Radiative Transfer

1
Radiative Transfer

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

2
Outline

• 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 Ter profile coherent incoherent
  approach
• Emissivity of dielectric Slab
• Emissivity of Rough surface

3
Radiative Transfer

• Some energy is transmitted
• some is scattered, some is absorbed

4
Outline

• Theory of Radiative Transfer
• Extinction Emission
• Equation of Transfer
• TAP of absorbing/scattering Media
• TAP of atmosphere Terrain
• upwelling-down-welling
• Emission and Scattering by Terrain
• Homogeneous terrain medium with
• uniform T profile
• non-uniform Ter profile coherent incoherent
  approach
• Emissivity of dielectric Slab
• Emissivity of Rough surface

5
Radiative Transfer Extinction Interaction
between radiation and matter

• Extinction (Change due to energy loss)
• dIextin ke I(r,r)dr
• ? two processes absorbtion scattering
• ke ka ks in Nepers/m
• where ke is the
• extinction or power
• attenuation coefficient,
• its due to absorption and
• scattering away
• in other direction.

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Radiative Transfer Emission Interaction
between radiation and matter

• Emission (Change due to energy gained)
• dIemit (kaJa ksJs )dr
• ? two processes or mechanisms emit scatter
• Ja Js account for thermal emission
  scattering
• ? Introduce the scattering albedo, a ks/ke
• dIemit (ke - ks)Ja ks Js )
• dIemit ke (1- a)Ja aJs ) keJdr
• where J (1- a)Ja aJs

7
albedo
http//www.sciencedaily.com/releases/2014/02/14021
9115110.htm
The 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.
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Outline

• Theory of Radiative Transfer
• Extinction Emission
• Equation of Transfer
• TAP of absorbing/scattering Media
• TAP of atmosphere Terrain
• upwelling-down-welling
• Emission and Scattering by Terrain
• Homogeneous terrain medium with
• uniform T profile
• non-uniform Ter profile coherent incoherent
  approach
• Emissivity of dielectric Slab
• Emissivity of Rough surface

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Radiative Transfer Equation of Transfer I

• dI I(rdr) - I(r)
• dIemit -dIextin
• ke (J-I)dr
• (J-I)dt
• where
• dt ke dr is the optical depth,
• therefore,
• and t is the optical thickness or opacity in Np.

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Radiative Transfer Equation of Transfer II

• dI(J-I)dt or dI/dt I J
• Multiplying by et(0,r)
• and integrating from 0 to r.

Horizontal layers (stratified atm)
B(r)
r
r
B(0)
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Outline

• Theory of Radiative Transfer
• Extinction Emission
• Equation of Transfer
• TAP of absorbing/scattering Media
• TAP of atmosphere Terrain
• upwelling-down-welling
• Emission and Scattering by Terrain
• Homogeneous terrain medium with
• uniform T profile
• non-uniform Ter profile coherent incoherent
  approach
• Emissivity of dielectric Slab
• Emissivity of Rough surface

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TAP of absorbing/scattering Media (1)
In the microwave region, where R-J applies,
theres a T ? to I
Similarly, under thermodynamic equilibrium
(emission absorption) the absorption source
function is Recall J (1- a) Ja aJs
where T is the physical temperature of the medium.
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TAP of absorbing/scattering Media (2)
Similarly, for the scattering source function, Js
where Y 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,
In eq. 6.24 Ulaby Long, the multiply by ½
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TAP of absorbing/scattering Media (3)
In terms of temperature,
ke ka ks

• For scatter-free medium
• ks( ake)0
• then a 0 and ke ka
• and the opacity is

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Brightness temperature
Define the 1-way atmospheric transmissivity
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TAP of absorbing/scattering Media III
Ex. For scatter-free medium An airborne
radiometer measuring ice.
TAP (0) Tice
e-t (0,H) attenuation of atmosphere up to
height H
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Scattering

