1 EE 543 Theory and Principles of Remote Sensing Derivation of the Transport Equation.
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Transcript of 1 EE 543 Theory and Principles of Remote Sensing Derivation of the Transport Equation.
1
EE 543Theory and Principles of
Remote Sensing
Derivation of the Transport Equation
O. Kilic EE 543
2
Theory of Radiative Transfer
We will be considering techniques to derive expressions for the apparent temperature, TAP of different scenes as shown below.
Atmosphere
Terrain
TA
TUP
TA
Terrain could be smooth, irregular, slab (such as layer of snow) over a surface.
STEP 1: Derive equation of radiative transferSTEP 2: Apply to different scenes
O. Kilic EE 543
3
Radiation and Matter
• Interaction between radiation and matter is described by two processes:– Extinction– Emission
• Usually we have both phenomenon simultaneously.
• Extinction: radiation in a medium is reduced in intensity (due to scattering and absorption)
• Emission: medium adds energy of its own (through scattering and self emission)
O. Kilic EE 543
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Mediums of Interest
• The mediums of interest will typically consist of multiple types of single scatterers (rain, vegetation, atmosphere,etc.)
• First we will consider a single particle and examine its scattering and absorption characteristics.
• Then we will derive the transport equation for a collection of particles in a given volume.
O. Kilic EE 543
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Apparent temperature distribution
Apparent Temperature (Overall Scene Effects)
antennaFn()
TA
Atmosphere
TAP()
TDN
TSC
TUP
TB
TB: Terrain emission
TDN: Atmospheric downward emissionTUP: Atmospheric upward emission
TSC: Scattered radiation
Terrain
SUMMARY
Brightness
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• In radiometry, both point and extended source of incoherent radiation (e.g. sky, terrain) are of interest.
• Brightness is defined as the radiated power per solid angle per unit area, as follows:
• The unit for brightness is Wsr-1m-2
t
t
UB
A
Power per solid angle (W/Sr)
Function of ,
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Apparent Temperature
• TAP() is the blackbody equivalent radiometric temperature of the scene.
2
2, ,i AP
kB T f
Incident brightnessConsists of several terms
SUMMARY
Similar in form to Planck’s blackbody radiation.
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Antenna Temperature (Overall Antenna Effects)
4
4
4
, ,
,
, ,
AP n
A
n
A A AP n
T F dT
F d
T T F d
We derived
Averaged temperature over the solid angle of receive antenna
Fn()
A
SUMMARY
TAP
TA
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Antenna Efficiency
• Radiation Efficiency
• Beam Efficiency: – Contributions due to sidelobes are undesired.– Ideally one would design a radiometer antenna
with a narrow pencil beam and no sidelobes.
' (1 )A a A a oT T T
SUMMARY
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Main Beam and Sidelobe Effects
4
4
1, ,
1, ,
1, ,
m
m
A AP n
r
AP n
r
AP n
r
A ML SL
T T F d
T F d
T F d
T T T
SUMMARY
O. Kilic EE 543
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Main Beam Efficiency
4
( , )
( , )M
n
M
n
F d
F d
Ratio of power contained within the main beam to total power.
SUMMARY
O. Kilic EE 543
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Effective Main Beam Apparent Temperature,
• Antenna temperature if the antenna pattern consisted of only the main beam.
, ,
,M
M
AP n
ML
n
ML ML ML
T F dT
F d
T T
MLT
SUMMARY
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Antenna Stray Factor
4
4
1
( , )
( , )S M
S M
n
n
F d
F d
SUMMARY
Ratio of power contained within the sidelobes to total power.
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Effective Sidelobe Antenna Temperature
, ,
,
(1 )
S
S
AP n
SL
n
SL SL SL ML SL
T F d
TF d
T T T
• Antenna temperature if the antenna pattern consisted of only the sidelobes.
