PC4262 Remote Sensing
Solution of the RTE in a Plane Parallel Atmosphere
Dr. S. C. Liew, Jan 2003
crslsc@nus.edu.sg
Formal Solution of the RTE
Consider the RTE for a plane parallel atmosphere,
dL (, u, )
u 
 L (, u, )  J  (, u, )
d
(symbols as defined in the previous chapter)
(1)
Using the integrating function exp(  / u ) , this equation can be written in the form:


d  / u
e  / u
e
L (, u, )  
J  (, u, )
d
u
(2)
Integrating from 1 to 2, we get the solution,
e   2 / u L ( 2 , u, )  e  1 / u L (1 , u, )  
1 2 t / u
J  (t , u, ) dt
 e
u
(3)
1
Thus, depending on the boundary conditions used, the solution can be expressed as:
L  (, u, )  L  (  0, u, )e  / u 
1  (  t ) / u
J  (t , u, ) dt (1  u  0)
e
u0
(4)
and
L  (, u, )  L  ( 0 , u, )e (0 ) / u 
1 0 (t ) / u
J  (t , u, ) dt (0  u  1)
 e
u 
(5)
Eqn. (4) is used for evaluating the downward radiance L (, u, ) (u negative) while Eqn. (5)
is for the upward radiance L (, u, ) (u positive).
This solution is known as the formal solution of the RTE. Note that the formal solution is
still not a complete solution of the RTE, since the source function J in general, contains the
radiance term L which is not yet known.
Thermal Emission in the Atmosphere
We now look at a case where only thermal emission is important. For longwave infrared
radiation, the scattering coefficient of the atmosphere is negligible, and only absorption is
important. The source function is
(6)
J (, u, )  L B (T ())


We will now find the upward radiance at the top of the atmosphere (e.g. detected by a
downward looking thermal sensor on-board a satellite), which can be expressed as (Eqn. 5),
(7)
1 0
L  (0, u, )  L  ( 0 , u, )e   0 / u   e  t / u L B (T (t )) dt
u 0
The first term on the right hand side of this equation is the upward radiance from the ground
surface attenuated by the atmosphere, while the second term is due to the emission from the
atmosphere itself.
The ground surface radiance is the sum of two terms: thermal emission from the surface, and
the downward emission from the atmosphere reflected upwards by the surface. We assume
that the atmosphere is bounded below by a Lambertian surface at a temperature Ts, with
thermal emissivity s(), and hemispherical reflectance s() = 1 - s(). The ground surface
upward radiance is,
(8)
[1   s ()]E  ( 0 )

B
L ( 0 , u , )   s () L (Ts ) 

where E  ( 0 ) is the downward irradiance at the ground due to atmospheric emission,
which can be expressed as (see Eqn. 4)
(9)
2 0  0 1

E ( 0 )      e (  0  t ) / u ' L B (T (t )) u' dt du ' d
0 1 0 u '
0 0
 2   L B (T (t )) e  (  0  t ) / u ' dt du '
1 0
In the thermal infrared region, the ground emissivity is typically close to one, and hence the
surface reflectance is  s  0 . The second term in Eqn. (8) can usually be neglected.
Hence, the upward radiance at top of atmosphere is

B
L (0, u, )   s () L (Ts )e
 0 / u
1 0 t / u B
  e
L (T (t )) dt
u 0
(10)
The transmission from a height z to the top of atmosphere along a path inclined at u = cos to
the zenith direction is defined as
(11)
( z, u )  e ( z ) / u
where (z) is the optical thickness from the height z to the top of atmosphere. In differential
form,
(12)
ud
d  

Eqn. (10) can be rewritten in terms of the transmission,

(13)
 d( z, u ) 
L  ( z  , u, )   s () L B (Ts )( z  0, u )   L B (T ( z )) 
 dz
 dz 
0
The derivative of transmission with respect to z is the instrumental weighting function for
the sensor used in measuring the upward radiance,
(14)
d( z , u )
W ( z, u ) 
dz
Hence,

(15)
L  ( z  , u, )   s () L B (Ts )( z  0, u )   L B (T ( z ))W ( z, u ) dz
0
The atmospheric pressure decreases monotonically with the height z above the ground. Since
pressure is a more easily measured quantity than the altitude, it is often more convenient to
express height in terms of the pressure P(z). The top of atmosphere radiance is expressed as
ps
(16)

