We evaluated a time-domain wave equation for modeling acoustic wave propagation in attenuating media. The wave equation was derived from Kjartansson’s constant-Q constitutive stress-strain relation in combination with the mass and momentum conservation equations. Our wave equation, expressed by a second-order temporal derivative and two fractional Laplacian operators, described very nearly constant-Q attenuation and dispersion effects. The advantage of using our formulation of two fractional Laplacians over the traditional fractional time derivative approach was the avoidance of time history memory variables and thus it offered more economic computations. In numerical simulations, we formulated the first-order constitutive equations with the perfectly matched layer absorbing boundaries. The temporal derivative was calculated with a staggered-grid finite-difference approach. The fractional Laplacians are calculated in the spatial frequency domain using a Fourier pseudospectral implementation. We validated our numerical results through comparisons with theoretical constant-Q attenuation and dispersion solutions, field measurements from the Pierre Shale, and results from 2D viscoacoustic analytical modeling for the homogeneous Pierre Shale. We also evaluated different formulations to show separated amplitude loss and dispersion effects on wavefields. Furthermore, we generalized our rigorous formulation for homogeneous media to an approximate equation for viscoacoustic waves in heterogeneous media. We then investigated the accuracy of numerical modeling in attenuating media with different Q-values and its stability in largecontrast heterogeneous media. Finally, we tested the applicability of our time-domain formulation in a heterogeneous medium with high attenuation.

Modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional Laplacians

We used a novel iterative estimation scheme for separation of blended seismic data from simultaneous sources. The scheme is based on an augmented estimation problem that can be solved by iteratively constraining the deblended data using shaping regularization in the seislet domain. We formulated the forward modeling operator in the common-receiver domain, in which two sources were assumed to be blended using a random time-shift dithering approach. The nonlinear shaping-regularization framework offered some freedom in designing a shaping operator to constrain the model in an underdetermined inverse problem. We designed the backward operator and the shaping operator for the shaping-regularization framework. The backward operator can be optimally chosen as half of the identity operator in the two-source case, and the shaping operator can be chosen as coherency-promoting operator. The high performance deblending effect of the iterative framework was tested on three numerically blended synthetic data s...

Iterative deblending of simultaneous-source seismic data using seislet-domain shaping regularization

A review of the current status of induced seismicity monitoring for hydraulic fracturing in unconventional tight oil and gas reservoirs

Simultaneous-source acquisition improves the efficiency of the seismic data acquisition process. However, direct imaging of simultaneous-source data may introduce crosstalk artifacts in the final image. Likewise, direct imaging of incomplete data avoids the step of data reconstruction, but it can suffer from migration artifacts. We have proposed to incorporate shaping regularization into least-squares reverse time migration (LSRTM) and use it for suppressing interference noise caused by simultaneous-source data or migration artifacts caused by incomplete data. To implement LSRTM, we have applied lowrank one-step reverse time migration and its adjoint iteratively in the conjugate-gradient algorithm to minimize the data misfit. A shaping operator imposing structure constraints on the estimated model was applied at each iteration. We constructed the shaping operator as a structure-enhancing filtering to attenuate migration artifacts and crosstalk noise while preserving structural information. We have carried out numerical tests on synthetic models in which the proposed method exhibited a fast convergence rate and was effective in attenuating migration artifacts and crosstalk noise.

https://library.seg.org/doi/pdf/10.1190/geo2014-0524.1

Seismic imaging of incomplete data and simultaneous-source data using least-squares reverse time migration with shaping regularization

S U M M A R Y Effective and efficient arrival picking plays an important role in microseismic and earthquake data processing and imaging. Widely used short-term-average long-term-average ratio (STA/LTA) based arrival picking algorithms suffer from the sensitivity to moderate-to-strong random ambient noise. To make the state-of-the-art arrival picking approaches effective, microseismic data need to be first pre-processed, for example, removing sufficient amount of noise, and second analysed by arrival pickers. To conquer the noise issue in arrival picking for weak microseismic or earthquake event, I leverage the machine learning techniques to help recognizing seismic waveforms in microseismic or earthquake data. Because of the dependency of supervised machine learning algorithm on large volume of well-designed training data, I utilize an unsupervised machine learning algorithm to help cluster the time samples into two groups, that is, waveform points and non-waveform points. The fuzzy clustering algorithm has been demonstrated to be effective for such purpose. A group of synthetic, real microseismic and earthquake data sets with different levels of complexity show that the proposed method is much more robust than the state-of-the-art STA/LTA method in picking microseismic events, even in the case of moderately strong background noise.

