1

EE 543Theory and Principles of

Remote Sensing

Introduction to Radiometry

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Outline• Introduction• Radiometric Quantities

– Brightness– Power, power density– Spectral Brightness– Spectral Power– Spectral Flux Density– Summary of Radiometric Definitions

• Thermal Radiation• Power-Temperature Correspondence• Non-Blackbody Radiation• Antenna Efficiency Considerations

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Summary

• So far we have discussed how waves interact with their surroundings:– Wave equation– Lossy medium– Plane waves, propagation– Reflection and transmission

• We have also discussed how waves are generated and received by antennas, studied the fundamental principles of antenna theory.

• Now we will build on our understanding of antennas and em radiation to investigate principles of radiometry.

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What is Radiometry?

• Radiometry is the field of science related to the measurement of electromagnetic radiation.

• It is concerned with the measurement of incoherent radiation.

• Passive form of remote sensing; i.e. a natural source, such as Sun, provides the available radiation.

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Introduction

• All matter radiate (=emit) electromagnetic energy.

• The bulk of energy received by earth is in the form of solar electromagnetic radiation.

• Part of the incident solar energy is scattered and absorbed by the atmosphere; the remainder is transmitted to the earth’s surface.

Characteristics of Solar Radiation

• The surface temperature of Sun is 5,750-6,000 K and radiates energy across a range of wavelengths.

• Its distance to earth is about 150 million km (vacuum), only 5x109 % reaches earth.

• It then travels 100 km long atmosphere to reach the surface.

• 99% of the solar energy is within 0.28-4.96 m waveband, most of it in infrared.

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Solar Energy Incident on Earth

• The solar energy incident on earth’s surface is either scattered outward or absorbed.

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Principles of Thermodynamics

Absorption of electromagnetic energy results in the transformation of energy into thermal energy.

The reverse process: the thermal emission serves to create a balance between absorbed and solar radiation and radiation emitted by earth and its atmosphere.

These transformation processes are treated by the theory of radiative transfer.

rise in temperature

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Radiometric Quantities • Brightness (Radiance) is radiation intensity per unit

projected area in a radial direction and expressed in the unit of W/(m2 sr).

• Power (Radiant flux) is energy transmitted as a radial direction per unit time and expressed in a unit of watt (W).

• Power Density (Irradiance) is power incident upon a surface per unit area and expressed in the unit of W/ m2.

• Radiation intensity (Radiant intensity) is power radiated from a point source per unit solid angle in a radiant direction and expressed in the unit of W/sr.

• Energy (Radiant energy) is defined as the energy carried by electromagnetic radiation and expressed in the unit of joule (J). (Integral of power over time)

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Remark: In microwave engineering a few of these quantities are given different names. For instance:

Brightness is often used to refer to radiance.

Energy means Radiant energy.

Power refers to radiant flux.

Power density refers to irradiance.

Summary of radiometric definitions

energy

power

brightness

radiation intensity

power density

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Brightness (W Sr-1 m2)

Consider the situation shown above, where a transmitting antenna of effective area At is at a distance R from a receiving antenna of effective area Ar.

Brightness

Transmitting Aperture

R

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Brightness

• Assume that:– The receiving antenna is lossless; r = 1

– The two antennas are oriented in the direction of maximum directivity,

– The separation R is large enough so that the power density, St due to the transmitting antenna may be considered constant over the solid angle r.

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Intercepted Power

• The power intercepted by the receiving antenna is given by

• Or equivalently by

t rP S A

2

2;t r

t t

U AP U R S

R

Radiation intensityFunction of directional properties of the transmit antenna

Power density

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Reminder…• Recall that the maximum effective aperture At and

radiation intensity Ut are related:

2 2

2

4 4

4

tt t

o

t o t

UA D

U

U U A

Finite areaRepresents the finite area of the transmit antenna as radiation from a point source. Function of angles and.

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Antenna as a Point Source

The transmitted power density St and radiation intensity Ut both depend on the input power to the antenna, as well as its effective area and other parameters that relate to the antenna efficiency.

Thus, although the transmitting antenna has a finite aperture At, the equation above treats it as a point source with a directional distribution function Ut ().

