Minnaert function explained

The Minnaert function is a photometric function used to interpret astronomical observations^{[1] [2]} and remote sensing data for the Earth.^[3] It was named after the astronomer Marcel Minnaert. This function expresses the radiance factor (RADF) as a function the phase angle (

\alpha ), the photometric latitude (\varphi ) and the photometric longitude (λ ).RADF = (I)/(F) = \pi~A _{M~\mu} ^{k~\mu} ^{0} ^{k-1} where A _{M} is the Minnaert albedo, k is an empirical parameter, I is the scattered radiance in the direction (\alpha,\varphi, λ ) , \pi F is the incident radiance, and\mu _{0} = \cos\varphi~\cos(\alpha -λ ) ~;~~ \mu = \cos\varphi~\cos λ ~. 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.

References

1. 10.1016/S0019-1035(03)00075-7 . 2003Icar..163..150C . 163 . Probing Titan's lower atmosphere with acousto-optic tuning . Icarus . 2003 . 150–163. Chanover . N.J. . Anderson . C.M. . McKay . C.P. . Rannou . P. . Glenar . D.A. . Hillman . J.J. . Blass . W.E. . 1 .
2. 10.1016/j.icarus.2006.05.006 . 2006Icar..184..401S . 184 . Martian phase function: Modeling the visible to near-infrared surface photometric function using HST-WFPC2 data . Icarus . 2006 . 401–423. Soderblom . J. . Belliii . J. . Hubbard . M. . Wolff . M. . 2 .
3. 10.1080/01431160500104194 . 26 . The use of the Minnaert correction for land-cover classification in mountainous terrain . International Journal of Remote Sensing . 2005 . 3831–3851. Blesius . L. . Weirich . F. . 17 . 2005IJRS...26.3831B . 129750287 .
4. The reciprocity principle in lunar photometry. 1941ApJ....93..403M. Minnaert. M.. The Astrophysical Journal. 1941. 93. 403. 10.1086/144279.