Minnaert function

The Minnaert function is a photometric function used to interpret astronomical observations ^[1]^[2] and remote sensing data for the Earth.^[3] This function expresses the radiance factor (RADF) as a function the phase angle ( $\alpha$ ) and 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,^[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.

Minnaert function

References