1 Microwave Interaction with Atmospheric Constituents Chris Allen ([email protected]) Course website URL people.eecs.ku.edu/~callen/823/EECS823.htm

1

Microwave Interaction with Atmospheric Constituents

Chris Allen ([email protected])

Course website URL people.eecs.ku.edu/~callen/823/EECS823.htm

2

Outline

Physical properties of the atmosphereAbsorption and emission by gases

– Water vapor absorption– Oxygen absorption

Extraterrestrial sourcesExtinction 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 atmosphereThe 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.

4

Atmospheric composition

5

Temperature, density, pressure profileAtmospheric 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

Stratosphereupper boundary ~ 47 kmdT/dz ~ 2.8 C km-1 above ~ 32 km

Mesosphereupper boundary 80 to 90 kmdT/dz ~ -3.5 C km-1 above ~ 60 km

6

Temperature modelOnly 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.

km32zkm20,)20z(11T

km20zkm11,11T

km11z0,zaT

zT0

7

U.S. Standard Atmosphere, 1962

8

Density and pressure modelsFor 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

km30z0for,Hzsin3.01e225.1z 2Hz

air2

km30z0for),mbar(zTz87.2zP air

km10z0for,ePzP 3Hz0

9

Water-vapor density modelThe 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.

km30z0for,ez 4Hz0v

10

Absorption and emission by gasesMolecular absorption (and emission) of electromagnetic energy may involve three types of energy states

whereEe = 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.

rve EEEE

11

Spectral line shapeFor 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.

12

Absorption spectrum modelThe absorption spectrum for transactions between a pair of energy states may be written as

wherea = 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, HzF = 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.

lmlmlma f,fFSc

f4f,f

13

Line-shape functionThere 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

22

lm

lmLff

1f,fF

22

lm22

lmlmlmVW

fffff

f1f,fF

14

Line-shape functionThe Gross function, FG, was developed using a different approach and shows better agreement with measured data further from the resonance frequency.

22222lm

lmlmG

f4ff

ff41f,fF

15

Water-vapor absorptionAbsorption due to water vapor can be modeled using

For each water-vapor absorption line the line strength is

whereSlm0 = 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

1lmGlmlmOH mNp,f,fFS

c

f4f,f

2

TkE25vlm0lmlm

leTfSS

lmGTkE25

vlm0lm3

lmOH f,fFeTffSc

41034.4f,f l

2

16

Water-vapor absorptionWater 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

f22,f rO2H

17

Water-vapor absorptionUsing 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

GHz,PT018.01T3001013P85.2 v626.0

1

121

2221

T64425v

2 kmdB,f4f4.494eT300f222,f

11

23v

26r kmdB,T300f104.2f

16

21

222

T644

123

v2

OH kmdB,102.1f4f4.494

e

T

300T300f2f

2

18

Water-vapor absorption

19

Oxygen absorptionMolecular 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 absorptionFor 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

1

22220

222

O kmdB,f

1

ff

1

T

300

1013

Pf101.1

2

GHz,T

300

1013

P85.0

0

mbar25P,18.1

mbar333P25,P333101.3159.0

mbar333P,59.03

0

21

Total atmospheric gaseous absorptionAs 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

1OOHg kmdB,fff

22

dB,dzz0 g0

22

Total atmospheric gaseous absorption

dB,sec0

Non-zenith optical thickness can be approximated as

for 70°.

23

Atmospheric gaseous emissionWe 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.

zdezTzsecT secz,0

0 gDN0

Np,dzzz,0z

0 g0

secEXTRADNSKY

0eTTT

GALCOSEXTRA TTT

24

Extraterrestrial sourcesTCOS 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.

K7.2TCOS

35.2GAL ftofT

25

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.

26

Effects of the sunThe 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.

27

Other radio starsTaken 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.

28

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

2iaa m,SPQ

2aa rQ

29


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

4

2ss dR,SP

2iss m,SPQ 2

ss rQ

sae QQQ sae

30


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

22

bis mW,

R4

SS

2

i

s2b m,

S

SR4

31

Mie scatteringGustov 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)

rb0b

r2r2

njnn cbcp

32

Mie scatteringNumerical solutions for spheres of various composition.

“optical” limit e = 2 for » 1

33

Mie scattering

For << 1, s << a

Strongly conducting sphere

34

Mie scatteringWeakly absorbing sphere

Again, for « 1, s « a

so e a

Also, as , a 1 and s 1 if 0 < n″ « 1

35

Backscattering efficiency, b

Mie’s solution also predicts the backscattering efficiency, b,

for a spherical particle

“optical” limit b = 1 for » 1

36

Rayleigh approximationFor 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.

24s K

3

8 KIm4a KIm4K

3

8 24ase

2262b2242

ss m,K3

2Kr

3

8rQ

232

a m,KImQ

2

1

2n

1nK

c

c2

2

37

Rayleigh approximation

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.

262b

s K3

2Q

and b

r2

so 46

s rQ

38

Rayleigh backscatteringAgain, 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.

