| The multiresolution nature of the wavelet transform is used to design adaptive wavelet 1-D Filters (AWFs) in the wavelet transform domain. In a partial overlap in frequency between signal and noise, the wavelet filter is broadband in frequency and effective in separating high energy spatially-aliased noise (ground roll and air-waves) from reflections. Where there is a full overlap in frequency between the signal and noise, an algorithm is developed for optimizing AWF. We empoly a correlation criterion and a grid search for an optimal solution. Field data examples illustrate the ability to reveal a reflection beneath the ground roll and the air wave, to improve signal-to-noise ratio.; A Multiscale Wavelet-Radon Algorithm (MWRA) is developed for separating the aliased energy and signal up to the Nyquist frequency or wavenumber in the wavelet transform domain. It is assumed that there is no aliasing in time and that the signal is consistent across wavelet scales. Interpolation of the aliased seismic data is performed by redefining the spatial sampling rate in the inverse slant stack. Compared to frequency-space (F-X) interpolation, MWRA accommodates an irregular geophone spacing and outperforms F-X interpolations in suppressing aliasing and noise in aliased curved events, as shown in both synthetic and field data.; Based on the wavelet decomposition theory and the Kirchhoff integral solution to the wave equation, an algorithm to migrate compressed wavelet coefficients is developed. Wavelet-based Prestack Multiresolution Kirchhoff Migration (WPMKM) involves 4 steps: wavelet decomposition, compression of the data volume, multiscale Kirchhoff migration and wavelet reconstruction. The conventional Kirchhoff migration is similar to WPMKM for one wavelet scale. Anti-aliasing of the migration operator is naturally implemented in the multi-scale space. Compared to the conventional Kirchhoff migration, WPMKM reduces the computation time by migrating wavelet coefficients rather than all the points, without sacrificing resolution. The time reduction is proportional to the compression ratios of the data and of the image. Successful tests are shown for migration of synthetic data from a point diffractor, a multi-layer, and the Marmousi model. |
