Multidimensional regularization and interpolation of seismic data using Matchin Pursuit, MWNI and ALFT algorithms
Seismic Processing, Seismic Regularization, Multidimensional Interpolation, Fourier Transform, Matching Pursuit
One of the great challenges in the seismic imaging is the data acquisition, since these are frequently sampled in an irregular way with many shots and missing traces due to logistical restrictions in the geophones distribution. This causes a serious problem of covering subsurface structures of interest, as well as making it difficult to apply processes to treat and modify the data for a better interpretation of these structures. Thus, regularization and interpolation are essential to obtain better results and facilitate subsequent steps in seismic data processing. The Fourier reconstruction theory makes use of the property that irregularly sampled data, with missing positions, has a regular representation in the Fourier domain. When finds the spectrum that best synthesizes the data in regular sampling in the original domain, it can be reconstructed in order to fill the empty spaces. Techniques based on the Fourier transform show the best results due to the ability to capture both correct amplitude and continuity of events. The Minimum Weighted Norm Interpolation (MWNI) technique estimates the Fourier coefficients that synthesize the spatial positions of the data through the regularized inversion of a desired regular grid. Unlike Matching Pursuit and Anti-Leakage Fourier Transform (ALFT), they use an iterative procedure to calculate the spectrum of irregularly sampled data for a regular grid. All the cited are band limited and interpolate beyond the aliasing, but the latter two have the disadvantage of being computationally expensive. All have the ability to regularize sparse data sets and deal with multidimensional seismic regularization. In this thesis it is proposed the implementation and comparative testing of the mentioned algorithms. Examples of regularization with harmonic functions 1D and 2D exemplify how the regularization in seismic data is performed, since by taking these functions as an approximation for the slices of temporal frequencies of the domain f-x or f-xy for 1D and 2D, respectively, it is demonstrated how the interpolation in spatial coordinates is performed. The results show the ability of these algorithms to regularize very sparse data sets. Also presented is an example of 2D regularization for a synthetic seismic section, with satisfactory results. Practical applications of these algorithms were applied to the 3D regularization of Marmousi data, which simulates (synthetic) 2D marine seismic data and to the real 2D land seismic data survey from the Parnaı́ba basin. ALFT and Matching Pursuit have similar results in all examples and applications made, with a slight advantage to ALFT. Both are superior to MWNI, with applications highlighted, where it became clear that MWNI failed to match the other two, as it had in the examples of 1D and 2D functions. However, its computational time is much lower and so it is still an option for 5D regularization. Matching Pursuit was an alternative to ALFT, as it has similar results and a computational cost of less than half of the ALFT.