• Rain and clouds produce a bit _at_ microwaves.
• Can be neglected for f under 10 GHz.
• Surface scattering - depends on the interface,
  dielectric properties, geometry.
• Volume scattering- occurs for l dia dist.

lgtgtd
appears homogeneous
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Outline

• Theory of Radiative Transfer
• Extinction Emission
• Equation of Transfer
• TAP of absorbing/scattering Media
• TAP of atmosphere Terrain
• upwelling-down-welling
• Emission and Scattering by Terrain
• Homogeneous terrain medium with
• uniform T profile
• non-uniform Ter profile coherent incoherent
  approach
• Emissivity of dielectric Slab
• Emissivity of Rough surface

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TAP of atmosphere Terrain Upwelling
rzsecq

• Upwelling (no ground)

All the upward radiation emitted by the entire
atmospheric path between the ground and the
observation point.
If Hgt20km
q
No ground contribution
20
TAP of atmosphere Terrain Downwelling

• Downwelling

rzsecq
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TAP of atmosphere Terrain downwelling

• Special case plane homogeneous atmosphere (or
  cloud) with T(z)To and kakao over the range z0
  to zH.

Used in programming codes.
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TAP of Atmosphere Terrain

• Upwelling Radiation (with ground)

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

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Radiative Transfer Theory

• Interaction between radiation and matter
  processes gt emission extinction (s a)
• Under clear sky conditions - no scattering

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Example Upward-looking radiometer like Arecibo
looks at...
Brightness Temperature K
Frequency GHz
where TC is the cosmic radiation
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Outline

• Theory of Radiative Transfer
• Extinction Emission
• Equation of Transfer
• TAP of absorbing/scattering Media
• TAP of atmosphere Terrain
• upwelling-down-welling
• Emission and Scattering by Terrain
• Homogeneous terrain medium with
• uniform T profile
• non-uniform Ter profile coherent incoherent
  approach
• Emissivity of dielectric Slab
• Emissivity of Rough surface

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Emission and Scattering by Terrainrelate TB
TSC to the medium properties.

• Flat surface vs. rough surface

Flat surface (Specular surface) hltltl Height
variations are much smaller than wavelength of
radiation. ?Snells Law applies.
Rough surface h?l Lambertian surface is
considered perfectly rough. Many times is
a combination of both.
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Properties of the Specular Surface
Fresnel reflection gives the power reflectivity
where,
The power transmissivity is,
and,
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Properties of the Specular Surface
Snells Law relates the angles of the incidence
and transmission.

• The field attenuation coefficient is

Np/m

• The power attenuation coefficient is
• ka 2a Np/m

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Homogeneous terrain medium Assuming uniform T
profile, T(z) Tg
The brightness temperature is
TDN
The upwelling temperature of homogeneous terrain
is
The scattered temperature is
The emissivity of such isothermal medium is
e(qp) TB/Tg 1-G1
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TB transmission across Specular boundary
Snells Law
(1)
Differentiating wrt q and multiplying by df on
both sides
(2)
Multiplying (1) and (2)
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TB transmission for Specular
Dividing P1/P2
This is the transmissivity
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TB transmission for Specular
Since power is proportional to TB
Where the reflectivities (ch.2)
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Emissivity at 10GHz for specular surface for H
and V polarizations
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Homogeneous terrain medium

• The apparent temperature from specular surface is
  then

Emission by the ground
Reflections from the atmosphere
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Example
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Probl. 4.5
where Let
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Assigned Problems
Ulaby Long 2013 6.1-7, 6.10, and 12
Exam next wed mar 5
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Emissivity 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.

Pattern of radiation transmitted across the
boundary from the thermal radiation incoming from
the medium.
air
medium
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Rough surface emissivity
Medium (b) with small irregularities on the order
of the wavelength.

