A compressive sensing framework for seismic source parameter estimation

Abstract

SUMMARY Simultaneous estimation of origin time, location and moment tensor of seismic events is critical for automatic, continuous, real‐time monitoring systems. Recent studies have shown that such systems can be implemented via waveform fitting methods based on pre‐computed catalogues of Green’s functions. However, limitations exist in the number and length of the recorded traces, and the size of the monitored volume that these methods can handle without compromising real‐time response. This study presents numerical tests using a novel waveform fitting method based on compressive sensing, a field of applied mathematics that provides conditions for sampling and recovery of signals that admit a sparse representation under a known base or dictionary. Compressive sensing techniques enable us to determine source parameters in a compressed space, where the dimensions of the variables involved in the inversion are significantly reduced. Results using a hypothetical monitoring network with a dense number of recording stations show that a compressed catalogue of Green’s functions with 0.004 per cent of its original size recovers the exact source parameters in more than 50 per cent of the tests. The gains in processing time in this case drop from an estimated 90 days to browse a solution in the uncompressed catalogue to 41.57 s to obtain an estimation using the compressed catalogue. For simultaneous events, the compressive sensing approach does not appear to influence the estimation results beyond the limitations presented by the uncompressed case. The main concern in the use of compressive sensing is detectability issues observed when the amount of compression is beyond a minimum value that is identifiable through numerical experiments. Tests using real data from the 2002 June 18 Caborn Indiana earthquake show that the presence of noise and inaccurate Green’s functions require a smaller amount of compression to reproduce the solution obtained with the uncompressed catalogue. In this case, numerical simulation enables the assessment of the amount of compression that provides a reasonable rate of detectability. Overall, the numerical experiments demonstrate the effectiveness of our compressed domain inversion method in the real‐time monitoring of seismic sources with dense networks of receivers. As an added benefit of the compression process, the size of the monitored volume can also be increased under specific restrictions while maintaining the real‐time response.

Compressive sensing imaging of microseismic events constrained by the sign-bit

“…As in this case there is only one source in the solution, the estimation of the source location from the compressed adjoint can be understood as solving equation 3 with a single iteration of block orthogonal matching pursuit (Eldar et al, 2010;Vera Rodriguez et al, 2012a), where sparsity is promoted when the largest coefficient Sign-bit constrained CS imaging KS3 in the output image is selected as the source location. In other words, the compressed imaging solution is also a full CS implementation for the single source monitoring problem.…”

“…In seismic source monitoring problems, CS has been proposed as an alternative to conventional seismic imaging to reduce processing time in the simultaneous estimation of origin time, location, and moment tensor of microseismic events when using full-waveform elastodynamic Green's functions. For example, in Vera Rodriguez et al (2012a), a synthetic simulation using a number of channels comparable to a surface microseismic monitoring job accomplished a reduction in response time that went from multiple days to less than a minute when CS was implemented. In this case, the dramatic reduction in processing time was related to the allocation of the inversion variables in the computing system.…”

Research Note: Sparsity‐Aware Multiple Microseismic Event Localization Blind to the Source Time‐Function

“…Motivation: In (Rodriguez, Sacchi, and Gu a; Rodriguez, Sacchi, and Gu b), the dictionary is highly dependent on the source time‐function , which means that the source in the run‐time phase should have the same source time‐function as the one which is considered to construct the dictionary denoted as . The situation gets even worse for the multi‐source case where (with being the source time‐function of the k ‐th source) should hold to avoid poor results.…”

“…Looking at the formulation in equation , we see an important phenomenon in the frequency domain where both the source origin‐time and the source time‐function (represented at ) are translated into two (complex) constant factors. For the sake of consistency with the time‐domain approach presented by Rodriguez, Sacchi, and Gu (a) and Rodriguez, Sacchi, and Gu (b), we also keep in and thus in our dictionary. The contribution of the origin‐time, however, can easily be accommodated in the newly defined sub‐vector of interest .…”

“…In Section II, we explain the acquisition setup and signal model under consideration. Section III briefly reviews the proposed approach of (Rodriguez, Sacchi, and Gu a; and Rodriguez, Sacchi, and Gu b). Next, our proposed frequency‐domain approach (blind to the source time‐function) is explained.…”