Report

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