Fractional Laplacian Viscoacoustic Wave Equation Research Articles

Source-independent Q-compensated viscoacoustic least-squares reverse time migration

The strong viscosity of the subsurface introduces amplitude absorption and phase-velocity dispersion. Incorrect compensation of the inherent attenuation (the strength of the seismic attenuation can be quantified by the inverse of the quality factor Q, which is defined as [Formula: see text] times the ratio of the stored energy to the lost energy in a single cycle of deformation) can significantly affect imaging quality. Although Q-least-squares reverse time migration ( Q-LSRTM) allows for the compensation of attenuation effects during the iterations, the traditional [Formula: see text]-norm minimization, which is highly sensitive to the source wavelet, poses a challenge in accurately estimating the source wavelet from the field data. Thus, we develop a source-independent Q-LSRTM, in which a convolutional objective function is introduced to replace the [Formula: see text]-norm constraint to mitigate the source wavelet effect. According to the Born approximation, we first linearize the constant-order decoupled fractional Laplacian viscoacoustic wave equation to derive the demigration operator and then construct the corresponding adjoint equation and gradient based on the convolutional objective function, iteratively estimating the reflectivity images. Our method relaxes the sensitivity to the wavelet compared with the conventional [Formula: see text]-norm scheme due to the convolutional objective function, which has the ability to construct the same new source for simulated and observed data. Numerical tests on a layered model, the Marmousi model, and field data demonstrate that our source-independent Q-LSRTM enables us to obtain high-quality reflectivity images even when using incorrect source wavelets.

Efficient modeling of fractional Laplacian viscoacoustic wave equation with fractional finite-difference method

The fractional viscoacoustic/viscoelastic wave equation, which accurately quantifies the frequency-independent anelastic effects, has been the focus of seismic industry in recent years. The pseudo-spectral (PS) method stands as one of the most widely used numerical methods for solving the fractional wave equation. However, the PS method often suffers from low accuracy and efficiency, particularly when modeling wave propagation in heterogeneous media. To address these issues, we propose a novel and efficient fractional finite-difference (FD) method for solving the wave equation with fractional Laplacian operators. This method develops an arbitrary high-order FD operator via the generating function of our fractional FD (F-FD) scheme, enhancing accuracy with L2-optimal FD coefficients. Similar to classic FD methods, our F-FD method is characterized by straightforward programming and excellent 3D extensibility. It surpasses the PS method by eliminating the need for Fast Fourier Transform (FFT) and inverse-FFT (IFFT) operations at each time step, offering significant benefits for 3D applications. Consequently, the F-FD method proves more adept for wave-equation-based seismic data processes like imaging and inversion. Compared with existing F-FD methods, our approach uniquely approximates the entire fractional Laplacian operator and stands as a local numerical algorithm, with an adjustable F-FD operator order based on model parameters for enhanced practicality. Accuracy analyses confirm that our method matches the precision of the PS method with a correctly ordered F-FD operator. Numerical examples show that the proposed method has good applicability for complex models. Finally, we have carried out reverse time migration on the Marmousi-2 model, and the imaging profiles indicate that the proposed method can be effectively applied to seismic imaging, demonstrating good practicability.

Visco-acoustic full waveform inversion using decoupled fractional Laplacian constant-Q wave equation and optimal transport-based misfit function

Full waveform inversion (FWI) has the potential to recover high-resolution physical parameters of the Earth from seismic data. Considering the attenuation effects of the subsurface, we develop a new visco-acoustic FWI method in the time domain with a known Q model. The seismic wavefield is modelled by solving a decoupled fractional Laplacian visco-acoustic wave equation, which can better describe amplitude attenuation and phase distortion during wave propagation. To weaken the initial model dependence of visco-acoustic FWI, we introduce the optimal transport-based misfit functional which can measure the data residual globally. Numerical examples on the 2D Marmousi model demonstrate the superiority of the proposed method compared with the L2 norm-based method which measures the misfit of observed and synthetic seismic data locally, generating cycle-skipping issue. The inversion results of seismic data without low-frequency components further demonstrate the validity and effectiveness of the proposed method, which is insensitive towards low-frequency components, and may achieve reasonable inversion results for field data application.