24b K4

2242b m,Kr4

46b r

39

Rayleigh backscatteringFor 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

4b 4

1«andnforK924

b

.sphereconductingaform,r9 242b

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, particle’s complex refractive index, n

particle radius, r

This requires an understanding of the dielectric properties of liquid water and ice.

42

Pure waterThe Debye equation describes the frequency dependence of the dielectric constant of pure water, w

wherew0 = 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

w0www f2j1

2w

w0www

f21

2w

w0www

f21

f2

43

Pure waterWhile 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

9.4w

3162141210w T10096.5T10938.6T10824.3101109.1T2

w0w 21f

35240w T10075.1T10295.6T4147.0045.88T

44

Pure waterRelative dielectric constant, real part, r′ vs. imaginary part, r″

45

Pure waterTo apply the solutions from Mie or Rayleigh requires the complex refractive index.

whererc is the complex relative dielectric constant

rcnjnn

rcRen

rcImn

46

Pure waterRefractive index, real part, n′

47

Pure waterRefractive index, imaginary part, n″

48

Pure waterRefractive index, magnitude |n|

49

Sea waterSaline 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.

wheresw 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

2sw

sw0swswsw

f21

0

i2

sw

sw0swswsw f2f21

f2

50

Sea waterThe 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

9.4wsw

sw0sw0sw S,Ta0,T

342210sw T10491.2T10276.1T10949.1134.870,T

3sw

72sw

5

sw3

sw5

sw

S10232.4S10210.3

S10656.3ST10613.10.1S,Ta

51

Sea waterThe relaxation time is also dependent on both salinity and temperature.

wheresw(T, 0) = w(T) that was given earlier

swswswsw S,Tb0,TS,T

3sw

82sw

6

sw4

sw5

sw

S10105.1S10760.7

S10638.7ST10282.20.1S,Tb

52

Sea waterFinally, 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

eS,25S,T swiswi

3sw

72sw

5sw

3swswi S10282.1S10093.2S104619.118252.0SS,25

2875

sw

2642

10551.210551.210849.1S

10464.210266.110033.2

53

Pure and sea waterRelative dielectric constant, real part, r′

54

Pure and sea waterRelative 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 2fi0 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.

15.3ii

56

Characteristics of iceThe 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.

f2j1

rrsrr 2

rrsrr

f21

2

rrsr

f21

f2

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

There is some variability in reported measured values for i′.

Recent work shows that


KelvininisT,15.273T00091.01884.3i

58

Pure and fresh-water iceSimilarly 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).

fff2 0ii0i

i

i0ii

59

Pure and fresh-water iceRelative dielectric constant, imaginary part, r″

60

Pure and fresh-water iceRelative dielectric constant, imaginary part, r″

Loss (dB/m) f·So for region where 1/f,Loss is frequency independent

61

Pure and fresh-water iceAn 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)

T025.002.2i 10

f10

1

T025.0

i

6

10c

10x955.0m/Np

62


63

Characteristics of ice

64


65


66


67


68

Liquid water hydrometeorsElectromagnetic scattering and absorption of a spherical particle depend on three parameters:

wavelength, particle’s complex refractive index, nparticle 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., rs

3 = ri3 / s and

26i4

0

5

bs m,r16

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 particle’s 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 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.”

whereQ(r) = scattering cross section of sphere of radius r, m2

r1 and r2 = lower and upper limits of drop radii within volume, m

1r

r ss mNp,drrQrp2

1

75

Volume scattering

The volume scattering coefficient, s, can also be found

using the scattering efficiency, s, since s = Qs/r2.

where = 2r/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.

1

0 s2

2

30

s mNp,dp8

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

1

0 a2

2

30

a mNp,dp8

1

0 e2

2

30

e mNp,dp8

1

0 b2

2

30

v m,dp8

77

Drop-size distribution – cloudsFor 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

Fog layer

Examples of cloud types

Cirrostratus Low-lying stratus

Haze, heavy Cumulus congestusFair-weather cumulus

79

Drop-size distribution – cloudsThe 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.

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

r0,rbexprarp

0v rdrpN

1b

aN 1

v

where ( ) is the standard gamma function and

1

1

80

Drop-size distribution – cloudsIn 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 4r3/3 and integrating which yields

brc

exprarp cc

32

6

v mg,b3

a104m

2

4

2where

81

Drop-size distribution – cloudsFinally, 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

1rrexprrrp ccn

expra1rrexprrrp ccc

crrexprarp

82

Volume extinction – cloudsFor 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

v1ec m

KIm6

434.0o

1

83

Volume backscattering – cloudsUnder 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.

22

40

65

b m,Kr64

vv N

1i

16i

2

40

5N

1iibvc m,rK

64r

6N

1i

6i m,dZ

v

84

Volume backscattering – cloudsThe 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

12

40

5

vc m,ZK

12

40

510

vc m,ZK10

62v

2w mm,m108.4Z

64v

3i mm,m1021.9Z

85

Volume backscattering – cloudsSo 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

12v

2

w9

40

vwc m,mK1047.1

14v

2

i4

40

vic m,mK1082.2

86

Extinction and backscattering – rainRaindrops 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 = 8.0106 m-4, and b (m-1) is related to rainfall rate Rr (mm hr-1) by

4db0 m,eNdp

21.0rR4100b

87

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

88

Volume extinction – rainThe volume extinction coefficient of rain (er) is

where = 2r/0. 1

0 e2

2

30

er mNp,dp8

89

Volume extinction – rain

90

Volume extinction – rainA 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).

