11

Microwave Interactions with Microwave Interactions with the Atmospherethe AtmosphereDr. Sandra Cruz PolDr. Sandra Cruz PolMicrowave Remote Sensing INEL 6669Microwave Remote Sensing INEL 6669Dept. of Electrical & Computer Engineering,Dept. of Electrical & Computer Engineering,UPRM, Mayagüez, PRUPRM, Mayagüez, PR

22

Atmosphere compositionAtmosphere compositionTypical Atmosphere in %

78

210.93

Ni

O2

Ar

Other components:

Carbon dioxide (CO2), Neon (Ne), Helium (He), Methane (CH4), Krypton (Kr), Hydrogen (H2) and Water vapor (highly variable)

Air Constituents Air Constituents in Troposphere and in Troposphere and StratosphereStratosphere

NN22 78.1%, O 78.1%, O22 20.9%, H 20.9%, H22O 0-2%O 0-2% Inert gases 0.938%Inert gases 0.938%Many of the least abundant have a disproportionally large Many of the least abundant have a disproportionally large

influence on atmospheric transmissioninfluence on atmospheric transmission.. COCO22 398ppm 398ppm absorbs 2.8, 4.3 & 15 absorbs 2.8, 4.3 & 15 mm CHCH44 1.7ppm 1.7ppm absorbs 3.3 & 7.8absorbs 3.3 & 7.8mm NN22O .35ppm O .35ppm absorbs 4.5, 7.8 & 17absorbs 4.5, 7.8 & 17mm OO33 ~10 ~10-8-8 absorbs UV-B, 9.6absorbs UV-B, 9.6mm CFClCFCl33, CF, CF22CLCL22 … … absorbs IRabsorbs IR

33

Atm. COAtm. CO22 Concentration Concentration

44

Last 200 years

55

66

Methane

77

88

HH22O is less than 2% yet has great O is less than 2% yet has great effect in climate & weathereffect in climate & weather

99

1010

Radiative Transfer in AtmosphereRadiative Transfer in Atmosphereduring Daytimeduring Daytime

During daytime only. Nighttime is another story

1111

Atm. Gases & Electromagnetic Atm. Gases & Electromagnetic propagationpropagation

Up to now, we have assumed lossless atm.Up to now, we have assumed lossless atm. For For 1 GHz< f< 15 GHz1 GHz< f< 15 GHz ~lossless ~lossless For higher frequencies, =>absorption bandsFor higher frequencies, =>absorption bands

H2O O2

•22.235 GHz•183.3 GHz•IR & visible

•50-70GHz•118.7GHz•IR & visible

1212

OutlineOutline

I. The atmosphere: composition, profileI. The atmosphere: composition, profileII. Gases: many moleculesII. Gases: many molecules

1. 1. ShapesShapes((G, VVW, LG, VVW, L): ): below 100GHz, up to 300GHzbelow 100GHz, up to 300GHz

e.g.e.g. HH22O , OO , O22

2. 2. Total AtmosphericTotal Atmospheric Absorption Absorption gg, , opacity opacity , , and atm-lossesand atm-losses L L

3. 3. TTBB: : Downwelling Emission by AtmosphereDownwelling Emission by Atmosphere

Sky Temp= cosmic + galaxySky Temp= cosmic + galaxy

1313

U.S. Standard AtmosphereU.S. Standard Atmosphere

Troposphere – clouds, weather

Stratosphere- no H2O or dustozone absorption of UV warms air to ~40oF

Mesosphereno aircrafts heretoo cold ~-90oF

Thermosphere(or Ionosphere) 1000-3000oF!

