1. Radiative Transfer Dr. X-Pol Microwave Remote Sensing INEL 6669
Dept. of Electrical & Computer Engineering,
UPRM, Mayagüez, PR
INEL 6069,

2. Outline Theory of Radiative Transfer TAP of absorbing/scattering Media
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&er profile: coherent & incoherent approach
Emissivity of dielectric Slab
Emissivity of Rough surface
INEL 6069,

3. Radiative Transfer Some energy is transmitted
some is scattered, some is absorbed
INEL 6069,

4. Outline Theory of Radiative Transfer TAP of absorbing/scattering Media
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 T&er profile: coherent & incoherent approach
Emissivity of dielectric Slab
Emissivity of Rough surface
INEL 6069,

5. Radiative Transfer,
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,
it’s due to absorption and
scattering away
in other direction. dA dr
Volume of
some matter
Incident
radiation
exiting
INEL 6069,

6. 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 dA dr Js Ja
INEL 6069,

7. albedo http://www.sciencedaily.com/releases/2014/02/140219115110.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.
albedo

8. Outline Theory of Radiative Transfer TAP of absorbing/scattering Media
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 T&er profile: coherent & incoherent approach
Emissivity of dielectric Slab
Emissivity of Rough surface
INEL 6069,

9. Radiative Transfer : Equation of Transfer I
dI= I(r+dr) - 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. dA dr
B(r)
B(r+dr)
INEL 6069,

10. 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)
INEL 6069,

11. Outline Theory of Radiative Transfer TAP of absorbing/scattering Media
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 T&er profile: coherent & incoherent approach
Emissivity of dielectric Slab
Emissivity of Rough surface
INEL 6069,

12. TAP of absorbing/scattering Media (1)
Radiative Transfer,
TAP 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 = (1- a) Ja + aJs ]
where T is the physical temperature of the medium.
INEL 6069,

13. TAP of absorbing/scattering Media (2)
Radiative Transfer,
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 Ulaby & Long, the multiply by ½
INEL 6069,

14. TAP of absorbing/scattering Media (3)
Radiative Transfer,
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
INEL 6069,

15. Brightness temperature
Define the 1-way atmospheric transmissivity:

16. TAP of absorbing/scattering Media III
Radiative Transfer,
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
INEL 6069,

17. Scattering Rain and clouds produce a bit @ microwaves.
Radiative Transfer,
Scattering
Rain and clouds produce a 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.
l>>d
appears homogeneous
INEL 6069,

18. Outline Theory of Radiative Transfer TAP of absorbing/scattering Media
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 T&er profile: coherent & incoherent approach
Emissivity of dielectric Slab
Emissivity of Rough surface
INEL 6069,

19. TAP of atmosphere & Terrain: Upwelling
Radiative Transfer,
TAP of atmosphere & Terrain: Upwelling
r’=z’secq
Upwelling (no ground)
All the upward radiation emitted by the entire atmospheric path between the ground and the observation point.
If H>20km q
No ground contribution
INEL 6069,

20. TAP of atmosphere & Terrain: Downwelling
Radiative Transfer,
TAP of atmosphere & Terrain: Downwelling
Downwelling
r’=z’secq
INEL 6069,

21. TAP of atmosphere & Terrain: downwelling
Radiative Transfer,
TAP of atmosphere & Terrain: downwelling
Special case: plane homogeneous atmosphere (or cloud) with T(z)=To and ka=kao over the range z=0 to z=H.
*Used in programming codes.
INEL 6069,

22. TAP of Atmosphere & Terrain
Radiative Transfer,
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.
INEL 6069,

23. Brightness Temperature
Radiative Transfer,
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
INEL 6069,

24. Radiative Transfer Theory
Interaction between radiation and matter : processes => emission & extinction (s & a)
Under clear sky conditions - no scattering
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25. Example: Upward-looking radiometer like Arecibo looks at...
Radiative Transfer,
Example: Upward-looking radiometer like Arecibo looks at...
Brightness Temperature [K]
Frequency [GHz]
where TC is the cosmic radiation
INEL 6069,