SUMMARY
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Antenna Temperature and Beam Efficiency
(1 )A ML SL
ML ML ML SL
T T T
T T
SUMMARY
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Overall Antenna Efficiency and Antenna Temperature
' (1 )
(1 ) (1 )
A a A a o
a ML ML ML SL a o
T T T
T T T
Combine beam efficiency and radiation efficiency:
SUMMARY
Desired value
Measured value
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Linear relation1 1 1M a
ML A SL o
a M M a M
T T T T
MLT
AT
Bias = B1 +B2
-( B1 +B2)
Slope
Depends on sidelobe levels, antenna efficiency and temperature Depends on
antenna efficiency
SUMMARY
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Summary
The accuracy of radiometric measurements is highly dependent on the radiation efficiency, and main beam efficiency, of the antenna.M
a
SUMMARY
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Theory of Radiative Transfer
We will be considering techniques to derive expressions for the apparent temperature, TAP of different scenes as shown below.
Atmosphere
Terrain
TA
TUP
TA
Terrain could be smooth, irregular, slab (such as layer of snow) over a surface.
STEP 1: Derive equation of radiative transferSTEP 2: Apply to different scenes
SUMMARY
O. Kilic EE 543
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Radiation and Matter
• Interaction between radiation and matter is described by two processes:– Extinction– Emission
• Usually we have both phenomenon simultaneously.
• Extinction: radiation in a medium is reduced in intensity (due to scattering and absorption)
• Emission: medium adds energy of its own (through scattering and self emission)
SUMMARY
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Terminology for Radiation/Scattering from a Particle
• Scattering Amplitude
• Differential Scattering Cross Section
• Scattering Cross Section
• Absorption Cross Section
• Total Cross Section
• Albedo
• Phase Function
SUMMARY
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Scattering Amplitudes and Cross Sections
• Brightness directly relates to power, and satisfies the transport equation.
• We will examine the effects of presence of scattering particles on brightness.
O
s
r
B(r,s) is a function of position and direction
Function of 5 parameters:r: x, y, zs:
SUMMARY
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Scattering AmplitudeConsider an arbitrary scatterer:
Imaginary, smallest sphere
D
i
Ei
o
Es
R
2
ˆˆ( , ) ; ,ikR
s o o i
e DE e f o i e E R
R
The scatterer redistributes the incident electric field in space:
SUMMARY
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Scattering Amplitude(2)
2
ˆˆ( , ) ; ,ikR
s o o i
e DE e f o i e E R
R
o: s, s
i: i, i
f(o,i) is a vector and it depends on four angles.
SUMMARY
Unit vectors along the incident and scattered directions
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Scattering Cross Section Definitions: Power Relations
• Differential Scattering Cross-section
• Scattering Cross-section
• Absorption Cross-section
• Total Cross-section
SUMMARY
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Differential Scattering Cross Section
2 2
ˆ ˆˆ ˆ, lim ,sd R
i
R So i f o i
S
o
i
RSi
SUMMARY
(m2/St)
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Scattering Cross-section
4
ˆ ˆˆ,ss d s
i
Pi o i d
S
SUMMARY
(m2)
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Absorption Cross-section
2
ˆ a Va
i i
E dvP
iS S
SUMMARY
(m2)
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Total Cross-section
t s a
tt s a
i
s
t
P P P
P
S
a
albedo
SUMMARY
(m2)
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Phase Function
ˆˆ ˆˆ ˆ, ,
4ˆ
ˆ ˆˆ ˆ, ,4
ˆ ˆˆ ˆ, ,
t
d
s
d
io i p o i
io i o i
p o i a o i
SUMMARY
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31
Phase Function (2)
4
4
1 ˆˆ,4
1 ˆˆ, 14
s
s
p o i d a
o i d
SUMMARY
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Derivation of the Radiative Transfer (Transport) Equation
v = a s
Consider a small cylindrical volume with identical scatterers inside.
The volume of the cylinder is given by:
Base area
0
a
s
Pout
Pin
s
s
length
rr + r
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Incident and Output Power
ˆIncident Brightness, ,
Incident Power:
ˆ ˆ , ,
Output Power:
ff
in f
out ff
B B r s
P B r s f a w B r s a w
P B B f a w
Change in brightness
(1)
(2)
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Conservation of Power
out in loss gainP P P P
Extinction:
off-scattering + absorption
self emission + scattering
(3)
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Extinction in the Cylindrical Volume Due to Scattering Phenomenon
N: # particles in volume V
: scattered power per particle
ˆ ˆ
s s
s
ins s i s
P N p
p
Pp i S i
a
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Scattering CoefficientLet denote the particle density in the volume.
