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.