ON A POSSIBLE WAY FORWARD FOR THE
GALACTIC RADIATION CORRECTION
Joe Tenerelli
SMOS Quality Working Group #15
ESA ESRIN
6-8 October 2014
Cosmic+Galactic
(up to 8 K after scattering)
(Tx+Ty)/2
Faraday rotation
atm absorption
(trans = 0.995)
atm emission
(2 K)
scattering by rough
surface
atm absorption
(trans = 0.995)
atm emission
(2 K)
Galactic radiation
incident from all
directions
specular (100 K) +rough (10 K)
surface emission
(Tx+Ty)/2 bias +1 K
SSS bias -2 psu
SCATTERING OF GALACTIC RADIATION
orientation angle
GEOMETRICAL OPTICS CROSS SECTIONS
Take high-frequency limit of the Kirchhoff approximation for scattering cross sections:
Obtain expression for cross sections in terms of slope probability distribution:
Assume Gaussian slope probability distribution. Then fit the slope variance to
the data:
GEOMETRICAL OPTICS CROSS SECTIONS
Take high-frequency limit of the Kirchhoff approximation for scattering cross sections:
• Note that the appropriateness of this high frequency limit becomes more
problematic as the incidence angle increases and the vertical component of
the wavenumber difference vector becomes small.
• The GO model differs greatly from Kirchhoff beyond about 15 deg from the
specular direction. Yet the contribution to reflectivity is significant beyond this
angle:
FITTING THE GEOMETRICAL OPTICS MODEL
Standard GO model:
FITTING THE GEOMETRICAL OPTICS MODEL
GO with Gaussian slope PDF:
FITTING THE GEOMETRICAL OPTICS MODEL
• GO with an adjusted Gaussian slope PDF can approach the data fairly well (to
within about 10% or so):
SMOS residuals
Example
geometrical
optics trial
solutions
FITTING THE GEOMETRICAL OPTICS MODEL
• At high incidence angles, optimal mean square slopes for descending
passes are lower than those for ascending passes. Discrepancy is largest
at 50o incidence angle (where the GO fit is most problematic).
• At low incidence angles, the opposite is true.
• Change occurs somewhere between 20o and 40o incidence angle.
REMAINING BIAS PROBLEM FOR SMOS
50o; Descending passes; 3-6 m/s; (Tx+Ty)/2
SMOS
GO MODEL
GO MODEL – SMOS RESIDUAL
Model prediction of
scattered brightness
is too low either side
of galactic plane
REMAINING BIAS PROBLEM FOR AQUARIUS (VERSION 261):
AQUARIUS v200
SMOS Asc Fit
AQUARIUS v261
SMOS Desc Fit
BACK TO THE BASIC EQUATION
The brightness temperature at polarization p of the scattered radiation at
the surface (before integration over any antenna pattern) is
BACK TO THE BASIC EQUATION
The brightness temperature at polarization p of the scattered radiation at
the surface (before integration over any antenna pattern) is
We know the sky brightness that appears in the integrand.
BACK TO THE BASIC EQUATION
The brightness temperature at polarization p of the scattered radiation at
the surface (before integration over any antenna pattern) is
We do not know the cross sections. We wish to find them.
BACK TO THE BASIC EQUATION
The brightness temperature at polarization p of the scattered radiation at
the surface (before integration over any antenna pattern) is
Although some highly localized strong sources exist, for practical
purposes, the sky brightness may be assumed to be a smooth function
and this integral is a Fredholm integral of the first kind.
BACK TO THE BASIC EQUATION
The brightness temperature at polarization p of the scattered radiation at
the surface (before integration over any antenna pattern) is
The inverse problem of finding the cross sections is notoriously ill-posed.
A simple way to see this is to note that if we have some solution for the
cross sections, we can obtain another very different solution that
produces brightness temperatures that are arbitrarily close to the correct
values by adding to the cross sections a sinusoidal function of sufficiently
high spatial frequency.
