Microwave Interaction with
Atmospheric Constituents
Chris Allen (callen@eecs.ku.edu)
Course website URL
people.eecs.ku.edu/~callen/823/EECS823.htm
1
Outline
Physical properties of the atmosphere
Absorption and emission by gases
– Water vapor absorption
– Oxygen absorption
Extraterrestrial sources
Extinction and emission by clouds and precipitation
– Single particle effects
• Mie scattering
• Rayleigh approximation
– Scattering and absorption by hydrometeors
– Volume scattering and absorption coefficients
– Extinction and backscattering
• Clouds, fog, and haze
• Rain
• Snow
– Emission by clouds and rain
2
Physical properties of the atmosphere
The gaseous composition, and variations of temperature,
pressure, density, and water-vapor density with altitude are
fundamental characteristics of the Earth’s atmosphere.
Atmospheric scientists have developed standard models
for the atmosphere that are useful for RF and microwave
models.
These models are representative and variations with
latitude, season, and region may be expected.
3
Atmospheric composition
4
Temperature, density, pressure profile
Atmospheric density, pressure,
and water-vapor density
decrease exponentially with
altitude.
The atmosphere is subdivided
based on thermal profile and
thermal gradients (dT/dz)
where z is altitude.
Troposphere
surface to about 10 km
dT/dz ~ -6.5 C km-1
Stratosphere
upper boundary ~ 47 km
dT/dz ~ 2.8 C km-1 above ~ 32 km
Mesosphere
upper boundary 80 to 90 km
dT/dz ~ -3.5 C km-1 above ~ 60 km
5
Temperature model
Only the lowermost 30 km of the atmosphere significantly
affects the microwave and RF signals due to the
exponential decrease of density with altitude.
For this region a simple piece-wise linear model for the
atmospheric temperature T(z) vs. altitude may be used.
 T0  a z ,

Tz    T11,
T11  (z  20) ,

0  z 11 km
11 km  z  20 km
20 km  z  32 km
Here T(z) is expressed in K, T0 is the sea-level temperature
and T(11) is the atmospheric temperature at 11 km. For the
1962 U.S. Standard Atmosphere, the thermal gradient term
a is -6.5 C km-1 and T0 = 288.15 K.
6
U.S. Standard Atmosphere, 1962
7
Density and pressure models
For the lowermost 30 km of the atmosphere a model that
predicts the variation of dry air density air with altitude is
air z   1.225 e  z H 2 1  0.3 sin z H 2 , for 0  z  30 km
where air has units of kg m-3, z is the altitude in km, H2 is
7.3 km.
Assuming air to be an ideal gas we can apply the ideal gas
law to predict the pressure P at any altitude (up to 30 km
above sea level) using
Pz   2.87 air z  Tz  (mbar ), for 0  z  30 km
Alternatively pressure can be found using
Pz   P0 e z H3 , for 0  z  10 km
where H3 = 7.7 km and Po = 1013.25 mbar
8
Water-vapor density model
The water-vapor content of the atmosphere is weather
dependent and largely temperature driven.
The sea-level water vapor density can vary from 0.01 g m-3
in cold dry climates to 30 g m-3 in warm, humid climates.
An average value for mid-latitude regions is 7.72 g m-3.
Using this value as the surface value at sea-level, we can
use the following model to predict the water-vapor density
v at any altitude using
 v z   0 e  z H 4 , for 0  z  30 km
where v has units of g m-3, 0 is 7.72 g m-3, and H4 is 2 km.
9
Absorption and emission by gases
Molecular absorption (and emission) of electromagnetic
energy may involve three types of energy states
where
E  Ee  E v  Er
Ee = electronic energy
Ev = vibrational energy
Er = rotational energy
Of the various gases and vapors in the Earth’s atmosphere,
only oxygen and water vapor have significant absorption
bands in the microwave spectrum.
Oxygen’s magnetic moment enables rotational energy
states around 60 GHz and 118.8 GHz.
Water vapor’s electric dipole enables rotational energy
states at 22.2 GHz, 183.3 GHz, and several frequencies
above 300 GHz.
10
Spectral line shape
For a molecule in isolation the absorption and emission
energy levels are very precise and produce well defined
spectral lines. Energy exchanges and interactions in the
form of collisions result in a spectral line broadening. One
mechanism that produces spectral line broadening is
termed pressure broadening as it results from collisions
between molecules.
11
Absorption spectrum model
The absorption spectrum for transactions between a pair of
energy states may be written as
where
4 f
 a f , f lm  
Slm Ff , f lm 
c
a = power absorption coefficient, Np m-1
f = frequency, Hz
flm = molecular resonance frequency for transitions between energy states
El and Em, Hz
c = speed of light, 3  108 m s-1
Slm= line strength of the lm line, Hz
F = line-shape function, Hz-1
The line strength Slm of the lm line depend on the number of
absorbing gas molecules per unit volume, gas temperature,
and molecular parameters.
12
Line-shape function
There are several different line-shape functions, F, used to
describe the shape of the absorption spectrum with respect
to the resonance frequency, flm.
The Lorentzian function, FL, is the simplest
1



FL f , f lm 
 f lm  f 2   2
here
 = linewidth parameter, Hz
The Van Vleck and Weisskopf function, FVW, takes into
account atmospheric pressures

1 f 


FVW f , f lm  


2
2
 f lm  f lm  f   
f lm  f 2   2 
13
Line-shape function
The Gross function, FG, was
developed using a different
approach and shows better
agreement with measured data
further from the resonance
frequency.
FG f , f lm  
1
4 f f lm 
2
2 2
 f lm
 f  4 f 2 2


