A horizontally extended surface layer of infinite 
thickness consists of scattering particles with mean 
scattering cross-section c, and (constant) particle 
density p,. Now consider a layer of thickness dZ in 
depth Z within the medium. This layer is illuminated 
by light with irradiance E, under angle i (see figure 2). 
  
  
  
  
Figure 2: Derivation of the Lommel-Seeliger law 
The probabilities for a photon to enter the layer and 
to emerge again towards the sensor can be described 
by two exponential attenuation factors: 
-T zT 
RUZ Sea (aquo 
The optical depth T is a means to describe how far 
incident radiation will penetrate into the medium. It is 
defined as 
© 
Ts J p, dZ = dT--p,c,dZ (4) 
Z zz 
The amount of radiation that 1) travels through the 
surface to depth Z, interacting with the particles 
within the layer, and 2) is scattered towards the sensor 
after attenuation by the particles lying above the layer 
is given by 
dL(Z) = -E,R - p(Z)- pz) d. (5) 
The factor R denotes the amount of light that is 
scattered towards the direction of the sensor by a 
single particle. R depends on the particle's scattering 
behaviour; if the particles within the surface layer are 
considered to be isotropic scatterers, i.e. if they scatter 
incident radiation equally in all directions, R can be 
646 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
replaced by the so-called single-particle scattering 
albedo w; the factor 1/4m (is introduced for 
normalization purposes, [Hapke 1993]. The total 
amount of radiance that is scattered towards the 
sensor is obtained by integration over all layers: 
© 
woipost © = dx 
L(LOZE i997 —— 
8 4T Jo cose (6) 
= E N * Cos! 
° 4m cosi «cose 
Substituting equation (6) into equation (1) leads to 
the Lommel-Seeliger law: 
W J cosi (7) 
4T cosi«cose 
r(i,e) - 
  
Figure 3: The Lommel-Seeliger law (top) in 
comparison to the Lambert law (bottom) 
This photometric function is a good description of the 
light scattering behaviour of low-albedo surfaces, 
[McEwen 1991, Hapke 1993], in contrast to the 
Lambert law, which is more valid for bright surfaces. 
Some of the most sophisticated reflectance models 
    
   
     
    
    
  
    
   
   
   
    
     
    
    
   
   
  
    
     
    
    
   
   
    
  
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