Representações espectrais de sistemas complexos: aplicações à síntese de superfícies brownianas fracionárias anisotrópicas, filtragem de sinais e identificação de correlações

In this thesis, we study the application of spectral representations to the solution of problems in seismic exploration, the synthesis of fractal surfaces and the identification of correlations between one-dimensional signals. We apply a new approach, called Wavelet Coherency, to the study of stratigraphic correlation in well log signals, as an attempt to identify layers from the same geological formation, showing that the representation in wavelet space, with introduction of scale domain, can facilitate the process of comparing patterns in geophysical signals. We have introduced a new model for the generation of anisotropic fractional brownian surfaces based on curvelet transform, a new multiscale tool which can be seen as a generalization of the wavelet transform to include the direction component in multidimensional spaces. We have tested our model with a modified version of the Directional Average Method (DAM) to evaluate the anisotropy of fractional brownian surfaces. We also used the directional behavior of the curvelets to attack an important problem in seismic exploration: the atenuation of the ground roll, present in seismograms as a result of surface Rayleigh waves. The techniques employed are effective, leading to sparse representation of the signals, and, consequently, to good resolutions