Radiative Transfer Dr. X-Pol Microwave Remote Sensing INEL 6669 Dept. of Electrical & Computer Engineering, UPRM, Mayagüez, PR

Radiative TransferRadiative TransferDr. X-PolMicrowave Remote Sensing INEL 6669Dept. of Electrical & Computer Engineering,UPRM, Mayagüez, PR

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Theory of Radiative Transfer– Extinction &Emission– Equation of Transfer

TAP of absorbing/scattering Media TAP of atmosphere & Terrain:

– upwelling and downwelling Emission and Scattering by Terrain Homogeneous terrain medium with

– uniform T profile– non-uniform T&r profile: coherent & incoherent approach

Emissivity of dielectric Slab Emissivity of Rough surface

Radiative TransferRadiative Transfer

Some energy is transmitted

some is scattered, some is absorbed

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– upwelling-down-welling Emission and Scattering by Terrain Homogeneous terrain medium with



Radiative Transfer :Radiative Transfer :Extinction Extinction Interaction between radiation and matterInteraction between radiation and matter

Extinction (Change due to energy loss)

dIextin= e I(r,r’)dr two processes: absorbtion & scattering

e= a + s in Nepers/m

where e is the

extinction or power

attenuation coefficient,

it’s due to absorption and

scattering away

in other direction.

dA

dr

Volume of some matter

Incident

radiatio

n

exiting

radiatio

n

Radiative Transfer : Radiative Transfer : EmissionEmission Interaction between radiation and matterInteraction between radiation and matter

Emission (Change due to energy gained)

dIemit= (aJa + sJs )dr

two processes or mechanisms: emit & scatter

[Ja & Js account for thermal emission & scattering]

Introduce the scattering albedo, a =s/e

dIemit= [(esJa + s Js )]

dIemit= e [(aJa + aJs )]= eJdr

where J = (aJa + aJs

dA

dr

Js

Ja

albedoalbedoThe retreat of sea ice in the Arctic Ocean is diminishing Earth's albedo, or reflectivity, by an amount considerably larger than previously estimated, according to a new study that uses data from instruments that fly aboard several NASA satellites.

The study, conducted by researchers at Scripps Institution of Oceanography, at the University of California, San Diego, uses data from the Clouds and Earth's Radiant Energy System, or CERES, instrument. There are CERES instruments aboard NASA's Tropical Rainfall Measurement Mission, or TRMM, satellite, Terra, Aqua and NASA-NOAA's Suomi National Polar-orbiting Partnership (Suomi NPP) satellites. The first CERES instrument was launched in December of 1997 aboard TRMM.As the sea ice melts, its white reflective surface is replaced by a relatively dark ocean surface. This diminishes the amount of sunlight being reflected back to space, causing Earth to absorb an increasing amount of solar energy.The Arctic has warmed by 3.6 F (2 C) since the 1970s.

http://www.sciencedaily.com/releases/2014/02/140219115110.htm

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Radiative Transfer : Radiative Transfer : Equation of Transfer IEquation of Transfer I

dI= I(r+dr) - I(r)

= dIemit -dIextin

=e (J-I)dr

= (J-I)d where

d = e dr is the optical depth,

therefore,

and is the optical thickness or opacity in Np.

dA

dr

B(r)

B(r+dr)

drrrr

r e2

1

),( 21

Radiative Transfer : Radiative Transfer : Equation of Transfer IIEquation of Transfer II

dI=(J-I)dor dI/d + I = J

– Multiplying by e(0,r’)

– and integrating from 0 to r.r

Horizontallayers(stratified atm)

B(0)

B(r)

r’

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TTAPAP of absorbing/scattering Media (1) of absorbing/scattering Media (1)

In the microwave region, where R-J applies, there’s a T to I

Similarly, under thermodynamic equilibrium (emission = absorption) the absorption source function is [Recall J = (aJa + aJs ]

where T is the physical temperature of the medium.