1br1er kmdB,R

vN

1i

13ii

4r hrmm,dv106R

91

Volume extinction – rain

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

92

Volume extinction – rainHorizontal-path extinction (attenuation) for various rainfall rates.

93

Volume backscattering – rainThe 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

12

w40

510

vr m,ZK10

6.1rR200Z

2

w5

10vr

40

eK

10Z

94

Volume backscattering – rain

95

Volume backscattering – rainIn 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.

010 ZZlog10dBZ

96

Volume backscattering – rainVCP denotes the vertical coverage pattern in use

97

Volume backscattering – rainPolarization

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 – snowIt 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.

10ri

40

6.1r

2es kmdB,R34.0R1022.2

1r

56.1r

7es kmdB,R1027.2R1038.4

99

Volume extinction – snowFor 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 – snowThe 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

1s

2

ds40

510

vs m,ZK10

36i2

s

N

1i

6i2

s

N

1i

6ss mmm,Z

1d

1dZ

vv

1i4

0

510

vs m,Z4

10

366.1ri mmm,R1000Z

101

Volume backscattering – snowComparison 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.”

1i4

0

510

vs m,Z4

10

12

w40

510

vr m,ZK10

6.1ri R1000Z 6.1

rR200Z

9.0K2

w

102

Impact on TSKY

Simulation results of TSKY() under three atmospheric conditions:

clear sky, moderate cloud cover, 4 mm hr-1 rain.

GHz10fforTeTTT DNsec

EXTRADNSKY0

am

z

0

seczamDN L11TzdeTsecT

1a

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)

Tm is mean temperature in atmosphere’s lower 2 to 3 km.

103

Application: space-based temperature soundingWe 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 soundingThe 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 atmosphere’s radiometric brightness temperature, Ts is the surface brightness temperature, and m is the optical thickness.

fsaAP

mefTfT)f(T

zdzdz,fexpzTz,ffT0 z aaa

0 am zdz,ff

feTfT physs

105

Application: space-based temperature soundingWe 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

z aa zdz,fexpz,fz,fW

0a zdzTz,fWfT

12222

0

2

26O mNp,

f

1

ff

1

zT

300

1013

zPf105.2z,f

2

mbar,ePzP Hz0

106

Application: space-based temperature soundingSo to first order

where

Substituting we get

where

1Hz0O mNp,efz,f

2

1Hzm0

Hz00

z

Hz0

Hz0

z OO

mNp,efH

zexpfz,fW

eHfH

zexpf

zdefexpef

zdz,fexpz,fz,fW22

Hff 0m

0,ff2O0

107

Application: space-based temperature sounding

108

Application: space-based temperature soundingFor a temperature weighting function of the form

we find

therefore

Hz

m0 efH

zexpfz,fW

1efH

z,fW

zd

Wd Hzm

melogHzfor0zd

Wd

eH

1

elogH,fWlogHzFor

0,fWzFor

ef0,fW0zFor

m

0meme

f0

m

point of local maximum

109

Application: space-based temperature soundingFrom 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


111

Application: space-based temperature soundingData 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

0a zdzTz,fWfT

N

1nnna zTz,fWfT

112

Application: space-based temperature soundingData 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

N

1nnnmam TWT

N

3

2

1

NMM3M2M1

3N332313

2N322212

1N312111

aM

3a

2a

1a

T

T

T

T

WWWW

WWWW

WWWW

WWWW

T

T

T

T

TWTa

113


Here Ta represents the M measured brightness temperatures, W is the MN 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.

TWTa

aTT TWWWT

1

aTWT 1

114


115


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.

116

Application: ground-based temperature soundingEstimating the temperature profile T(z) for a scatter-free atmosphere using data from an up-looking ground-based radiometer.

117

Application: ground-based temperature soundingAs 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

fEXTRAaAP

mefTfT)f(T

zdzdz,fexpzTz,ffT0

z

0 aaa

0 am zdz,ff

fTTfT GALCOSEXTRA Note a change in the integration limits for the up-looking case.

118

Application: ground-based temperature soundingWe 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

z

0 aa zdz,fexpz,fz,fW

0a zdzTz,fWfT

1Hz0O mNp,efz,f

2

1Hzmm0

Hz000

z

0

Hz0

Hz0

mNp,effH

zexpfz,fW

eHfHfH

zexpf

zdefexpefz,fW

0,ff2O0

119

Application: ground-based temperature sounding

120

Application: ground-based temperature soundingFor a weighting function of the form

we find

therefore

Hz

mm0 effH

zexpfz,fW

Hzm ef1

H

z,fW

zd

Wd

0,fWzFor

1H

dzdWf0,fW0zFor m0

0z0

121


122


Documents

1 Microwave Interaction with Atmospheric Constituents Chris Allen ([email protected]) Course website URL people.eecs.ku.edu/~callen/823/EECS823.htm