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Antennas as Point Sources of Radiation

E

r

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What about extended sources?• In radiometry, both point and extended

source of incoherent radiation (e.g. sky, terrain) are of interest.

• Brightness is defined as the radiated power per solid angle per unit area, as follows:

• The unit for brightness is Wsr-1m-2

t

t

UB

A

Power per solid angle (W/Sr)

Function of ,

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Received power from extended sources

2

tr

AP BA

R

Noting that the solid angle t subtended by the transmitting antenna area is

2

tt

A

R

Using Ut = B At and P = St Ar = Ut Ar/R2

r tP BA For extended sources, we cannot assume that radiation originates from a point.Brightness defines the radiation from extended objects.

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Received Power from an Extended Source

( ) ( )

rP dP B A d

,rA

B()

dFn()

Receiving Antenna

Maximum effective area along antenna peak

Incident brightness

along direction

Normalized radiation intensity of receive antenna

( , ) ( , )r nP A F B d

Extended source

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Spectral Brightness, Bf() unit: W/(Sr. m2 Hz)

• Antennas operate over a certain frequency range.

• Spectral Brightness, Bf() is brightness per unit bandwidth.

• Unit: W/(Sr. m2 Hz)

4

( , ) ( , )

( , ) ( , )

f f

ff

f f

r n ff

B B df

P A df F B d

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Unpolarized Brightness

• If Bf() is unpolarized (such as in atmospheric emission), the antenna detects only half of the power incident on its surface.

• This is because the antenna is polarized, and favors that polarization versus others.

4

1( , ) ( , )

2

f f

unp r n ff

P A df F B d

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Spectral Power, Pf unit: (W/Hz)

Power received by an antenna in a bandwidth of 1 Hz.

4

1( , ) ( , )

2

f f

ff

f r n f

P P df

P A F B d

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Spectral Flux Density, Sf unit: (Wm-2Hz-1)

4

( , ) ( , )

1

2

f n f

f r f

S F B

P A S

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Discrete Source• If the antenna is observing radiation from a discrete

source such as a star, subtending a solid angle s such that s << r

s

Fn()

Receiving Antenna

, 1n

f fs s

F

S B

Over the subtended angle s:

Spectral brightness of the source can be assumed constant

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Planck’s Blackbody Radiation

• In general, when radiation is incident on a substance, a certain fraction is absorbed and the remainder is scattered.

• A blackbody is defined as an idealized, perfectly opaque material that absorbs all incident radiation at all frequencies, and reemits it to stay in in thermodynamic equilibrium at temperature T.

• Although this is a theoretical concept, the Sun and Earth are often modeled as blackbodies.

• At microwave frequencies, good approximations to ideal blackbody are the highly absorbing material used in anechoic chambers.

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Planck’s Radiation Law

"Blackbody radiation" or "cavity radiation" refers to an object or system which absorbs all radiation incident upon it and re-radiates energy which is characteristic of this radiating system only, not dependent upon the type of radiation which is incident upon it.

3

2

2 1

1bb hff

kT

hfB

c e

W/(m2 Sr Hz)

h= Planck’s constant = 6.63x10-34 Joulesk = Boltzman’s constant = 1.38x10-23 JK-1

T = absolute temperature, Kf = frequency, HzC = speed of light = 3x108 m/s

Uniform in all directions

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Planck Law

•The Planck Law gives a distribution that peaks at a certain wavelength.

•The peak shifts to shorter wavelengths (i.e. higher frequency) for higher temperatures, and

•The area under the curve grows rapidly with increasing temperature.

1 angstrom = 1.0 × 10-10 meters

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Properties of Planck’s Law

• The Wien Displacement Law– gives the wavelength of the peak of the radiation

distribution – explains the shift of the peak to shorter wavelengths

as the temperature increases

• The Stefan-Boltzmann Law – gives the total energy being emitted at all

wavelengths by the blackbody (which is the area under the Planck Law curve)

– explains the growth in the height of the curve as the temperature increases

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Stefan-Boltzmann Law

3

20 0

2

1hffkT

h fB B df df

c e

41B T

Stefan-Boltzmann constant = 5.673x10-8 Wm-2K-4Sr-1

Total brightness B of a blackbody increases as the fourth power of its temperature T.

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Derivation of Stefan-Boltzman Law

3

20

3

3 3

34 4

2 2 2 20

2

1

43 3

41 1 10 0

2

1

Let

,

2 2

1

1Using 1

1 16

690

hfkT

x

xx x x nx

x xn

nx nx

n n n

h fB df

c ehf

xkT

kT kTdf dx f x

h h

xB kT dx kT

c h e c h

ee e e e

e e

I x e dx x e dxn

I

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Wien Displacement Law

10

3

m

0

5.87 10 (Hz)

2.879 10 (m)

m

f

f

m

dB

df

f T

T

Where does the curve peak?

fm increases with T.

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Wien law for three different stars

For convenience in plotting, these distributions have been normalized to unity at the respective peaks; by the Stefan-Boltzmann Law, the area under the peak for the hot star Spica is in reality 2094 times the area under the peak for the cool star Antares.

Visible

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Rayleigh-Jeans Law• Low frequency approximation for Planck’s Law.• hf/kT << 1

• Useful in microwave region:

2

1 1 1 ... 12

xhf xx e x x

kT

2

2 2

2 2f

f kT kTB

c

For a blackbody at room temperature, Rayleigh Jeans Law error is 3% at 300 GHz

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Comparison of Approximations to Planck Curve

Rayleigh-Jeans Low frequencyWien High frequency

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Power Temperature Correspondence

Pbb

T

absorber

Consider a lossless microwave antenna placed inside a chamber maintained at a constant temperature, T.

4

1, ,

2 bb

ff

bb r f nf

P A df B F d

Power received by the antenna due to emission by the chamber:

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Power Temperature Correspondence: Rayleigh-Jeans Assumption

2

24

fbb

24

2

4

2

1 2,

2

Assume of antenna is narrow so that B is constant

over the bandwidth:

,

where , . Thus:

bbf

ff

bb r nf

rbb n

A n

r

kTB

kTP A df F d

f

AP kT f F d

F dA

bbP kT f

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Recall That…

max

max

max

m

max

ax

2max

max max

2 22max max max

max

2max

max2

, where 4

4

Also,

Thus

r

o

r total

o total r

tot

A

A

A

r

r

al

UA D D

U

U U UA P

U A

P

UA

P

U

UA

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Implications on Microwave Remote Sensing

• The result of the power-temperature relation is of fundamental significance in microwave remote sensing.

• Received power is directly related to the temperature of the environment.

The physical temperature of the antenna has no bearing on its output power (as long as it is lossless.)

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Non-Blackbody Radiation

• A blackbody is an idealized body, which absorbs all incident energy.

• It also emits all to be in thermal equilibrium.

• Real materials (usually referred to as “grey bodies”) emit less energy than a blackbody. They also do not absorb all energy incident on them.

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Grey Body Brightness

B()

Consider the semi-infinite surface below.

Brightness of a grey body is possibly a function of direction:

2

2bbf

kTB

Recall that for a blackbody there is no angular dependence, all uniform. In -wave region;

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Brightness Temperature of Grey Bodies

• The brightness temperature, TB of a grey body is defined as a “blackbody equivalent radiometric temperature.”

• This allows B() to be represented in a form similar to that of a blackbody.

2

2, ,

2

2

B

bb

kB T f

kB T f

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Emissivity• The brightness B() of a material relative to

that of a blackbody at the same temperature is defined as emissivity, e().

• Remark: Analogy in antenna theory– e() D()– blackbody isotropic source– Brightness, B() radiation intensity, U()

( , ) ( , )( , ) B

bb

B Te

B T

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EmissivityNote that since B() <= Bbb

Thus, the brightness temperature TB() of a material is always less than or equal to its physical temperature T.

0 , 1e

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Remarks

• Note that we are assuming homogenous material of uniform temperature.

• Convention: To avoid confusion between physical and radiometric temperatures, we will use uppercase subscripts (e.g. TA) to denote radiometric temperature, and either no subscript or lowercase subscript (e.g. Ta) for physical temperatures.

