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8/14/2019 ET 4357 Lecture 2 Particle Scattering Spheres
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
1
Particle scattering:spheres
Observation Systems
Lecture 2
8/14/2019 ET 4357 Lecture 2 Particle Scattering Spheres
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Source: www.everythingweather.com
The received power is related tot rainfall rate.
The question is: how?
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Question 1: what is the definition of rainfall rate?
( ) ( ) ( )R N D v D Vol D dD
Answer: the volume of rain watercollected per unit of areaand per unit of time
Fall speed of raindrop
Volume of raindrop
Dropsize distribution
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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What is the dropsize distribution?
( )N D dD
the number of raindrops with a diameter between
D dDD and
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Taken fromMatzler, Univ. Bern
Fall speed of raindrops depends on size and air pressure
( ) 10.3 9.75exp( 0.6 )v D D
Fall speed at sea level
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Drop volume follows from size
Fall speed follows from data and model
Remains:
What is the dropsize distribution?
Radar can give this information
( ) ( ) ( )R N D v D Vol D dD
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Source: www.everythingweather.com
Describe scattering by one particle
Estimate scattering by ensemble of particles
Calculate radar observables
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Scattering by one particle
IncidentEM-wave
Radiation pattern scattered wave
Side scatter
Forward scatter
Radar: back, side scatterPropagation: forward
mono bi static
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Definitions to describe scattering by one particle
Incident power density
Si [W/m2]
Scattered power PsAbsorbed power Pa
Absorption cross-section ai
P
a SQ
Scattering cross-sections
i
P
s SQ Extinction cross-section e a sQ Q Q
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Scattering in all directionsRadar: backscattering only
Definitions to describe scattering by one particle:
ss
Q
A aa
Q
A ee
Q
A
Efficiency factors
Cross-section of particle
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Definition radar backscattering cross-section
2
4
i br
SS
R
R
iS r i b
P S
b
Radar cross-section (RCS):cross-section of equivalent isotropic radiator with power Pr
b
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Two basic types of particles
spheres
spheroids
Mie-theoryRayleigh approximation
Modified Mie-theoryRayleigh approximationPolarization dependence
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Scattering by a homogeneous dielectric spherein a medium
Background medium
2r
Normalized radius
'2 2
o
r r
' ''j
2
nn
permittivity
refractive index
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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The formulas for Mie-scattering
2 2
2
2( , ) (2 1)( )s l l
ln l a b
2
2( , ) (2 1)(Re( ))e l l
l
n l a b
( , ) ( , ) ( , )a e sn n n
2
2 2
1
( , ) ( 1) (2 1)( )lb
e l l
ln l a br
al, bl:Bessel, Hankel functions
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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The Mie-formulation is exact for all particle sizes and wavelengthsbut quite intractable,
therefore: approximations!
Rayleigh approximation
Particle small compared to wavelengthSmall phase shift of wave inside particle
1n
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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25226 6
4
2 2
3 3s
KQ K D
2522 6 6
4b
KQ K D
2 2 33
3 Im( ) Im( )a
DQ K K
2
21 12 2
nKn
Real permittivity:no absorption
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Physical interpretation Rayleigh approximation
Iz
Iy r
24 2
2
cos ( )yparallel
I kI
r
24
2
z
perpendicular
I k
I r
Excited dipoles
plane of reference
perpendicular
parallel
Polarizibility of particle (~volume)Unpolarized wave:
24 22
1 cos ( )
2
oparallel perpendicular
unpolarized
I kI II
r
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Physical interpretation Rayleigh approximation
Dipole field
Side view
Helicopter view
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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r
cosr r
E1
E2
1 2 1 1 exp( )totE E E E j
10 2
totE E
0
0
2 r
02
cosr
2 1 cosr
Independent of separation between particles
Forward scattering is always coherent: constructive interferenceScattering in other directions is always (partially) incoherent:(partially) destructive interference
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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bluered
The sky is blue
because of this
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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rayleigh resonance
optical
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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0.3 mm
3 mm
30 mm
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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0.3 mm 3 mm
30 mm
6D
decade
3Ddecade
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Source: www.everythingweather.com
Estimate scattering by ensemble of particles
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Scattering by N particles
( )
1 1
( ) ( ) nN N j t
tot n n
n n
E t E t e
R
rn
2 2n
n
r vt t
E0
En
Doppler shift
Er
22
, 0 2
vj t
r n nr RE t E t ec
dR
dR R
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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2
0
0ijT N N
j f t
i i j
dt e
2 ijf t
Incoherent addition requires:
large Nsufficient integration time
2 21 1
2 2
N
tot n
n
P E
Total power is sum of powers of signals from individual particles
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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2 2
,
1 1
2 2
N N
tot n b nn nP E Q
P is proportional to sum of backscatter cross sections of individual particles,in case of Rayleigh scattering:
25
6
, , 4
N N
b tot b n n
n n
KQ Q D
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Definition of radar reflectivity factor:
46
,25
N
b tot n
n
Z Q DK
Or in integral form
max
min
6( )D
DZ N D D dD
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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max
min
6
( )
D
DZ N D D dD
( ) ( ) ( )R N D v D Vol D dD
Observed:
Wanted
( )N D
Needed
A parametrization of the dropsize distribution
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Based on field experiments 0( ) exp( )N D D DN
Three unknowns, one observation: fix twoparameters. For instance:
6 4
0 8 10 [ ]N m 0
Marshall-Palmer distribution for stratiform rain0
3.67
D
0.21
0
4.1
3.67
D R
Other values for other rain typesmedian
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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max
min
6 6
0
0
3.67
( ) exp
D
DZ N D D dD N D dD f RD
200 1.6Z R
Approximated by power law
But the coefficients can change from rain type to rain type
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Radar not only for rain.
Also ice particles: Mie/Rayleighcalculations with differentrefractive index and size distribution
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Error overview
Stratiform rain
rain
Snow/ice
melting snow
Remember:fall speed of rain drops depends on air pressure (height)
Z-R relationship not valid anymore
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November 16, 2013 Observation Technology: scattering by spherical particles
International Research Centre for Telecommunications and Radar
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Error overview Attenuation of radar waves
2522 6 6
4b
KQ K D
2 2 33
3 Im( ) Im( )aD
Q K K
Higher frequency:more absorption andmore backscattering
1
1.6 1 11.6 1.6
1.6exp 2 200
exp 2 exp 2200
observed true
trueestimated true
Z Z r R
ZR r R r
Above 3 GHz attenuation becomes significant
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November 16, 2013 Observation Technology: scattering by spherical particles 41
Error overview Model error
11
Z R R R
ZZ
RZ R R R
R
1R Z
R Z
1.5
2
3
R Z
R Z
If required accuracy of R ~ 10 %,
then needed accuracy of Z~15%