Tropopause

Stratopause

Mesopause

8/15km

P= 1013 mbars = 1013 HPaT= 300K

50/60km

95/120km

1414

Atmospheric ProfilesAtmospheric ProfilesUS Standard Atmosphere 1962US Standard Atmosphere 1962

TemperatureTemperature

Density in kg/mDensity in kg/m33

Pressure Pressure P= P= nRT/V=nRT/V=airairRT/M or PRT/M or Pooee-z/H-z/H33

height scale Pressure 7.7 where 3 kmH

km 320km2 )20(

km 20km 11 km 110

)(

)11(

)11(

zzTzTzazT

zTo

1/225.1)( Hzair ez

)]3.7/sin(3.01[225.1)( 3.7/ zez zair or

height scaledensity 5.9 where 1 kmH

Rair= 2.87

1515

Water Vapor ProfileWater Vapor Profile

Depends on factors like weather, seasons, time of the day.It’s a function of air temperature.•Cold air can’t hold water•Hot air can support higher humidities.(P dependence)

v(z)oe-z/H4 [g/m3]where o averages 7.72 in mid latitudesand the total mass of water vaportotal mass of water vapor in a in a columncolumn of unit cross section is of unit cross section is

40

)( HdzzM o

height scalevapor - water5.22between where 4 kmH

Relative HumidityRelative Humidity

Dew point Dew point temperature (dew=rocío)temperature (dew=rocío)– is the T below which the WV in a volume of humid is the T below which the WV in a volume of humid

air at a constant barometric P will condense into air at a constant barometric P will condense into liquid water.liquid water.

– Is the Is the TT as which as which fogfog forms forms Relative HumidityRelative Humidity

– When When TTairair is close to is close to TTdewdew => => high %RHhigh %RH Absolute HumidityAbsolute Humidity, the mass of water per unit , the mass of water per unit

volume of air. volume of air.

1616

Equations for RHEquations for RH

1717Where e = pressure and exp means exponential ex

Relative Humidity (RH) Relative Humidity (RH) simplified equationssimplified equations

1818

T is in Celsius

1919

Relative Humidity, RHRelative Humidity, RHvapor in airvapor in air

Air Air TemperatureTemperature

TT

Vapor air can Vapor air can holdhold

Actual Vapor in Actual Vapor in the airthe air

[gr per kg dry air][gr per kg dry air]

Relative Relative humidityhumidity

RHRH8686ooFF 27.627.6 10.8310.83 39%39%

7777ooFF 20.420.4 10.8310.83 53%53%

6868ooFF 14.914.9 10.8310.83 72%72%

5959ooFF 10.810.8 10.8310.83 100%100%

2020

Relative Humidity, RHRelative Humidity, RHdew Temperaturedew Temperature

Air Air TemperatureTemperature

TT

Dew Dew TemperatureTemperature

TTdpdp

Actual Vapor in Actual Vapor in the airthe air

[gr per kg dry air][gr per kg dry air]

Relative Relative humidityhumidity

RHRH

8686ooFF 6464ooFF 10.8310.83 39%39%

7777ooFF 6060ooFF 10.8310.83 53%53%

6868ooFF ooFF 10.8310.83 72%72%

5959ooFF ooFF 10.8310.83 100%100%

2121

Quantum of energyQuantum of energy

2222

EM interaction with MoleculesEM interaction with Molecules Total internal energy state for a moleculeTotal internal energy state for a molecule

– electronicelectronic energy corresponding to atomic level energy corresponding to atomic level– vibrationvibration of atoms about their equilibrium position of atoms about their equilibrium position – rotationrotation of atoms about center of molecule of atoms about center of molecule – EE = = EEee + + EEvv + + EErr

Bohr conditionBohr condition ff lm lm= (= (EEll – – EEmm ) /h ) /h Values for energy differences forValues for energy differences for

– electronicelectronic: : 22 to to 1010 eV eV– vibrational-rotationalvibrational-rotational: : 0.10.1 to to 22 eV eV– pure rotationalpure rotational: : 1010-4-4 to to 5 x 105 x 10-2-2 eV ( eV ( microwavesmicrowaves))

2323

Line ShapesLine Shapes

where,where,– SSlmlm is the line strength is the line strength

– F(f,fF(f,flmlm)) is the line shape is the line shape

LINE SHAPESLINE SHAPES– LorentzLorentz– GrossGross– Van-Vleck-WeisskoptVan-Vleck-Weisskopt