s
Then
scattering coeffDefine icient " "
ins s s in
s
N NV a s
N a s
PP a s s P
a
N
ps
Unit: #/m
(4)
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Loss Due to Scattering
ˆ,
s s in
s s f
P s P
P sB r s f a w
(5)
Using (1) in (5)
(6)
s s
where
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Extinction in the Cylindrical Volume Due to Absorption Phenomenon
;
N: # particles in volume V
: absorbed power per particle
ˆ ˆ
ˆ
a a
a
ina a i a
ina a a in
P N p N a s
p
Pp i S i
aP
P a s i s Pa
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Absorption Coefficient
ˆ,a
a
f
a
aP sB r s f a w
Define “absorption coefficient: Unit: #/m
(7)
(8)
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Total Power Loss
ˆ,
ˆ,
loss s a
s a f
los
t s
s t f
a
P P P
B r s s a f w
P B r s s a f w
Define “total coefficient” or “extinction coefficient”
Total power loss is given by:
(9)
(10)
ˆ,s s fP sB r s f a w
ˆ,a a fP sB r s f a w
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Loss in Power - Summary
ˆ,loss t f
t a s
s s
a a
P B r s s a f w
Particle density in v
Incident Brightness
Due to scattering and absorption.
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Gain in the Cylindrical Volume Due to Scattering Phenomenon
s0
a
s
Pout
Pin
s
An increase in power is experienced when the particles scatter energy along s direction when they are illuminated from other directions; i.e. ˆ ˆ's s
ˆ 's
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Scattering of Incident Radiation Along s’ Towards s
2
Incident power density:
ˆ ˆ ' , ' '
Scattered power density per particle:
ˆ ˆ, 'ˆ ˆ '
Collective increase in power:
ˆ ˆ ˆ '
i f
d
r i
scatgain r r r r
r
S s B r s f w
s sS s S s
R
P s NS s A a s S s A
a s A
2
ˆ ˆ, 'ˆ, ' 'd
f
s sB r s f w
R
(11)
(12)
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44
Collective Increase in Power (2)
scatgain 2
scat scatgain gain
4
ˆ ˆ ˆ ˆ P ' , ' , ' '
ˆ ˆ ˆ , ' , ' '
Collective increase in power, from
possible incidence angles:
ˆ P P '
ˆ = ,
all
rd f
d f
d
As a s s s B r s f w
Ra s w s s B r s f w
s
s
4
ˆ ˆ' , ' 'fs B r s dw a s w f
(13)
O. Kilic EE 543
45
Collective Increase in Power
scatgain
4
scatgain
4
Using:
ˆ ˆ, 'ˆ ˆ ˆ ˆ , ' , ' and
4
ˆ ˆ ˆP , ' , ' '4
ˆ ˆ ˆ ˆor equivalently from , ' , '
ˆ ˆ ˆP , ' , ' '4
t
d
t t
tf
tf
s ss s p s s
p s s B r s dw a s w f
a s s p s s
a s s B r s dw a s w f
(14a)
(14b)
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Gain in the Cylindrical Volume Due to Self Emission
ˆ,emissiongainP r s s a f w
Define emission source function as power emitted per
(Volume Steradian Hertz) as ˆ,r s
(15)
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47
Total Increase in Power
4
ˆ ˆ ˆ, ' , ' '4
ˆ ,
scat emissiongain gain gain
tgain f
P P P
P p s s B r s dw a s w f
r s s a f w
(16)
Self emission
scattering
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Power Conservationout in loss gainP P P P From (3) i.e.