Automatic microseismic event picking via unsupervised machine learning

Conventional full-waveform inversion (FWI) using the least-squares norm as a misfit function is known to suffer from cycle-skipping issues which increases the risk of computing a local rather than the global minimum of the misfit. The quadratic Wasserstein metric has been proved to have many ideal properties with regards to convexity and insensitivity to noise. When the observed and predicted seismic data are considered to be two density functions, the quadratic Wasserstein metric corresponds to the optimal cost of rearranging one density into the other, where the transportation cost is quadratic in distance. Unlike the least-squares norm, the quadratic Wasserstein metric measures not only amplitude differences but also global phase shifts, which helps to avoid cycle-skipping issues. We propose a new way of using the quadratic Wasserstein metric trace-by-trace in FWI and compare it to the global quadratic Wasserstein metric via the solution of the Monge-Ampere equation. We incorporate the quadratic Wasser...

Application of optimal transport and the quadratic Wasserstein metric to full-waveform inversion

A constant-Q wave equation involving fractional Laplacians was recently introduced for viscoacoustic modeling and imaging. This fractional wave equation has a convenient mixed-domain space-wavenumber formulation, which involves the fractional-Laplacian operators with a spatially varying power. We have applied the low-rank approximation to the mixed-domain symbol, which enables a space-variable attenuation specified by the variable fractional power of the Laplacians. Using the proposed approximation scheme, we formulated the framework of the Q-compensated reverse time migration (Q-RTM) for attenuation compensation. Numerical examples using synthetic data demonstrated the improved accuracy of using low-rank wave extrapolation with a constant-Q fractional-Laplacian wave equation for seismic modeling and migration in attenuating media. Low-rank Q-RTM applied to viscoacoustic data is capable of producing images comparable in quality with those produced by conventional RTM from acoustic data.

https://www.jsg.utexas.edu/tyzhu/files/SEG2014_lowrank_CQ.pdf

Viscoacoustic modeling and imaging using low-rank approximation

Attenuation of seismic waves needs to be taken into account to improve the accuracy of seismic imaging. In viscoacoustic media, reverse time migration (RTM) can be performed with Q-compensation, which is also known as Q-RTM. Least-squares RTM (LSRTM) has also been shown to be able to compensate for attenuation through linearized inversion. However, seismic attenuation may significantly slow down the convergence rate of the least-squares iterative inversion process without proper preconditioning. We have found that incorporating attenuation compensation into LSRTM can improve the speed of convergence in attenuating media, obtaining high-quality images within the first few iterations. Based on the low-rank one-step seismic modeling operator in viscoacoustic media, we have derived its adjoint operator using nonstationary filtering theory. The proposed forward and adjoint operators can be efficiently applied to propagate viscoacoustic waves and to implement attenuation compensation. Recognizing that, ...

https://library.seg.org/doi/pdf/10.1190/geo2015-0520.1

Q-compensated least-squares reverse time migration using low-rank one-step wave extrapolation

From the seismic imaging point of view, the difficulty in locating passive seismic sources lies in their unknown start times. In other words, the source model has an additional dimension of time, which leads to an extended model space. Without proper preconditioning, the computational cost of directly inverting for the source functions can be intractable. Using the recently proposed cross-correlation time-reversal imaging condition, we formulate the imaging task as an inverse problem, and use a sparse weighting function calculated from the cross-correlation of back-propagated events to constrain the model space. We demonstrate that the proposed approach can effectively reduce the number of model parameters, leading to a rapid convergence rate using preconditioned conjugategradient iterations. The least-squares imaging of passive seismic sources can be further incorporated into full waveform inversion for Earth properties using the variable projection method. Synthetic examples verify the proposed method.

Full-waveform inversion of passive seismic data for sources and velocities

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