Abs

orpt

ion

frequency

frequency

One molecule

Many molecules:pressure broaden*

*caused by collision between molecules

2424

Line shapesLine shapes

LorentzLorentz

GrossGross

Van-Vleck-WeisskoptVan-Vleck-Weisskopt

22)(1),(

lmlmL ff

ffF

22222 4)(41),(

fffffffF

lm

lmlmG

2222

2

)()(1),(

ffffffffF

lmlmlmlmvw

2525

Absorption BandsAbsorption Bands Mainly water and oxygen for microwavesMainly water and oxygen for microwaves

Brig

htne

ss T

empe

ratu

re [K

]B

right

ness

Tem

pera

ture

[K]

Frequency [GHz]Frequency [GHz]

2727

Total AtmosphericTotal Atmospheric

AbsorptionAbsorptiongg, ,

Opacity Opacity , [Np] , [Np]

Loss factorLoss factor L L

[L en dB] [L en dB]

22 OOHg

o

e dzz

sec

sec)(0

0sec)(

secdzzg

o eeL

To convert from Np/km to dB/km multiply by 4.343 for 1-way propagation

2828

Atmospheric EmissionAtmospheric Emission For clear atmosphereFor clear atmosphere

wherewhere

Also there is some background radiationAlso there is some background radiation

TTcoscos=2.7K from the Big Bang and =2.7K from the Big Bang and TTgalgal~0 above 5GHz~0 above 5GHz

0

sec)',0( ')'()'(sec dzezTzT zaDN

0

)()',0( dzzz a

gallacticmicextra TTT cos

2929

AvirisAviris

Latent Heat Latent Heat – to understand radiation budget – to understand radiation budget need to monitor water content in atmosphereneed to monitor water content in atmosphere

3131

Scattering from Scattering from Hydrometeors:Hydrometeors:

Clouds, Snow, RainClouds, Snow, Rain

3232

Outline: Clouds & RainOutline: Clouds & Rain1.1. Single sphere (Single sphere (Mie vs. RayleighMie vs. Rayleigh))2.2. Sphere of rain, snow, & ice (Sphere of rain, snow, & ice (HydrometeorsHydrometeors))

Find their Find their cc, n, ncc, , bb

3.3. Many spheres together : Clouds, Rain, SnowMany spheres together : Clouds, Rain, Snowa. Drop size distributiona. Drop size distributionb. Volume Extinction= Scattering+ Absorptionb. Volume Extinction= Scattering+ Absorptionc. Volume Backscatteringc. Volume Backscattering

Radar Equation Radar Equation forfor Meteorology Meteorology TTBB Brightness by Clouds & Rain Brightness by Clouds & Rain

3333

Clouds Types on our Clouds Types on our AtmosphereAtmosphere

3434

%

Cirrus Clouds Composition

3535

EM interaction with EM interaction with Single Spherical Particles Single Spherical Particles

Absorption Absorption – Cross-Section, Cross-Section, QQa a =P=Pa a /S/Sii

– Efficiency, Efficiency, aa== QQa a //rr22

Scattered Scattered – Power, Power, PPs s

– Cross-section , Cross-section , QQs s =P=Ps s /S/Sii

– Efficiency,Efficiency, ss== QQs s //rr22

Total power removedTotal power removed by sphere from the incident EM by sphere from the incident EM wave, wave, e e = = ss+ + aa

BackscatterBackscatter, , SSss(() = S) = Siibb/4/4RR22

Si

Definitions:

3636

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

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

– Speed ratio on both media:Speed ratio on both media:

coλπrr

2 2

p

oc

cb

cp

b kj

nn

n )( p

[Index of Refraction and [Index of Refraction and RefractivityRefractivity]]

The Propagation constant The Propagation constant depends on the relative depends on the relative complex permittivitycomplex permittivity

Where the index of Where the index of refraction is refraction is

But But n’n’airair≅≅1.0003 1.0003 So we define So we define NN

3737

So… So…

Propagation in terms of Propagation in terms of NN is is

And the attenuation and And the attenuation and phase is phase is

And the power density carried And the power density carried by wave traveling in the by wave traveling in the zz--direction is :direction is :

– With With ff in GHz in GHz

3939

Mie SolutionMie Solution

Mie solutionMie solution

Where Where aamm & b & bmm are the are the Mie coefficientsMie coefficients given by given by eqs 8.33a to 8.33b in the textbook.eqs 8.33a to 8.33b in the textbook.

