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UBC Theses and Dissertations

Aperture compensated radon and fourier transforms Sacchi, Mauricio Dino


In seismic data analysis, recorded data often are transformed to various domains to discriminate against coherent and incoherent noise. For instance, by mapping a shot record from time-space domain to frequency-wavenumber domain, coherent linear noise can be attenuated. Similarly, by mapping a common-midpoint gather from time-space to time-velocity domain (velocity stacks) multiples are separated from primaries based on moveout discrimination. In these procedures the correct identification of seismic events with similar moveout can be severely affected by the aperture of the array and the discrete sampling of the wavefield. Economic and/or logistic reasons usually dictate the cable length and spatial sampling of the seismic experiment. This thesis examines how the resolution (the ability to distinguish close events) of slant stack and parabolic stack operators deteriorates under limited aperture. An algorithm is developed to increase the resolution of the aforementioned operators. This procedure constructs an operator that collapses each seismic signal in the transform domain, thus diminishing truncation artifacts. The overall procedure is equivalent to the simulation of a longer array of receivers. Slant stacks and the parabolic stacks are linear operations used to map the seismic data into another domain, the transform domain (r — p or r — q). In this thesis an inverse problem is posed. This is accomplished by considering the data as the result of a linear operation onto the transform domain. This approach permits one to incorporate prior information into the problem which is utilized to attenuate truncation artifacts. The prior information is incorporated into the inverse problem by means of the Bayesian formalism. The observational errors and the prior information are combined through Bayes' rule using the likelihood function and a long tailed distribution, respectively. The posteriori probability is then used to induce the objective function of the problem. Finally, minimizing the objective function leads to the solution of the inverse problem. The advantage of incorporating a long tailed distribution to model the transform domain is that the solution is constrained to be sparse which is a desired feature for highly resolved models. The method is also used to design an artifacts-reduced 2-D discrete Fourier transform. A by-product of the method is a high resolution periodogram. This periodogram coincides with the periodogram that would have been computed with a longer array of receivers if the data consist of a limited superposition of linear events.

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