Illumination Analysis of Wave-equation Imaging with “curvelets”

We present a comprehensive framework for wave-equation illumination analysis and introduce a target-oriented illumination correction that simultaneously accounts for limited acquisition aperture and, locally, compensates for the so-called normal operator in inverse scattering to yield a “true-amplitude” image of reflectivity or reflection coefficient, while minimizing (orientation dependent) phase distortions and artifacts. The corrections are nested while the most significant correction is typically the one associated with the limited acquisition aperture. To carry out the analysis we make use of higher-dimensional “curvelets”, which provide the means of extracting directional information, and introduce associated matrix representations for the component operators, including the tapers associated with the acquisition aperture, that make up wave-equation migration; we essentially exploit the properties of these matrices. Curvelets can be viewed as “fat”, optimally localized plane waves and hence form a natural candidate to generalize geophysical diffraction tomography, which is at the basis of our approach. Our approach admits the formation of caustics and, hence, is valid in complex velocity models, though these need to be known for the illumination compensation to be effective.