Similarly, for the scattering source function, Js

where is the phase function and accounts for the portion of incident

radiation scattered from direction ri into direction r,

where we have defined a scattered radiometric temperature as,

iiAPisc drTrrrT )():(4

1)(

4

In eq. 6.24 Ulaby & Long, the multiply by ½


In terms of temperature,

For scatter-free medium:• s(= ae)=0•then a= 0 and e= a

•and the opacity is

sa aJJaJ )1(

e= a + s

Brightness temperature Brightness temperature

Define the 1-way atmospheric transmissivity:

TTAPAP of absorbing/scattering Media III of absorbing/scattering Media III

Ex. For scatter-free medium: An airborne radiometer measuring ice.

TAP (0)= Tice

e-(0,H) = attenuation of atmosphere up to height H

ScatteringScattering

Rain and clouds produce a bit @ microwaves.Can be neglected for f under 10 GHz.Surface scattering - depends on the interface,

dielectric properties, geometry.Volume scattering- occurs for dia & dist.

>>d

appears homogeneous

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TTAPAP of atmosphere & Terrain: Upwelling of atmosphere & Terrain: Upwelling

Upwelling (no ground)

')'()'(sec)(sec),(

0

' dzezTzrTdzfz

H

aUP

H

z

a

r’=z’sec

If H>20km

No ground contribution

All the upward radiation emitted by the entire atmospheric path between the ground and the observation point.

TTAPAP of atmosphere & Terrain: Downwelling of atmosphere & Terrain: Downwelling

Downwelling r’=z’sec

TTAPAP of atmosphere & Terrain: downwelling of atmosphere & Terrain: downwelling

Special case: plane homogeneous atmosphere (or cloud) with T(z)=To and a=ao over the range z=0 to z=H.

]1[

'sec)(

sec),0(

sec'

0

H

o

zo

H

aoDN

eT

dzeTHT ao

*Used in programming codes.

TTAPAP of Atmosphere & Terrain of Atmosphere & Terrain

Upwelling Radiation (with ground)

Where TB(0) is the contribution from the surface emissions and reflections (fromdownwelling and cosmic radiation)which is treated in more detail in the following chapter.

Brightness TemperatureBrightness Temperature

Self-emitted radiation from

the surface (e.g. terrain, ice, ocean)

upward radiation from the atmosphere

downward-emitted atmospheric radiation that is reflected by the surface in the antenna direction

virtually no solar contamination

Radiative Transfer TheoryRadiative Transfer Theory

Interaction between radiation and matter : processes => emission & extinction (s & a)

Under clear sky conditions - no scattering

0

')',(

),0(

),(])()1[(

)0()(

dzezfaTzTa

eTrTr

ze dzzf

esc

rA

Frequency [GHz]Frequency [GHz]

Bri

ghtn

ess

Te

mp

era

ture

[K]

Bri

ghtn

ess

Te

mp

era

ture

[K]

),0(

0

')',(

),()( 0

eTdzezfzTT C

dzzf

aB

z

a

Example: Upward-looking Example: Upward-looking radiometer like Arecibo looks at...radiometer like Arecibo looks at...

where TC is the cosmic radiation

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Emission and Scattering by Terrain:Emission and Scattering by Terrain:relaterelate TTBB & & TTSCSC to the medium properties. to the medium properties.

Flat surface vs. rough surface

Rough surfaceh

Lambertian surface is considered “perfectly”

rough. Many times is acombination of both.

Flat surface(Specular surface)

h<<Height variations aremuch smaller than

wavelength of radiation.Snell’s Law applies.

