Rapid Seismic Inversion for Isolated Singularities
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Thin layers can cause problems in inversion algorithms as the material parameters and the thickness of the layer are inseparable. Here a thin layer model is developed where a thin sand layer embedded in shale is considered as a singularity with all information gathered in the midpoint of the layer. The aim is to find the singularity's contribution to the seismic data. An algorithm is developed using Bayesian inversion. Assuming that the variables involved are Gaussian, an analytical answer to inversion problem can be found. The spatial coupling of the parameters is represented by a spatial correlation function. This spatial variables can be decoupled in the Fourier domain, enabling independent inversion of each of the frequency components. The model includes variation in the parameters, spatial coupling and angular dependency. Results of the inversion in two dimensions are found for incidence angles of $0$, $9$, $21$ and $33$ degrees, but the angular coupling is here omitted. The inversion is accurate for the two lowest angles, but fails when the angles are $21$ and $33$ degrees.