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The power can also be expressed in terms of the radiation intensity as

2R

AFP rt= W, (4.3)

The brightness can then be defined as

t

t

A

FB= W/m2/sr, (4.4)

and the solid angle tsubtended by the source of the radiation is given by

2

R

Att= sr. (4.5)

Substituting into the power equation

trBAP = W. (4.6)

For a differential solid angle

= ),(),( nr FBAP , (4.7)

where B(,) Source brightness as a function of solid angle (W/m2/sr),

Fn(,) Normalised radiation pattern of antenna as a function of solid angle,

If this is integrated over all 4steradians and over the frequency bandf1tof2,

=2

1 4),(),(

2

f

f n

r fFBA

P

W. (4.8)

This allows for the calculation of the power incident on the antenna in terms of the

brightness of the source of the radiation and the gain pattern of the antenna.

This received power is reduced by one half in this case because the direct polarisation

from the source is random and it is being received by a linearly polarised antenna.

Considering that this antenna is placed within a blackbody, and if the detected power

is limited to a small bandwidth such that the brightness is constant with frequency,

then the Rayleigh-Jeans approximation can be substituted for B(,) to obtain the

= ),()(

42

12

nr

bb FAffkT

P W. (4.9)

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From basic antenna theory it can be shown that the integral above equates to the

pattern solid angle pwhich is given by

r

pA

2

= . (4.10)

This is substituted into the power equation to give the fundamental equation of

)( 12 ffkTPbb = W. (4.11)

Certain points are worth noting here:

The detected power is independent of the antenna gain because the source ofradiation is extended and uniform and not a point source

The equation is independent of the distance from the radiating target

The temperature of the antenna structure has no effect on the output power Temperature and power are interchangeable so all the gain calculations can be

applied directly to the measured temperature

The power detected is directly proportional to the bandwidth.

Example

Consider the power received by an antenna operating at 100GHz with a bandwidth of

2GHz observing a blackbody with a temperature 310K

P = 1.3810-23

3102109

= 8.5610-12

W

= -80.68dBm

4.3. Brightness TemperatureTb(,) is defined as the brightness temperature of the thermal sourceB.

All real bodies are to some extent grey, as they radiate less than a black body. In

addition the brightness temperature, Tb(,) for a grey body can also angle dependent

because of variations in its emissivity.

Tis defined such that the brightness of the grey body is the same as a blackbody at the

brightness temperature. It can be obtained from the physical temperature

( ) ( )TTb .,, = , (4.12)

where (,) Emissivity,

T Physical temperature of the radiating element (K).

In the example above, if the target has an emissivity = 0.8, then the brightness

temperature Tb = 0.8310 = 248K, and the received power is reduced accordingly(-81.64dBm).

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4.4. Apparent TemperatureIn radiometry the apparent antenna temperature TAPreplaces the received powerWas

the measure of signal strength, where TAPis defined as the temperature of a matched

resistor with noise power output equal to W; that is W= k.TAPat the antenna port

The apparent antenna temperature TAP is calculated from the brightness temperature

including atmospheric and antenna losses.

Figure 4.1: Radiometer configuration showing effects

The radiation from the main lobe of the antenna is made up of two components:

The brightness temperature TBfrom terrain emissions The scatter temperature TSCwhich is the radiation reflected from terrain in the

main lobe but not generated by it. Radiation from both the atmosphere TDN

(the downward or downwelling temperature) and galactic radiation may be

reflected. At frequencies greater than 10GHz only the downward radiation

from the atmosphere need be considered.

These contributors to the total radiation are then attenuated by the atmosphere

before they reach the antenna.

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In addition to this there is the upward (or upwelling) radiation from the

atmosphere.

[ ]),(),(1

),(),( SCBA

UPAP TTL

TT ++= , (4.13)

where TAP Apparent Temperature (K),

TUP Upwelling temperature from the atmosphere (K),

LA Atmospheric loss factor,

TB Brightness of the observation area (K),

TSC Brightness of the radiation scattered from observation area (K).

4.5. Atmospheric Effects4.5.1. AttenuationAtmospheric attenuation is a function of the air density, and, for horizontal or oblique

paths through the atmosphere, it must be calculated by integration. The graph below

shows the attenuation right through the atmosphere

Figure 4.2: Attenuation through the atmosphere

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The upward and downward brightness temperatures of the atmosphere vary with

frequency, and will obviously be higher where the attenuation is higher as the

atmosphere is more opaque.

At 94GHz, the attenuation through the entire atmosphere can be calculated as follows:

oAL 06.017.0 += , (4.14)

where LA Atmospheric attenuation (dB),

o Water vapour concentration (g/m3).

For aircraft based radiometers, the attenuation is far more complex, and will be dealt

with in some detail later in this chapter.

Figure 4.3: Downwelling brightness temperature as a function of frequency with water

vapour concentration as a parameter

For operation at 94GHz, typical values of the downwelling temperature as a functionof the atmospheric conditions are shown in the following table

Table 4.1: Downwelling temperature under different weather conditions

Conditions Downwelling

Temperature (K)

Clear Sky 10-60

Thick Fog 120

Overcast 150

Fog 180

Thick Clouds 180

Moderate rain 240

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and is equal to the downwelling radiation

For aircraft, only part of the atmosphere contributes to the upwelling radiation, and asthe atmosphere can be stratified and complex, it is easiest to treat it as an attenuator

and calculate the upwelling radiation in those terms.

To maintain thermal equilibrium, any medium that absorbs radiation (attenuates) must

also radiate. As the atmosphere can be modelled as an attenuator, it can be shown that

its effective temperature is

TL

TA

e )1

1( = , (4.15)

where Te Effective temperature of attenuator (atmosphere),LA Attenuator loss factor = 10

/10,

T Physical temperature of the attenuator (K).

4.6. Terrain BrightnessVarious forms of terrain have completely different brightness temperatures

Metallic Objects: These are lossless and opaque and so are perfectly reflecting. As a

result their brightness will be the same as the downwelling radiation.

Water: The brightness of water is dependent on polarisation, angle of view, and to alesser extent, temperature, purity and surface conditions. Because it is also reflective,

its brightness is also dependent on the downwelling temperature. At 94GHz the

reported brightness for water (vertical polarisation) varies between 150 and 300K.

Soil: As with water, it is dependent on polarisation and angle of view. It is also

dependent on moisture content and surface roughness. At 94GHz the reported

brightness for soil (vertical polarisation) varies between 160 and 280K.

Vegetation: Brightness of vegetation depends on its type and moisture content. At

94GHz it is reported to vary between 230 and 300K.

Built-Up Areas: This will be complex, however at 94GHz, asphalt is given to be 260

to 300K.

Though there is a significant overlap between the brightness temperatures in these

cases, this is due to the fact that the data were taken under a variety of weather

conditions. In general, there will be a significant contrast between different materials

under the same weather conditions.

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4.7. ExampleA space based radiometer operating at 94GHz with a bandwidth of 2GHz looks

di

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