Properties of the Properties of the SpecularSpecular Surface Surface

c2

r2

c1 =’ -j”

r1

1

2

Fresnel reflection givesthe power reflectivity

where,

The power transmissivity is,and,

2

12

122

2,12,1 ZZ

ZZR

HPfor sec

VPfor cos

1

1

i

iiZ

2,11,2 1 12,11,2

2

2

a2

c1

r1

1

2

Properties of the Specular SurfaceProperties of the Specular Surface

122

112 sinsin

•The field attenuation coefficient is

[Np/m]

•The power attenuation coefficient is

a= 2 [Np/m]

Snell’s Law relates the anglesof the incidence and transmission.

1)'/"(12

2 2 rr

o

Homogeneous terrain medium; Homogeneous terrain medium; Assuming Assuming uniformuniform T profile, T profile, T(z) = TT(z) = Tgg

Snell’s Law:

2

2

a2=

c1

r1

T(z) =Tg

TSC

TB

The brightness temperature is

The upwelling temperature of homogeneous terrain is

The scattered temperature is

TDN

The emissivity of such isothermal medium is e(;p) =TB/Tg = 1-1

g

za

APg

T

dzezT

T

0

sec)',0(22 ')'(sec 2

TTBB transmission across transmission across

Specular boundarySpecular boundary

Differentiating wrt and multiplying by don both sides

Snell’s Law:

Multiplying (1) and (2)

(1)

(2)

TTBB transmission for Specular transmission for SpecularDividing P1/P2:

This is the transmissivity

TTBB transmission for Specular transmission for SpecularSince power is proportional to TB:

Where the reflectivities (ch.2):

Emissivity at 10GHz for specular Emissivity at 10GHz for specular surface for H and V polarizationssurface for H and V polarizations

Homogeneous terrain mediumHomogeneous terrain medium

The apparent temperature from specular surface is then

Reflections fromthe atmosphere

Emissionby the ground

Example Example

Probl. 4.5Probl. 4.5TAP=TUP + TDN*/L + T\ *emiss/L

0

50

100

150

200

250

0 5 10 15 20 25 30

freq, [GHz]

Ap

par

ent

Tem

per

atu

re

at A

irp

lan

e ke

lvin

s

TAP=TUP + TDN*R/L + T\ *emiss/L

where L=e

Assigned Problems Assigned Problems

Ulaby & Long 20136.1-7, 6.10, and 12

Exam next wed mar 5

Emissivity of Rough surfaceEmissivity of Rough surface

When the surface is rough in terms of wavelength, there are small height irregularities on the surface that scatter power in many directions.

There will be an angular dependency.

air

medium

Pattern of radiation transmitted across the boundary from the thermal radiation incoming from the medium.

Rough surface emissivityRough surface emissivity

Part of the scatter power is reflected in specular direction and it’s mostly phase-coherent.•The remainder is diffuse or phase-incoherent.

• Part is the same polarization as incident

• Rest is orthogonal polLaw of thermodynamic equilibrium requires absorptivity=emissivity:

Medium (b) with small irregularities on the order of the wavelength.

Rough surface emissivityRough surface emissivity For wave incident in medium 1 upon medium 2, the power

absorbed by medium 2

Where the scattered fields at a distance Rr are related to the incident fields by:

• Same polarization as incident

• orthogonal polarization

air

medium

Rough surface emissivityRough surface emissivity Find Reflectivity Substitute and get emissivity = 1-reflectivitty

where Spq are FSA scattering amplitudes & we used the backscattering coefficient or RCS defined by:

Then the scattered power is given by:

FSA= forward scattering alignment

Rough surface emissivityRough surface emissivity Find Reflectivity Substitute and get emissivity = 1-reflectivitty

Where the backscattering coefficient consists of coherent component along the specular direction and incoherent component along all directions. (Chapter 5)

Rough surface emissivityRough surface emissivity Substitute and get emissivity = 1-reflectivitty

Where the coherent component is given by: (Chapter 5)

where is the roughness parameter, given by , s=rms height

Rough surface emissivityRough surface emissivity The eq. for h can also be used to relate surface scattered

brightness temperature, TSS to the unpolarized emitted atmospheric temperature TDN coming down from all directions in the upper hemisphere.

