```EE/Ae 157 a
Passive Microwave Sensing
EE/Ge 157b Week 6
6-1
TOPICS TO BE COVERED
•
•
•
Rayleigh-Jeans Approximation
Power-Temperature Correspondence
– Bare Surfaces
– Vegetation Covered Surfaces
•
– Total Power Radiometers
•
Applications
– Polar Ice Mapping
– Soil Moisture Mapping
EE/Ge 157b Week 6
6-2
•
•
•
Heat energy is a special case of EM radiation
The random motion (due to collisions) of the molecules due to
kinetic energy results in exitation (electronic, vibrational and
rotational) followed by random emissions during decay
This leads to radiation over a large bandwidth according to
Planck’s law for an ideal source (called a black body)
S   
•
2  hc
5
2
1
e ch
 kT
1
Thermal emission is usually unpolarized
EE/Ge 157b Week 6
6-3
Rayleigh-Jeans Approximation
•
When ch   kT we can approximate the exponential term in
Planck’s law by the first two terms in its Taylor series expansion,
e
•
ch  kT
 1  1  ch  kT  1  ch  kT
Substituting this into Planck’s formula, we find
S  , T  
•
2  ckT

4
This approximation shows less than 1% deviation from Planck’s
law as long as
 T  0 . 77 mK
EE/Ge 157b Week 6
6-4
Rayleigh-Jeans Approximation
EE/Ge 157b Week 6
6-5
Relationship between Surface Brightness
and Spectral Radiant Emittance
•
The surface spectral radiant emittance is the integral over all
angles of a quantity known as the surface brightness
S f 
2  2
 B q , f  cos q
d 

•
0
dqd
0
If the brightness is independent of q, (a Lambertian surface)
S  f   B  f
•
  B q , f  cos q sin q

The surface brightness is therefore given by
B f  
EE/Ge 157b Week 6
S f



2 kT

2

2 kT
c
2
f
2
6-6
Power-Temperature Correspondence
•
The power per unit bandwidth
radiated into a solid angle d   by
a surface element ds with
emissivity  q  is
P  f    q  B  f dsd   
•
2 kT

2
2 kT

EE/Ge 157b Week 6
2
 q 
1
2
ANTENNA
 q dsd  
The antenna receives the energy
with different amounts of gain
from different angles. If the
normalized gain pattern is g q ,  
the received power over a narrow
bandwidth df would be
Pr  f  
A
d 
R
GROUND ELEMENT
ds
g q ,  dsd  df
6-7
Power-Temperature Correspondence
•
If the antenna has a receiving area A , the solid
angle subtended by the antenna is
d  
A
R
•
ANTENNA
2
Therefore, the received power is
Pr
•
A
f
2 kT

2
 q

1
2
g q ,  
A
R
2
d 
dsdf
GROUND ELEMENT
ds
The receiver integrates the energy received by
the antenna from all angles. The solid angle
subtended by the surface element when viewed
from the antenna is
d 
ds
R
EE/Ge 157b Week 6
R
ANTENNA
d
R
2
ds
GROUND ELEMENT
6-8
Power-Temperature Correspondence
•
Therefore, we can write the received power as
Pr  f  
•
AkT

2
 q  g q ,   d  df
To find the total power received by the
radiometer, we now have to integrate over the
antenna angles and the bandwidth:
A
ANTENNA
d 
R
GROUND ELEMENT
Pr  f   AkT
•
 
f
1


2
ds
 q  g q ,   d  df
If  f  f this becomes
Pr  f  
AkT  f

2
  q  g q ,   d 

ANTENNA
d
R
ds
EE/Ge 157b Week 6
GROUND ELEMENT
6-9
Power-Temperature Correspondence
•
The received power is usually written as
Pr  kT eq  f
•
where the equivalent temperature is given by
T eq 
•
AT

2
  q  g q ,   d 

The effective temperature observed by the radiometer is therefore
the physical temperature of the surface, multiplied by a factor that
is a function of the surface emissivity and the antenna pattern.