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
Consider once again the general expression for the contribution of the scattered
galactic radiation to the antenna temperature Stokes vector (neglecting Faraday
rotation):
The GO expression for the cross sections is
We expand the slope PDF is a series in terms of slope magnitude and
azimuth relative to the incidence plane (not azimuth relative to the wind
direction):
radial basis functions
azimuthal functions
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
Consider once again the general expression for the contribution of the scattered
galactic radiation to the antenna temperature Stokes vector (neglecting Faraday
rotation):
The GO expression for the cross sections is
We expand the slope PDF is a series in terms of slope magnitude and
azimuth relative to the incidence plane (not azimuth relative to the wind
direction):
Surface slope magnitude
Azimuth relative to incidence plane
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
This expansion in azimuth may seem strange, but we are implicitly
supposing that this ‘slope PDF’ actually incorporates a correction factor for
any errors in the model (e.g. associated with diffraction effects), and that
these errors may depend upon the scattering geometry (including direction
of the incident wave with respect to the incidence plane).
We assume that the surface is isotropic and so do not consider any effect of
wind direction on the cross sections. Moreover, we only consider a small
portion of the entire Mueller scattering matrix:
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
This expansion in azimuth may seem strange, but we are implicitly
supposing that this ‘slope PDF’ actually incorporates a correction factor for
any errors in the model (e.g. associated with diffraction effects), and that
these errors may depend upon the scattering geometry (including direction
of the incident wave with respect to the incidence plane).
Yueh (Radio Science 1994) showed that these four cross sections inside
the red box (but not all 16) must be reflection symmetric about some
vertical plane if the surface is reflection symmetric about that plane.
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
This expansion in azimuth may seem strange, but we are implicitly
supposing that this ‘slope PDF’ actually incorporates a correction factor for
any errors in the model (e.g. associated with diffraction effects), and that
these errors may depend upon the scattering geometry (including direction
of the incident wave with respect to the incidence plane).
We assume that the surface is isotropic and therefore reflection symmetric
about every vertical plane, including the incidence plane.
A consequence of this assumption of surface isotropy is that the azimuthal
variation above must be an even function of the azimuth with respect to the
incidence plane).
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
This expansion in azimuth may seem strange, but we are implicitly
supposing that this ‘slope PDF’ actually incorporates a correction factor for
any errors in the model (e.g. associated with diffraction effects), and that
these errors may depend upon the scattering geometry (including direction
of the incident wave with respect to the incidence plane).
There are many possible choices for the radial basis functions (e.g. Bessel
functions, Chebyshev polynomials, Zernike polynomials, shifted Gaussians
of varying width etc.) but for this example I will choose very simple basis
functions that are highly local in surface slope. These simple functions
provide some intuition for what is going on in the method.
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
Chapeau basis functions in slope:
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
Chapeau basis functions in slope:
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
Chapeau basis functions in slope:
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
Chapeau basis functions in slope:
RESULTING CROSS SECTIONS
Chapeau basis functions in slope:
RESULTING CROSS SECTIONS
Chapeau basis functions in slope:
RESULTING CROSS SECTIONS
RESULTING CROSS SECTIONS
RESULTING CROSS SECTIONS
RESULTING CROSS SECTIONS
RESULTING CROSS SECTIONS
RESULTING CROSS SECTIONS
RESULTING CROSS SECTIONS
RESULTING CROSS SECTIONS
RESULTING CROSS SECTIONS
RESULTING CROSS SECTIONS
RESULTING CROSS SECTIONS
RESULTING CROSS SECTIONS
RESULTING CROSS SECTIONS
RESULTING CROSS SECTIONS
RESULTING CROSS SECTIONS
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
ADJUSTING THE GO MODEL: SERIES
EXPANSION METHOD
The orientation angle is important for both Aquarius
and SMOS and is associated with different levels of
scattered radiation for descending and ascending
passes at a given boresight specular direction in the
sky:
SETTING UP THE LEAST SQUARES PROBLEM
Build a large matrix B whose columns contain
the pre-integrated basis function sky maps.
SETTING UP THE LEAST SQUARES PROBLEM
We seek this vector of
coefficients for the series
expansion of the slope
PDF.
Pre-integrated
basis functions
Brightness temperatures
of
scattered
galactic
radiation deduced from the
Aquarius data.
