Minnaert function

The Minnaert function is a photometric function used to interpret astronomical observations and remote sensing data for the Earth. 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 ($\lambda$).

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

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, 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.