Polarization mixing

Example: A sloping, specular surface

The definition of the four Stokes components are, in a fixed basis:

: \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 \operatorname{Re}\left\langle E_v E_h^* \right\rangle \ 2 \operatorname{Im}\left\langle E_v E_h^* \right\rangle \end{array} \right],

where Ev and Eh are the electric field components in the vertical and horizontal directions respectively. The definitions of the 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: \theta is the instrument viewing angle with respect to nadir, \theta_\mathrm{eff} is the viewing angle with respect to the surface normal and \alpha 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:

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

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

:\mathbf{\hat{n}}=(\cos \psi \sin \mu,\sin \psi \cos \mu,\cos \mu),

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

:\theta_\mathrm{eff} = \cos^{-1}(\mathbf{\hat{n}} \cdot \mathbf{\hat{v}}),

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

: \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),

where \mathbf{\hat{j}} is the unit vector defining the y-axis.

The angle, \alpha, 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. | display-authors = 4 | name-list-style = amp

Correcting or removing bad data

To test the calibration of the EMIRAD II radiometer | name-list-style = amp

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:

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

where eh and ev are the emissivities calculated via the Fresnel or similar equations and eU is the emissivity in U—that is, U = e_U T, where T is physical temperature—for the rotated 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 | name-list-style=amp

References