Astronomy: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 ([math]\displaystyle{ \alpha }[/math]), the photometric latitude ([math]\displaystyle{ \varphi }[/math]) and the photometric longitude ([math]\displaystyle{ \lambda }[/math]).
- [math]\displaystyle{ \text{RADF} = \frac{I}{F} = \pi~A_M~\mu_0^k~\mu^{k-1} }[/math]
where [math]\displaystyle{ A_M }[/math] is the Minnaert albedo, [math]\displaystyle{ k }[/math] is an empirical parameter, [math]\displaystyle{ I }[/math] is the scattered radiance in the direction [math]\displaystyle{ (\alpha,\varphi,\lambda) }[/math], [math]\displaystyle{ \pi F }[/math] is the incident radiance, and
- [math]\displaystyle{ \mu_0 = \cos\varphi~\cos(\alpha-\lambda) ~;~~ \mu = \cos\varphi~\cos\lambda ~. }[/math]
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 [math]\displaystyle{ k }[/math], 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
- ↑ 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. doi:10.1016/S0019-1035(03)00075-7. Bibcode: 2003Icar..163..150C.
- ↑ 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. doi:10.1016/j.icarus.2006.05.006. Bibcode: 2006Icar..184..401S.
- ↑ 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.
- ↑ Minnaert, M. (1941). "The reciprocity principle in lunar photometry". The Astrophysical Journal 93: 403. doi:10.1086/144279. Bibcode: 1941ApJ....93..403M. 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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