14
Water-vapor absorption
Absorption due to water vapor can be modeled using
4 f
 H 2O f , f lm  
Slm FG f , f lm , Np m 1
c
For each water-vapor absorption line the line strength is
Slm  Slm0 f lm  v T 5 2 e  E l
kT
where
Slm0 = constant characteristic of the lm transition
flm = the resonance frequency
v = water-vapor density
El = lower energy state’s energy level
k = Boltzmann’s constant (1.38  10-23 J K-1)
T = thermodynamic temperature (K)
Thus (f, flm) expressed in dB km-1 is
 4 
5 2  E l
 H 2O f , f lm   4.34  10 
e
 Slm0 f f lm  v T
 c 
3
kT
FG f , f lm 
15
Water-vapor absorption
Water vapor has resonant frequencies at
22.235 GHz, 183.31 GHz, 323 GHz, 325.1538 GHz,
380.1968 GHz, 390 GHz, 436 GHz, 438 GHz, 442 GHz, …
For frequencies below 100 GHz we may consider the
water-vapor absorption coefficient to be composed of two
factors
 H 2O   f , 22   r f 
Where
(f, 22) = absorption due to 22.235-GHz resonance
r(f) = residual term representing absorption due to all higherfrequency water-vapor absorption lines
16
Water-vapor absorption
Using data for the 22.235-GHz resonance we get
f , 22  2 f  v 300 T  e
52
2
 644 T
1
494.4  f

2 2

 4 f 2 12 , dB km 1
where the linewidth parameter 1 is
 1  2.85 P 1013300 T
0.626
1  0.018 vT P , GHz
f and 1 are expressed in GHz, T is in K, v is in g m-3, and
P is in millibars.
The residual absorption term is
r f   2.4 106 f 2 v 300 T 1 , dB km1
32
Therefore the total water vapor absorption below 100 GHz is
 H 2O f   2 f 2  v 300 T 
32
 300

e 644 T
6
1
1 

1
.
2

10
,
dB
km

2 2
2 2
 T 494.4  f   4 f 1

17
Water-vapor absorption
18
Oxygen absorption
Molecular oxygen has numerous absorption lines between
50 and 70 GHz (known as the 60-GHz complex) as well as
a line at 118.75 GHz.
Around 60 GHz there
are 39 discrete resonant
frequencies that blend
together due to
pressure broadening
at the lower altitudes.
Complex models are
available that predict
the oxygen absorption
coefficient throughout
the microwave
Resonant frequencies (GHz) in the 60-GHz complex: 49.9618, 50.4736, 50.9873, 51.5030, 52.0212,
spectrum.
52.5422, 53.0668, 53.5957, 54.1300, 54.6711, 55.2214, 55.7838, 56.2648, 56.3634, 56.9682, 57.6125,
58.3239, 58.4466, 59.1642, 59.5910, 60.3061, 60.4348, 61.1506, 61.8002, 62.4863, 62.4112, 62.9980,
63.5685, 64.1278, 64.6789, 65.2241, 65.7647, 66.3020, 66.8367, 67.3964, 67.9007, 68.4308, 68.9601, 69.4887
19
Oxygen absorption
For frequencies below 45 GHz a low-frequency approximation
model may be used that combines the effects of all of the
resonance lines in the 60-GHz complex with a single
resonance at 60 GHz, and that neglects the effect of the
118.75-GHz resonance.
O
2
2

P
300
1
1 




2 2
1
 1.110 f 

,
dB
km

 
2
2
2
2
1013
T
f




  f  f 0   

where f is in GHz, f0 = 60 GHz, and
 P   300 
  0 


 1013   T 
0.85
, GHz
P  333 mbar
0.59 ,

 0  0.59 1  3.1103 333  P  , 25  P  333 mbar
 1.18 ,
P  25 mbar



20
Total atmospheric gaseous absorption
As water vapor and oxygen are the dominant sources for
atmospheric absorption (and emission), the total gaseous
absorption coefficient is the sum of these two components
 g f    H O f    O f , dB km
2
2
1

 0    g z  dz , dB
0
21
Total atmospheric gaseous absorption
Non-zenith optical thickness
can be approximated as
    0 sec  , dB
for   70°.
22
Atmospheric gaseous emission
We know that for a non-scattering gaseous atmosphere

