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

Microwave Remote Sensing INEL 6069Sandra Cruz PolProfessor, Dept. of Electrical & Computer Engineering,UPRM, Mayagüez, PR

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Outline: Clouds & RainOutline: Clouds & Rain

1. Single sphere (Mie vs. Rayleigh)

2. Sphere of rain, snow, & ice (Hydrometeors) Find their c, nc, b

3. Many spheres together : Clouds, Rain, Snowa. Drop size distribution

b. Volume Extinction= Scattering+ Absorption

c. Volume Backscattering

4. Radar Equation for Meteorology

5. TB Brightness by Clouds & Rain

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Clouds Types on our AtmosphereClouds Types on our Atmosphere

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0

10

20

30

40

50

60

70

Ice Crystals

hexagonalplatesbullet rosettes

dendrites

others

%

Cirrus Clouds Composition

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EM interaction with EM interaction with Single Spherical ParticlesSingle Spherical Particles

Absorption – Cross-Section, Qa =Pa /Si

– Efficiency, a= Qa /r2

Scattered – Power, Ps

– Cross-section , Qs =Ps /Si

– Efficiency, s= Qs /r2

Total power removed by sphere from the incident EM wave, e = s+ a

Backscatter, Ss() = Sib/4R2

Si

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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. :

– Speed ratio on both media:

b

2

r

bn

nn p

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Mie SolutionMie Solution

Mie solution

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

}Re{)12(2

),(

)|||)(|12(2

),(

12

2

1

22

mm

ma

mm

ms

bamn

bamn

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Mie coefficientsMie coefficients

"'

1

1

1

1

cossin

}Re{}Re{

}Re{}Re{

jnnn

jWwhere

WWm

nA

WWm

nA

b

WWm

n

A

WWm

n

A

a

o

mmm

mmm

m

mmm

mmm

m

coλ

πrr 2

2

p

oc

cb

cp

b k

j

n

nn

)( p

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Non-absorbing Non-absorbing sphere or dropsphere or drop((n”=n”=0 for 0 for a a perfect dielectricperfect dielectric, , which is awhich is anon-absorbingnon-absorbing sphere) sphere)

oook

k

jjnnn

call

o

)("'

Re

=.06

Rayleigh region |n|<<1

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Conducting (absorbing) sphereConducting (absorbing) sphere

=2.4

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Plots of Mie Plots of Mie ee versus versus

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

Optical limit (r >>) is e =2.

Crossover for – Hi conducting sphere at =2.4

– Weakly conducting sphere is at =.06

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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Rayleigh Approximation |Rayleigh Approximation |nn|<<1|<<1

Scattering efficiency

Extinction efficiency

where K is the dielectric factor

...||3

8}Im{4 24 KKe

...||3

8 24 Ks

2

1

2

12

2

c

c

n

nK

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Absorption efficiency in Rayleigh Absorption efficiency in Rayleigh regionregion

esea K }Im{4

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

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Scattering from HydrometeorsScattering from Hydrometeors

Rayleigh Scattering Mie Scattering

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

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Single Particle Cross-sections vs.Single Particle Cross-sections vs.

Scattering cross section

Absorption cross section

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

][m ||3

2 2262

KQs

][m }Im{ 232

KQa

as

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

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Rayleigh-Mie-GeometricOpticsRayleigh-Mie-GeometricOptics 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/) of the radiation. When this ratio is less than about one-tenth (d/), 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/), the laws of geometric optics begin to apply.

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Mie Scattering Mie Scattering (d/(d/), ),

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 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/ 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.

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Backscattering Cross-sectionBackscattering Cross-sectionFrom Mie solution, the backscattered field by a

spherical particle is

Observe that perfect dielectric

(nonabsorbent) sphere

exhibits large

oscillations for >1. Hi absorbing and perfect

conducting spheres show

regularly damped oscillations.

