Sparse Bayesian Learning-Based Time-Variant Deconvolution

In seismic exploration, the wavelet-filtering effect and <inline-formula> <tex-math notation="LaTeX">${Q}$ </tex-math></inline-formula>-filtering (amplitude attenuation and velocity dispersion) effect blur the reflection image of subsurface layers. Therefore, both wavelet- and <inline-formula> <tex-math notation="LaTeX">${Q}$ </tex-math></inline-formula>-filtering effects should be reduced to retrieve a high-quality subsurface image, which is significant for fine reservoir interpretation. We derive a nonlinear time-variant convolution model to sparsely represent nonstationary seismograms in time domain involving these two effects and present a time-variant deconvolution (TVD) method based on sparse Bayesian learning (SBL) to solve the model to obtain a high-quality reflectivity image. The SBL-based TVD essentially obtains an optimum posterior mean of the reflectivity image, which is regarded as the inverted reflectivity result, by iteratively solving a Bayesian maximum posterior and a type-II maximum likelihood. Because a hierarchical Gaussian prior for reflectivity controlled by model-dependent hyper-parameters is adopted to approximately represent the fact that reflectivity is sparse, SBL-based TVD can retrieve a sparse reflectivity image through the principled sequential addition and deletion of <inline-formula> <tex-math notation="LaTeX">${Q}$ </tex-math></inline-formula>-dependent time-variant wavelets. In general, strong reflectors are acquired relatively earlier, whereas weak reflectors and deep reflectors are imaged later. The method has the capacity to avoid false artifacts represented by sequential positive or negative reflectivity spikes with short two-way travel time, which typically occur within stationary deconvolution outcomes. Synthetic, laboratorial, and field data examples are used to demonstrate the effectiveness of the method and illustrate its advantages over SBL-based stationary deconvolution and TVD using an <inline-formula> <tex-math notation="LaTeX">$l_{2}$ </tex-math></inline-formula>-norm or an <inline-formula> <tex-math notation="LaTeX">$l_{1}$ </tex-math></inline-formula>-norm regularization. The results show that SBL-based TVD is a potentially effective, stable, and high-quality imaging tool.