Multisnapshot imaging for LIIFE: Along-the-track inversion of the Van Cittert-Zernike theorem for unfolding and denoising

The Soil Moisture and Ocean Salinity (SMOS) satellite [3] produces images of L-band brightness temperature using interferometry. The brightness temperatures that gave rise to a set of observed visibilities are inferred by solving an inverse problem based on the Van Cittert-Zernike theorem. The images are generated from visibilities one correlation period at a time, even though a given patch of Earth’s surface may be visible to the satellite for 100 consecutive periods. The recovered brightness temperature images are noisy and, due to the antenna array’s undersampling of the u-v frequency plane, aliased. A simultaneous inversion of multiple snapshots (call it "satellite-rotation synthesis" or a "global inversion") could take advantage of these multiple looks to improve resolution, field of view, and noise level, but appeared prohibitively expensive computationally and ill-posed: due to directional emissivity, many brightness temperatures must be recovered for each pixel. However, this directional emissivity can be sparsely modeled; accordingly, for satellites like SMOS, the observation model becomes shift-invariant across an orbital segment when expressed in geodetic coordinates relative to the trace. The observation model performs a convolution on the image of parameters (uniform in geodetic coordinates) governing observed brightness as a function of satellite position, and we can use the "Fourier transform trick" introduced in [6], to enable a global inversion.For the proposed L-Band Interferometer for Fluxes and Interfaces in the Environment (LIIFE) [7], [11], we present a realistic global inversion of the observation model across the orbital trace, which simultaneously unfolds and denoises the aliased snapshots. By using sparse spline models of the brightness temperature parameters, we can ensure this inversion is accurate and stable, even with a realistic simulation that accounts for the difficulty of modeling the Van Cittert-Zernike theorem (expressed in satellite-centered direction-cosine coordinates) using geodetic coordinates.