METO 621
Lesson 16
Transmittance
• For monochromatic radiation the transmittance, T, is
given simply by
T ( ;  )  e  / 
and
dT ( ;  )
1
  e  / 
d

• The solution for the radiative transfer equation was:
Transmittance
I ( ,  ,  )  I ( *,  ,  )e


( * ) / 
*
1
  S  e ( ' ) /  d '


Which is equivalent to
*
I ( ,  ,  )  I ( *,  ,  )T ( * , ;  )  

dT ( ' , ;  ) 
S d '
d '
similarly

dT ( , ' ;  ) 
S d '
d '
0
I ( ,  ,  )  I (0,  ,  )T ( ;  )  
Transmittance
• If we switch from the optical depth coordinate to
altitude, z, then we get
dT ( z, z ' ;  ) 
I ( z,  ,  )  I (0,  ,  )T (0, z;  )  
S dz '
dz '
0


z
dT ( z , z ' ;  ) 
I ( z ,  ,  )  I ( zT ,  ,  )T ( zT , z;  )  
S dz '
dz '
z


zT
• The change of sign before the integral is because
the integral range is reversed when moving from  to z
Transmission in spectrally complex media
• So far we have defined the monochromatic transmittance,
T(). We can also define a beam absorptance ab() , where
ab() =1- T().
• But in radiative transfer we must consider transmission over
a broad spectral range, D. (Often called a band)
• In this case the mean band absorptance is defined as
1
a b  
D
 (1  e
 k u
)d
D
• Here u is the total mass along the path, and k is the
monochromatic mass absorption coefficient.
Schematic for A
Transmission in spectrally complex media
• Similarly for the transmittance
TD
1
 1  a D  
D
  k u 
  e d 


 D

Transmission in spectrally complex media
• If in the previous equation we replace the frequency
with wavenumber then the product
W  a b D is called the equivalent width
• The relationship of W to u is known as the curve
of growth
Absorption in a Lorentz line shape
Transmission in spectrally complex media
• Consider a single line with a Lorentz line shape.
• As u gets larger the absorptance gets larger. However as u
increases the center of the line absorbs all of the radiation,
while the wings absorb only some of the radiation.
• At this point only the wings absorb.
• We can write the band transmittance as;
TD
1

D
 

S
a
u
L
 
d
exp

2
D   ((  0 )  a L )  


Transmission in spectrally complex media
• Let D be large enough to include all of the line,
then we can extend the limits of the integration to
±∞.
• Landenburg and Reichle showed that
1 x
TD  1 
e J 0 (ix )  iJ 1 (ix )
D
Su
where x 
J 0 and J1 are Bessel functions
2aL
Transmission in spectrally complex media
• For small x , known as the optically thin condition,
Su
T  1
D
for large x, optically thick,
2 Su
2
T  1
 1
a L Su
D
D 2x
Elsasser band model
Elsasser Band Model
• Elsasser and Culbertson (1960) assumed evenly
spaced lines (d apart) of equal S that overlapped.
They showed that the transmittance could be
expressed as

T   exp  ycoth(  )J 0 (iy sinh(  )) dy

2aL
Su
where  
and y 
d
d sinh(  )
optically thin
optically thick
Su
T  1d
2
T  1a L Su
d
Statistical Band Model (Goody)
• Goody studied the water-vapor bands and noted the
apparent random line positions and band strengths.
• Let us assume that the interval D contains n lines
of mean separation d, i.e. D=nd
• Let the probability that line i has a line strength Si
be P(Si) where

 P(S )dS  1
i
0
• P is normally assumed to have a Poisson
distribution
Statistical Band Model (Goody)
• For any line the most probable value of S is
given by

0 SP(S )dS
S 
 P(S )dS
0
• The transmission T over an interval D is given
by
1
Ti 
D

 d  P(S )e
i
D
0
 ki u
dSi
Statistical Band Model (Goody)
• For n lines the total transmission T is given by
T=T1T2T3T4…………..Tn
1
T
(D ) n
 d  d .................  d
1
D
1
D
n
D



0
0
0
x  P( S1 )e  k1u dS1  P( S 2 )e  k2u dS 2 ........ P( S n )e  k nu dS1

1 
 ku
T
d  P( S )e dS 
n 
(D ) D 0



1
 1 
 D

 d  P(S )(1  e
D
0
 ku
n

)dS 

n
Statistical Band Model (Goody)
n
 x
The equation has the form 1   which in
 n
the limit equals e  x

 1

ku
T  exp   d  P( S )(1  e )dS 
 d D o

assuming a Lorentz line shape, one gets
1 / 2
 Su 
Su  
1 
 
T  exp 
 d  aL  
where S is the mean line strength.
Statistical Band Model (Goody)
• We have reduced the parameters needed to calculate T
to two
S
d
and
aL
d
• These two parameters are either derived by
fitting the values of T obtained from a line-by-line
calculation, or from experimental data.
weak line
 Su 

T  exp  
 d 
strong line


a
S
u
L

T  exp  


d


Summary of band models
K distribution technique
K distribution technique