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Apparent Temperature (Overall Scene Effects)

antennaFn()

TA

Atmosphere

TAP()Apparent temperature distribution

TDN

TSC

TUP

TB

TB: Terrain emission

TDN: Atmospheric downward emissionTUP: Atmospheric upward emission

TSC: Scattered radiation

Terrain

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Apparent Temperature

• TAP() is the blackbody equivalent radiometric temperature of the scene.

2

2, ,

i AP

kB T f

Incident brightnessConsists of several terms

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Remarks on Apparent Temperature

• If the atmosphere is lossless (i.e. no absorption no emission) the only contribution to TAP is the emission from the terrain.

• In that case TAP = TB

• For f: [1-10] GHz, one can assume lossless atmosphere under clear sky conditions.

• In general TAP is not equal to TB.

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Antenna Temperature (Overall Antenna Effects)

Recall that we derived power at the output terminals of a lossless antenna placed inside a blackbody enclosure as:

4

1, ,

2 bb

ff

bb r f nf

P A df B F d

For narrowband applications,

4

1, ,

2 bbbb r f nP A f B F d

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Overall Antenna Effects

24

1 2, ,

2 r AP n

kP A f T F d

AP kT f

For a non-blackbody brightness distribution defined in terms of the apparent temperature, the power is

The power measured at the antenna terminals is (Nyquist result for a resistor):

( , )B

Antenna radiometric temperature,NOT a physical temperature!

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Antenna TemperatureTherefore, the antenna radiometric temperature

relates to the apparent temperature of the environment:

24

4 4

4

, ,

1, , ; ,

, ,

rA AP n

AP n r n

r

A r AP n

AT T F d

T F d F d

T T F d

An average measure:

TA

TAP

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Antenna Temperature Measured from a Scene

4

4

4

, ,

,

, ,

AP n

A

n

A A AP n

T F dT

F d

T T F d

Averaged temperature over the solid angle of receive antenna

Fn()

A

TAP

TA

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Some Observations• For a blackbody enclosure:

– TAP() = T TA = T

– (antenna temperature = ambient temperature)

• For a discrete source (e.g. sun) whose subtended solid angle is much less than the main beam solid angle, i.e. s << r

TAP()=TAP (constant over the main beam)

4

4

,,

,s

sA

nAP n

A AP

n r

AP

r

F dT F dT T

F d

T T

s

Fn() =1over s

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General Solution (Lossless Antenna)

Atmosphere

La

TDN TB

TSC

TUP TAP() 4

, ,AP n

r

T F d

TA

1,AP UP B sc

a

T T T TL

TUP: Atmospheric self emissionTsc: radiative temperature of scattered brightnessTB: Background emission

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Example 1

From the earth, the angle subtended by the Sun is 0.5o. A 1cm wavelength radio-telescope antenna pointed at the Sun measured an antenna temperature of 1174K. The antenna effective area is 0.4 m2. What is the apparent temperature of the Sun?

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Solution

Assume discrete source approximation for the Sun.

4

4

,,

,s

sun

nAP n

A AP

n r

sA AP

r

F dT F dT T

F d

T T

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Solution

2

2

2

2

2

2

4

2 44

4

4

tan2

tan2

tan ; 0.52

0.6 10

102.5 10

0.4

2.5 101174 4891.67

0.6 10sun

s

s

s

ss s

r

r

rAP A

s

dsR

ds r R

R

R

A

T T K

R

Earth

r

s

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Background Effects

To

T1

A

t

Case A

To

T1

A

t

Case B

Target is not resolved, and antenna’s field of view includes some background.

Target is resolved.

Assume that the power pattern of the antenna is uniform within the solid angle A, and zero outside.

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Case A

4

1

1

1, ,

1

A AP n

A

A t o t

A

t to

A A

T T F d

T T

T T

TA is the linearly weighted average of the brightness temperatures of the components in the antenna’s field of view.

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Case B

• If the antenna is made physically larger, thus reducing the size of the beam solid angle, the weighting will shift in favor of T1.

• However, once the target becomes fully resolved (i.e. when A ≤ t), the antenna temperature is just TA = T1

• Further increase in the effective area does not result in detecting more power, since more radiation is collected from a smaller range of direction.

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Antenna Efficiency Considerations

• So far we have assumed lossless antennas.

• The antenna temperature TA represented a lossless antenna; i.e. radiation efficiency = 1

• However, the efficiency of an antenna is always less than 1, since there are losses in the circuitry.

• We will define TA’ as the temperature of a lossy antenna.

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Antenna Temperature – Radiation Efficiency Effects

• Based on the power-temperature correspondence:

'

rec a in

A a A

P P

T T

Temperature measured if antenna was lossless

Temperature measured with a lossy antenna

Radiation efficiency

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Antenna Radiation Efficiency and Temperature

• But, a lossy device is also a radiator!

• The emitted power is related to the physical temperature:

'

1

1

N a o

A aA oa TT T

T T

drop increase

Physical temperature

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Example 2

• From the earth, the angle subtended by the Sun is 0.5o. A 1 cm wavelength radio telescope antenna pointed at the Sun measured an antenna temperature of 1174K. The antenna efficiency is 0.8 and its effective area is 0.4 m2. What is the apparent temperature of the Sun? (Same as Example 1, with a lossy antenna)

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Solution

TA

T’A

A lossless antenna would measure this:

' 1

300

0.8

sun

A a A a o

o

a

sA AP

r

T T T

T K

T T

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Solution4

2 44

0.5 0.6 10

102.5 10

0.4

s s

r

rA

From Example 1:

'

'

1

1

0.61174 0.8 0.2 * 300

2.55802.08

sun

sun

sun

sun

A a A a o

sA AP

r

sA a AP a o

r

AP

AP

T T T

T T

T T T

T

T

Recall that the apparent temperature was less in Example 1. Thus, if one assumes the antenna to be lossless, the apparent temperature would be underestimated.

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Antenna Efficiency

• Radiation Efficiency:

• Beam Efficiency: – In reality, in addition to the thermal emission received

through the antenna main beam, antenna receives (undesired) contributions through the sidelobes.

– Ideally one would design a radiometer antenna with a narrow pencil beam and no sidelobes.

' (1 )A a A a oT T T

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Main Beam and Sidelobe Effects

4

4

1, ,

1, ,

1, ,

m SL

m

m

A AP n

r

AP n

r

AP n

r

A ML SL

T T F d

T F d

T F d

T T T

SL

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Main Beam Efficiency

4

( , )

( , )M

n

M

n

F d

F d

Ratio of power contained within the main beam to total power.

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Effective Main Beam Apparent Temperature,

• Antenna temperature if the antenna pattern consisted of only the main beam.

, ,

,M

M

AP n

ML

n

ML ML ML

T F dT

F d

T T

MLT

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Antenna Stray Factor

4

4

1

( , )

( , )S M

S M

n

n

F d

F d

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Effective Apparent Temperature of the Sidelobe Contribution

, ,

,

(1 )

S

S

AP n

SL

n

SL SL SL ML SL

T F d

TF d

T T T

• Antenna temperature if the antenna pattern consisted of only the sidelobes.

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Antenna Temperature and Beam Efficiency

(1 )A ML SL

ML ML ML SL

T T T

T T

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Overall Antenna Efficiency and Antenna Temperature

' (1 )

(1 ) (1 )

A a A a o

a ML ML ML SL a o

T T T

T T T

Combine beam efficiency and radiation efficiency:

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Radiometer Signal Detection

• The objective in radiometric remote sensing is to relate the radiometer receiver output voltage, Vout to the effective main beam antenna temperature,

• The output voltage Vout can be calibrated to read temperature.

• is the measured quantity, and is the quantity of interest.

'AT

MLT

MLT

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Linear Relation

1 1 1M aML A SL o

a M M a M

T T T T

MLT

AT

Bias = B1 +B2

-( B1 +B2)

Slope

Depends on sidelobe levels, antenna efficiency and temperature Depends on

antenna efficiency

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Remarks on Accuracy

• The desired value, is known if are known.

• are very critical as they determine the slope of the curve and offset. Therefore the antenna needs to be accurately characterized.

• The first bias term is a function of antenna

pattern and emission from the scene. Therefore, it can have a wide range of variation.

• To limit variations due to the bias terms, antenna should be designed for the highest possible main beam efficiency; i.e.