TDN   sec   g z Tze  0 0, z secdz
0
where
z
0 0, z    g z  dz , Np
0
An upward-looking radiometer would receive the downwelling radiation, TDN, plus a small energy component from
cosmic and galactic radiation sources.
TSKY   TDN   TEXTRA e   sec 
where
TEXTRA  TCOS  TGAL
0
TCOS and TGAL are the cosmic and galactic brightness
temperatures, and TEXTRA is the extraterrestrial brightness
temperature.
23
Extraterrestrial sources
TCOS is independent of frequency and direction.
TCOS  2.7 K
TGAL is both frequency and direction dependent.
Frequency dependence
Depending on the specific region of the galaxy,
TGAL  f 2.5 to f 3
Above 5 GHz, TGAL « TDN and TGAL may be neglected.
Below 1 GHz TGAL may not be ignored.
TGAL plus man-made emissions limit the usefulness of
Earth observations below 1 GHz.
Direction dependence
TGAL(max) in the direction of the galactic center while
TGAL(min) is the direction of the galactic pole.
24
Extraterrestrial sources
The galactic center is located in the
constellation Sagittarius. Radiation
from this location is associated with
the complex astronomical radio
source Sagittarius A, believed to be a
supermassive black hole.
25
Effects of the sun
The sun’s brightness temperature TSUN is frequency
dependent as well as dependent on the “state” of the sun.
For the “quiet” sun (no significant sunspots or flares) TSUN
decreases with increasing frequency.
At 100 MHz, TSUN is about 106 K, while at 10 GHz it is 104 K,
and above 30 GHz TSUN is 6000 K.
When sunspots and flares are present, TSUN can increase by
orders of magnitude.
Jupiter, a star wannabe, also emits significant energy though
it is smaller than the active sun by at least two orders of
magnitude.
26
Other radio stars
Taken from: Preston, GW; “The Theory of Stellar Radar,” Rand Corp. Memorandum RM-3167-PR, May 1962.
The radio stars (Cassiopeia A, Cygnus A, Centaurus A, Virgo,
etc.) are astounding sources of RF energy, not only because
of their great strength, but also because of their remarkable
energy spectra.
These spectra reach their maxima at
about 10 m wavelength (30 MHz in
frequency) and fall off rather sharply
at higher frequencies (~ 10 dB/decade).
The flux density from Cassiopeia exceeds
the solar flux at longer wavelengths.
Compared to Cassiopeia, Cygnus is
2 dB weaker, Centaurus is 8 dB weaker,
and Virgo is 10 dB weaker.
27
Extinction and emission by
clouds and precipitation
Electromagnetic interaction with individual spherical
particles
A spherical particle with a radius r is illuminated by an
electromagnetic plane wave with power density Si [W m-2], a
portion of which is absorbed, Pa.
The absorption cross-section, Qa is
Q a  Pa Si , m 2
The absorption efficiency factor, a, is the ratio of Qa to the
geometrical cross-section, A, is
a  Qa  r 2
28
Electromagnetic interaction with
individual spherical particles
If the incident wave were traveling along the +z axis, and
Ss(, ) is the power density radiation scattered in the (, )
direction at distance R, then the total power scattered by
the particle is
Ps   Ss ,  R 2 d 
4
The scattering cross section, Qs and the scattering
efficiency factor, s are
Q s  Ps Si , m 2
s  Qs  r 2
Thus Pa + Ps represent the total power removed from the
incident wave and the extinction cross section Qe and
extinction efficiency e are
Q e  Q a  Qs
e  a  s
29
Electromagnetic interaction with
individual spherical particles
For monostatic radar applications, the radar backscattering
cross-section b is of interest and this is that portion of
Ss(, ) directed back toward the radiation source, i.e.,
Ss( = ) or Ss().
Note: Incident wave travels along the +z axis,
so  =  corresponds to backscatter direction.
Also, when  = ,  has no significance.
b is defined as
Ss   
or
Si  b
2
,
W
m
4R2
Ss  
 b  4 R
, m2
Si
2
30
Mie scattering
Gustov Mie, in 1908, developed the complete solution for
the scattering and absorption of a dielectric sphere of
arbitrary radius, r, composed of a homogeneous, isotropic
and optically linear material irradiated by an infinitely
extending plane wave.
Key terms are the Mie particle size parameter  and the
refractive index n (refractive contrast?)
2r 2r


rb
b
0
n   cp  cb  n   j n 
where
′rb = real part of relative dielectric constant of background medium
cb = complex dielectric constant of background medium (F m-1)
cp = complex dielectric constant of particle medium (F m-1)
0 = free-space wavelength (m)
b = wavelength in background medium (m)
31
Mie scattering
Numerical solutions for spheres of various composition.
“optical” limit
e = 2 for
»1
32
Mie scattering
Strongly conducting sphere
For  << 1,
s << a
33
Mie scattering
Weakly absorbing sphere
Again, for  « 1, s « a
so e  a
Also, as   ,  a  1
and s  1 if 0 < n″ « 1
34
Backscattering efficiency, b
Mie’s solution also predicts the backscattering efficiency, b,
for a spherical particle
“optical” limit
b = 1 for
»1
35
Rayleigh approximation
For particles much smaller than the incident wave’s
wavelength, i.e., |n | « 1, the Mie solution can be
approximated with simple expressions known as the
Rayleigh approximations.
For |n | < 0.5 (Rayleigh region)
8 4 2
s   K
3
where
a  4  Im K
8 4 2
 e  s   a   K  4  Im K
3
n 2  1 c  1
K 2

