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should be m dual variables

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In [[optics]], '''polarization mixing''' refers to changes in the relative strengths of the [[Stokes parameters]] caused by [[reflection (physics)|reflection]] or [[scattering]]—see [[vector radiative transfer]]--or by changes in the radial orientation of the detector.

==Example: A sloping, specular surface==

[[Image:U geo.png|thumb|upright=2|center|Geometry of a polarimetric detector relative to a sloping surface.<ref name="smos_final"/>]]

The definition of the four Stokes components
are, in a fixed [[basis (linear algebra)|basis]]:

:<math>
\left [
\begin{array}{c}
I \\ Q \\ U \\ V
\end{array}
\right ]
=
\left [
\begin{array}{c}
|E_v|^2 + |E_h|^2 \\
|E_v|^2 - |E_h|^2 \\
2 \mathrm{Re}<E_v E_h^*> \\
2 \mathrm{Im}<E_v E_h^*>
\end{array}
\right ],
</math>

where ''E''v and ''E''h are the [[electric field]] components
in the vertical and horizontal directions respectively.
The definitions of the [[basis (linear algebra)|coordinate bases]] are
arbitrary and depend on the orientation of the instrument.
In the case of the [[Fresnel equations]], the bases are defined in terms
of the surface, with the horizontal being parallel to
the surface and the vertical in a plane [[perpendicular]] to
the surface.

When the bases are rotated by 45 degrees around the viewing axis,
the definition of the third Stokes component becomes equivalent
to that of the second, that is the difference in field intensity
between the horizontal and vertical polarizations.
Thus, if the instrument is rotated out of plane from the
surface upon which it is looking, this will give rise to a
signal. The geometry is illustrated in the above figure:
<math>\theta</math> is the instrument viewing angle with respect
to nadir, <math>\theta_{\mathrm{eff}}</math> is
the viewing angle with respect to the surface normal
and <math>\alpha</math> is the angle
between the polarisation axes defined by the instrument
and that defined by the Fresnel equations, i.e., the surface.

Ideally, in a [[polarimetric]] [[radiometer]], especially a satellite mounted one, the polarisation axes are
aligned with the Earth's surface, therefore we define the instrument viewing direction using the following vector:

:<math>
\mathbf{\hat{v}}=(\sin \theta, ~0, ~\cos \theta).
</math>

We define the slope of the surface in terms of the normal vector,
<math>\mathbf{\hat{n}}</math>, which can be calculated in a number of ways.
Using angular slope and azimuth, it becomes:

:<math>
\mathbf{\hat{n}}=(\cos \psi \sin \mu,~\sin \psi \cos \mu,~\cos \mu),
</math>

where <math>\mu</math> is the slope and <math>\psi</math> is the azimuth relative
to the instrument view. The effective viewing angle can be
calculated via a dot product between the two vectors:

:<math>
\theta_{eff}=\cos^{-1}(\mathbf{\hat{n}} \cdot \mathbf{\hat{v}}),
</math>

from which we compute the reflection coefficients,
while the angle of the polarisation plane can be calculated
with cross products:

:<math>
\alpha=\mathrm{sgn}(\mathbf{\hat{n}} \cdot \mathbf{\hat{j}})
\cos^{-1}\left ( \frac{\mathbf{\hat{j}} \cdot
\mathbf{\hat{n}} \times \mathbf{\hat{v}}}
{| \mathbf{\hat{n}} \times \mathbf{\hat{v}} |} \right ),
</math>

where <math>\mathbf{\hat{j}}</math> is the unit vector defining the y-axis.
<ref name="smos_final"/>

The angle, <math>\alpha</math>, defines the rotation of the polarization axes between those defined for the Fresnel equations versus those of the detector. It can be used to correct for polarization mixing caused by a rotated detector, or to predict what the detector "sees", especially in the third Stokes component. See [[Stokes parameters#Relation to the polarization ellipse]].