• Part of the scatter power is reflected in
  specular direction and its mostly
  phase-coherent.
• The remainder is diffuse or phase-incoherent.
• Part is the same polarization as incident
• Rest is orthogonal pol
• Law of thermodynamic equilibrium requires
  absorptivityemissivity

41
Rough surface emissivity

• For wave incident in medium 1 upon medium 2, the
  power absorbed by medium 2

air
medium

• Same polarization as incident
• orthogonal polarization

Where the scattered fields at a distance Rr are
related to the incident fields by
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Rough surface emissivity

• Find Reflectivity
• Substitute and get emissivity 1-reflectivitty

FSA forward scattering alignment
where Spq are FSA scattering amplitudes we used
the backscattering coefficient or RCS defined by
Then the scattered power is given by
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Rough 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)
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Rough surface emissivity

• Substitute and get emissivity 1-reflectivitty

Where the coherent component is given by
(Chapter 5)
where y is the roughness parameter, given by
, srms height
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Rough surface emissivity

• The eq. for Gh 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.

For h pol
Similar for v pol
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Lambertian surface emissivity

• For Lambertian surface (perfectly rough)

Which is polarization and angle independent, and
so is a constant related to the permittivity of
the surface.
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Emissivity of 2-layer Composite
In thermodynamic equilibrium we have

• Example a 20GHz nadir looking radiometer maps
  the thickness, d, of oil spill over ocean at
  To293K.
• Plot increase in TB as function of oil thickness.

Where the emissivity is related to the
reflectivity as (for v or h)
Ch.2
Both medium 2 and 3 are lossy.
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Online modules
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subroutine to compute Tdownwelling where ka
water oxygen attenuations Begin computation
of radiative transfer integral Integration of
Tb_dn, Tb_up, opacity DO 45 ifr 1 ,3 freq
fre(ifr) Compute the DOWNWELLING
BRIGHTNESS TEMPERATURE TBdn(ifr)
0. opa 0.0 TAU 1.0 DO 130 J 1,
JJ vap v(j) pre p(j) tem
t(j) absorpVAPOR(freq,Vap,Pre,Tem,CL,CW,CC)
OXYGEN(freq,Vap,Pre,Tem,CX) LTMP0
EXP(-DZabsorp) DTB Tem(1.0 - LTMP0)TAU
TAU TAULTMP0 TBdn(ifr) TBdn(ifr)
DTB 130 CONTINUE tauf(ifr) tau Add on
freq dependent cosmic background term
TBdn(ifr) TBdn(ifr) (2.7570.00379(freq-18))
TAU COMPUTE THE UpWELLING BRIGHTNESS
TEMPERATURE TBup(ifr) 0.
TAU 1.0 DO 135 J JJ,1,-1 vap
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 TAULTMP0 TBup(ifr)
TBup(ifr) DTB 135 CONTINUE 45 CONTINUE
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Code for Tdn
Compute the DOWNWELLING BRIGHTNESS
TEMPERATURE TBdn 0. TAU 1.0
jj1000 dZ30000/jj DO 130 J 1, JJ vap
v(j) pre p(j) tem t(j) KaVAPOR(freq,Vap,Pre
,Tem,CL,CW,CC) OXYGEN(freq,Vap,Pre,Tem,CX) LTMP
0 EXP(-dZKa) DTB Tem(1.0 - LTMP0)TAU
TAU TAULTMP0 TBdn TBdn DTB 130
CONTINUE Add on freq dependent cosmic
background term TBdn(ifr) TBdn(ifr)
(2.7570.00379(freq-18))TAU
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Code for Tup
COMPUTE THE UpWELLING BRIGHTNESS
TEMPERATURE TBup(ifr) 0.
TAU 1.0 DO 135 J JJ,1,-1 vap
v(j) pre p(j) tem t(j)
LTMP0 EXP(-DZka(j)) DTB Tem(1.0 -
LTMP0)TAU TAU TAULTMP0 TBup(ifr)
TBup(ifr) DTB 135 CONTINUE 45 CONTINUE