PC4262: Remote Sensing
Radiometric Quantities
Dr. S. C. Liew, Jan. 2002
Radiant Energy Q [J]
The amount of energy carried by electromagnetic radiation.
Flux 
Area A
dQ
[W]
dt
The time rate at which radiant energy passes through a surface.
Radiant Flux  
d
[W m-2]
dA
Radiant flux per unit area intercepted by a surface at a spatial location.
Flux Density F 
d
[W m-2]
dA
Flux density incident onto a surface.
Irradiance E 
d
[W m-2]
dA
Flux density emerging out of a surface.
Exitance, Emittance M 
d 2
cos  dAd
Flux density along a given direction s passing through a unit area of a surface normal to the
direction of propagation s per unit solid angle. If the normal of the surface area of interest (dA)
Radiance L 
makes an angle  with the direction of propagation s, then the projected area |cos|dA is used in
the calculation.
The radiance field L(x, y, z, , ) is a function of the spatial location r = (x, y, z) (the cartesian
coordinates of the spatial location) and the direction of radiant flux propagation s = (, ) (the
angular coordinates of the propagation vector).
Relation between radiance and irradiance
Emittence: M 

2 
 L(, ) cos  d    L(, ) cos  sin  d d
0 
outer half
hemisphere
Irradiance: E 
2 
 L(, ) cos  d    L(, ) cos  sin  d d
0 

inner half
hemisphere
Note: In the above equations, the outward normal of the surface of interest is taken as the z-axis
(the upward direction if the surface is horizontal). For emittence calculation, all the flux emerging
outwards is considered, i.e. all flux with  ranging from 0 (normal to surface, along z-direction)
to /2 (parallel to surface). For irradiance calculation, the flux going into the surface has 
ranging from /2 (parallel to surface) to  (along the –z axis, into the surface).
Example: Irradiance of a parallel beam
z
s
y
s
x
The radiance of a parallel beam of sunlight illuminating a horizontal plane can be modeled by,
L  F(cos   cos  s )(   s )  F(u  u s )(   s )
where F is the flux density of the solar beam (W m-2 through a surface normal to the beam), s is
the solar zenith angle, and s is the solar azimuthal angle.
The irradiance on a horizontal plane is:
2 
2 0
0 

0 1
E    L(, ) cos  sin  d d    L(, ) u du d  Fu s ; where u s  cos  s
Example: Irradiance of an isotropic diffused radiance distribution
Suppose that the radiance field illuminating a horizontal surface is isotropic, i.e. independent of
direction of propagation, i.e. L(, ) = constant. The irradiance on the surface is:
2 
2 0
0 

0 1
E    L cos  sin  d d  L   u du d  L
Example: A non-isotropic radiance distribution
Consider a surface emitting radiation with a radiance field:
L(, )  L0 cos 
The emittance of the surface is

2 
2 1
0 0
0 0
M    L(, ) cos  sin  d d  L0   u 2 du d 
2
L0

d
[W sr-1]
d
Radiant flux emitted from a point source per unit solid angle
Intensity of a Point Source I 
Spectral Quantities
So far, the radiant quantities considered above are broadband quantities, i.e. quantities integrated
over a wide wavelength band. The distribution of a given radiant quantity within a wavelength
band is characterized by the respective spectral quantities. If X is a wideband radiant quantity,
then the corresponding spectral quantity is often denoted by X. They are related by
2
X ( to  2 )   X  d
1
Thus, we have the following spectral radiant quantities:
Spectral radiant energy, Q [J m-1]
Spectral radiant flux,  [W m-1]
Spectral flux density, F [W m-2 m-1]
Spectral irradiance, E [W m-2 m-1]
Spectral exitance, Spectral emittence, M [W m-2 m-1]
Spectral radiance, L [W m-2 sr-1 m-1]
A simplified geometry of satellite imaging
Detector element area, a
Focal length,
f
Aperture diameter, D
Altitude, H
Ground Pixel Area, A
(Image of detector element
formed on the ground)
In this simple imaging geometry, a detector element of area a forms an image on the ground, with
a ground pixel area
A  H 2a / f 2 .
The optical axis is oriented along the vertical direction. Suppose that the ground leaving radiance
is L0. The flux from the ground pixel area reaching the telescope aperture (neglecting effects of
the atmosphere) is
  L0 A ,  
D 2
4H 2
where  is the solid angle subtended by the aperture on the ground. Assuming a lossless optical
system, this flux is transported to the detector element. If the effective exposure time for the pixel
is t, the radiant energy received by the detector element is
Q  t  L0
H 2 a D 2
t
D 2 a
L0 t .
f 2 4H 2
4f 2
From this equation, it is seen that the possible ways to increase the signal level are:
1. increase a, i.e. use a large detector;
2. increase D, i.e. use a large aperture telescope,
3. decrease f, i.e. use a short focal length imaging optics,
4. increase t, i.e. increase the exposure time.
Each of these steps will affect the performance in other aspects. Increasing a and decreasing f will
increase the ground pixel size, resulting in a poorer ground resolution. The exposure time is
limited by the motion of the satellite in orbit. Increasing the aperture will increase the collection
area, but it will also add to the weight of the instrument to be carried by the spacecraft.
Sensor Response