}Re{)12(2),(

)|||)(|12(2),(

12

2

1

22

mm

ma

mm

ms

bamn

bamn

4040

Mie coefficientsMie coefficients

"'

1

1

1

1

cossin

}Re{}Re{

}Re{}Re{

jnnn

jWwhere

WWmnA

WWmnAb

WWmnA

WWmnA

a

o

mmm

mmm

m

mmm

mmm

m

coλπrr

2 2

p

oc

cb

cp

b kj

nn

n )( p

4141

Non-absorbing Non-absorbing sphere or dropsphere or drop

((nn””==0 for 0 for a a perfect dielectricperfect dielectric, ,

which is awhich is anon-absorbingnon-absorbing sphere) sphere)

oook

kjjnnn

call

o

)("'

Re

=.06

Rayleigh region |n|<<1

4242

Conducting (absorbing) sphereConducting (absorbing) sphere

=2.4

4343

Plots of Mie Plots of Mie ee versus versus

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

Optical limit (Optical limit (r r >>>>) is ) is ee =2. =2. Crossover Crossover forfor

– Hi conducting sphere at Hi conducting sphere at =2.4=2.4

– Weakly conducting sphere is at Weakly conducting sphere is at =.06=.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

4444

Rayleigh Approximation |Rayleigh Approximation |nn|<<1|<<1 Scattering efficiencyScattering efficiency

Extinction efficiencyExtinction efficiency

where K is the where K is the dielectric factordielectric factor

...||38}Im{4 24 KKe

...||38 24 Ks

21

21

2

2

c

c

nnK

4545

Absorption efficiency in Absorption efficiency in Rayleigh regionRayleigh region

esea K }Im{4

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

|n|<<1

4646

Scattering from HydrometeorsScattering from Hydrometeors

Rayleigh Scattering Mie Scattering

>> particle size comparable to particle size--when rain or ice crystals are present. 33GHz (9mm)

95GHz (3mm)

4747

Rayleigh scattering (λ >d)

Mie scattering (λ ~ d)

Rayleigh Approximation Rayleigh Approximation for ice crystalsfor ice crystals

4848

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

Scattering cross sectionScattering cross section

Absorption cross sectionAbsorption cross section

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

][m ||32 226

2

KQs

][m }Im{ 232

KQa

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

4949

Gas molecules = much smaller than visible => Rayleigh approx. is OK.

Red 700nm

Violet 400nm

5050

Mie ScatteringMie Scattering

Mie scatt. is almost independent of frequencyMie scatt. is almost independent of frequency Cloud droplets ~20mm compare to 500nmCloud droplets ~20mm compare to 500nm Microwaves have Microwaves have ~cm or mm (large) – Rayleigh for most ~cm or mm (large) – Rayleigh for most

atmospheric constituentsatmospheric constituents Laser have Laser have ~nm - Mie~nm - Mie

dependent] [almost independent]

5151

Observe scattering in Visible EMObserve scattering in Visible EM; ; forward scattering vs. backscatteringforward scattering vs. backscattering

Mie scattering by dust particles and aerosols

Rayleigh scattering by water vapor molecules and gases.

5252

5353

Mie forward scattering nos impide ver bien a menos que haya alto contraste.

5454

5555

5656

5757

Rayleigh-Mie-GeometricOpticsRayleigh-Mie-GeometricOptics Along with absorption, scattering is a major cause of the Along with absorption, scattering is a major cause of the

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

to the wavelength (to the wavelength (d/d/) of the radiation.) of the radiation. When this ratio is less than about one-tenth (When this ratio is less than about one-tenth (d/d/), ), RayleighRayleigh

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

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

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

5858

Mie Scattering Mie Scattering (necessary if d/(necessary if d/), ), Mie theory : A complete mathematical-physical theory Mie theory : A complete mathematical-physical theory

of the scattering of electromagnetic radiation by of the scattering of electromagnetic radiation by spherical particles, developed by G. Mie in 1908. spherical particles, developed by G. Mie in 1908.

