We present a new method to stabilize the three-term AVO inversion using Curvelet and Wavelet transforms. Curvelets are basis functions that effectively represent otherwise smooth objects having discontinuities along smooth curves. The applied formalism explores them to make the most of the continuity along reflectors in seismic images. Combined with Wavelets, Curvelets are used to denoise the data by penalizing high frequencies and small contributions in the AVO-cube. This approach is based on the idea that rapid amplitude changes along the ray-parameter axis are most likely due to noise. The AVO-inverse problem is linearized, formulated and solved for all (x, z) at once. Using densities and velocities of the Marmousi model to define the fluctuations in the elastic properties, the performance of the proposed method is studied and compared with the smoothing along the ray-parameter direction only. We show that our method better approximates the true data after the denoising step, especially when noise level increases.

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Amplitude Variation with Oset (AVO) inversion is one of the techniques that is being used to estimate subsurface physical parameters such as P-wave velocity, S-wave velocity, and density or their attributes. AVO inversion is an ill-conditioned problem which has to be regularized in order to obtain a stable and unique solution. In this thesis, a Bayesian procedure that uses a Multivariate Cauchy distribution as a prior probability distribution is introduced. The prior includes a scale matrix that imposes correlation among the AVO attributes and induces a regularization that provokes solutions that are sparse and stable in the presence of noise. The performance of this regularization is demonstrated by both synthetic and real data examples using linearized approximations to the Zoeppritz equations. / Geophysics