Microwave Interaction with Atmospheric Constituents

1
Microwave Interaction with Atmospheric
Constituents
Chris Allen (callen_at_eecs.ku.edu) Course website
URL people.eecs.ku.edu/callen/823/EECS823.htm
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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

3
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 Earths 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.

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Atmospheric composition
5
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 kmdT/dz 2.8 ?C km-1 above
  32 km
• Mesosphere
• upper boundary 80 to 90 kmdT/dz -3.5 ?C km-1
  above 60 km

6
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.
• 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.

7
U.S. Standard Atmosphere, 1962
8
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
• 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
• Alternatively pressure can be found using
• where H3 7.7 km and Po 1013.25 mbar

9
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
• where ?v has units of g m-3, ?0 is 7.72 g m-3,
  and H4 is 2 km.

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Absorption and emission by gases

• Molecular absorption (and emission) of
  electromagnetic energy may involve three types of
  energy states
• where
• Ee electronic energy
• Ev vibrational energy
• Er rotational energy
• Of the various gases and vapors in the Earths
  atmosphere, only oxygen and water vapor have
  significant absorption bands in the microwave
  spectrum.
• Oxygens magnetic moment enables rotational
  energy states around 60 GHz and 118.8 GHz.
• Water vapors electric dipole enables rotational
  energy states at 22.2 GHz, 183.3 GHz, and several
  frequencies above 300 GHz.

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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.

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Absorption spectrum model

• The absorption spectrum for transactions between
  a pair of energy states may be written as
• where
• ?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.

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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
• here
• linewidth parameter, Hz
• The Van Vleck and Weisskopf function, FVW, takes
  into account atmospheric pressures

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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.

15
Water-vapor absorption

• Absorption due to water vapor can be modeled
  using
• For each water-vapor absorption line the line
  strength is
• where
• Slm0 constant characteristic of the lm
  transition
• flm the resonance frequency
• ?v water-vapor density
• El lower energy states energy level
• k Boltzmanns constant (1.38 ? 10-23 J K-1)
• T thermodynamic temperature (K)
• Thus ?(f, flm) expressed in dB km-1 is

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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
• Where
• ?(f, 22) absorption due to 22.235-GHz resonance
• ?r(f) residual term representing absorption due
  to all higher- frequency water-vapor absorption
  lines

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Water-vapor absorption

• Using data for the 22.235-GHz resonance we get
• where the linewidth parameter ?1 is
• f and ?1 are expressed in GHz, T is in K, ?v is
  in g m-3, andP is in millibars.
• The residual absorption term is
• Therefore the total water vapor absorption below
  100 GHz is

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Water-vapor absorption
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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
spectrum.
Resonant frequencies (GHz) in the 60-GHz complex
49.9618, 50.4736, 50.9873, 51.5030, 52.0212,
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
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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.
• where f is in GHz, f0 60 GHz, and

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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

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Total atmospheric gaseous absorption

• Non-zenith optical thickness can be approximated
  as
• for ? ? 70.

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Atmospheric gaseous emission

• We know that for a non-scattering gaseous
  atmosphere
• where
• An upward-looking radiometer would receive the
  down-welling radiation, TDN, plus a small energy
  component from cosmic and galactic radiation
  sources.
• where
• TCOS and TGAL are the cosmic and galactic
  brightness temperatures, and TEXTRA is the
  extraterrestrial brightness temperature.

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Extraterrestrial sources

• TCOS is independent of frequency and direction.
• TGAL is both frequency and direction dependent.
• Frequency dependenceDepending on the specific
  region of the galaxy,
• 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 dependenceTGAL(max) in the direction
  of the galactic center while TGAL(min) is the
  direction of the galactic pole.

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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.
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Effects of the sun

• The suns 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.

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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.

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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
• The absorption efficiency factor, ?a, is the
  ratio of Qa to the geometrical cross-section, A,
  is

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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
• The scattering cross section, Qs and the
  scattering efficiency factor, ?s are
• Thus Pa Ps represent the total power removed
  from the incident wave and the extinction cross
  section Qe and extinction efficiency ?e are

30
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
• or

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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?)
• 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)

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Mie scattering

• Numerical solutions for spheres of various
  composition.

optical limit ?e 2 for ? 1
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Mie scattering

• Strongly conducting sphere

For ? ltlt 1, ?s ltlt ?a
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Mie scattering

• Weakly absorbing sphere
• Again, for ? 1, ?s ?aso ?e ? ?a
• Also, as ? ? ?, ? a ? 1 and ?s ? 1 if 0 lt n? 1

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Backscattering efficiency, ?b

• Mies solution also predicts the backscattering
  efficiency, ?b, for a spherical particle

optical limit ?b 1 for ? 1
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Rayleigh approximation

• For particles much smaller than the incident
  waves wavelength, i.e., n ? 1, the Mie
  solution can be approximated with simple
  expressions known as the Rayleigh approximations.
• For n ? lt 0.5 (Rayleigh region)
• where
• and
• Unless the partical is weakly absorbing (i.e.,
  n? n') such that Im-K K2, Qa Qs since
  Qs varies as ?6 and Qa varies as ?3.