26. Outline Theory of Radiative Transfer TAP of absorbing/scattering Media
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 T&er profile: coherent & incoherent approach
Emissivity of dielectric Slab
Emissivity of Rough surface
INEL 6069,

27. Flat surface vs. rough surface
Radiative Transfer,
Emission and Scattering by Terrain: relate TB & TSC to the medium properties.
Flat surface vs. rough surface
Flat surface
(Specular surface)
h<<l
Height variations are
much smaller than
wavelength of radiation.
Snell’s Law applies.
Rough surface
hl
Lambertian surface is
considered “perfectly”
rough. Many times is a
combination of both.
INEL 6069,

28. Properties of the Specular Surface
Radiative Transfer,
Properties of the Specular Surface
ec2
mr2
ec1 =e’ -je”
mr1 q1 q2
Fresnel reflection gives
the power reflectivity
where,
The power transmissivity is,
and,
INEL 6069,

29. Properties of the Specular Surface
Radiative Transfer,
Properties of the Specular Surface e2 m2
ka2
ec1
mr1 q1 q2
Snell’s 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]
INEL 6069,

30. Homogeneous terrain medium; Assuming uniform T profile, T(z) = Tg
Radiative Transfer,
Homogeneous terrain medium; Assuming uniform T profile, T(z) = Tg
Snell’s Law: e2 m2
Ka2=2a2
ec1
mr1
T(z) =Tg
TSC TB
TDN
The brightness temperature is
The upwelling temperature
of homogeneous terrain is
The scattered temperature is
The emissivity of such
isothermal medium is
e(q;p) =TB/Tg = 1-G1
INEL 6069,

31. TB transmission across Specular boundary
Radiative Transfer,
TB transmission across Specular boundary
Snell’s Law:
(1)
Differentiating wrt q and multiplying by df on both sides
(2)
Multiplying (1) and (2)
INEL 6069,

32. TB transmission for Specular
Radiative Transfer,
TB transmission for Specular
Dividing P1/P2:
This is the transmissivity
INEL 6069,

33. TB transmission for Specular
Radiative Transfer,
TB transmission for Specular
Since power is proportional to TB:
Where the reflectivities (ch.2):
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34. Emissivity at 10GHz for specular surface for H and V polarizations

35. Homogeneous terrain medium
Radiative Transfer,
Homogeneous terrain medium
The apparent temperature from specular surface is then
Emission
by the ground
Reflections from
the atmosphere
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36. Example

37. Radiative Transfer,
Probl. 4.5
where L=et
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38. Assigned Problems Ulaby & Long 2013 6.1-7, 6.10, and 12
Exam next wed mar 5

39. Emissivity of Rough surface
Radiative Transfer,
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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40. 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 it’s 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 absorptivity=emissivity:

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:

42. 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:

43. 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)

44. 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 , s=rms height

45. 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:
Accounts for the incoherent part of the scattering by the surface
Similar for v pol

46. 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.

47. 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 To=293K.
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.

48. Online modules

49. Radiative Transfer,
*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)
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) + ( *(freq-18))*TAU *
*COMPUTE THE UpWELLING BRIGHTNESS TEMPERATURE***********
TBup(ifr) = 0.
DO 135 J = JJ,1,-1
LTMP0 = EXP(-DZ*(VAPOR(freq,Vap,Pre,Tem,CL,CW,CC)+ OXYGEN(freq,Vap,Pre,Tem,CX)) )
TBup(ifr) = TBup(ifr) + DTB
CONTINUE
45 CONTINUE
INEL 6069,

50. Radiative Transfer,
Code for Tdn
** Compute the DOWNWELLING BRIGHTNESS TEMPERATURE *********
TBdn = 0.
TAU = 1.0
jj=1000
dZ=30000/jj
DO 130 J = 1, JJ
vap =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) + ( *(freq-18))*TAU
INEL 6069,

51. Code for Tup *COMPUTE THE UpWELLING BRIGHTNESS TEMPERATURE***********
Radiative Transfer,
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(-DZ*ka(j))
DTB = Tem*(1.0 - LTMP0)*TAU
TAU = TAU*LTMP0
TBup(ifr) = TBup(ifr) + DTB
CONTINUE
45 CONTINUE
INEL 6069,