Accounts for the incoherent part of the scattering by the surface

For h pol:

Similar for v pol

Lambertian surface emissivityLambertian surface emissivity For Lambertian surface (perfectly rough):

Which is polarization and angle independent, and o is a constant

related to the permittivity of the surface.

Emissivity of Emissivity of 2-layer Composite2-layer CompositeIn thermodynamic equilibrium we have:

Where the emissivity is related to the reflectivity as (for v or h):

Both medium 2 and 3 are lossy.

Example: a 20GHz nadir looking radiometer maps the thickness, d, of oil spill over ocean at To=293K. Plot increase in TB as function of oil thickness.

Ch.2

Online modulesOnline modules

*subroutine to compute Tdownwelling where ka= water + oxygen attenuations* Begin computation of radiative transfer integral *Integration of Tb_dn, Tb_up, opacityDO 45 ifr = 1 ,3

freq =fre(ifr)** Compute the DOWNWELLING BRIGHTNESS TEMPERATURE *********

TBdn(ifr) = 0.opa = 0.0TAU = 1.0

DO 130 J = 1, JJvap =v(j)pre =p(j)tem =t(j)

absorp=VAPOR(freq,Vap,Pre,Tem,CL,CW,CC)+ OXYGEN(freq,Vap,Pre,Tem,CX)LTMP0 = EXP(-DZ*absorp) DTB = Tem*(1.0 - LTMP0)*TAU TAU = TAU*LTMP0 TBdn(ifr) = TBdn(ifr) + DTB

130 CONTINUE tauf(ifr) = tau

* Add on freq dependent cosmic background term: TBdn(ifr) = TBdn(ifr) + (2.757+0.00379*(freq-18))*TAU **COMPUTE THE UpWELLING BRIGHTNESS TEMPERATURE*********** TBup(ifr) = 0. TAU = 1.0

DO 135 J = JJ,1,-1vap =v(j)pre =p(j)tem =t(j)

LTMP0 = EXP(-DZ*(VAPOR(freq,Vap,Pre,Tem,CL,CW,CC)+ OXYGEN(freq,Vap,Pre,Tem,CX)) )DTB = Tem*(1.0 - LTMP0)*TAU TAU = TAU*LTMP0 TBup(ifr) = TBup(ifr) + DTB

135 CONTINUE45 CONTINUE

** Compute the DOWNWELLING BRIGHTNESS TEMPERATURE *********

TBdn = 0.TAU = 1.0 jj=1000dZ=30000/jj

DO 130 J = 1, JJvap =v(j); pre =p(j); tem =t(j)

Ka=VAPOR(freq,Vap,Pre,Tem,CL,CW,CC)+ OXYGEN(freq,Vap,Pre,Tem,CX)

LTMP0 = EXP(-dZ*Ka) DTB = Tem*(1.0 - LTMP0)*TAU TAU = TAU*LTMP0 TBdn = TBdn + DTB

130 CONTINUE* Add on freq dependent cosmic background term: TBdn(ifr) = TBdn(ifr) + (2.757+0.00379*(freq-18))*TAU

Code for TdnCode for Tdn

Code for TupCode for Tup*COMPUTE THE UpWELLING BRIGHTNESS TEMPERATURE*********** TBup(ifr) = 0. TAU = 1.0

DO 135 J = JJ,1,-1vap =v(j)pre =p(j)tem =t(j)

LTMP0 = EXP(-DZ*ka(j))DTB = Tem*(1.0 - LTMP0)*TAU TAU = TAU*LTMP0 TBup(ifr) = TBup(ifr) + DTB

135 CONTINUE45 CONTINUE

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Radiative Transfer Dr. X-Pol Microwave Remote Sensing INEL 6669 Dept. of Electrical & Computer Engineering, UPRM, Mayagüez, PR