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Rayleigh approximation
and
so

• Therefore the scattering cross section increases
  quite rapidly with particle radius and with
  increasing frequency.
• ExampleFor ? 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.

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Rayleigh backscattering

• Again, for the Rayleigh region (n ? lt 0.5), a
  simplified expression for the backscattering
  efficiency is found, Rayleighs backscattering
  law
• or
• And as was the case for the scattering cross
  section,
• Therefore in the Rayleigh region, the
  backscattering cross section is very sensitive to
  particle size relative to wavelength.

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Rayleigh backscattering

• For large n, K ? 1 yielding
• However for the case of n ? (perfect
  conductor) which violates the Rayleigh condition
  (n ? lt 0.5) for finite particle sizes, the
  backscattering cross section can be found for
  ? 1 using Mies solution
• or

40
Rayleigh backscattering
41
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, ?
• particles complex refractive index, n
• particle radius, r
• This requires an understanding of the dielectric
  properties of liquid water and ice.

42
Pure water

• The Debye equation describes the frequency
  dependence of the dielectric constant of pure
  water, ?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

43
Pure water

• While ?w? is apparently temperature independent,
  temperature affects ?w0 and ?w causing ?'w and
  ??w to be dependent on temperature, T.
• The relaxation time for pure water is
• where T is expressed in ?C.
• The corresponding relaxation frequency fw0 of
  pure water is
• which varies from 9 GHz at 0 ?C to 17 GHz at 20
  ?C.
• The temperature-dependent static dielectric of
  water is

44
Pure water
Relative dielectric constant, real part, ?r' vs.
imaginary part, ?r?
45
Pure water

• To apply the solutions from Mie or Rayleigh
  requires the complex refractive index.
• where
• ?rc is the complex relative dielectric constant

46
Pure water
Refractive index, real part, n'
47
Pure water
Refractive index, imaginary part, n?
48
Pure water
Refractive index, magnitude n
49
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.
• where
• ?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

50
Sea water

• The high-frequency (or optical) limit of ?sw is
  independent of salinity.
• The static relative dielectric constant of saline
  water depends on salinity () and temperature
  (?C).
• where

51
Sea water

• The relaxation time is also dependent on both
  salinity and temperature.
• where
• ?sw(T, 0) ?w(T) that was given earlier

52
Sea water

• Finally, the ionic conductivity for sea water,
  ?i, depends on salinity () and temperature (?C)
  as
• where the ionic conductivity at 25 ?C is
• and
• where ? 25 T, T is in ?C

53
Pure and sea water
Relative dielectric constant, real part, ?r'
54
Pure and sea water
Relative dielectric constant, imaginary part, ?r?
55
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.

56
Characteristics of ice

• The dielectric properties of ice can be predicted
  by the Debye equation
• Multiple relaxation frequencies exist for pure
  ice, some in the kHz, others in the THz.

Complex Real part Imaginary part
Multiple relaxation frequencies exist for pure
ice, some in the kHz, others in the THz. In the
kHz band20 ?s ? 40 ms In the THz band6 fs
? 30 fs
57
Pure and fresh-water ice

• There is some variability in reported measured
  values for ?i'.
• Recent work shows that

58
Pure and fresh-water ice

• Similarly the Debye expression for the imaginary
  part (?i?) simplifies to
• 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).

59
Pure and fresh-water ice
Relative dielectric constant, imaginary part, ?r?
60
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
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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
• 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)

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Pure and fresh-water ice
63
Characteristics of ice
64
Characteristics of ice
65
Characteristics of ice
66
Characteristics of ice
67
Characteristics of ice
68
Liquid water hydrometeors

• Electromagnetic scattering and absorption of a
  spherical particle depend on three parameters
• wavelength, ?
• particles 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
  ?mlow-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.

69
Drop-size distribution for cloud types
70
Drop-size distribution by rain rate
71
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.

72
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

73
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 particles
  contribution. This applies to scattering,
  absorption, extinction, and backscattering.

74
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
  volumetherefore 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.
• where
• Q(r) scattering cross section of sphere of
  radius r, m2
• r1 and r2 lower and upper limits of drop radii
  within volume, m

75
Volume scattering

• The volume scattering coefficient, ?s, can also
  be found using the scattering efficiency, ?s,
  since ?s Qs/?r2.
• where ? 2?r/?0.
• Note that while the limits go from 0 to ?, in
  reality
• p(?) 0 for r lt r1 and r gt 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.