B
L ( p  0, u, )   s () L (Ts )( p s , u )   L B (T ( p))W ( p, u ) dp
0
where
(17)
d( p, u )
dp
is the rate of change of transmission with respect to pressure, and ps is the atmospheric
pressure at ground surface (z = 0).
W ( p, u )  
We will see in a later chapter that Eqn. (15) can be used to retrieve the temperature profile
T ( p) and the amount of water vapour in the atmosphere, given the measurements of the
upward radiance at several different wavelength bands corresponding to the water vapour
absorption bands.
Isothermal Atmosphere
We now consider a hypothetical atmosphere where the temperature is constant, independent
of the height. For this atmosphere, the upward radiance at the top of atmosphere (Eqn. 15)
can be evaluated to get
(18)
L  ( p  0, u, )   s () L B (Ts )( ps , u)  L B (Ta )[1  ( ps , u)]
Note: If the top of atmosphere radiance is measured in a wavelength band where gaseous
absorption is small (i.e. high transmission from surface to top of atmosphere), the surface
emission term in Eqn. (11) dominates. On the other hand, if the measurement is made in an
absorption band, then the atmospheric emission term dominates.
Scattering in an Atmosphere Illuminated by Solar Radiation
Sun
Sensor
z
us
u
Top of atmosphere
=0
 us
s

Ground level

x
We now consider the RTE equation in the shortwave region where scattering is important and
thermal emission can be neglected. The illumination source is the solar radiation at the top of
atmosphere. Hence, the downward radiance at the top of the atmosphere is
(19)
L  (0, u, )  F, s (u  u s )(   s  )
where F , s is the solar spectral flux density at the top of atmosphere, us is the cosine of the
solar zenith angle and s is the solar azimuthal angle.
At any level within the atmosphere, the downward radiance (u < 0) is (Eqn. 4)
1
L  (, u, )  F, s e  / u (u  u s )(   s  )   e (  t ) / u J  (t , u, ) dt
u0
(20)
where the scattering source function is
 2 1
J  (t , u, ) 
  P(u, ; u ' , ' ) L (t , u ' , ' ) du ' d'
 0 1
(21)
Note that the first term on the right hand side of the equation (20) is the direct solar radiance
attenuated by the column of atmosphere, and the second term is the diffuse radiance due to
scattering of the solar flux by the atmosphere.
The upward radiance at any level within the atmosphere is (Eqn. 5)


L (, u, )  L ( 0 , u, )e
 (  0  ) / u
1  0  (t   ) / u
  e
J  (t , u, ) dt
u 
(22)
The upward radiance at the ground L (0 , u, ) is the result of the reflection of the
downward irradiance by the ground surface. Assuming a Lambertian surface, the upward
radiance leaving the ground is
L  ( 0 , u, ) 
E  ( 0 )

(23)
where  is the ground hemispherical reflectance and E  ( 0 ) is the downward irradiance at
the ground.
From Eqn. (20), the downward radiance at the ground level is
L  ( 0 , u, )  F, s e  0 / u (u  u s )(   s  ) 
1  0 ( 0  t ) / u
J  (t , u, ) dt
 e
u 0
(24)
Thus, the downward irradiance is
E  ( 0 )  F, s u s e   0 / u s
(25)
2 0  0

e ( 0  t ) / u J
  
0 1 0
 (t , u , ) dtdud
The downward irradiance at the ground consists of two terms, direct and diffused irradiance:
(26)
E  direct  F, s u s e   0 / u s
(27)
2 0  0

E diffused     e (  0  t ) / u J  (t , u, ) dtdud
0 1 0
Hence, the upward radiance at the ground is
 / u
E  ( 0 ) F,s u s e 0 s  2 0 0 ( 0 t ) / u '
L ( 0 , u, ) 

    e
J  (t; u ' , ' ) dt du ' d'


 0 1 0

(28)
We are particularly interested in finding the upward radiance at the top of atmosphere:
0 e t / u
L  (0; u, )  e 0 / u L  ( 0 ; u, )  
J  (t; u, ) dt
(29)
u
This upward radiance is the radiance detected by a sensor on-board a satellite. It is the sum of
of two terms. The first term is the ground leaving radiance attenuated by the column of
atmosphere while the second term is the path radiance, which is due to the radiance
scattered by the atmosphere into the sensor, without reaching the ground.
0
Substituting Eq. (28) into Eq. (29), we have
() F, s u s    (1 / u 1 / u ) E  diffuse
s 
L  (0; u, ) 
e 0


F, s u s

 0 et / u

J  (t ; u, ) dt
 
u
 0
(30)
The apparent top of atmosphere reflectance is then given by
L  (0; u, )
  
 T   path
F, s u s
(31)
where  path is due to the path radiance, and T is a transmission factor for the direct solar
irradiance and the diffuse irradiance. In remote sensing of the ground or ocean surface, the
top-of-atmosphere reflectance TOA is measured by the sensor. However, the actual ground or
water surface reflectance  is the parameter of interest. The reflectance to be retrieved is
given by
TOA   path
(32)

