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New Reconstruction Algorithms for the Improvement of SMOS L1c Images: Preliminary Results
[16-Apr-13] Turiel, A. and Gonzalez, V.
Presented at the 2013 SMOS-Aquarius Science Workshop
Due to its interferometric design, the direct measurements by SMOS payload, MIRAS, are the visibilities of the signal defined on an hexagonal grid. The visibility field can be expressed as the Fourier transform of brightness temperature in real space modulated by the antenna gain pattern. Retrieving brightness temperatures (the actual physical variable of interest) from visibilities requires to implement an appropriate reconstruction algorithm. Different approaches to this reconstruction algorithm have been proposed so far, trying to improve the quality of the signal and to diminish side-lobes and other effects associated to incorrect spatial phases. One important difficulty when reconstructing the signal is that visibilities are not evenly sampled by MIRAS on a translational invariant cell but on a star-shaped subset, so no tiling can be defined and hence any inverse Fourier transform will require to introduce zero coefficients at high frequencies. Those lacking coefficients may generate Gibbs-like phenomena and spread side-lobes from any sharp transition present in the original scene.
The simplest reconstruction algorithm consists in calculating the inverse Fourier coefficients associated to the minimum hexagon embedding the MIRAS visibility star by direct application of the Fourier series formulas. However, this approach is very sensitive to the presence of high amplitude, small spatial extent sources in the original scene, that will spread side lobes that may eventually corrupt all the retrieved scene. Even if the coefficients for the full embedding hexagon were known and even if the version of Fourier series adapted to hexagonal grids to avoid introducing spurious zeros at high frequencies were applied, any delta-like high-amplitude signal in the original scene will significantly corrupt the scene retrieved with this algorithm. This makes RFI sources, land-sea transitions and even the presence of Sun very disturbing, as they may generate so large side-lobes that scenes get irredeemably corrupted over large areas.
An alternative reconstruction algorithm, able to eliminate side-lobes associated to delta-like sources, can be proposed by accepting some degree of representativity error in the retrieved signal. This new algorithm is based on the observation that when the visibility star is embedded in hexagons of larger scales and the inverse Fourier coefficients are calculated, the nodal points - the zeros of the sinus function associated to side-lobes- remain invariant under changes in scale. We call the grid formed by the nodal points the nodal grid. As scale is increased nodal points are representative of pixels with smaller footprint, so they become less and less representative of the associated geographical area; but if the geophysical signal to retrieve has a slow-enough spatial pattern of variation the retrieved value is still very representative of the actual geophysical average on the coarser grid. Nodal grids sampling hexagons of large enough scales lead to a maximum reduction of side-lobes on synthetic images.
Direct application of uniform nodal grids on SMOS visibilites corrupted by high levels of RFI-induced noise leads to highly attenuated side-lobes, too. However, due to imperfections in the antenna distribution the spatial pattern of nodal points associated to MIRAS is not completely uniform. Even worse, changes in local phase between consecutive calibrations make the nodal grid not constant with time. The implementation of a fast, adaptive algorithm capable to retrieve a non-uniform nodal grid is hence in order. Non-uniform adaptive nodal grids lead to the highest reduction of side-lobes while preserving a good enough representativity of the retrieved geophysical signal.
Our results show a promising way to diminish the impact of side-lobes on SMOS images, but they are just a starting point. Future work include the design of appropriate metrics for evaluating the final quality of a given adaptive algorithm, to use those metrics to optimize the algorithm and to include the effect of new fusion schemes to enhance even further the retrieved brightness temperatures.

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