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

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

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

5959

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

spherical particle isspherical particle is

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

2

2

12 ))(12(11),(

rbamn b

mm

mm

b

Backscattering from metal Backscattering from metal spheresphere

5.0nfor

||4 24

Kb

Rayleigh Region defined asRayleigh Region defined as

For conducting sphere (|For conducting sphere (|nn|= )|= )

49 b

Where we define the Dielectric Factor:

6161

Scattering by HydrometeorsScattering by HydrometeorsHydrometeors (water particles)Hydrometeors (water particles) In the case of In the case of waterwater, the index of refraction is a function , the index of refraction is a function

of of T & fT & f. (fig 5.16). (fig 5.16)

@T=20C@T=20C

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

GHz 300 @ 47.4.2GHz 30 @ 5.22.4

GHz 1 @ 25.9'''

jjj

jnnnw

78.1' in

6262

Liquid water refractivity, nLiquid water refractivity, n’’

6363

Liquid water refractivity, nLiquid water refractivity, n””

6464

Sphere pol signatureSphere pol signature

Co-pol

Cross-pol

6565

Mie Efficiency at 3GHz and 30GHzMie Efficiency at 3GHz and 30GHz

6666

At 300GHzAt 300GHz

6767

6868

Sizes for cloud and rain dropsSizes for cloud and rain drops

6969

SnowflakesSnowflakes

Snow is mixture of ice crystals and airSnow is mixture of ice crystals and air

The relative permittivity of The relative permittivity of dry snowdry snow

The The KKdsds factor for dry snow factor for dry snow

0a3g/cm3.005.0 s

''

'

'

'

21

31

dsi

ds

i

s

ds

ds

5.01.1

i

i

ds

ds KK

21

i

iiK

24

652

4

652 ||

4D ||D i

ods

osbbs KKr

7070

Volume ScatteringVolume Scattering

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

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

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

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

DdDDN bb )()(

7171

Total number of drops per unit Total number of drops per unit volumevolume

DdDNrdrpNv )()(

oDDo

c

eNDN

earrp/

/

)(

)(

in units of mm-3

7272

Volume ScatteringVolume Scattering

ItIt’’s also expressed ass also expressed as

or in dB/km units,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.]

24

652

322

24

652

||D

)Im(D

||3

D 2

wbb

waa

wss

Kr

KrQ

KrQ

7373

For Rayleigh approximationFor Rayleigh approximation

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

D=2r =diameter

7474

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

7575

Volume extinction from cloudsVolume extinction from clouds Total attenuation is due to gases,cloud, and rainTotal attenuation is due to gases,cloud, and rain

cloud volume extinction is (eq.5.98)cloud volume extinction is (eq.5.98)

Liquid Water Content Liquid Water Content LWCLWC or or mmvv ) )

water density = 10water density = 1066 g/m g/m33

epcega

dDDKdDQ wo

ace3

2

}Im{

dDDdrrm wv363

610

34

w

7676

Relation with Cloud water Relation with Cloud water contentcontent

This means extinction increases with cloud This means extinction increases with cloud water content.water content.

wherewhere

and wavelength is in cm.and wavelength is in cm.

][ )Im(6434. 3111 mgdBkmK

o

vce m1

7777

Raindrops symmetryRaindrops symmetry

Differential Reflectivity

Zdr

7878

Volume backscattering from Volume backscattering from CloudsClouds

Many applications require the modeling of the Many applications require the modeling of the radar return.radar return.

