Microwave RadiometryMicrowave Radiometry

2

OutlineOutline

Introduction Thermal Radiation Black body radiation

– Rayleigh-Jeans

Power-Temperature correspondence Non-Blackbody radiation

– TB, brightness temperature

– TAP, apparent temperature

– TA, antenna temperature

More realistic Antenna– Effect of the beam shape– Effect of the losses of the antenna

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Thermal RadiationThermal Radiation

All matter (at T>0K) radiates electromagnetic energy!

Atoms radiate at discrete frequencies given by the specific transitions between atomic energy levels. (Quantum theory)– Incident energy on atom can be absorbed by it to move

an e- to a higher level, given that the frequency satisfies the Bohr’s equation.

– f = (E1 - E2) /h

where,

h = Planck’s constant = 6.63x10-34 J

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Thermal RadiationThermal Radiation

absorption => e- moves to higher levelemission => e- moves to lower level

(collisions cause emission)

Absortion Spectra = Emission SpectraAbsortion Spectra = Emission Spectra

atomic gases have (discrete) line spectra according to the allowable transition energy levels.

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Molecular Radiation SpectraMolecular Radiation Spectra

Molecules consist of several atoms. They are associated to a set of vibrational and

rotational motion modes.

Each mode is related to an allowable energy level.

Spectra is due to contributions from; vibrations, rotation and electronic transitions.

Molecular Spectra = many lines clustered together; not discrete but continuous.

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Atmospheric WindowsAtmospheric Windows

Absorbed(blue area)

Transmitted(white)

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Radiation by bodies Radiation by bodies (liquids - solids)(liquids - solids)

Liquids and solids consist of many molecules which make radiation spectrum very complex, continuous; all frequencies radiate.

Radiation spectra depends on how hot is the object as given by Planck’s radiation law.

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CommonTemperature -CommonTemperature -conversionconversion

90oF = 305K = 32oC70oF = 294K = 21oC32oF = 273K=0oC0oF = 255K = -18oC -280oF = 100K = -173oC

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Spectral brightness Spectral brightness BBff [Planck][Planck]

]Hzsr[W/m 1

12 22/2

3

kThff ec

hfB

10

SunSun

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Solar Radiation Solar Radiation TTsunsun= = 5,800 K5,800 K

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Properties of Planck’s LawProperties of Planck’s Law

fm = frequency at which the maximum

radiation occurs

fm = 5.87 x 1010 T [Hz]

where T is in KelvinsMaximum spectral Brightness Bf (fm)

Bf (fm) = c1 T3

where c1 = 1.37 x 10-19 [W/(m2srHzK3)]

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Stefan-Boltzmann Stefan-Boltzmann Total brightness of body at Total brightness of body at TT

Total brightness is 4

0

TdfBB f

20M W/m2 sr

13M W/m2 sr67%

where the Stefan-Boltzmannconstant is= 5.67x10-8 W/m2K4sr

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Blackbody Radiation -Blackbody Radiation -given by Planck’s Lawgiven by Planck’s Law

Measure spectral brightness Bf [Planck]

For microwaves, Rayleigh-Jeans Law,

condition hf/kT<<1 (low f ) , then ex-1 x

]Hzsr [W/m 2kT2 2

22

2

c

kTfB f

At T<300K, the error < 1% for f<117GHz),and error< 3% for f<300GHz)

]Hzsr [W/m 1

12 2/2

3

kThff ec

hfB

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Rayleigh-Jeans ApproximationRayleigh-Jeans Approximation

Rayleigh-Jeansf<300Hz>2.57mm)T< 300K

TB f

Mie Theory

frequency

Bf

Wien

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Total power measured due Total power measured due to objects Brightness, to objects Brightness, BBff

dfdFfBAP n

ff

f

frbb

,,, 2

1

4

B=Brightness=radiance [W/m2 sr]Bf = spectral brightness (B per unit Hz)B = spectral brightness (B per unit cm)

Fn= normalized antenna radiation pattern= solid angle [steradians]Ar=antenna aperture on receiver

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Power-Temperature correspondencePower-Temperature correspondence

fkTP

A

dFAfP

fB

dfdFAP

bb

rp

nrbb

f

n

ff

f

rbb

2

42

42

is angle solidpattern but the

,2kT

2

1

overconstant ely approximat is if

,2kT

2

1

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Analogy with a resistor noise Analogy with a resistor noise

Antenna Pattern

T

fkTPbb

R

T

fkTPn

Analogous to Nyquist;noise power from R

Direct linear relationpower and temperature

*The blackbody can be at any distance from the antenna.

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Non-blackbody radiationNon-blackbody radiation

But in nature, we find variationswith direction, B()

ffBB fbb 2

2kT

Isothermal medium at physical temperature T

TB()

=>So, define a radiometric temperature (bb equivalent) TB

f

kTBeB B

bb 2

,2),(),(

For Blackbody,

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Emissivity, Emissivity, ee

The brightness temperature of a material relative to that of a blackbody at the same temperature T. (it’s always “cooler”)

10 where

,T),(),( B

e

TB

Be

bb

TB is related to the self-emitted radiation from theobserved object(s).

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Quartz versus BB at same TQuartz versus BB at same T

Emissivity depends alsoon the frequency.

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Ocean colorOcean color

Pure Water is turquoise blue The ocean is blue because it absorbs all the other

colors. The only color left to reflect out of the ocean is blue.

“Sunlight shines on the ocean, and all the colors of the rainbow go into the water. Red, yellow, green, and blue all go into the sea. Then, the sea absorbs the red, yellow, and green light, leaving the blue light. Some of the blue light scatters off water molecules, and the scattered blue light comes back out of the sea. This is the blue you see.”

Robert Stewart, Professor

Department of Oceanography, Texas A&M University

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Apparent Temperature, Apparent Temperature, TTAPAP

Is the equivalent T in connection with the power incident upon the antenna

TAP()

fBinc

2AP ,2kT

),(

dFTAfP nAPr

4

2,,

2k

2

1

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Antenna Temperature, Antenna Temperature, TTAA

dFTA

T

fkTP

dFTAf

P

nAPr

rA

An

nAPr

r

42

42

,,

,,2k

2

1

Noise power received at antenna terminals.

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Antenna Temperature Antenna Temperature (cont…)(cont…)

Using we can rewrite as

for discrete source such as the Sun.

dF

dFT

Tn

nAP

A

4

4

,

,,

sunp

sunA TT

prA

2

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Antenna Beam EfficiencyAntenna Beam Efficiency, , MM

Accounts for sidelobes & pattern shape

dF

dFT

dF

dFT

Tn

ml

nAP

n

ML

nAP

A

44

,

,,

,

,,

dF

dF

dF

dFT

dF

dF

dF

dFT

Tn

ml

n

ml

n

ml

nAP

n

ML

n

ML

n

ML

nAP

A

44

,

,

,

,,

,

,

,

,,

TA= M TML +(1- M)TSL

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Radiation Efficiency, Radiation Efficiency, ll

Heat loss on the antenna structure produces a noise power proportional to the physical temperature of the antenna, given as

TN= (1-l)To

The l accounts for losses in a real antenna

TA’= l TA +(1-l)To

TATA’

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Combining both effectsCombining both effects

Combining both effects

TA’= l M TML + l(1- M)TSL+(1-l)To

*where, TA’ = measured,

TML = to be estimated

TML= 1/(l M) TA’ + [(1- M)/ M]TSL+(1-l)To/ lM