2

2

12

))(12(11

),(r

bamn bm

mm

mb

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Backscattering from metal sphereBackscattering from metal sphere

5.0nfor

||4 24

Kb

Rayleigh Region defined as

For conducting sphere (|n|= )49 b

Kwhere,

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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

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

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

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Sphere pol signatureSphere pol signature

Co-pol

Cross-pol

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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

The Kds factor for dry snow

0a3g/cm3.005.0 s

''

'

'

'

2

1

3

1

dsi

ds

i

s

ds

ds

3g/cm 916.0i

5.01.1

i

i

ds

ds KK

2

1

i

iiK

24

652

4

652 ||

4

D ||

D i

ods

osbbs KKr

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Volume ScatteringVolume Scattering

Two assumptions:– particles randomly distributed in volume--

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

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

rdrQrp ss )()( [Np/m]rdrrp bb )()( 222 / / / rrQrQ bbaass

DdDDN bb )()(

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

DdDNrdrpNv )()(

oDDo

c

eNDN

earrp/

/

)(

)(

in units of mm-3

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Volume ScatteringVolume Scattering

It’s also expressed as

or in dB/km units,

0

,,2

2

3

,, )()(8

dp beso

bes

[dB/km]

[Np/m]

DdDDN bbdB

0

3 )()(1034.4

ddrrQr o

sso 2 and / , /2 2 Using...

[s,e,b stand for scattering, extinction and backscattering.]

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For Rayleigh approximationFor Rayleigh approximation

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

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652

322

24

652

||D

)Im(D

||3

D 2

wbb

waa

wss

Kr

KrQ

KrQ

D=2r =diameter

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

Liquid Water Content LWC or mv )

water density = 106 g/m3

epcega

dDDKdDQ wo

ace3

2

}Im{

dDDdrrm wv363

610

3

4

w

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Relation with Cloud water contentRelation with Cloud water content

This means extinction increases with cloud water content.

where

and wavelength is in cm.

][ )Im(6

434. 3111 mgdBkmK

o

vce m1

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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 drop

For many drops (cloud)

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652 ||

D wbb Kr

ZK

dDKdDDN

w

wbvc

24

5

624

5

||

N(D)D||

)(

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Reflectivity Factor, ZReflectivity Factor, Z

Is defined as

so that

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

Z is also used for rain and ice measurements.

dDDNZ )(D6 ZKwo

vc2

4

5

||

ZdBZ log10

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Reflectivity in other references…Reflectivity in other references…

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1-

24

512

/mmmin expressed is

and cmin is where

||

10

Z

ZKwo

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Reflectivity & Reflectivity FactorReflectivity & Reflectivity FactorR

efle

ctiv

ity,

[cm

-1]

dBZ

for

1g/

m3

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

Z (in dB)

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Cloud detection vs. Cloud detection vs. frequencyfrequency

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Rain dropsRain drops

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Precipitation (Rain)Precipitation (Rain)

Volume extinction

where Rr is rain rate in mm/hr

[dB/km] and b are given in Table 5.7can depend on polarization since large drops

are not spherical but ~oblong.

0

22

3

)()(8

dp eo

er

Mie coefficients

brR1

1

[dB/km]

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W-band UMass CPRS radarW-band 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

rainfall rate is depth of water per unit time

a useful formula

dDDDNDvR tr3)()(6/

wDm 3

6

0

3 )()6/()()( dDvDNDdAdtdDDmDN tw

4.88D)(-6.8D2

e-19.25)( Dvt

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Volume Backscattering for RainVolume Backscattering for Rain

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

Marshall and Palmer developed

but need Mie for f>10GHz.

dDbrvr

ewvr ZK 24

5

||

6.1200 rRZ

ZKdDK ww2

4

562

4

5

||

D||

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Rain retrieval AlgorithmsRain retrieval AlgorithmsSeveral types of algorithms used to retrieve rainfall

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

band Sfor 62.11)(ˆ

85.0

937.0

dpdp

dpdp

KKR

KKR

)(2

)()(

12

12

rr

rrK dpdp

dp

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Snow extinction coefficientSnow extinction coefficient

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

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.

dDQdDQ sase 31034.4

se

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SnowSnow Volume Backscattering Volume Backscattering

Similar to rain

sds

o

dsvs ZKdDK 24

562

4

5

||

D||

iss

s ZdDdDDNZ2

6i2

6s

1D

1)(D

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Radar equation for MeteorologyRadar equation for Meteorology

For weather applications

for a volume

2

43

22

4 e

R

GPP ootr dr

R

o

epceg

22

2pcR

V

vpoot

rR

ecGPP

2

2222

432

Vv

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Radar EquationRadar Equation

For power distribution in the main lobe assumed to be Gaussian function.

22

2

22

2ln1024 RL

LcGPP vrpooootr

22

as here defined are losses catmospheriway - two theAndeL

lossesreceiver and

tyreflectiviradar

where,

r

v

L