For a single dropFor a single drop

For many drops (cloud)For many drops (cloud)

24

652 ||D wbb Kr

ZK

dDN(D)DKdDDN

w

wbvc

24

5

624

5

||

|| )(

7979

Reflectivity Factor, ZReflectivity Factor, Z Is defined asIs defined as

so thatso that

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

Z is also used for rain and ice measurements.Z is also used for rain and ice measurements.

dDDNZ )(D6 ZKwo

vc2

4

5

||

8080

Reflectivity in other booksReflectivity in other books

36

1-

24

512

/mmmin expressed is

and cmin is where

|| 10

Z

ZKwo

8181

Reflectivity & Reflectivity FactorReflectivity & Reflectivity FactorR

efle

ctiv

ity,

[cm

-1]

dBZ

for 1

g/m

3

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

Z (in dB)

8282

Cloud detection vs. Cloud detection vs. frequencyfrequency

8383

Rain dropsRain drops

8484

Precipitation (Rain)Precipitation (Rain)

Volume extinctionVolume extinction

where where RRrr is rain rate in mm/hr is rain rate in mm/hr [dB/km] and [dB/km] and bb are given in Table 5.7 are given in Table 5.7 can depend on polarization since large drops are can depend on polarization since large drops are

not spherical but ~oblong.not spherical but ~oblong.

0

22

3

)()(8

dp eo

er

Mie coefficients

brR1

1

[dB/km]

8585

W-band UMass CPRS radarW-band UMass CPRS radar

8686

Rain Rate [mm/hr]Rain Rate [mm/hr] If know the rain drop size distribution, each drop If know the rain drop size distribution, each drop

has a liquid water mass of has a liquid water mass of

total mass per unit area and timetotal mass per unit area and time

rainfall rate is depth of water per unit timerainfall rate is depth of water per unit time

a useful formulaa useful formula dDDDNDvR tr

3)()(6/

wDm 3

6

0

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

4.88D)(-6.8D2

e-19.25)( Dvt

ZKdDK ww2

4

562

4

5

|| D||

8787

VolumeVolume Backscattering for Backscattering for RainRain

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

Marshall and Palmer developedMarshall and Palmer developed

but need Mie for but need Mie for ff>10GHz.>10GHz.

dDbrvr

ewvr ZK 24

5

||

6.1200 rRZ

8888

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

with polarimetric radars; mainly with polarimetric radars; mainly R(ZR(Zhh), ), R(ZR(Zhh, Z, Zdrdr)) R(KR(Kdpdp)) R(KR(Kdpdp, Z, Zdrdr))where where RR is rain rate, is rain rate, ZZhh is the horizontal co-polar radar reflectivity factor, is the horizontal co-polar radar reflectivity factor, ZZdrdr is the is the differential reflectivitydifferential reflectivity KKdpdp is the is the differentialdifferential specific specific phasephase shift a.k.a. shift a.k.a.

differential propagation phase, defined asdifferential propagation phase, defined as

band Xfor 5.40)(ˆband Sfor 62.11)(ˆ

85.0

937.0

dpdp

dpdp

KKR

KKR

)(2)()(

12

12

rrrr

K dpdpdp

8989

Snow extinction coefficientSnow extinction coefficient

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

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

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

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

dDQdDQ sase 31034.4

se

9090

SnowSnow Volume Backscattering Volume Backscattering

Similar to rainSimilar to rain

sdso

dsvs ZKdDK 24

562

4

5

|| D||

iss

s ZdDdDDNZ 26

i26

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Radar equation for MeteorologyRadar equation for Meteorology For weather applicationsFor weather applications

for a volume for a volume

2

43

22

4 e

RGPP oot

r drR

oepceg

22

2pcRV

vpoot

r RecGP

P

2

2222

432

Vv

9292

Radar EquationRadar Equation

For power distribution For power distribution in the main lobe in the main lobe assumed to be assumed to be Gaussian function.Gaussian function. 2

22

22

2ln1024 RL

LcGPP vrpoooot

r

22

as here defined are losses catmospheriway - two theAndeL

lossesreceiver andtyreflectiviradar

where,

r

v

L

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Radar EquationRadar Equation222

22

2ln1024 RLLcGP

P v

recatm

pooootr

RLLc

GPP

atmrec

prad

orad

oodBo

dBt

dBr

log20)2ln1024log(10log20log10log10log10

log10)log(10)log(10)log(2022

RRPP dBcp

dBt

dBr log20log10log10

RcdB=radar constant (including atmospheric

attenuation)

For calibrated target