T
The procedure of retrieving  from TOA is known as atmospheric correction. It involves the
calculation of the path and the transmission factors for every pixel in the remotely sensed
image.
Single Scattering Approximation
We will now consider only the rays that undergo at most one scattering event in the
atmosphere when calculating the radiances.
The scattering source function (from Eq. 21) is given by,
 ) 2 1
(33)
J  (; u, ) 
  P(u,  u ' , ' ; ) L (; u ' , ' ) du ' d
4 0 1
For single scattering events, we can use the incident solar radiance (Eq. 19) in the integrand,
() 2 1
 / us
(34)
J  (; u, ) 
F,s (u'u s )(' s  ) du' d
  P(u,  u ' , ' ; ) e
4 0 1
which can be integrated to get,
F,s e   / us ()
(35)
J  (; u, ) 
P(u, u s ,  s  ; )
4
Downward Radiance
Assuming that the phase function and the single scattering albedo is the same at all height in
the atmosphere, we can integrate Eq. 24 to get the downward radiance at the ground,
L  ( 0 ; u, )  e   0 / u s Fs (u  u s )(   s  )

F, s e
0 / u
P(u, u s ,  s
4u
(36)
 )  0
 t (1 / u 1 / u s )
dt
 e
0
i.e.
L  ( 0 ; u, )  e   0 / u s F, s (u  u s )(   s )
F, s   u s    / u

 e 0 s  e  0 / u P(u, ;u s ,  s - )

4  u s  u 


(37)
Upward radiance
The downward irradiance at the ground is

2 0 0
E  ( 0 )  e 0 / us F,s u s     e (0 t ) / u J  (t; u, ) dt du d
0 1 0
(25)
Using the single scattering source function (Eq. 35), we can integrate the diffuse irradiance
part of Eq. (25)
E   diffuse

2 0 0
    e ( 0 t ) / u J  (t ; u, ) dt du d
(38)
0 1 0
F,s  2 0 0 t (1/ u 1/ u )  / u
s e 0 P (u , ;u ,   ) dt du d

   e
s s
4 0 1 
F,s  2  uu s

1  e 0 (1/ u 1/ us ) e 0 / u P(u, ;u s ,  s  ) du d
 
4 0  u s  u


In general, numerical integration is needed to evaluate the integral in Eq. (38). If  << 1, then
we can use the approximation, exp(–x) = 1 – x, to get
E  diffuse 
F,s   0 2 
  P(u, ;u s ,  s  ) du d
4
0 
(39)
If the phase function is approximately isotropic, then the integral of the phase function in Eq.
(39) has a value of 2, then we have
E   diffuse 
1
F,s  0
2
(40)
The upward radiance at the ground is
 / u
E  ( 0 ) F,s u s e 0 s E  diffuse
L ( 0 , u, ) 





(41)

At the top of atmosphere, the upward radiance is given by
0 e t / u
L  (0; u, )  e 0 / u L  ( 0 ; u, )  
(42)
J  (t; u, ) dt
u
The ground leaving radiance is given by Eq. (41), and the path radiance will be
determined using the single scattering source function.
0

F,s u s 
e t / u
J  (t; u, ) dt 
P(u, ;u s ,  s  ) 1  e 0 (1 / u 1 / us )
u
4(u  u s )
0
0
L, path  

(43)
The apparent top of atmosphere reflectance is given by
  
L  (0; u, )
 T   path
F, s u s
In the single scattering approximation, in the limit 0 << 1,
 0
 path 
P(u, ;u s ,  s  )
4uu s
and
    0
T  1  0  
 u s  2u s
(44)
(45)
(46)
Multiple Scattering
Multiple scattering effects can be included using the successive order of scattering
approximation. In this method, the radiance is evaluated as a series
N
L (, u, )  L (0) (, u,    L (n) (, u, )
(47)
n1
where L (0) (, u ,  is the direct radiance from the sun without undergoing scattering in the
atmosphere, and L ( n) (, u , ) is the higher order radiance that has undergone n scattering
events.
The n-th order scattering source function is evaluated by integrating the (n-1) order radiance.
(48)
 2 1
( n 1)
J  (n) (t , u, ) 
(t , u ' , ' ) du ' d'
  P(u, ; u ' , ' ) L
 0 1
The n-th order radiance is evaluated by using the n-th order source function in the formal
solution of the RTE (Eq. 4 and Eq. 5). Thus, starting from the solution for the zero-th order
radiance, the higher order solutions can be successively obtained.