1 ee 543 theory and principles of remote sensing derivation of the transport equation
TRANSCRIPT
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
4
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
5
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
O. Kilic EE 543
6
• 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 ,
O. Kilic EE 543
7
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.
O. Kilic EE 543
8
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
O. Kilic EE 543
9
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
O. Kilic EE 543
10
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
11
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
12
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
O. Kilic EE 543
13
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.
O. Kilic EE 543
14
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
O. Kilic EE 543
15
Antenna Temperature and Beam Efficiency
(1 )A ML SL
ML ML ML SL
T T T
T T
SUMMARY
O. Kilic EE 543
16
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
O. Kilic EE 543
17
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
O. Kilic EE 543
18
Summary
The accuracy of radiometric measurements is highly dependent on the radiation efficiency, and main beam efficiency, of the antenna.M
a
SUMMARY
O. Kilic EE 543
19
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
20
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
O. Kilic EE 543
21
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
O. Kilic EE 543
22
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
O. Kilic EE 543
23
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
O. Kilic EE 543
24
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
O. Kilic EE 543
25
Scattering Cross Section Definitions: Power Relations
• Differential Scattering Cross-section
• Scattering Cross-section
• Absorption Cross-section
• Total Cross-section
SUMMARY
O. Kilic EE 543
26
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)
O. Kilic EE 543
27
Scattering Cross-section
4
ˆ ˆˆ,ss d s
i
Pi o i d
S
SUMMARY
(m2)
O. Kilic EE 543
28
Absorption Cross-section
2
ˆ a Va
i i
E dvP
iS S
SUMMARY
(m2)
O. Kilic EE 543
29
Total Cross-section
t s a
tt s a
i
s
t
P P P
P
S
a
albedo
SUMMARY
(m2)
O. Kilic EE 543
30
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
O. Kilic EE 543
31
Phase Function (2)
4
4
1 ˆˆ,4
1 ˆˆ, 14
s
s
p o i d a
o i d
SUMMARY
O. Kilic EE 543
32
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
O. Kilic EE 543
33
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)
O. Kilic EE 543
34
Conservation of Power
out in loss gainP P P P
Extinction:
off-scattering + absorption
self emission + scattering
(3)
O. Kilic EE 543
35
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
O. Kilic EE 543
36
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)
O. Kilic EE 543
37
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
O. Kilic EE 543
38
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
O. Kilic EE 543
39
Absorption Coefficient
ˆ,a
a
f
a
aP sB r s f a w
Define “absorption coefficient: Unit: #/m
(7)
(8)
O. Kilic EE 543
40
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
O. Kilic EE 543
41
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.
O. Kilic EE 543
42
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
O. Kilic EE 543
43
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)
O. Kilic EE 543
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)
O. Kilic EE 543
46
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)
O. Kilic EE 543
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
O. Kilic EE 543
48
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)
O. Kilic EE 543
49
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
O. Kilic EE 543
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
O. Kilic EE 543
51
Self Emission Function
ˆ, ( ) ( )
(1 ) ( );
a f a f
st f
t
r s B T B T
a B T a
(18)
O. Kilic EE 543
52
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)
O. Kilic EE 543
55
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
O. Kilic EE 543
56
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
O. Kilic EE 543
57
Solution to Transport Equation (1)
Solution to Transport Equation (2)
O. Kilic EE 543
58
Solution to Transport Equation (3)
O. Kilic EE 543
59
Transport Equation Scaled to Temperature (1)
O. Kilic EE 543
60
Transport Equation Scaled to Temperature (2)
O. Kilic EE 543
61
Transport Equation Scaled to Temperature (3)
O. Kilic EE 543
62
Solution to Transport Equation (Temperature Form)
O. Kilic EE 543
63
Solution to Transport Equation
O. Kilic EE 543
64
'
0
ˆ ˆ ˆ, 0, ', 'AP AP TT s T s e J s e d
Low Albedo Case, a<<1
O. Kilic EE 543
65
Solution for Low Albedo
O. Kilic EE 543
66
Upwelling Radiation (Observe the Medium from Above)
O. Kilic EE 543
67
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
O. Kilic EE 543
68
Upwelling Radiation
O. Kilic EE 543
69
Solution: Upwelling Radiation
O. Kilic EE 543
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
O. Kilic EE 543
73
Solution
O. Kilic EE 543
74
Solution
O. Kilic EE 543
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
O. Kilic EE 543
76
Solution
O. Kilic EE 543
77
Downwelling Radiation from a Layer (Observe the Medium from Below)
O. Kilic EE 543
78
cos
sec
seca a
H z s
ds
dzd ds dz
Downwelling Radiation
O. Kilic EE 543
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
O. Kilic EE 543
80
Downwelling Radiation
O. Kilic EE 543
81
'
'
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
O. Kilic EE 543
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
O. Kilic EE 543
83
O. Kilic EE 543
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
O. Kilic EE 543
85
Solution
O. Kilic EE 543
86
Solution
O. Kilic EE 543
87
O. Kilic EE 543
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)
O. Kilic EE 543
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
O. Kilic EE 543
90
Downwelling Radiation (Low Albedo)
O. Kilic EE 543
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)
O. Kilic EE 543
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
O. Kilic EE 543
92
SPECIAL CASES:Constant Medium Temperature, T(z) = To
Uniform Particle Distribution, a(z) = ao
Application to Homogenous Half Space
O. Kilic EE 543
93
(eg. Soil, sea, etc.)
Recall Conservation of Power
O. Kilic EE 543
94
12 12
12 12
2
12 12
1i r t i iP P P P P
R
Homogenous Terrain Contributions
O. Kilic EE 543
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
O. Kilic EE 543
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
O. Kilic EE 543
97
Atmospheric Contribution
O. Kilic EE 543
98
1 12 1 1atm
sc DNT T
Downwelling atmospheric temperature
Total Apparent Temperature for a Homogenous Half Space
O. Kilic EE 543
99
1 12 10 atmAP g DNT e T T
Apparent Temperature at Altitude, H
O. Kilic EE 543
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
O. Kilic EE 543
101
References
• Microwave Remote Sensing, F. T. Ulaby, et.al. Addison-Wesley