EE/Ge 157b Week 6
6-10
Bare Surface
•
In practice, the radiometer receive power not only from the
surface radiation, but also from energy radiated by the sky and
reflected by the surface
EE/Ge 157b Week 6
6-11
Bare Surface
•
The total power radiated by the surface is therefore
P  f    q  B g  f    q  B s  f
•
Following the same derivation as before, we find the
equivalent temperature to be
T eq 
•
dsd  
A

2
  q T

g

  q T s g q ,   d 
Therefore, the equivalent microwave temperature is
T i   q T g   q T s
EE/Ge 157b Week 6
6-12
Bare Surface
•
Since
  1 
•
we can rewrite the microwave temperature as
Ti  T g   q T s  T g 
•
Note that the refection coefficient is a function of polarization, we
will measure different microwave temperatures for different
polarizations
EE/Ge 157b Week 6
6-13
Reflection Coefficient
•
From Maxwell’s equations, one
finds that
e=3, Rh

2
Rv
2
cos q 
  sin
cos q 
  sin 2 q

q
e=80 , Rh
e=80 , Rv
1
0.8
 cos q 
  sin q
 cos q 
  sin q
2
2
R e fle c tion C oe ffi c ie nt
Rh
2
e=3, Rv
2
0.6
0.4
0.2
2
0
0
10
20
30
40
50
60
70
80
90
Inc idence Angle
EE/Ge 157b Week 6
6-14
Microwave Temperature
T s  40 K ;
350
T g  300 K
Microwave Temperature, K
300
250
200
150
Th,3
100
Tv,3
Th,20
Tv,20
50
0
0
EE/Ge 157b Week 6
20
40
60
80
100
6-15
Effects of Polarization
Tv  q   T s
Th  q   Ts

v
h
1 .8
1 .7
E m m is s iv ity R a tio
1 .6
1 .5
1 .4
1 .3
1 .2
1 .1
1
0
10
20
30
40
50
60
70
80
D ie le c tric C o n s ta n t
EE/Ge 157b Week 6
6-16
Applications: Polar Ice Mapping
EE/Ge 157b Week 6
6-17
Ice Concentration Mapping: Arctic
EE/Ge 157b Week 6
6-18
Ice Concentration Mapping: Arctic
EE/Ge 157b Week 6
6-19
Sea Ice Concentration: Arctic
EE/Ge 157b Week 6
6-20
Vegetation Cover
4
1
Ti  e

 q T s e
EE/Ge 157b Week 6

  q T g e


 Tc 1  e
3

2
   q T 1  e e


c
6-21
Applications: Soil Moisture
EE/Ge 157b Week 6
6-22
Applications: Soil Moisture
EE/Ge 157b Week 6
6-23
Circular Antenna Beam

D

D
D
h
h
h
D cos q
h
h
D
D cos q
h
D cos q
2
EE/Ge 157b Week 6
Side-Looking View
6-24
Conical Scan Geometry
Scan Direction
EE/Ge 157b Week 6
6-25
d
q
V1  q

V 2 q
B



V q
EE/Ge 157b Week 6

6-26
1
0 .8
0 .6
0 .4
0 .2
0
-9 0
-7 5
-6 0
-4 5
-3 0
-1 5
0
15
30
45
60
75
90
-0 .2
-0 .4
-0 .6
-0 .8
-1
EE/Ge 157b Week 6
6-27
1
0 .8
0 .6
0 .4
0 .2
0
-9 0
-7 5
-6 0
-4 5
-3 0
-1 5
0
15
30
45
60
75
90
-0 .2
-0 .4
-0 .6
-0 .8
-1
EE/Ge 157b Week 6
6-28
1
0 .8
0 .6
0 .4
0 .2
0
-9 0
-7 5
-6 0
-4 5
-3 0
-1 5
0
15
30
45
60
75
90
-0 .2
-0 .4
-0 .6
-0 .8
-1
EE/Ge 157b Week 6
6-29
1
0 .8
0 .6
0 .4
0 .2
0
-9 0
-7 5
-6 0
-4 5
-3 0
-1 5
0
15
30
45
60
75
90
-0 .2
-0 .4
-0 .6
-0 .8
-1
EE/Ge 157b Week 6
6-30
1
0 .8
0 .6
0 .4
0 .2
0
-9 0
-7 5
-6 0
-4 5
-3 0
-1 5
0
15
30
45
60
75
90
-0 .2
-0 .4
-0 .6
-0 .8
-1
EE/Ge 157b Week 6
6-31
1
0 .8
0 .6
0 .4
0 .2
0
-9 0
-7 5
-6 0
-4 5
-3 0
-1 5
0
15
30
45
60
75
90
-0 .2
-0 .4
-0 .6
-0 .8
-1
EE/Ge 157b Week 6
6-32
1 .0 0
R e a l P a rt o f V is ib ility
0 .5 0
0 .0 0
-0 .5 0
-1 .0 0
-9 0
-6 0
-3 0
0
30
60
90
O ff-A x is A n g le
EE/Ge 157b Week 6
6-33
 2
2 2
3 2
 2
4 2
5 2
6 2
7 2
EE/Ge 157b Week 6
6-34
1 .0
R e la tiv e A m p litu d e
0 .8
6 0 D e g R ig h t
0 .6
4 0 D e g R ig h t
2 0 D e g R ig h t
N a d ir
2 0 D e g L e ft
4 0 D e g L e ft
0 .4
6 0 D e g L e ft
0 .2
0 .0
-9 0
-6 0
-3 0
0
30
60
90
A n g le fro m N a d ir
EE/Ge 157b Week 6
6-35
SMOS Mission
EE/Ge 157b Week 6
6-36
10
5
0
-5
-10
-10
EE/Ge 157b Week 6
-5
0
5
10
6-37
•
•
The function of a radiometer is to measure the equivalent
temperature of the scene, based on the amount of power delivered
by the antenna to the receiver
The measurement process is characterized by two important
attributes
– accuracy
– precision
•
•
The accuracy of the measurement depends on how well the
The precision of the measurement defines the smallest change in
temperature that the radiometer can measure reliably, and is
driven by radiometer stability
EE/Ge 157b Week 6
6-38
Calibration
•
To calibrate the transfer function of a radiometer, the output
voltage is measured as a function of noise temperature of a
source connected to the input terminals of the receiver
EE/Ge 157b Week 6
6-39
•
The output power is
Pout  Pantenna  Preceiver  k T a  T rec  B  kT sys B
EE/Ge 157b Week 6
6-40
•
•
•
•
Since the input power consists of thermal noise, the
instantaneous voltage at the output of the IF amplifier has a
Gaussian distribution with zero mean.
