Adaptive wavelet filtering of seismic data in the wavelet transform domain

The wavelet transform coefficients of a seismic trace describe the temporal and frequency distributions of the energy in the trace. We use this time/frequency separation produced by the wavelet transform to design adaptive wavelet filters (AWFs) in the wavelet transform domain. Depending on the (data-dependent) definition of a threshold filter function in time-frequency-amplitude space, the AWF performs as a time-varying nonlinear filter. A translation-invariant wavelet transform minimizes phase shift errors induced by filtering With a traditional wavelet transform. The computation time of AWFs is of the same order as frequency bandpass filtering. We illustrate filter design for two situations. Given a partial overlap in frequency between signal and noise,the wavelet filter is effective in separating high-energy spatially-aliased noise (ground roll and air-waves) from reflections. Where there is full overlap in frequency between the signal and noise, we employ an optimized adaptive, wavelet filter with a correlation criterion and a grid search for an optimal solution in the solution space generated by a recursive wavelet filter, this is a target-oriented method for a single reflection with a data-dependent wavelet. Field data examples illustrate the ability to reveal a reflection beneath ground roll and to improve the signal-to-noise ratio.