SETTING UP THE LEAST SQUARES PROBLEM
BASIC LEAST SQUARES SOLUTION WITH
TIKHONOV REGULARIZATION
We seek a solution to the regularized problem
General Tikhonov:
Seek solutions with small 2-norm:
Phillips 1962, Twomey 1963 (seek
solutions with small second
differences):
BASIC LEAST SQUARES SOLUTION WITH
TIKHONOV REGULARIZATION
Solve for the coefficients:
Once we have the coefficients, we can compute
the scattering cross sections
and the contribution of scattered galactic radiation to the
antenna temperature Stokes vector
BASIC LEAST SQUARES SOLUTION WITH
TIKHONOV REGULARIZATION
Another method involves subtracting a reference model from the
Aquarius data and then solving for the coefficients of a correction for the
error in this reference model:
In this method the total cross sections and scattered radiation brightness
temperatures become
AN EXAMPLE
AQUARIUS BEAM 2; ASC PASSES; WIND SPEEDS = 6-8 m/s
Aquarius v261
No Reference
Tikhonov coefficient = 0.0 (no regularization)
SMOS Asc GO Ref
SMOS Desc GO Ref
AN EXAMPLE
AQUARIUS BEAM 2; ASC PASSES; WIND SPEEDS = 6-8 m/s
No Reference
Tikhonov coefficient = 0.0 (no regularization)
SMOS Asc GO Ref
SMOS Desc GO Ref
AN EXAMPLE
AQUARIUS BEAM 2; ASC PASSES; WIND SPEEDS = 6-8 m/s
No Reference
Tikhonov coefficient = 0.0 (no regularization)
SMOS Asc GO Ref
SMOS Desc GO Ref
AN EXAMPLE
AQUARIUS BEAM 2; ASC PASSES; WIND SPEEDS = 6-8 m/s
No Reference
Tikhonov coefficient = 0.0 (no regularization)
SMOS Asc GO Ref
SMOS Desc GO Ref
AN EXAMPLE
AQUARIUS BEAM 2; ASC PASSES; WIND SPEEDS = 6-8 m/s
No Reference
Tikhonov coefficient = 0.0 (no regularization)
SMOS Asc GO Ref
SMOS Desc GO Ref
IMPACT OF REGULARIZATION
AQUARIUS BEAM 2; ASC PASSES; WIND SPEEDS = 6-8 m/s
No Reference
Tikhonov coefficient = 0.4 (some regularization)
SMOS Asc GO Ref
SMOS Desc GO Ref
IMPACT OF REGULARIZATION
AQUARIUS BEAM 2; ASC PASSES; WIND SPEEDS = 6-8 m/s
No Reference
Tikhonov coefficient = 0.4 (some regularization)
SMOS Asc GO Ref
SMOS Desc GO Ref
IMPACT OF REGULARIZATION
AQUARIUS BEAM 2; ASC PASSES; WIND SPEEDS = 6-8 m/s
The solution coefficients very near the specular
direction (the innermost two coefficients) can be
varied strongly with little impact on the resulting
brightness temperature maps (solid angles very
small there). It seems that for these coefficients the
regularisation parameter plays a strong role.
Tikhonov coefficient = 0.4 (some regularization)
SMOS Asc GO Ref
SMOS Desc GO Ref
IMPACT OF REGULARIZATION
AQUARIUS BEAM 2; ASC PASSES; WIND SPEEDS = 6-8 m/s
No Reference
Tikhonov coefficient = 3.0 (strong regularization)
SMOS Asc GO Ref
SMOS Desc GO Ref
CONCLUSIONS
• We have introduced a methodology by which we can obtain
corrections to the existing theoretical models for the bistatic
scattering of galactic radiation.
• Initial tests with Aquarius data suggest that the method can provide
solutions that match the data resonably well, and possible better
than the existing GO model with an isotropic gaussian slope PDF.
• The method can be adapted to SMOS as well.
REMAINING ISSUES
• Discrete problem is ill-conditioned an can be very sensitive to changes
and biases in the data used for fitting the model.
• Some form of regularization is necessary, but how do we best do this?
Ideally we can introduce constraints that are, as much as possible,
based upon the physics (e.g. smoothness). For the Tikhonov
regularization the solutions depend upon the regularization parameter,
and the L-plot approach for finding this parameter is somewhat arbitrary.
• Choice of basis functions and choice of grid may be important or even
critical. Stretched grid in radius? Shifted Gaussian radial basis
functions? There are many possibilities.
• Can we account for the impact of anisotropy of the surface, or the
impact of swell?
EXTRA SLIDES
SINGULAR VALUES OF THE BASIS MATRIX