The output of the square law detector has an exponential
distribution. Such a distribution has a standard deviation that is
equal to its mean value.
Th output of the square law detector will therefore be a signal with
a mean value, and a fluctuating part that has a standard deviation
equal to the mean
It is this fluctuating part that limits the precision of the radiometer,
and will be interpreted as random fluctuations in the measured
system temperature T sys
EE/Ge 157b Week 6
6-41
•
•
The effect of the low-pass filter is to smooth out the fluctuations in
time. If the filter has an equivalent integration time  the
fluctuations at the output of the filter will have a standard
deviation that is reduced by a factor B 
Therefore, at the output of the low-pass filter, we have
 T sys
T sys
•

1
B
From an observational point of view,  T sys is the smallest change
in temperature that the radiometer can measure reliably:
 T IDEAL   T sys 
EE/Ge 157b Week 6
T a  T rec
B
6-42
Effect of System Gain Variations
•
Th previous analysis assumes the system to be perfect. Changes
in receiver gain will also cause the output power to fluctuate. This
will be interpreted as a temperature fluctuation equal to
 Gs
 T g  T sys 
 Gs
•
Since the noise fluctuations, and the gain fluctuations are
uncorrelated, the resulting uncertainty in the system temperature
is
T 
•




 T noise   T g  T sys
2
2
 Gs
 
B  Gs
1




2
In many cases, the gain variations are the largest error source
EE/Ge 157b Week 6
6-43
•
•
•
Experimental results show that the bulk of the gain fluctuations
are at frequencies lower than 1 Hz
A Dicke radiometer uses modulation techniques to reduce the
effects of system gain variations
A Dicke radiometer is basically a total power radiometer with two
– A switch connexted to the receiver input (as close to the antenna as
possible) that modulates the input signal
– A synchronous demodulator placed between the square law detector
and the low-pass filter
•
The modulation consists of periodically switching the receiver
input between the antenna and a constant (reference) noise
source
EE/Ge 157b Week 6
6-44
Dicke Radiometer Block Diagram
EE/Ge 157b Week 6
6-45
•
•
The switching rate is chosen so that over a period of one
switching cycle is essentially constant, and therefore identical for
the half cycle during which the receiver is connected to the
antenna and the half cycle during which the receiver is connected
to the reference source
The output of the square law detector is
V a  CGk T a  T rec  B
V ref  CGk T ref  T rec B
•
for
for
0  t s 2
s 2  t s
Superimposed on these average values are fluctuations due to
noise and gain fluctuations
EE/Ge 157b Week 6
6-46
•
•
•
The synchronized demodulator is consists of a switch operated
synchronously with the input Dicke switch, followed by parallel
amplifiers with opposite polarity
The output of these amplifiers are summed and fed to the lowpass filter
The output of the low-pass filter is
Pout 
•
1
2
G s k T a  T rec  B 
2
G s k T ref  T rec B
Which can be written as
Pout 
•
1
1
2
G s k T a  T ref B
Note that the output is independent of the receiver noise
temperature
EE/Ge 157b Week 6
6-47
•
The fluctuating part of the radiometer output consists of three
parts:
– Gain variations that lead to an uncertainty
 T g  T a  T ref
 G
s
Gs 
– Noise variations, which after integrating over half the cycle lead to an
uncertainty of
 T nant 
2 T a  T rec

B
– Noise on the second half of the integration cycle equal to
 T nref 
EE/Ge 157b Week 6
2 T ref  T rec 
B
6-48
•
Assuming the uncertainties to be statistically independent, the
total uncertainty is
T 
•
2
2
2
This can be written as
T 
•
 T g   T nant   T nref
2 T a  T rec
2  2 T ref
B
 T rec

2
 Gs
 
 Gs
2

 T a  T ref



2
This is known as the sensitivity of an unbalanced Dicke
EE/Ge 157b Week 6
6-49
•
•
•
Of particular importance is the case where T a  T ref
This is a balanced Dicke radiometer
The sensitivity of the balanced Dicke radiometer becomes
T 
•
•
2 T a  T rec 
B
 2  T IDEAL
The factor of 2 comes from the fact that the antenna is observed
for only half the time
Several different approaches are used for balancing Dicke
radiometers. The simplest (conceptually) involves using a
feedback loop to control the reference temperature
EE/Ge 157b Week 6
6-50