METO 621
LESSON 8
Thermal emission from a surface
• Let
ˆ ) cos d
I (

c
be the emitted
energy from a flat surface of temperature
Ts , within the solid angle din the
direction .A blackbody would emit
 B (T )cosd.The spectral directional
 s
emittance is defined as
 ˆ
ˆ
I () cos d Ie ()
ˆ
 ( , , TS ) 

B (TS ) cos d B (TS )

e
Thermal emission from a surface
• In general  depends on the direction of
emission, the surface temperature, and the
frequency of the radiation. A surface for
which  is unity for all directions and
frequencies is a blackbody. A hypothetical
surface for which  = constant<1 for all
frequencies is a graybody.
Flux emittance
• The energy emitted into 2psteradians relative to
a blackbody is defined as the flux or bulk
emittance
 ( ,2p , TS ) 
ˆ)
d cos Ie (

 d cos B (T



1
p


ˆ ,T )
d cos  ( , 
S
S
)



ˆ , T ) B (T )
d cos  ( , 
S

S
p B (TS )
Absorption by a surface
• Let a surface be illuminated by a downward
intensity I. Then a certain amount of this energy
will be absorbed by the surface. We define the
spectral directional absorptance as:
 ˆ
 ˆ
I
(

'
)
cos

'
d

'
I
a
a (' )
ˆ
 ( ,' , TS )  
 
ˆ ' ) cos ' d ' I (
ˆ ')
I (


The minus sign in - emphasizes the
downward direction of the incident radiation
•
Absorption by a surface
• Similar to emission, we can define a flux
absorptance
 ˆ
ˆ
 d ' cos ' ( ,' , TS )I (' )
 ( ,2p , TS )  1
•

F
Kirchoff showed that for an opaque surface
ˆ ,T )  (,
ˆ ,T )
(,
S
S
• That is, a good absorber is also a good
emitter, and vice-versa

Surface reflection : the BRDF
ˆ ).
I (
ˆ ) cos  d '.
I  (
Consider a downward beam with intensity
The energy incident on a flat surface is

Let the intensity of the reflected light around the
ˆ within a solid angle d be dI  then
direction 
r
 ˆ
dI
)
r (
ˆ
ˆ
 ( ,', )  ˆ
I (' )cos ' d '
ˆ ', 
ˆ ) is the bidirectional relfectance
where  ( ,
distribution function,
or BRDF.
BRDF
ˆ,
The total reflected intensity in the direction 
from all beams is
ˆ )  d I  (
ˆ ) d ' cos  '  ( ,
ˆ ', 
ˆ ) I  (
ˆ ')
I  (
r

r




If a reflecting surface has a BRDF which is
independen t of both the incidence and observatio n
directions , then it is called a Lambert surface.
ˆ ', 
ˆ )   ( ), and
In this case  ( ,
L
ˆ ' ) ( ) F
Ir  L ( )  d ' cos  'I (
L

Surface reflectance - BRDF
Collimated incidence
Collimated Incidence - Lambert
Surface
• If the incident light is direct sunlight then
I  ()  F S ( 0 )  F S (cos   cos  0 ) (   0 )
The incident flux is given by
Hence
F   F S cos 0
Ir   L cos  0 F S
For a collimated beam the intensity reflected from
a Lambert surface is proportional to the cosine of
the angle of incidence.
Collimated Incidence - Specular
reflectance
• Here the reflected intensity is directed along
the angle of reflection only.
• Hence ’and ’+p
• Spectral reflection function S(,)
ˆ )   ( )F S(cos   cos )(    p )
Ir (
S
0
0
•
and the reflected flux:
Fr   S (0 )F S cos  0
Absorption and Scattering in Planetary
Media
• Kirchoff’s Law for volume absorption and
Emission
 ( ,T)
 ( ,T) 
k( )
The volume emittance is proportional to the
absorption coefficient
Differential equation of Radiative
Transfer
• Consider conservative scattering - no change in
frequency.
• Assume the incident radiation is collimated
• We now need to look more closely at the secondary
‘emission’ that results from scattering. Remember
that from the definition of the intensity that
ˆ ' )dAdtd d
d E I (
4

Differential Equation of Radiative
Transfer
• The radiative energy scattered in all
directions is
 ds d E'
4
• We are interested in that fraction of the
scattered energy that is directed into the solid
angle d centered about the direction .

• This fraction is proportional to
ˆ ',
ˆ ) d /4p
p(
Differential Equation of Radiative
Transfer
• If we multiply the scattered energy by this
fraction and then integrate over all incoming
angles, we get the total scattered energy
emerging from the volume element in the
direction ,
ˆ
ˆ
p(

'
,

) ' ˆ
4
d E   ( ) dV dt d d  d '
I (' )
4p
4p
• The emission coefficient for scattering is
jSC
d4 E
d ' ˆ ˆ
ˆ ')

  ( ) 
p(', )I (
dV dt d d
4 p 4p
Differential Equation of Radiative
Transfer
• The source function for scattering is thus
SC
j
 ( ) d ' ˆ ˆ
SC

ˆ
ˆ ')
S ( rˆ, ) 

p(', ) I (

k( ) k( ) 4 p 4 p
• The quantity ()/k() is called the
single-scattering albedo and given the
symbol a().
• If thermal emission is involved, (1-a) is
the volume emittance .
Differential Equation of Radiative
Transfer
• The complete time-independent radiative
transfer equation which includes both multiple
scattering and absorption is
dI
a( )
ˆ ', 
ˆ )I
 I 1  a( )B (T )
d

'
p
(



d s
4p 4p