Using (1), (2), (10) and (16)
4
4
ˆ ˆ, ,
ˆ ˆ ˆ , ' , ' '4
ˆ ˆ + , ,
Divide by , take lim 0
ˆ,ˆ ˆ ˆ ˆ ˆ, , ' , ' ' ,
4
f t f
tf
ff
f tt ff
B r s f a w B r s a w f s
p s s B r s dw a w f s
r s a w f s B r s B a w f
s s
dB r sB r s p s s B r s dw r s
ds(17)
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Scalar Transport Equation
4
ˆ,ˆ,
ˆ ˆ ˆ, ' , ' '4
ˆ,
f
t f
tf
dB r sB r s
ds
p s s B r s dw
r s
Loss due to scattering and absorption
Gain due to scattering of other incident energy along s direction
Gain due to self emission
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50
Remarks on Emission Source Function
ˆ, : Power emitted/(Vol. St. Hz)
Under thermodynamic eq
thermal emission = absorption
uilibrium,
ˆ, ( )a f
r s
r s B T
a; absorption coefficient
#/m
Brightness of each particle inside the mediumPower/(St. Area. Hz)
Physical temperature
Directly proportional to absorption
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51
Self Emission Function
ˆ, ( ) ( )
(1 ) ( );
a f a f
st f
t
r s B T B T
a B T a
(18)
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Scalar Transport Equation – Based on Extinction Coefficient Only
4
ˆ,ˆ ˆ ˆ ˆ, , ' , ' '
4
(1 ) ( )
f tt f f
t f
dB r sB r s p s s B r s dw
ds
a B T
(19)
O. Kilic EE 543
53
Optical Distance
1
1,o
s
o ts
t
s s ds
d ds
0
s
s0
s1
ds
Dimensionless#
Loss factor per length
#t m
(20)
O. Kilic EE 543
54
Transport Equation as a Function of Optical Distance
t r
ˆ ˆ, ,ffB r s B s
Divide the transport equation in (19) by and express as a
function of ; i.e.
4
ˆ, 1ˆ ˆ ˆ ˆ, , ' , ' '
4
(1 ) ( )
f
f f
f
dB sB s p s s B s dw
d
a B T(21)
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Transport Equation as a Function of Optical Distance and Albedo
4
ˆ,ˆ,
ˆ ˆ ˆ, ' , ' '4
1 ( )
f
f
f
dB sB s
da
s s B s dw
a B T
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Transport Equation as a Function of Optical Distance
4
4
ˆ ˆMultiply by f, and let , ,
ˆ,ˆ ˆ, ,
1ˆ ˆ ˆ ˆ, , ' , ' ' 1 ( )
4
or
ˆ ˆ ˆ ˆ, , ' , ' ' 1 ( )4
Define
ˆ ˆ, ,
f
s a
B s B s f
dB sB s J s
d
J s p s s B s dw a B T
aJ s s s B s dw a B T
J s J s J T
Js Ja
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Solution to Transport Equation (1)
Solution to Transport Equation (2)
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Solution to Transport Equation (3)
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Transport Equation Scaled to Temperature (1)
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Transport Equation Scaled to Temperature (2)
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Transport Equation Scaled to Temperature (3)
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Solution to Transport Equation (Temperature Form)
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Solution to Transport Equation
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'
0
ˆ ˆ ˆ, 0, ', 'AP AP TT s T s e J s e d
Low Albedo Case, a<<1
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Solution for Low Albedo
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Upwelling Radiation (Observe the Medium from Above)
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Low Albedo Case, a <<1
cos
sec
sec
z s
s z
ds dz
sec
sec
t a a
a
d ds ds dz
d
dz
Upwelling Radiation
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Upwelling Radiation
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Solution: Upwelling Radiation
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70
0
'
sec ' '
sec '' ''
0
, 0,
( ') sec ' '
z
a
z
a
z
z dz
AP AP
z z dz
a
T z T e
T z e z dz
O. Kilic EE 543
71
O. Kilic EE 543
72
Example
Consider a downward looking, nadir pointing radiometer observing the ocean surface from an airborne platform above a 2 km thick cloud with water content of 1.5 g/m3. The absorption coefficient of the cloud is approximately given by:
where f is in GHz and mv is the water content in g/m3. Assuming that the ocean has an apparent temperature, TAP(0,0) of 150 K, calculate the apparent temperature observed by the radiometer at f = 1 GHz. The cloud may be assumed to have a physical temperature of 275K.