n  2 c  2
and
8 4 2 2 22b 6 2
Qs  s  r    r K 
 K , m2
3
3
2
2 3
Q a   Im K, m 2

Unless the partical is weakly absorbing (i.e., n″« n′) such that
Im{-K} « |K|2, Qa » Qs since Qs varies as 6 and Qa varies as 3.
36
Rayleigh approximation
2r
22b 6 2
Qs 
 K and  
3
b
so
Q s  r 6 4
Therefore the scattering cross section increases quite
rapidly with particle radius and with increasing frequency.
Example
For  held constant, a 12% increase in radius r (a 40%
volume increase) doubles the scattering cross section.
For a constant radius r, an octave increase in frequency
(factor of 2) results in a 16-fold increase (12 dB) in the
scattering cross section.
37
Rayleigh backscattering
Again, for the Rayleigh region (|n | < 0.5), a simplified
expression for the backscattering efficiency is found,
Rayleigh’s backscattering law
b  4  K
4
or
2
b  4  r 2 4 K , m2
2
And as was the case for the scattering cross section,
 b  r 6 4
Therefore in the Rayleigh region, the backscattering cross
section is very sensitive to particle size relative to
wavelength.
38
Rayleigh backscattering
For large |n|, |K|  1 yielding
b  4 4
However for the case of |n| =  (perfect conductor) which
violates the Rayleigh condition (|n | < 0.5) for finite particle
sizes, the backscattering cross section can be found for
|| «1 using Mie’s solution
b  9 4 K
2
for n   and  « 1
or
 b  9  r 2  4 , m 2 for a conducting sphere .
39
Rayleigh backscattering
40
Scattering and absorption by hydrometeors
Now we consider the interaction of RF and microwave
waves with hydrometeors (i.e., precipitation product, such
as rain, snow, hail, fog, or clouds, formed from the
condensation of water vapor in the atmosphere).
Electromagnetic scattering and absorption of a spherical
particle depend on three parameters:
wavelength, 
particle’s complex refractive index, n
particle radius, r
This requires an understanding of the dielectric properties
of liquid water and ice.
41
Pure water
The Debye equation describes the frequency dependence
of the dielectric constant of pure water, w
 w   w 
 w 0   w
1 j2  f w
where
w0 = static relative dielectric constant of pure water, dimensionless
w = high-frequency (or optical) limit of w, dimensionless
w = relaxation time of pure water, s
f = electromagnetic frequency, Hz
Algebraic manipulation yields
w   w 
 w 0   w
2
1  2  f  w 
w 
2  f  w  w 0   w 
2
1  2  f  w 
42
Pure water
While w is apparently temperature independent,
temperature affects w0 and w causing ′w and ″w to be
dependent on temperature, T.
 w  4.9
The relaxation time for pure water is
2   w T   1.1109 10 10  3.824 10 12 T  6.938 10 14 T 2  5.096 10 16 T 3
where T is expressed in C.
The corresponding relaxation frequency fw0 of pure water is
f w 0  1 2  w
which varies from 9 GHz at 0 C to 17 GHz at 20 C.
The temperature-dependent static dielectric of water is
 w 0 T   88.045  0.4147 T  6.295 10 4 T 2  1.075 10 5 T 3
43
Pure water
Relative dielectric constant, real part, r′ vs. imaginary part, r″
44
Pure water
To apply the solutions from Mie or Rayleigh requires the
complex refractive index.
n  n  j n   rc
where
rc is the complex relative dielectric constant
n  Re
n  Im
 
rc
 
rc
45
Pure water
Refractive index, real part, n′
46
Pure water
Refractive index, imaginary part, n″
47
Pure water
Refractive index, magnitude |n|
48
Sea water
Saline water is water containing dissolved salts.
The salinity, S, is the total salt mass in grams dissolved in
1 kg of water and is typically expressed in parts per
thousand (‰) on a gravimetric (weight) basis.
The average sea-water salinity, Ssw, is 32.54 ‰
The following expressions for the real and imaginary parts
of the relative dielectric constant of saline water are valid
over salinity range of 4 to 35 ‰ and the temperature range
from 0 to 40 C.
sw  sw 
where
sw 0  sw
2
1  2  f sw 
 
sw
2  f sw sw0  sw 
i

2
2  f 0
1  2  f sw 
sw is the relaxation time of saline water, s
i is the ionic conductivity of the aqueous soluiton, S m-1
0 is the free-space permittivity, 8.854  10-12 F m-1
49
Sea water
The high-frequency (or optical) limit of sw is independent of
salinity.
sw   w  4.9
The static relative dielectric constant of saline water
depends on salinity (‰) and temperature (C).
sw0  sw0 T, 0 a T, Ssw 
where
 sw 0 T, 0  87.134  1.949 10 1 T  1.276 10 2 T 2  2.49110 4 T 3
a T, Ssw   1.0  1.613 105 T Ssw  3.656 103 Ssw
2
 3.210 105 Ssw
 4.232 107 S3sw
50
Sea water
The relaxation time is also dependent on both salinity and
temperature.
sw T, Ssw   sw T, 0 b T, Ssw 
where
sw(T, 0) = w(T) that was given earlier
b T, Ssw   1.0  2.282 105 T Ssw  7.638 104 Ssw
2
 7.760 106 Ssw
 1.105 108 S3sw
51
Sea water
Finally, the ionic conductivity for sea water, i, depends on
salinity (‰) and temperature (C) as
i T, Ssw   i 25, Ssw  e  
where the ionic conductivity at 25 C is

2
i 25, Ssw   Ssw 0.18252  1.4619 10 3 Ssw  2.093 10 5 Ssw
 1.282 10 7 S3sw

and

  2.033 102  1.266 104   2.464 106 2

 Ssw 1.849 105  2.551107   2.551108 2

where  = 25 – T, T is in C
52
Pure and sea water
Relative dielectric constant, real part, r′
53
Pure and sea water
Relative dielectric constant, imaginary part, r″
54
Pure and fresh-water ice
As water goes from its liquid state to its solid state, i.e., ice,
its relaxation frequency drops from the GHz range to the
kHz range.
At 0 C the relaxation frequency of ice, fi0, is 7.23 kHz and
at -66 C it is only 3.5 Hz.
At RF and microwave frequencies the term 2fi0 or f/fi0 is
much greater than one. Therefore the real part of the
relative dielectric of pure ice (i′) should be independent of
frequency and temperature (below 0 C) at RF and
microwave frequencies.
i  i  3.15
55
Characteristics of ice
The dielectric properties of ice can be predicted by the
Debye equation
 rs   r
2  f  rs   r 
   r


