1 scattering from hydrometeors: clouds, snow, rain microwave remote sensing inel 6069 sandra cruz...
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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