo

r 1

g/m

Reflectivity and reflectivity factor produced by 1g/m3 liquid water Divided into drops of same diameter. (from Lhermitte, 2002).

Cloud detection vs. Cloud detection vs. frequencyfrequency

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Rain dropsRain drops

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Precipitation (Rain)Precipitation (Rain)

� Volume extinction

χχξχχπλκ ∫

∞

= 22

3

)()(8

dp eo

erbrR1κ= [dB/km]

40

� where Rr is rain rate in mm/hr� [dB/km] and b are given in Table 5.7� can depend on polarization since large

drops are not spherical but ~oblong.

χχξχχπ

κ ∫=0

2)()(

8dp eer

Mie coefficients

rR1κ=

1κ

[dB/km]

WW--band UMass CPRS radarband UMass CPRS radar

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Rain Rate [mm/hr]Rain Rate [mm/hr]

� If know the rain drop size distribution, each drop has a liquid water mass of

� total mass per unit area and time

wDm ρπ 3

6=

∫ ∫∞

42

� rainfall rate is depth of water per unit time

� a useful formula

∫= dDDDNDvR tr3)()(6/π

∫ ∫∞

=0

3 )()6/()()( dDvDNDdAdtdDDmDN tw πρ

[ ]4.88D)(-6.8D2

e-19.25)( +=Dvt

Volume Backscattering for RainVolume Backscattering for Rain

� For many drops in a volume, if we use Rayleigh approximation

∫ == dDσσ ZKdDK 25

625

||

D|| ππ =∫

43

� Marshall and Palmer developed

� but need Mie for f>10GHz.

∫ == dDbrvr σσ

ewvr ZK 24

5

||

λ

πσ =

6.1200 rRZ =

ZKdDK ww2

462

4|| D||

λλ=∫

Rain retrieval AlgorithmsRain retrieval AlgorithmsSeveral types of algorithms used to retrieve rainfa ll

rate with polarimetric radars; mainly � R(Zh), � R(Zh, Zdr)� R(Kdp)� R(Kdp, Zdr)where

band Xfor 5.40)(ˆ

band Sfor 62.11)(ˆ

85.0

937.0

dpdp

dpdp

KKR

KKR

=

=

44

where R is rain rate, Zh is the horizontal co-polar radar reflectivity factor , Zdr is the differential reflectivity Kdp is the differential specific phase shift a.k.a.

differential propagation phase, defined as

band Xfor 5.40)( dpdp KKR =

)(2

)()(

12

12

rr

rrK dpdp

dp −−

=φφ

Snow extinction coefficientSnow extinction coefficient

� Both scattering and absorption ( for f < 20GHz --Rayleigh)

[ ]dDQdDQ sase ∫∫ +×= 31034.4κ

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� for snowfall rates in the range of a few mm/hr, the scattering is negligible.

� At higher frequencies,the Mie formulation should be used.

� The is smaller that rain for the same R, but is higher for melting snow.

∫∫

seκ

SnowSnow Volume BackscatteringVolume Backscattering

� Similar to rain

sdsdsvs ZKdDK 24

562

4

5

||

D||

λπ

λπσ == ∫

46

sds

o

dsvs 44 λλ ∫

iss

s ZdDdDDNZ2

6i2

6s

1D

1)(D

ρρ=== ∫∫

Radar equation for MeteorologyRadar equation for Meteorology

� For weather applications

( )τσ

πλ 2

43

22

4−= e

R

GPP oot

r( )dr

R

o

epceg∫ ++= κκκτ

47

� for a volume

=22

2pcR

Vτβπ

( ) vpoot

rR

ecGPP σ

πτβλ τ

2

2222

432

−

=

Vvσσ =

Radar EquationRadar Equation

� For power distribution in the main lobe assumed to be

22

2

22

2ln1024 RL

LcGPP vrpoooot

r

σπ

τφθλ=

48

assumed to be Gaussian function.

2ln1024 Rπ

τ22

as here defined are losses catmospheriway - two theAnd−= eL

lossesreceiver and

tyreflectiviradar

where,

===

r

v

L

ησ