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 ($\phi$) 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 $(\alpha ,\phi ,\lambda )$, $\pi F$ is the incident radiance, and

${\mu }_0=\cos \phi \cos (\alpha -\lambda );\mu =\cos \phi \cos \lambda .$

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.