Recently, a decoupled fractional Laplacian viscoacoustic wave equation has been developed based on the constant-$Q$ model to describe wave propagation in heterogeneous media. We have developed two efficient modeling schemes to solve the decoupled fractional Laplacian viscoacoustic wave equation. Both schemes can cope with spatial variable-order fractional Laplacians conveniently, and thus are applicable for modeling viscoacoustic wave propagation in heterogeneous media. Both schemes are based on fast Fourier transform, and have a spectral accuracy in space. The first scheme solves a modified wave equation with constant-order fractional Laplacians instead of spatial variable-order fractional Laplacians. Due to separate discretization of space and time, the first scheme has only first-order accuracy in time. Differently, the second scheme is based on an analytical wave propagator, and has a higher accuracy in time. To increase computational efficiency of the second modeling scheme, we have adopted the low-rank decomposition in heterogeneous media. We also evaluated the feasibility of applying an empirical approximation to approximate the fractional Laplacian that controls amplitude loss during wave propagation. When the empirical approximation is applied, our two modeling schemes become more efficient. With the help of numerical examples, we have verified the accuracy of our two modeling schemes with and without applying the empirical approximation, for a wide range of seismic quality factor ($Q$). We also compared computational efficiency of our two modeling schemes using numerical tests.

Accurate estimation of source wavelet is crucial in a successful full-waveform inversion (FWI); however, it cannot be guaranteed in the case of real seismic data. We have developed time-domain source-independent elastic FWI using the convolution-based objective function that was originally developed for acoustic FWI. We have applied a new time window on the reference traces in the objective function to suppress the noises induced by the convolution and crosscorrelation operations. Also, we have adopted ${\ell }_1$-, Huber-, and hybrid-norm objective functions to improve the antinoise ability of our algorithm. We implemented a multiscale inversion strategy to conduct the tests with a quasi-Newton limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method to reduce the sensitivity to initial models and to improve the quality of inversion results. Synthetic tests verified that the new added time window can not only improve the inversion results, but also accelerate the convergence rate. Our method can be implemented successfully without a priori knowledge or accurate estimation of the source wavelet and can be more robust to Gaussian and spike noises, even for a Dirac wavelet. Finally, we applied our method to real seismic data. The similarity between the observed and modeled seismic data, the higher resolution of the migration image, and flatter common image gathers corresponding to the inverted models proved the relevance of our algorithm.

Simultaneous source acquisition, which allows a temporal overlap between shot records, has significant advantages to improve data quality (e.g., denser shooting) and reduce acquisition cost (e.g., efficient wide-azimuth shooting). We have developed a novel shot-domain deblending approach for a wide-azimuth simultaneous shooting survey based on an inversion scheme. The inverse problem is formulated by introducing two convolution operators that can respectively destruct the signal and interference and is solved by a conjugate-gradient algorithm in the least-squares sense. Our approach breaks down the limit that source separation can only be implemented in some domains other than the shot domain based on the incoherency principle. Our shot-domain approach does not require the random dithering time; thus, it is very flexible to use in a wide-azimuth simultaneous shooting survey. Three numerically blended synthetic examples were developed to demonstrate the excellent performance of our method. The feasibility has been further validated via a field-data example that is numerically blended from two realistic marine towed streamers.

Two staggered-grid finite-difference (SGFD) schemes with fourth- and sixth-order accuracies in time have been developed recently based on new SGFD stencils. The SGFD coefficients of the two schemes are determined by a Taylor-series expansion (TE) approach, which is accurate only near a zero wavenumber. We have adopted the same SGFD stencils and determined the SGFD coefficients by minimizing the errors between the wavenumber responses of the SGFD operators and the first-order $k$ (wavenumber)-space operator in a least-squares (LS) sense. We solved the LS problems by performing a weighted pseudoinverse of nonsquare matrices to obtain the SGFD coefficients and to yield LS-based SGFD methods. We have developed an efficient LS implementation by using a small number of representative wavenumbers, which makes the computational time for estimating the coefficients negligible. Dispersion analysis and numerical examples demonstrate that our LS-based SGFD methods can preserve the original temporal accuracy and achieve better spatial accuracy than the existing TE-based SGFD methods. Extensive numerical tests proved that our LS-based methods do not damage the stability of the corresponding TE-based methods significantly.

Seismic waves are attenuated and distorted during the propagation because of the conversion of acoustic energy to heat energy. We focus on intrinsic attenuation, which is caused by Q. Amplitude attenuation can decrease the energy of the amplitude, and dispersion effects distort the phase of seismic waves. The Attenuation and dispersion effects can both reduce the resolution of image, and especially distort the real position of interfaces. On the basis of the viscoacoustic wave equation consisting of single standard linear solid, we derive a new viscoacoustic wave equation with decoupled amplitude attenuation and phase dispersion. Subsequently, we propose a theoretical framework of viscoacoustic reverse time migration that can compensate both the amplitude loss and phase dispersion. Compared to the other variable fractional Laplacian viscoacoustic wave equations with amplitude attenuation and phase dispersion terms decoupled, the order of the Laplacian operator in our equation is a constant. The amplitude attenuation term is solved by pseudospectral method, and only one fast Fourier transform is required in each time step. The phase dispersion term can be computed using a finite difference method. Numerical examples prove that our equation can accurately simulate the attenuation effects very well. Simulation of the new viscoacoustic equation shows high efficiency because only one constant fractional Laplacian operator exists in this new viscoacoustic wave equation, which can reduce the time of Inverse Fourier Transforms to improve the computation efficiency of forward modeling and Q-compensated reverse time migration (Q-RTM). We test the Q-RTM by using Marmousi and BP gas models and compare the Q-RTM images with those without compensation and attenuation (the reference image). Q-RTM results match well with the reference images. We also compare the field data migration images with and without compensation. Results demonstrate the accuracy and efficiency of the presented new viscoacoustic wave equation.

Q-compensated reverse time migration (Q-RTM) can compensate seismic attenuation caused by the anelastic behavior of subsurface media. Although, the traditional Q-RTM has high computational efficiency, it is instable because the high frequency or wavenumber ambient noise is exponentially boosted during forward and backward seismic wavefield propagation. The existing stable Q-RTM method costs twice as much computing time and memory compared to the traditional Q-RTM. In this letter, we propose a new Q-RTM method to address the above issues simultaneously. First, a new viscoacoustic wave equation is derived based on a wavefield decomposition method to obtain the velocity-dispersion-only and viscoacoustic wavefields efficiently. Then, a theoretical framework of stable and high-efficiency Q-RTM method is proposed based on the velocity-dispersion-only and viscoacoustic wavefields. The synthetic example shows that the new stable Q-RTM results match well with the reference images (without attenuation images). Moreover, the field data images also demonstrate the stability and high-efficiency of our proposed Q-RTM method.