r   r 


 r   r  rs
r
2
2
1  2 f  
1  2 f  
1  j 2 f 
Complex
Real part
Imaginary part
Multiple relaxation frequencies exist for pure ice, some in
the kHz, others in the THz.
Multiple relaxation
frequencies exist for
pure ice, some in the
kHz, others in the THz.
In the kHz band
20 s ≤  ≤ 40 ms
In the THz band
6 fs ≤  ≤ 30 fs
56
Pure and fresh-water ice
There is some variability in reported measured values for i′.
Recent work shows that
i  3.1884  0.00091T  273.15, T is in Kelvin
57
Pure and fresh-water ice
Similarly the Debye expression for the imaginary part (i″)
simplifies to
i 
 i 0   i
  i 0   i  f i 0 f
2  f i
where i0 = 91.5 at 0 C.
However while the Debye equation predicts that i″ should
decrease monotonically with increasing frequency,
experimental data do not agree.
The relatively small value for the loss factor i″ makes
accurate measurement difficult.
Possible cause for this discrepancy is a resonant frequency
in the infrared band (5 THz and 6.6 THz).
58
Pure and fresh-water ice
Relative dielectric constant, imaginary part, r″
59
Pure and fresh-water ice
Relative dielectric constant, imaginary part, r″
Loss (dB/m)  f·
So for region where   1/f,
Loss is frequency independent
60
Pure and fresh-water ice
An empirical fit of the data presented in Fig. E.3 (previous
slide) relating  to frequency and temperature resulted in
1  2.02 0.025T


 i  10
10f
where T is the physical ice temperature in C (always a
negative value) and f is the frequency expressed in GHz.
Strictly speaking, this relationship is only valid for
frequencies from 100 MHz to about 700 MHz and
temperatures from -1 C and -20 C.
This yields the following expression for ice attenuation
which is independent of frequency (up to around 700 MHz)
0.955x106  0.025T
Np / m 
10
c i
61
Pure and fresh-water ice
62
Characteristics of ice
63
Characteristics of ice
64
Characteristics of ice
65
Characteristics of ice
66
Characteristics of ice
67
Liquid water hydrometeors
Electromagnetic scattering and absorption of a spherical
particle depend on three parameters:
wavelength, 
particle’s complex refractive index, n
particle radius, r
Now consider the various sizes of water particles naturally
found in the atmosphere.
The radius of particles in clouds range from 10 to 40 m
cirrostratus: 40 m, cumulus congestus: 20 m
low-lying stratus & fair-weather cumulus: 10 m
Particles in a fog layer have a radius around 20 m.
Particles forming “heavy haze” conditions have a radius
around 0.05 m.
Rain clouds may have particles with radii as large as a few
millimeters.
68
Drop-size distribution for cloud types
69
Drop-size distribution by rain rate
70
Liquid water hydrometeors
At 3 GHz, Rayleigh approx. is valid
for rain clouds while at 30 GHz it is
valid for water clouds and at
300 GHz for fair-weather clouds.
71
Ice particles and snow
For ice particles (e.g., sleet, hail) the Rayleigh and Mie
solutions are applicable recognizing that |ni| = 1.78 and
using the appropriate particle dimensions.
For snowflakes, the radius, rs, and density, s, of the
snowflake must be known. Snow is a mixture of air and ice
crystals so the snow density can vary from that of air to that
of ice, i = 1 g cm-3.
It has been shown that the backscattering cross section of
a snowflake can be approximated using an equivalent
radius for an ice particle, ri, i.e., rs3 = ri3 / s and
16 5 6
 bs  4 ri , m  2
0
72
Volume scattering and absorption coefficients
Consider now the situation were we have multiple particles
within a volume (e.g., cloud or rain) such that as a plane
wave propagates through this volume it experiences
scattering, absorption, extinction, and backscatter.
Some reasonable assumptions used to simplify the
analysis of this problem include:
– the particles are randomly distributed with the volume
(permitting the application of incoherent scattering theory)
– the volume density is low
(may ignore shadowing of one particle on another)
With these assumptions the effects of the ensemble of
particles is simply the algebraic summation of the effects of
each particle’s contribution. This applies to scattering,
absorption, extinction, and backscattering.
73
Volume scattering
The volume scattering coefficient, s, will be the sum of the
scattering cross section of each particle in the volume.
It is the total scattering cross section per unit volume;
therefore its units are (Np m-3)(m2)=Np m-1
Since the particles are not of a uniform size, the particle
size distribution must be a factor in the calculation. We use
the drop-size distribution, p(r), which defines the “partial
concentration of particles per unit volume per unit
increment in radius.”
 s   pr  Q s r  dr, Np m 1
r2
where
r1
Q(r) = scattering cross section of sphere of radius r, m2
r1 and r2 = lower and upper limits of drop radii within volume, m
74
Volume scattering
The volume scattering coefficient, s, can also be found
using the scattering efficiency, s, since s = Qs/r2.
30
s  2
8


0
 2 p  s   d , Np m 1
where  = 2r/0.
Note that while the limits go from 0 to , in reality
p() = 0 for r < r1 and r > r2
The scattering efficiency term, s, comes from the Mie
solution, however if the conditions for use of the Rayleigh
approximations are satisfied, the s may be the simplier
expressions.
75
Volume absorption, extinction, and backscattering
Similarly, the volume absorption coefficient, a, is
30
a  2
8


0
 2 p  a   d , Np m 1
And the volume extinction coefficient, e, is
30
e  2
8


 2 p  e   d , Np m 1
0
The volume backscattering coefficient, v, also known as
the radar reflectivity with units of (m-3)(m2) = m-1, is
30
v  2
8


0
 2 p   b   d , m 1
76
Drop-size distribution – clouds
For clouds, fog, and haze, key parameters and
characterizations of various cloud models include:
–
–
–
–
Water content, mv (g m-3)
Drop-size distribution, p(r)
Particle composition – ice, water, or rain
Height (above groud) of the cloud base (m)
77
Examples of cloud types
Cirrostratus
Low-lying stratus
Fog layer
Haze, heavy
Fair-weather cumulus
Cumulus congestus
78
Drop-size distribution – clouds
The drop size distribution is given by


pr   a r  exp  b r  , 0  r  
and p(0) = p() = 0. The variables a, b, , and  are
positive, real constants related to the cloud’s physical
properties. Furthermore,  must be an integer.
Values for both  and  are listed in the previously shown
table.
Given p(r), the total number of particles per unit volume, Nv,
can be found by integrating p(r) over all values of r

N v   pr  dr
0
which simplifies to
Nv 
a  1 
 b1
where ( ) is the standard
gamma function and
1 
 1

79
Drop-size distribution – clouds
In addition, the mode radius of the distribution, rc, is
rc   b 
[Note: mode = the most frequent value assumed by a random variable]
So the maximum density in the distribution is
prc   a rc exp    
The total water content per unit volume, mv (g m-3), also
known as the mass density, is the product of the volume
occupied by the particles, Vp, and the density of water
(106 g m-3) where Vp is obtained by multiplying p(r) by 4r3/3
and integrating which yields
4 106 a 
3


mv 


,
g
m
2
2
3b
where
 4
2 

80
Drop-size distribution – clouds
Finally, a normalized drop-size distribution, pn(r) can be
found where pn(r) is the ratio of p(r) to p(rc).
p n r   r rc 

So p(r) = pn(r)  p(rc)
pr   r rc 



 


exp  r rc   1 
 



 
 

exp  r rc   1  a rc exp    
 

or
 

pr   a r exp  r rc  
 


81
Volume extinction – clouds
For ice clouds the Rayleigh
approximation is valid for
frequencies up to 70 GHz while for
water clouds it is valid up to about
50 GHz.
For both cloud types, the absorptive
cross section Qa is much greater
than the scattering cross section Qs.
The extinction due to clouds ec (dB
km-1) can be expressed as
 ec  1  m v
where 1 (dB km-1 g-1 m3) is the
extinction for mv= 1 g m-3 and
6
1  0.434
Im  K
o
with o in cm
82
Volume backscattering – clouds
Under the Rayleigh assumption
64 5 r 6 2
2
b 
K
,
m
40
For the case of Nv particles per unit volume, the cloud
volume backscattering coefficient, vc is
Nv
 vc    b ri 
i 1
64 5 2

K
4
0
Nv
6
1
r
,
m
i
i 1
Now define the reflectivity factor Z to be
Nv
Z   d i6 , m6
i 1
where di is the diameter of the ith particle expressed in m.
83
Volume backscattering – clouds
The cloud volume backscattering coefficient now becomes
5
2
 vc  4 K Z , m 1
0
When Z is expressed in mm6 and 0 is in cm,
5

