PC4262 Remote Sensing (Semester II 2002/2003)
Assignment 2
Question 1.
If the atmosphere can be considered as an ideal gas, the pressure P(z), temperature T(z)
and molecular number density N(z) are related by the ideal gas equation
P( z )
N ( z) 
kT ( z )
where k is the Boltzmann’s constant. Using the hydrostatic equation, it can be shown
that
dP ( z )
  MgN ( z )
dz
where M is the average mass of a gas molecule in the atmosphere, and g is the
acceleration due to gravity.
(a) From the two equations above, show that the atmospheric pressure varies with the
altitude z according to
 Mg z dz ' 

P( z )  P0 exp  


k
T
(
z
'
)
0


(b) In the troposphere, the temperature falls off approximately linearly with the
altitude, according to the equation
z 

T ( z )  T0 1  
 H
where T0 is the temperature at the ground level (z = 0) and H is a constant that has the
dimension of a length whose value is greater than the height of the tropopause (i.e.
upper limit of troposphere). Use this relation in the above equation for P(z) to obtain
the solution
 MgH  H  
P( z )  P0 exp  
ln 
 
 kT0  H  z  
Question 2.
The angular scattering cross-section for Rayleigh scattering in the atmosphere for
unpolarized light of wavelength  can be expressed as:
d r (; )
2 

[n()  1]2 (1  cos 2 ) [m 2 sr -1 ]
2
4
d
N 
where N is the molecular number density (number of molecules per unit volume) of
the atmosphere, n is the refractive index of the atmosphere and  is the scattering
angle.
(a) Derive expressions for
(i) Scattering cross-section  r ()
(ii) Scattering phase function Pr (; )
(iii) Angular scattering coefficient
(iv) Scattering coefficient  r ()
d r (; )
d
(b) The refractive index of air varies with the wavelength and number density
according to the relation
n()  1  b1 N (1  b2 2  b34 )
where b1 , b2 , b3 are constants. Use this relation together with the results of the
previous problem to derive an expression for the optical thickness of atmosphere due
to Rayleigh scattering at altitude z,

 r ( ; z )    r ( ; z ' ) d z '
z
Show that the expression for the Rayleigh scattering optical thickness can be reduced
to the form:
 P( z )  a1  a2 a3 
 1 
 r (; z )  
 
4

4 
 P0    
where
32 2 b12
a1 
3
 P0 

 , a2  2b2 , a3  2b3  b2 2
 Mg 
When the wavelength  is in micrometers, the values of these coefficients are
a1  0.008569, a2  0.0113, a3  0.00013 .
Generate a graph of the Rayleigh scattering optical thickness  r () at sea level (z=0)
as a function of the wavelength .
Question 3.
Refer to the satellite sensor described in Question 3 of Problem set 1.
(a) Calculate the radiance and flux collected by channel 2 of the sensor, assuming that
the ground is dark and only molecular scattering (Rayleigh scattering) as described in
Question 2 above is significant.
(b) Repeat (a), but now the ground reflectance is    .