, , ,SL o M aT T MLT

,M a

1

1 MSL

M

B T

11 0M B

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Summary

The accuracy of radiometric measurements is highly dependent on the radiation efficiency, and main beam efficiency, of the antenna.M

a

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Example 3

A radiometer is used to observe terrain surfaces with apparent temperatures in the range of [100-300] K. Antenna radiation efficiency is 0.9 and its physical temperature, To is always monitored. The sidelobe apparent temperature is unknown, but it’s between [100-200] K. Corresponding to this uncertainty, what should be the minimum value of main-beam efficiency so that % error in the estimated value of apparent terrain temperature is less than 3%?

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Solution – Example 3

3%ML

ML

Te

T

MLT

Requirement:

Need to derive an expression for

Given:

0.9

: 100 300

: 100 200

?

a

A

SL

ML

T K

T K

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Solution

' (1 )

(1 ) (1 )

where

0.9

: known

: 100 300

: 100 200

: unknown quantity to be solved for

A a A a o

a ML ML ML SL a o

a

o

ML

SL

ML

T T T

T T T

T

T K

T K

MLT

AT -( B1 +B2)

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Solution

'

'

'

'

0.1 0.9 0.9(1 )

0.1 (1 )

0.9

0.1 (1 )

0.9

1 0.1 (1 )

0.9

(1 )

A o ML ML ML SL

A o MLML SL

ML ML

A o MLML SL

ML ML

A o MLSL

ML ML

MLSL

ML ML

T T T T

T TT T

T TT T

T TT

AT

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Solution(1 )

(1 )

3%

(1 ) (1 )0.03

o o

o o

o

o

o o

MLML SL

ML ML

MLML ML ML SL SL

ML

ML

ML

SL SL SLML ML

ML ML ML ML

AT T

T T T T T

Te

T

T T TT T

Correct value

Worst case when has minimum possible value (= 100K) and

has maximum possible value ( = 200-100 = 100K)SLT

oMLT

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Solution

(1 ) (1 ) 1000.03

100

(1 ) 10.03 1.03

10.97

1.03

o

SLML ML

ML ML ML

ML

ML ML

ML

TT

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Theory of Radiative Transfer

We will be considering techniques to derive expressions for the apparent temperature, TAP of different scenes as shown below.

Atmosphere

Terrain

TA

TUP

TA

Terrain could be smooth, irregular, slab (such as layer of snow) over a surface.

STEP 1: Derive equation of radiative transferSTEP 2: Apply to different scenes

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Radiation and Matter

• Interaction between radiation and matter is described by two processes:– Extinction– Emission

• Usually we have both phenomenon simultaneously.

• Extinction: radiation in a medium is reduced in intensity (due to scattering and absorption)

• Emission: medium adds energy of its own (through scattering and self emission)

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Scattering Amplitudes and Cross Sections

• Brightness directly relates to power, and satisfies the transport equation.

• We will examine the effects of presence of scattering particles on brightness.

O

s

r

B(r,s) is a function of position and direction

Function of 5 parameters:r: x, y, zs:

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Terminology for Radiation/Scattering from a Particle• Scattering Amplitude

• Differential Scattering Cross Section

• Scattering Cross Section

• Absorption Cross Section

• Total Cross Section

• Albedo

• Phase Function

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Scattering AmplitudeConsider an arbitrary scatterer:

Imaginary, smallest sphere

D

i

Ei

o

Es

R

2

ˆˆ( , ) ; ,ikR

s o o i

e DE e f o i e E R

R

The scatterer redistributes the incident electric field in space:

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Scattering Amplitude(2)

2

ˆˆ( , ) ; ,ikR

s o o i

e DE e f o i e E R

R

o: s, s

i: i, i

f(o,i) is a vector and it depends on four angles.

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Scattering Cross Section Definitions: Power Relations

• Differential Scattering Cross-section

• Scattering Cross-section

• Absorption Cross-section

• Total Cross-section

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Differential Scattering Cross Section

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Scattering Cross-section

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Absorption Cross-section

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Total Cross-section

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Phase Function

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Derivation of the Radiative Transfer (Transport) Equation

O

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References

• Microwave Remote Sensing, F. T. Ulaby, et.al. Addison-Wesley

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References

• Microwave Remote Sensing, F. T. Ulaby, et.al. Addison-Wesley