libradtran_devel:tests:surface_polarisation:polarization_of_surface []

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====== Polarization of surface ======

Polarization by surface reflection has been included into MYSTIC so that arbitrary reflectance matrices
can be handled.
A program by Mishchenko (~/literature/mishchenko1997.pdf) (freely availabe at
[[http://www.giss.nasa.gov/staff/mmishchenko/brf/]]) has been integrated to compute the reflectance
matrix for rough water surfaces. This program is based on the Fresnel equations, on Cox and Munk
(1954) to describe the wind-speed dependent slope of the waves, and on Tsang et al. (1985) to account
for shadowing effects.

The following equations are from Mishchenko 1997:
{{ :surface_polarisation:reflectance_matrix1.png |}}
{{ :surface_polarisation:reflectance_matrix2.png |}}
{{ :surface_polarisation:reflectance_matrix3.png |}}
{{ :surface_polarisation:reflectance_matrix4.png |}}

[{{ :surface_polarisation:bpdf_tsang.png |Two elements of the polarized reflectance matrix by Mishchenko. R11 corresponds to the BRDF. The element R12 can be related to the polarization in the principle plane.}}]

==== Implementation in MYSTIC ====

<code>
mc_polarisation           # enable polarisation
mc_escape                 # enable local estimate, should always be switched on for radiances 
bpdf_tsang_u10            # enable BPDF and specify wind speed   
mc_surface_reflectalways  # reflect all photons that reach the surface
</code>


libsrc_f/ocean.shadow.f : Code by Mishchenko

libsrc_c/ocean.c (bpdf_tsang) : Wrapper function for Mishchenko's code

libsrc_c/mystic.c (reflection_polarized, reflect): Function is called here each time when a photon hits the surface. BPDF must be used in combination with mc_reflectalways.


==== Check Fresnel equations ====


[{{:surface_polarisation:fresnel_eq.png|Fresnel equations if magnetic permeability equals 0.}}]
As a first test, we checked whether MYSTIC calculates the [[http://de.wikipedia.org/wiki/Fresnelsche_Formeln|Fresnel equations]] correctly.
For this we computed the polarized reflected radiance in the principle plane for the specular reflection, where the incident angle is equal to the reflected angle. This also gives us the [[http://en.wikipedia.org/wiki/Brewster%27s_angle|Brewster angle]] where the polarization parallel to the reference plane is 0. 

With MYSTIC we get the Frenel formula with an additional factor contributed by the waves (see formula of reflectance matrix). Deviding by the wave contribution to the reflection (which of course depends on the wind speed) we get the Fresnel equations. At shallow angles the shadowing function reduces again the reflectance matrix.

[{{ :surface_polarisation:fresnel_test.png?500 |Specular reflection calculated by MYSTIC (no wave and shadow modification of reflectance matrix). The solid lines show the analytical Fresnel formulas and the dashed lines are MYSTIC results. One can clearly see the Brewster angle arctan(m_w/m_a)*180/pi∼53°, refractive index of water m_w=1.33, refractive index of air m_l=1.0.}}]

[{{ :surface_polarisation:fresnel_test_shadow.png?500 |Same as left but with shadowing.}}]



==== Comparison with Cox and Munk BPDF ====

Calculations in the principle plane have been performed for various wind speed and compared to the Cox and Munk BRDF as implemented in libRadtran. One result for a wind speed of 2 m/s is shown below, more wind speeds are shown   {{:surface_polarisation:all_ws.pdf|here}}.
The major difference is that Cox and Munk apparently includes some additional constant reflection term outside the reflection peak which might be the parameterization of the white wave caps that are not included in Mishchenko's function.

[{{ :surface_polarisation:result_ws02.png |Comparison against BRDF by Cox and Munk.}}]


==== Reflection in all directions, Stokes components ====

[{{ :surface_polarisation:tsangi.png?750 |Reflected total intensity I.}}]
[{{ :surface_polarisation:tsangq.png?750 |Reflected Q (parallel - perpendicular polarization).}}]
[{{ :surface_polarisation:tsangu.png?750 |Reflected U (135°-45° polarization).}}]


 ===== Rayleigh scattering atmosphere above surface =====

Tests for a BPDF below a scattering atmosphere (Rayleigh, cloud) are in the directory 
       /home/users/emde/libRadtran_tests/mystic_polarisation/surface_20100511

The results look reasonable after a bug fix in May 2010 (in ESA release the BPDF does not work correctly because of numerical problems which occur for multiple reflections with the surface, when photons hit the surface with small incident angle).
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