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 ( $\alpha$ ), 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,^[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

↑ "Probing Titan's lower atmosphere with acousto-optic tuning". Icarus. 163: 150–163. Bibcode:2003Icar..163..150C. doi:10.1016/S0019-1035(03)00075-7.
↑ "Martian phase function: Modeling the visible to near-infrared surface photometric function using HST-WFPC2 data". Icarus. 184: 401–423. Bibcode:2006Icar..184..401S. doi:10.1016/j.icarus.2006.05.006.
↑ "The use of the Minnaert correction for land‐cover classification in mountainous terrain". International Journal of Remote Sensing. 26: 3831–3851. doi:10.1080/01431160500104194.
↑ http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1941ApJ....93..403M&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf

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