==Application: Aircraft radiometry data==

The [[Pol-Ice 2007]] campaign included measurements over [[sea ice]] and open water from a fully polarimetric, aeroplane-mounted, L-band (1.4 GHz) [[radiometer]].
<ref name="smos_final">
{{cite techreport
| author = G. Heygster, S. Hendricks, L. Kaleschke, N. Maass, P. Mills, D. Stammer, R. T. Tonboe and C. Haas
| title=L-Band Radiometry for Sea-Ice Applications
| institution=Institute of Environmental Physics, University of Bremen
| year=2009
| number=ESA/ESTEC Contract N. 21130/08/NL/EL
}}</ref>
Since the radiometer was fixed to the aircraft, changes in [[aircraft attitude]] are equivalent to changes in surface slope. Moreover, [[emissivity]] over calm water and to a lesser extent, sea ice, can be effectively modelled using the [[Fresnel equations]]. Thus this is an excellent source of data with which to test the ideas discussed in the previous section. In particular, the campaign included both circular and [[zig-zag]]ging overflights which will produce strong mixing in the Stokes parameters.

===Correcting or removing bad data===

[[File:Val ow2a.png|thumb|upright=2|center|Comparison of aircraft radiometry data over water with an emissivity model based on the [[Fresnel equations]].]]

[[File:Val ow2a.lf.png|thumb|upright=1.5|right|Comparison of aircraft radiometry data with an emissivity model based on the [[Fresnel equations]]. All points with significant polarization mixing have been removed.<ref name="Mills_Heygster2011"/>]]

To test the calibration of the EMIRAD II radiometer
<ref name="Skou_etal2007">
{{cite techreport
| author = N. Skou, S. S. Sobjaerg and J. Balling
| title=EMIRAD-2 and its use in the CoSMOS Campaigns
| institution=Electromagnetic Systems Section Danish National Space Center, Technical University of Denmark
| year=2007
| number=ESTEC Contract No. 18924/05/NL/FF
}}</ref>
used in the Pol-Ice campaign, measurements over open water were compared with model results based on the Fresnel equations.
<ref name="Mills_Heygster2011">
{{cite journal
| journal=IEEE Transactions on Geoscience and Remote Sensing
| year=2011
| title=Sea ice emissivity modelling at L-band and application to Pol-Ice campaign field data
| issue=in press
| doi=10.1109/TGRS.2010.2060729
| volume=49
| page=612
| url=http://peteysoft.users.sourceforge.net/smos_ieee.pdf
| last1=Mills
| first1=Peter
| last2=Heygster
| first2=Georg
}}</ref>
The plot above, which compares the measured data with the model, shows that vertically polarized channel is too high, but more importantly in this context, are the smeared points in between the otherwise relatively clean function for measured vertical and horizontal [[brightness temperature]] as a function of [[viewing angle]]. These are the result of polarization mixing caused by changes in the attitude of the aircraft, particularly the [[Flight dynamics|roll angle]].
Since there are plenty of data points, rather than correcting the bad data, we simply exclude points for which the angle, <math>\alpha</math>, is too large. The result is shown at right.

===Predicting U===

Many of the [[radiance]] measurements over sea ice included large signals in the third Stoke component, ''U''. It turns out that these can be predicted to fairly high accuracy simply from the aircraft attitude. We use the following model for emissivity in ''U'':

:<math>
e_U=\sqrt{e_v^2 - e_h^2} \sin (2 \alpha)
</math>

where ''e''h and ''e''v are the emissivities calculated
via the Fresnel or similar equations and
''e''U is the emissivity in ''U''--that is, <math>U = e_U T</math>, where
''T'' is physical temperature—for the roated polarization axes.
The plot below shows the dependence on surface-slope and [[azimuth]] angle for a [[refractive index]] of 2 (a common value for sea ice
<ref name="Vant_etal1978">
{{cite journal
| author=M. R. Vant, R. O. Ramseier and V. Makios
| journal=Journal of Applied Physics
| year=1978
| title=The complex-dielectric constant of sea ice at frequencies in the range 0.1-4.0 GHz
| volume=49
| pages=1246–1280
| doi = 10.1063/1.325018
| issue=3
}}</ref>)
and a nominal instrument pointing-angle of 45 degrees. Using the same model, we can simulate the ''U''-component of the Stokes vector for the radiometer.

[[File:U dep.png|thumb|upright=2|center|Dependence of ''e''U on surface-slope and azimuth-angle for a [[refractive index]] of 2 and a nominal instrument pointing-angle of 45 degrees.
<ref name="smos_final"/>]]
[[File:Ugotit.png|thumb|upright=2|center|Modelled ''U'' versus Pol-Ice field data for a circular overflight over sea ice.<ref name="smos_final"/>]]

==References==

{{reflist}}

==See also==

*[[Stokes parameters]]

[[Category:Optics]]
[[Category:Radiometry]]

Weak duality - Revision history

en>Zfeinst: should be m dual variables