The Hilbert–Huang Transform (HHT) has been sparsely applied to problems in seismology, although previous studies have pointed to its broad scope. In this maiden attempt, we use the HHT to represent earthquake energy release duration and frequency content and compare the results with two conventional inversion methods. By selecting examples from interplate, intraplate, and intraslab settings, we demonstrate that the HHT has the power to discriminate energy release of earthquakes with different tectonic affiliations. We observe that the dominant frequencies for energy release are higher for intraslab earthquakes than for interplate and intraplate events. We use the empirical mode decomposition-based HHT and introduce a new parameter, which we name the energy rate function (ERF), to quantify the energy release. By employing empirical Green’s functions to remove the path and site effects and using a linear combination of a select set of intrinsic mode functions, we generate the station-specific relative measure of energy that we refer to as relative ERFs (RERFs). Averaged over RERFs from multiple stations, the ERF represents a measure of the total relative energy release, comparable to the moment rate functions (MRFs) and SCARDEC source time functions (STFs). Results for six of the seven earthquakes we analyzed show high cross correlation with the STFs (0.84 ± 0.03) and MRFs (0.79 ± 0.06), but there are mismatches between ERFs and MRFs or STFs when the energy release is complex and involves multisegment or bilateral ruptures. The proposed method is computationally efficient, requiring only 3.46 ± 2.62 s on average, compared to ~20 min (~1200 s) for the teleseismic inversion method we employ. With its ability to represent the seismic source in terms of energy release, the ERF method has the potential to evolve not as an alternative to waveform inversion but as a rapid time–frequency analysis tool, useful for earthquake hazard assessment.
Sparse coding methods are iterative and typically rely on proximal gradient methods. While the commonly used sparsity promoting penalty is the ℓ1 norm, alternatives such as the minimax concave penalty (MCP) and smoothly clipped absolute deviation (SCAD) penalty have also been employed to obtain superior results. Combining various penalties to achieve robust sparse recovery is possible, but the challenge lies in parameter tuning. Given the connection between deep networks and unrolling of iterative algorithms, it is possible to unify the unfolded networks arising from different formulations. We propose an ensemble of proximal networks for sparse recovery, where the ensemble weights are learnt in a data-driven fashion. We found that the proposed network performs superior to or on par with the individual networks in the ensemble for synthetic data under various noise levels and sparsity conditions. We demonstrate an application to image denoising based on the convolutional sparse coding formulation.
We solve the problem of sparse signal deconvolution in the context of seismic reflectivity inversion, which pertains to high-resolution recovery of the subsurface reflection coefficients. Our formulation employs a nonuniform, non-convex synthesis sparse model comprising a combination of convex and non-convex regularizers, which results in accurate approximations of the 0 pseudo-norm. The resulting iterative algorithm requires the proximal average strategy. When unfolded, the iterations give rise to a learnable proximal average network architecture that can be optimized in a data-driven fashion. We demonstrate the efficacy of the proposed approach through numerical experiments on synthetic 1-D seismic traces and 2-D wedge models in comparison with the benchmark techniques. We also present validations considering the simulated Marmousi2 model as well as real 3-D seismic volume data acquired from the Penobscot 3D survey off the coast of Nova Scotia, Canada.
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