Minnaert function

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 (${\displaystyle \alpha }$), the photometric latitude (${\displaystyle \varphi }$) and the photometric longitude (${\displaystyle \lambda }$).

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

where ${\displaystyle A_{M}}$ is the Minnaert albedo, ${\displaystyle k}$ is an empirical parameter, ${\displaystyle I}$ is the scattered radiance in the direction ${\displaystyle (\alpha ,\varphi ,\lambda )}$, ${\displaystyle \pi F}$ is the incident radiance, and

${\displaystyle \mu _{0}=\cos \varphi ~\cos(\alpha -\lambda )~;~~\mu =\cos \varphi ~\cos \lambda ~.}$

The phase angle is the angle between the light source and the observer with the object as the center.

The assumptions made are:

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

Minnaert's contribution is the introduction of the parameter ${\displaystyle k}$, 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.