2
 vc  10 10 4 K Z , m 1
0
The Z factor can be related to the liquid water content
mv (g m-3) as
Z w  4.8 10 2 m 2v , mm 6
Similarly a Z factor for the liquid water content of an ice
cloud is found
Zi  9.21103 m 4v , mm 6
84
Volume backscattering – clouds
So while the |K|2 term is larger for water particles, the
backscattering from ice clouds is larger since ice particles
are typically an order of magnitude larger than water
particles. Consequently ice clouds are therefore more
readily detected.
water
1.47
2
 vwc  4 10 9 K w m 2v , m 1
0
ice
2.82
2
 vic  4 10  4 K i m 4v , m 1
0
At microwave frequencies,
0.89  |Kw|2  0.93 (0 C  T  20 C)
|Ki|2  0.2
85
Extinction and backscattering – rain
Raindrops are typically two orders of magnitude larger than
water droplets in clouds.
Therefore while the Rayleigh approximation is valid for water
clouds at frequencies up to 50 GHz, for rainfall rates of 10 mm
hr-1 it is valid up to only about 10 GHz.
Knowledge of the drop-size distribution is required to predict
the extinction and backscattering parameters for rain.
For rainfall rates between 1 and 23 mm hr -1 the following
model may be used
pd   N 0 e  b d , m 4
Where p(d) is the number of drops of diameter d (m) per unit
volume per drop-diameter interval, N0 = 8.0106 m-4, and
b (m-1) is related to rainfall rate Rr (mm hr-1) by
b  4100 R r 0.21
86
Drop-size distribution by rain rate
Measured drop-size data for various rainfall rates
87
Volume extinction – rain
The volume extinction coefficient of rain (er) is
30  2
 er  2   p  e   d  , Np m 1 where  = 2r/0.
8 0
88
Volume extinction – rain
89
Volume extinction – rain
A direct relationship between the volume extinction
coefficient of rain (er) and the rainfall rate Rr involves
1 (dB km-1 per mm hr-1)
 er  1 R br , dB km 1
where b is a dimensionless parameter.
Both 1 and b are wavelength dependent and determined
experimentally.
The rainfall rate, Rr (mm hr-1), is related to the drop-size
distribution, p(d), as well as the raindrop’s terminal velocity,
vi (m s-1) and the number of drops per unit volume, Nv (m-3).
R r  6  104
Nv
3
1
v
d
,
mm
hr
 i i
i 1
90
Volume extinction – rain
The polarization dependence arises from
the oblate spheriod (i.e., non-spherical)
raindrop shape.
91
Volume extinction – rain
Horizontal-path extinction
(attenuation) for various
rainfall rates.
92
Volume backscattering – rain
The volume backscattering coefficient for rain, vr (m-1), can
be found using the same expressions developed for clouds
that use the Rayleigh approximation
 vr  10
10
5
2
1
K
Z
,
m
w
40
where 0 is expressed in cm.
For frequencies below 10 GHz, the reflectivity factor,
Z (mm6 m-3), is related to the rainfall rate, Rr (mm hr-1) by
Z  200 R1r.6
For f > 10 GHz, an effective reflectivity factor, Ze, is used
Ze 
40  vr 1010
 Kw
5
2
93
Volume backscattering – rain
94
Volume backscattering – rain
In weather radar applications, such as the
WSR-88D, the parameter dBZ is used where
dBZ  10 log 10 Z Z0 
where
Z0 corresponds to a rainfall rate of 1 mm hr-1 (0.04 in hr-1)
Reflectivities in the range between 5 and 75 dBZ are
detected when the radar is in precipitation mode.
Reflectivities in the range between -28 and +28 dBZ
are detected when the radar is in clear air mode.
95
Volume backscattering – rain
VCP denotes the vertical
coverage pattern in use
96
Volume backscattering – rain
Polarization
Spherical targets tend to preserve the polarization during
backscattering.
For example, when the illumination is horizontally polarized, the
backscattered wave is also horizontally polarized with minimal verticallypolarized backscatter.
Thus weather radars use transmitters and receivers with the same
polarization.
For applications where backscatter from rain represents clutter
(e.g., air traffic control radars) so to suppress backscatter from
rain radar designers often employ circular polarization.
Transmit right circular, receive left circular thus minimizing rain
backscatter (as long as the raindrop remains spherical).
While the backscatter from the desired target is reduced, the rain
backscatter suppression is even greater yielding a net improvement in
the signal-to-clutter ratio.
97
Volume extinction – snow
It can be shown that for a precipitation rate, Rr, expressed in
mm of melted water per hour and a free-space wavelength 0
expressed in cm the snow extinction coefficient, es, is
 es  2.22 10 2 R 1r.6 40  0.34 i R r  0 , dB km 1
This expression is valid for frequencies up to about 20 GHz.
Here the first term represents the scattering component while
the second term represents absorption.
Note that i˝ varies with both temperature and frequency.
At -1 °C and 2 GHz (0 = 15 cm), i˝  10-3,
 es  4.38 10 7 R 1r.6  2.27 10 5 R r , dB km 1
Here the extinction coefficient is dominated by absorption for
snowfall rates up to a few mm hr-1.
98
Volume extinction – snow
For the same
precipitation rate Rr, the
extinction rate for rain is
20 to 50 times greater
than that of dry snow.
However, observations
show that the extinction
rate for melting snow is
substantially larger than
that of rain.
99
Volume backscattering – snow
The volume backscattering coefficient for dry snow, vs, is
 vs  10 10
where
Nv
Zs  
i 1
1
d  2
s
6
s
5
2
1
K
Z
,
m
ds
s
40
Nv

i 1
1
d  2 Zi , mm 6 m 3
s
6
i
and the snowflake diameter, ds, has been replaced by the
ice particle diameter, di, containing the same mass.
Therefore recognizing that |Kds|2/s2  ¼, the expression for
vs becomes
5
10 
1
 vs  10
Z
,
m
i
4
4 0
and for Rr expressed in mm of water per hour
Zi  1000 R1r.6 , mm 6 m 3
100
Volume backscattering – snow
Comparison of volume backscattering for rain and snow
Rain
Snow
 vr  10
10
5
2
1
K
Z
,
m
w
40
Z  200 R1r.6
Kw
2
5
1
 vs  10
Z
,
m
i
4 40
Zi  1000 R1r.6
10
 0.9
The expressions are comparable in magnitude.
However the terminal velocity of snowflakes (vs) are
relatively small (1 m s-1) compared to raindrops, the snow
precipitation rates are typically much smaller than rainfall
rates (2 to 9 m s-1).
Therefore the volume backscattering from snow is typically
smaller than that of rain, unless the snow is melting in
which case the backscattering from snow is substantially
larger. These are termed “bright bands.”
101
Impact on TSKY
TSKY   TDN   TEXTRA e  0 sec   TDN  for f  10 GHz
TDN   sec  Tm

z1
 a e  a z sec  d z  Tm 1  1 L a 
Tm is mean temperature in
0
atmosphere’s lower 2 to 3 km.
Simulation results of TSKY() under three atmospheric conditions:
clear sky, moderate cloud cover, 4 mm hr-1 rain.
0 = 3 cm (10 GHz), 1.8 cm (16.7 GHz), 1.25 cm (24 GHz), 0.86 cm (35 GHz), 0.43 cm (70 GHz), 0.3 cm (100 GHz)
102
Application: space-based temperature sounding
We seek to estimate the temperature profile T(z) for a
scatter-free atmosphere using data from a down-looking
spaceborne radiometer.
103
Application: space-based temperature sounding
The temperature profile will be derived in the lower
atmosphere using the brightness temperature around an
resonance frequency for an atmospheric constituent that is
homogenously distributed, i.e., oxygen.
We know that
TAP (f )  Ta f   Ts f  e  m f 
where Ta is the atmosphere’s radiometric brightness
temperature, Ts is the surface brightness temperature, and
m is the optical thickness.
Ta f   

0


 a f , z  Tz  exp    a f , z d z  d z
 z

Ts f   Tphys ef 
 m f   

0
 a f , z d z
104
Application: space-based temperature sounding
We define a temperature weighting function W(f,z) as


