Phase and amplitude tracking for seismic event separation

Abstract

We have developed a method to decompose seismic records into atomic events, each defined by a smooth phase function and a smooth amplitude function. This decomposition is intrinsically nonlinear and calls for a nonconvex least-squares optimization formulation, along the lines of full-waveform inversion. To overcome the lack of convexity, we have developed an iterative refinement-expansion scheme to initialize and track the phase and amplitude for each atomic event. For short, we called the method phase tracking. The initialization is carried out by applying multiple signal classification to a few seed traces in which events can be separated and identified by their arrival times and amplitudes. We then construct the initial solution at the seed traces using linear phase functions from the arrival times and constant amplitude functions, assuming the medium is mostly dispersion free. We refine this initial solution to account for dispersion and imperfect knowledge of the wavelet at the seed traces by fitting the observed data using a gradient descent method. The resulting phase and amplitude functions are then carefully expanded across the traces in an adequately smooth way to match the whole data record. We have evaluated the proposed method on two synthetic records and a field record. Because the parametrization of the seismic events is physically meaningful, it also enables a simple form of bandwidth extension of the observed shot record to unobserved low and high frequencies. We tested this procedure on the same shot records. Bandwidth extension is in principle helpful to initialize full-waveform inversion with frequency sweeps and enhanced its resolution.

Conditioning of Partial Nonuniform Fourier Matrices with Clustered Nodes

“…The problem of (sparse) super-resolution is to recover discrete, point-like objects from their noisy and bandlimited spectral measurements. It arises in many fields such as frequency estimation, sampling theory, array processing, astronomical imaging, seismic imaging, nonuniform FFT, statistics, radar signal detection, error correction codes, and others [4,12,13,16,10,20,28,25,35]. Our main results have direct implications for the problem of super-resolution under sparsity constraints, in the so-called "on-grid" model 2 .…”

Full-waveform inversion with extrapolated low-frequency data

“…Yet, by posing it as a data processing problem, the nonconvexity can be empirically overcome with an explicit initialization scheme using multiple signal classification (MUSIC), coupled with a careful trust-region "expansion and refinement" scheme to track the smooth phase and amplitude. We refer the reader to the detailed algorithm in the previous paper (Li and Demanet, 2015).…”

“…In a previous paper (Li and Demanet, 2015), we demonstrated that there exist interesting physical scenarios, in which low-frequency data can be synthesized from the band-limited field recordings using nonlinear signal processing. This processing step is performed before FWI in the frequency domain.…”

“…The premise of this paper is that the phase-tracking method, proposed in Li and Demanet (2015), is a reasonably effective algorithm for extrapolating the low-frequency data based on the phases and amplitudes in the observed frequency band. A tracking algorithm is able to separate each seismic record into atomic events, the amplitude and phase functions of which are smooth in space and frequency.…”

Compressing Large-Scale Wave Propagation Models via Phase-Preconditioned Rational Krylov Subspaces

“…However, to achieve it for particular applications one can try to incorporate the underlying physics and asymptotic analysis to arrive at PPRKS. Our approach is related to other known approaches in the field of oscillatory wave problem computation, such as preconditioners for Helmholtz solvers [14,17], Filon quadrature [19], and a recent approach to data compression using phase-tracking [21].…”