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.

References

1. Chanover, N.J.; Anderson, C.M.; McKay, C.P.; Rannou, P.; Glenar, D.A.; Hillman, J.J.; Blass, W.E. (2003). "Probing Titan's lower atmosphere with acousto-optic tuning". Icarus. 163 (1): 150–163. Bibcode:2003Icar..163..150C. doi:10.1016/S0019-1035(03)00075-7.
2. Soderblom, J.; Belliii, J.; Hubbard, M.; Wolff, M. (2006). "Martian phase function: Modeling the visible to near-infrared surface photometric function using HST-WFPC2 data". Icarus. 184 (2): 401–423. Bibcode:2006Icar..184..401S. doi:10.1016/j.icarus.2006.05.006.
3. Blesius, L.; Weirich, F. (2005). "The use of the Minnaert correction for land‐cover classification in mountainous terrain". International Journal of Remote Sensing. 26 (17): 3831–3851. doi:10.1080/01431160500104194.
4. Minnaert, M. (1941). "The reciprocity principle in lunar photometry" (PDF). The Astrophysical Journal. 93: 403. Bibcode:1941ApJ....93..403M. doi:10.1086/144279.