Minnaert function

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The Minnaert function is a photometric function used to interpret astronomical observations [1][2] and remote sensing data for the Earth.[3] This function expresses the radiance factor (RADF) as a function the phase angle (\alpha) and the photometric latitude (\varphi) and the photometric longitude (\lambda).

Failed to parse (unknown function "\text"): \text{RADF} = \frac{I}{F} = \pi~A_M~\mu_0^k~\mu^{k-1}

where A_M is the Minnaert albedo, k is an empirical parameter, I is the scattered radiance in the direction (\alpha,\varphi,\lambda), \pi F is the incident radiance, and

   \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:

  • the surface is illuminated by a distant point source.
  • the surface is isotropic and flat.

Minnaert's contribution is the introduction of the parameter k, having a value between 0 and 1,[4] originally for a better interpretation of observations of the Moon. In remote sensing the use of this function is referred to as Minnaert topographic correction, a necessity when interpreting images of rough terrain.


  1. http://dx.doi.org/10.1016/S0019-1035(03)00075-7
  2. http://dx.doi.org/10.1016/j.icarus.2006.05.006
  3. http://dx.doi.org/10.1080/01431160500104194
  4. http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1941ApJ....93..403M&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf