Minnaert function

The Minnaert function is a <a href="/facts/Photometry_(astronomy)/fnV9JdSa">photometric</a> function used to interpret <a href="/facts/Astronomical/0ArJVM1p">astronomical</a> observations and <a href="/facts/Remote_sensing/pj2crmGs">remote sensing</a> data for the <a href="/facts/Earth/HLLmDfj0">Earth</a>. It was named after the astronomer <a href="/facts/Marcel_Minnaert/CBCa7TzK">Marcel Minnaert</a>. This function expresses the radiance factor (RADF) as a function the <a href="/facts/Phase_angle_(astronomy)/rPhG45gN">phase angle</a> (
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
), the photometric latitude (
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
) and the photometric longitude (
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
).

RADF
        
        =
        
          
            I
            F
          
        
        =
        π
         
        
          A
          
            M
          
        
         
        
          μ
          
            0
          
          
            k
          
        
         
        
          μ
          
            k
            −
            1
          
        
      
    
    {\displaystyle {\text{RADF}}={\frac {I}{F}}=\pi ~A_{M}~\mu _{0}^{k}~\mu ^{k-1}}

where 
 
 
 
 
 A
 
 M
 
 
 
 
 {\displaystyle A_{M}}
 
 is the Minnaert <a href="/facts/Albedo/TJDRdKEB">albedo</a>, 
 
 
 
 k
 
 
 {\displaystyle k}
 
 is an empirical parameter, 
 
 
 
 I
 
 
 {\displaystyle I}
 
 is the scattered <a href="/facts/Radiance/5kS1FhpD">radiance</a> in the direction 
 
 
 
 (
 α
 ,
 φ
 ,
 λ
 )
 
 
 {\displaystyle (\alpha ,\varphi ,\lambda )}
 
, 
 
 
 
 π
 F
 
 
 {\displaystyle \pi F}
 
 is the incident radiance, and

μ
          
            0
          
        
        =
        cos
        ⁡
        φ
         
        cos
        ⁡
        (
        α
        −
        λ
        )
         
        ;
         
         
        μ
        =
        cos
        ⁡
        φ
         
        cos
        ⁡
        λ
         
        .
      
    
    {\displaystyle \mu _{0}=\cos \varphi ~\cos(\alpha -\lambda )~;~~\mu =\cos \varphi ~\cos \lambda ~.}

The phase angle is the angle between the light source and the observer with the object as the center.
The assumptions made are:

<ul><li>the surface is illuminated by a distant point source.</li>
<li>the surface is isotropic and flat.</li></ul>
Minnaert's contribution is the introduction of the parameter 
 
 
 
 k
 
 
 {\displaystyle k}
 
, having a value between 0 and 1, originally for a better interpretation of observations of the <a href="/facts/Moon/zKHqEMPb">Moon</a>. In remote sensing the use of this function is referred to as Minnaert topographic correction, a necessity when interpreting images of rough terrain.

Minnaert function open-in-new

Minnaert function