Microwave Interaction with Atmospheric Constituents

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

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

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.

Atmospheric composition

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

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 = K.

U.S. Standard Atmosphere, 1962

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 = mbar

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.

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 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 GHz. Water vapor’s electric dipole enables rotational energy states at 22.2 GHz, GHz, and several frequencies above 300 GHz.

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.

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.

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

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.

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 state’s energy level k = Boltzmann’s constant (1.38  J K-1) T = thermodynamic temperature (K) Thus (f, flm) expressed in dB km-1 is

Water vapor has resonant frequencies at GHz, GHz, 323 GHz, GHz, 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 GHz resonance r(f) = residual term representing absorption due to all higher- frequency water-vapor absorption lines

Using data for the 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, and P is in millibars. The residual absorption term is Therefore the total water vapor absorption below 100 GHz is

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 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: , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

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 GHz resonance. where f is in GHz, f0 = 60 GHz, and

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

Total atmospheric gaseous absorption
Non-zenith optical thickness can be approximated as for   70°.

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. TCOS and TGAL are the cosmic and galactic brightness temperatures, and TEXTRA is the extraterrestrial brightness temperature.

Extraterrestrial sources
TCOS is independent of frequency and direction. TGAL is both frequency and direction dependent. Frequency dependence Depending 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 dependence TGAL(max) in the direction of the galactic center while TGAL(min) is the direction of the galactic pole.

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.

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.

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.

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

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

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

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)

“optical” limit e = 2 for  » 1
Mie scattering Numerical solutions for spheres of various composition. “optical” limit e = 2 for  » 1

For  << 1, s << a
Mie scattering Strongly conducting sphere For  << 1, s << a

Mie scattering Weakly absorbing sphere
Again, for  « 1, s « a so e  a Also, as   ,  a  1 and s  1 if 0 < n″ « 1

Backscattering efficiency, b
Mie’s solution also predicts the backscattering efficiency, b, for a spherical particle “optical” limit b = 1 for  » 1

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) where and 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.

Rayleigh approximation
and so 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.

Rayleigh backscattering
Again, for the Rayleigh region (|n | < 0.5), a simplified expression for the backscattering efficiency is found, Rayleigh’s 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.

For large |n|, |K|  1 yielding 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 or

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.

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

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

Pure water Relative dielectric constant, real part, r′ vs. imaginary part, r″

Pure water To apply the solutions from Mie or Rayleigh requires the complex refractive index. where rc is the complex relative dielectric constant

Pure water Refractive index, real part, n′

Pure water Refractive index, imaginary part, n″

Pure water Refractive index, magnitude |n|

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 ‰ 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,  F m-1

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

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

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

Pure and sea water Relative dielectric constant, real part, r′

Pure and sea water Relative dielectric constant, imaginary part, r″

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.

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 band 20 s ≤  ≤ 40 ms In the THz band 6 fs ≤  ≤ 30 fs

There is some variability in reported measured values for i′. Recent work shows that

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).

Relative dielectric constant, imaginary part, r″

Relative dielectric constant, imaginary part, r″ Loss (dB/m)  f· So for region where   1/f, Loss is frequency independent

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)

Characteristics of ice

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.

Drop-size distribution for cloud types

Drop-size distribution by rain rate

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.

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

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.

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.” 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

p() = 0 for r < r1 and r > r2
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 < 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.

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

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)

Examples of cloud types
Cirrostratus Low-lying stratus Fog layer Haze, heavy Fair-weather cumulus Cumulus congestus

The drop size distribution is given by 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. [b is related to the rainfall rate – covered later] 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

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

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

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

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.

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

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 ice At microwave frequencies, 0.89  |Kw|2  0.93 (0 C  T  20 C) |Ki|2  0.2

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

Drop-size distribution by rain rate
Measured drop-size data for various rainfall rates

Volume extinction – rain
The volume extinction coefficient of rain (er) is where  = 2r/0.

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 raindrop’s terminal velocity, vi (m s-1) and the number of drops per unit volume, Nv (m-3).

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

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

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 > 10 GHz, an effective reflectivity factor, Ze, is used

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.

VCP denotes the vertical coverage pattern in use

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.

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.

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.

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 |Kds|2/s2  ¼, the expression for vs becomes and for Rr expressed in mm of water per hour

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.”

Tm is mean temperature in atmosphere’s lower 2 to 3 km.
Impact on TSKY Tm is mean temperature in 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)

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.

The temperature profile will be derived in the lower atmosphere using the brightness temperature around a resonance frequency for an atmospheric constituent that is homogenously distributed, i.e., oxygen. We know that where Ta is the atmosphere’s radiometric brightness temperature, Ts is the surface brightness temperature, and m is the optical thickness.

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

So to first order where Substituting we get

For a temperature weighting function of the form we find therefore point of local maximum

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 GHz) atmospheric temperatures at various altitudes can be sensed.

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

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

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 The least-squares solution for T where N < M requires where WT denotes a matrix transpose and W-1 denotes a matrix inverse.

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.

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.

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.

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

For a weighting function of the form we find therefore

Microwave Interaction with Atmospheric Constituents

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Microwave Interaction with Atmospheric Constituents

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Presentation on theme: "Microwave Interaction with Atmospheric Constituents"— Presentation transcript:

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