4 1.952.4 10a vf m
Solution
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73
Solution
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74
Solution
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75
0
'
sec ''
0
0
,0 0,0
sec '
where
' 275
(0,0) 150
H
a
z
AP AP UP
H dz
UP o a a
AP
T H T e T
T T e dz
T z T
T
Solution
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76
Solution
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Downwelling Radiation from a Layer (Observe the Medium from Below)
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cos
sec
seca a
H z s
ds
dzd ds dz
Downwelling Radiation
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79
0
0
, ' sec '
, , sec '
s z
a a
s z H
z
a
H
r r ds dz
z H z H z dz
Downwelling Radiation
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Downwelling Radiation
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'
'
sec '
sec ''
sec '
sec ''
, ,
' sec '
,
' sec '
z
a
H
z
a
z
H
a
z
z
a
z
dz
AP AP
z dz
a
H
dz
AP
H dz
a
z
T z T H e
T z e dz
T H e
T z e dz
Summary
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82
'
' sec '
'' sec ''
, ,
' sec '
H
a
z
z
a
z
z dz
AP AP DN
H z dz
DN a
z
T z T H e T
T T z e dz
Atmospheric Radiation
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84
Example
Atmospheric water vapor absorption coefficient at 22 GHz is
where v=moe-0.5z is the water vapor density (g/m3), and T=To-6.5z is temperature (K) and z is the altitute (km). Assuming that the most of the atmospheric absorption will be for the lowermost 10 km, calculate the downwelling and upwelling radiation temperatures for nadir direction. Let To = 300K and o = 7.5 g/m3
2
3 3006 10 Np/kma v T
Solution
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Solution
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86
Solution
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87
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88
Summary: Upwelling and Downwelling Radiation
',
0
, '
, sec ( ') ' '
, sec ( ') ' '
, sec ' '
zz z
UP a
Hz z
DN az
b
aa
T z T z e z dz
T z T z e z dz
a b z dz
Low Albedo Case
Upwelling Radiation (Low Albedo)
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89
SPECIAL CASES:Constant Medium Temperature, T(z) = To
Uniform Particle Distribution, a(z) = ao
0 0
0'
0 0
0 0 0
0 0
0
0
0
( ', )0
0 '
sec ''( ', )
0 0
0 0
sec ( ') sec sec '
0 0
0 0
sec
0
sec '; ', sec ''
sec ' sec '
sec ' sec '
1sec
s
H
a
z
a a a
a
H Hz H
UP a a
z
H H dzz H
a a
H HH z H z
a a
H
aa
T T e dz z H dz
T e dz T e dz
T e dz T e e dz
T e
0
0
sec
sec
0
1ec
1
a
a
H
H
e
T e
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90
Downwelling Radiation (Low Albedo)
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90
SPECIAL CASES:Constant Medium Temperature, T(z) = To
Uniform Particle Distribution, a(z) = ao
0 0
'
00
0
0
0
0
0
0
0
(0, ')0
0 0
sec ''
0
0
sec '
0
0
sec
0
sec
0
sec '; 0, ' ' sec ''
sec '
sec '
1sec 1
sec
1
z
a
a
a
a
H Hz
DN a a
H dz
a
Hz
a
H
aa
H
T T e dz z z dz
T e dz
T e dz
T e
T e
Special Cases: (Low Albedo)
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91
SPECIAL CASES:Constant Medium Temperature, T(z) = To
Uniform Particle Distribution, a(z) = ao
0sec
0 1 a H
DN UPT T T e
If layer H is several optical depths thick; aoH>>1
0DN UPT T T
Apparent Temperature Inside the Medium
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SPECIAL CASES:Constant Medium Temperature, T(z) = To
Uniform Particle Distribution, a(z) = ao
Application to Homogenous Half Space
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93
(eg. Soil, sea, etc.)
Recall Conservation of Power
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94
12 12
12 12
2
12 12
1i r t i iP P P P P
R
Homogenous Terrain Contributions
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95
There are two contributing factors to radiation observed from above:(1)Scattering of downward radiation from the atmosphere(2)Refraction of upward radiation from ground (terrain contribution)
Terrain Contribution
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96
1 21 2
21 2
0 ,
1 0 ,
B AP
AP
T T
T
Upwelling radiation from a half space with uniform temperature
Recall that for an infinite layer of uniform temperature:
0DN UPT T T
Terrain cont’ed
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Atmospheric Contribution
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1 12 1 1atm
sc DNT T
Downwelling atmospheric temperature
Total Apparent Temperature for a Homogenous Half Space
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99
1 12 10 atmAP g DNT e T T
Apparent Temperature at Altitude, H
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100
0,1 1
0,1 12 1
, 0, atm
atm
H atmAP AP UP
Hatmg DN
atmUP
T H T e T H
e T T e
T H
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References
• Microwave Remote Sensing, F. T. Ulaby, et.al. Addison-Wesley