76
Volume absorption, extinction, and backscattering

• Similarly, the volume absorption coefficient, ?a,
  is
• And the volume extinction coefficient, ?e, is
• The volume backscattering coefficient, ?v, also
  known as the radar reflectivity with units of
  (m-3)?(m2) m-1, is

77
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)

78
Examples of cloud types
Low-lying stratus
Fog layer
Cirrostratus
Fair-weather cumulus
Haze, heavy
Cumulus congestus
79
Drop-size distribution clouds

• The drop size distribution is given by
• and p(0) p(?) 0. The variables a, b, ?, and
  ? are positive, real constants related to the
  clouds 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
• which simplifies to

where ?( ) is the standard gamma function and
80
Drop-size distribution clouds

• In addition, the mode radius of the distribution,
  rc, is
• Note mode the most frequent value assumed by
  a random variable
• So the maximum density in the distribution is
• 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

where
81
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).
• So p(r) pn(r) ? p(rc)
• or

82
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
• where ?1 (dB km-1 g-1 m3) is the extinction for
  mv 1 g m-3 and
• with ?o in cm

83
Volume backscattering clouds

• Under the Rayleigh assumption
• For the case of Nv particles per unit volume, the
  cloud volume backscattering coefficient, ?vc is
• Now define the reflectivity factor Z to be
• where di is the diameter of the ith particle
  expressed in m.

84
Volume backscattering clouds

• The cloud volume backscattering coefficient now
  becomes
• When Z is expressed in mm6 and ?0 is in cm,
• The Z factor can be related to the liquid water
  content mv (g m-3) as
• Similarly a Z factor for the liquid water content
  of an ice cloud is found

85
Volume backscattering clouds

• So while the K2 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
• ice
• At microwave frequencies,
• 0.89 ? Kw2 ? 0.93 (0 ?C ? T ? 20 ?C)
• Ki2 ? 0.2

86
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
• 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

87
Drop-size distribution by rain rate

• Measured drop-size data for various rainfall rates

88
Volume extinction rain

• The volume extinction coefficient of rain (?er)
  is
• where ? 2?r/?0.

89
Volume extinction rain
90
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)
• 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
  raindrops terminal velocity, vi (m s-1) and the
  number of drops per unit volume, Nv (m-3).

91
Volume extinction rain

• The polarization dependence arises from the
  oblate spheriod (i.e., non-spherical) raindrop
  shape.

92
Volume extinction rain

• Horizontal-path extinction (attenuation) for
  various rainfall rates.

93
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
• 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
• For f gt 10 GHz, an effective reflectivity factor,
  Ze, is used

94
Volume backscattering rain
95
Volume backscattering rain

• In weather radar applications, such as the
  WSR-88D, the parameter dBZ is used where
• 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.

96
Volume backscattering rain
VCP denotes the vertical coverage pattern in use
97
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
  vertically-polarized 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.

98
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
• 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,
• Here the extinction coefficient is dominated by
  absorption for snowfall rates up to a few mm hr-1.

99
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.

100
Volume backscattering snow

• The volume backscattering coefficient for dry
  snow, ?vs, is
• where
• and the snowflake diameter, ds, has been replaced
  by the ice particle diameter, di, containing the
  same mass.
• Therefore recognizing that Kds2/?s2 ? ¼, the
  expression for ?vs becomes
• and for Rr expressed in mm of water per hour

101
Volume backscattering snow

• Comparison of volume backscattering for rain and
  snow
• Rain Snow
• 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.

102
Impact on TSKY
Tm is mean temperature in atmospheres lower 2 to
3 km.

• Simulation results of TSKY(?) under three
  atmospheric conditionsclear 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)
103
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.

104
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
• where Ta is the atmospheres radiometric
  brightness temperature, Ts is the surface
  brightness temperature, and ?m is the optical
  thickness.

105
Application space-based temperature sounding

• We define a temperature weighting function W(f,z)
  as
• so that the atmospheric component Ta(f) is
• we know that for O2 the absorption coefficient
  depends on the pressure and the temperature as
• where
• and H 7.7 km , P0 1013 mbar

106
Application space-based temperature sounding

• So to first order
• where
• Substituting we get
• where

107
Application space-based temperature sounding
108
Application space-based temperature sounding

• For a temperature weighting function of the form
• we find
• therefore

point of local maximum
109
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.

110
Application space-based temperature sounding
111
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.
• 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

112
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
• where Wnm W(fm, zn) and Tam Ta(fm).
• So that
• or

113
Application space-based temperature sounding

• 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 gt M, there is no unique solution for this
  ill-posed problem.
• For the case where N M
• The least-squares solution for T where N lt M
  requires
• where WT denotes a matrix transpose and W-1
  denotes a matrix inverse.

114
Application space-based temperature sounding
115
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 sensors performance.

116
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.

117
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
• Where TEXTRA is the extraterrestrial brightness
  temperature

Note a change in the integration limits for the
up-looking case.
118
Application ground-based temperature sounding

• We again define a temperature weighting function
  W(f,z) as
• so that the atmospheric component Ta(f) is
• So to first order
• where
• Substituting we get

119
Application ground-based temperature sounding
120
Application ground-based temperature sounding

• For a weighting function of the form
• we find
• therefore

121
Application ground-based temperature sounding
122
Application ground-based temperature sounding