Wf , z    a f , z  exp    a f , z d z 
 z

so that the atmospheric component Ta(f) is
Ta f   

0
W f , z  Tz  d z
we know that for O2 the absorption coefficient depends on
the pressure and the temperature as

1
1 
6 2  Pz    300 
1
  
O2 f , z   2.5 10 f 

,
Np
m
 

2
2
f 2  2 
 1013   Tz    f  f 0   
2
where
Pz   P0 e  z H , mbar
and H = 7.7 km , P0 = 1013 mbar
105
Application: space-based temperature sounding
So to first order
O2 f , z  0 f  ez H , Np m1
where  0 f    O f ,0
Substituting we get
2


W f , z    O 2 f , z  exp    O 2 f , z d z 
 z


z H

  0 f  e
exp   0 f   e  z H d z


z
 z
z H 




  0 f exp    0 f H e 
 H

 z




W f , z   0 f exp 
 H
z H 


 m f e  ,

Np m 1
where
m f    0 f  H
106
Application: space-based temperature sounding
107
Application: space-based temperature sounding
For a temperature weighting function of the form
 z

W f , z    0 f  exp    m f  e  z H 
 H

we find
d W W f , z 

 m f  e  z H  1
dz
H


dW
 0 for z H  log e  m
dz
therefore
For z  0
point of local maximum
W f ,0   0 f  e  m f 
For z  
W f ,    0
For z  H log e  m
0
1
W f , H log e  m  

m e H e
108
Application: space-based temperature sounding
From this analysis it is clear that:
The temperature weighting function causes most of the contribution
to be from a limited range of altitudes.
By selecting the proper frequency (and thus m(f )) the altitude for
the region of peak contribution can be selected.
By selecting an oxygen resonance frequency, known
absorption characteristics are available throughout the
atmosphere.
And by selecting a series of frequencies near resonance
(the 60-GHz complex or 118.75 GHz) atmospheric
temperatures at various altitudes can be sensed.
109
Application: space-based temperature sounding
110
Application: space-based temperature sounding
Data inversion to extract the temperature profile
Previously we adopted the following form to relate the
atmospheric temperature at altitude z, T(z), to the apparent
temperature atmospheric, Ta.
Ta f   

0
W f , z  Tz  d z
Now let us divide the atmosphere into N layers where each
has a constant temperature and equal thickness z such
that the nth layer is centered at altitude zn and has
temperature Tn.
The equation above can be rewritten as
N
Ta f    W f , z n  Tn  z
n 1
111
Application: space-based temperature sounding
Data inversion to extract the temperature profile
Also, if brightness temperature measurements are made
for M unique frequencies fm, then
N
Tam   Wnm Tn
n 1
where Wnm = W(fm, zn) and Tam = Ta(fm).
So that
 Ta1   W11
T   W
 a 2   12
 Ta 3    W13
  
    
TaM   W1M
or
W21
W22
W23

W2 M
W31
W32
W33

W3M
 WN1   T1 
 WN 2   T2 
 WN 3   T3 
 

   
 WNM  TN 
Ta  W T
112
Application: space-based temperature sounding
Ta  W T
Here Ta represents the M measured brightness temperatures,
W is the MN matrix of temperature weighting functions, and T
is the N-element vector representing the unknown atmospheric
temperature profile.
Various techniques are available to find T given W and Ta.
For N > M, there is no unique solution for this ill-posed problem.
For the case where N = M
T  W 1 Ta
The least-squares solution for T where N < M requires

T W W
T

1
W T Ta
where WT denotes a matrix transpose and W-1 denotes a matrix
inverse.
113
Application: space-based temperature sounding
114
Application: space-based temperature sounding
Derived atmospheric temperature profiles show good agreement
with radiosonde data.
Using a similar approach, other atmospheric
properties can be sensed.
Examples include the precipitable water
vapor distribution and the concentration
of certain gases such as ozone (O3).
A radiosonde is a balloon-borne instrument
platform with radio transmitting capabilities.
Comparison with “ground truth” is important when characterizing
a sensor’s performance.
115
Application: ground-based temperature sounding
Estimating the temperature profile T(z) for a scatter-free
atmosphere using data from an up-looking ground-based
radiometer.
116
Application: ground-based temperature sounding
As was done previously, the temperature profile will be
derived in the lower atmosphere using the brightness
temperature around an resonance frequency for oxygen.
We know that
TAP (f )  Ta f   TEXTRA f  e  m f 
Where TEXTRA is the extraterrestrial brightness temperature
Ta f   

0
z

 a f , z  Tz  exp    a f , z d z  d z
 0

TEXTRA f   TCOS  TGAL f 
 m f   

0
 a f , z d z
Note a change in the
integration limits for the
up-looking case.
117
Application: ground-based temperature sounding
We again define a temperature weighting function W(f,z) as
z

Wf , z   a f , z  exp   a f , z d z 
 0

so that the atmospheric component Ta(f) is
Ta f   

0
W f , z  Tz  d z
So to first order
O2 f , z  0 f  ez H , Np m1
where  0 f    O f ,0
Substituting we get
2
z

Wf , z    0 f  e
exp   0 f   e  z H d z


0
 z

  0 f  exp    0 f  H   0 f  H e  z H 
 H

z H
 z

Wf , z    0 f  exp    m f    m f  e  z H  , Np m 1
 H

118
Application: ground-based temperature sounding
119
Application: ground-based temperature sounding
For a weighting function of the form
 z

W f , z    0 f  exp    m f    m f  e  z H 
 H

we find
dW
W f , z 

1   m f  e  z H
dz
H


therefore
For z  0
Wf ,0   0 f 
For z  
Wf ,    0
dW dz
z 0
0
1  m 

H
120
Application: ground-based temperature sounding
121
Application: ground-based temperature sounding
122