ET 4357 Lecture 2 Particle Scattering Spheres

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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Particle scattering:spheres

Observation Systems

Lecture 2


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Source: www.everythingweather.com

The received power is related tot rainfall rate.

The question is: how?


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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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What is the dropsize distribution?

( )N D dD

the number of raindrops with a diameter between

D dDD and


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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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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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Describe scattering by one particle

Estimate scattering by ensemble of particles

Calculate radar observables


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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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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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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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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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Two basic types of particles

spheres

spheroids

Mie-theoryRayleigh approximation

Modified Mie-theoryRayleigh approximationPolarization dependence


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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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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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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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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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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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Physical interpretation Rayleigh approximation

Dipole field

Side view

Helicopter view


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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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bluered

The sky is blue

because of this


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rayleigh resonance

optical


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0.3 mm

3 mm

30 mm


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0.3 mm 3 mm

30 mm

6D

decade

3Ddecade


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Estimate scattering by ensemble of particles


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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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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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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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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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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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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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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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Radar not only for rain.

Also ice particles: Mie/Rayleighcalculations with differentrefractive index and size distribution


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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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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

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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%

ET 4357 Lecture 2 Particle Scattering Spheres

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