Source Wavefield Reconstruction in Fractional Laplacian Viscoacoustic Wave Equation-Based Full Waveform Inversion

We develop a fractional Laplacian viscoacoustic wave equation-based full waveform inversion (FWI) method. The main novelty is efficient reconstruction of the source wavefields in gradient computation by the adjoint state method. Our FWI is based on Fourier pseudo-spectral time-domain (PSTD) numerical solutions of the fractional Laplacian viscoacoustic wave equation, which can describe the frequency independent <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${Q}$ </tex-math></inline-formula> (quality factor) behaviors of seismic waves accurately. The presented wavefield reconstruction strategy utilizes reverse time-marching formulae to compute the source wavefields backward in time, including an implicit formula in the interior domain and an explicit one in the perfectly matched layer (PML) absorbing domains. Since the wavefields in the entire domain are recovered, our method does not require storing massive boundary values like conventional reconstruction methods. To avoid numerical instability in reverse time reconstruction, we design a global-local checkpointing technique. When reverse time reconstruction encounters numerical instability, the forward propagation restarts from the nearest global checkpoint and proceeds to the unstable time step. Then, the reverse time reconstruction continues. The local checkpoints help to delay the numerical instability in the PML domains. Numerical examples verify the feasibility of our reconstruction strategy and the efficiency gain achieved by using this strategy in FWI.

Solving fractional Laplacian visco-acoustic wave equations on complex-geometry domains using Grünwald-formula based radial basis collocation method

We propose a radial basis function collocation method (RBF method) to solve fractional Laplacian visco-acoustic wave equation for the Earth media having heterogeneous velocity model and complex geometry. Unlike the fractional Laplacian wave equation proposed in Zhu and Harris (2014), the wave equation we consider has a different definition for the fractional Laplacian. Specifically, spectral and Riesz fractional Laplacians are considered in Zhu and Harris (2014) and the present paper, respectively. Accordingly, the Fourier pseudospectral method (FPS method) and the RBF method are employed to solve the spectral and the Riesz fractional Laplacian wave equations. The two wave equations are observed to produce obviously different wavefields. We demonstrate the validity and flexibility of the proposed RBF method by considering five benchmarks of seismic forward modeling: (1) two-dimensional Earth media with four types of velocity models (homogeneous, two-layer, homogeneous but complex-geometry, and heterogeneous models) and (2) a spherical medium with homogeneous velocity model. We make a three-way comparison among numerical solutions to the Riesz fractional Laplacian, the spectral fractional Laplacian, and the integer-order visco-acoustic wave equations, and observe that when wave attenuation is weak the Riesz wave equation yields more similar wavefield to that of the integer-order wave equation than the spectral wave equation does. Furthermore, uniform and quasi-uniform layouts for collocation points of the RBF method are considered, and the latter layout turns out to be economical since it can preserve the solution accuracy with the minimum number of collocation points. The RBF method is truly mesh-free and dimension-free and can easily handle high-dimensional, irregular domains. Additionally, the method is easier to implement than element-based methods, such as finite element and spectral element methods, for discretizing the Riesz fractional Laplacian.

A matrix-transform numerical solver for fractional Laplacian viscoacoustic wave equation

We have developed a matrix-transform method (MTM) to numerically solve the fractional Laplacian constant-[Formula: see text] viscoacoustic wave equation. The new method is based on a matrix representation of the fractional Laplacians, and it is different from the traditional Fourier-spectrum representation. With the MTM approach, the viscoacoustic wave equation is discretized in the space domain; thus, the periodic boundary caused by the Fourier-transform method can be avoided. Spatial discretization offers great convenience for handling various boundary conditions. In the MTM scheme, the fractional power of a matrix needs to be computed, and it can be realized with the help of eigenvalue decomposition (EVD). However, the application of EVD to a huge matrix is uneconomical and impractical. To avoid forming any huge matrix explicitly, we adopted the Lanczos approximation to compute the matrix-vector product directly. The key step in the Lanczos approximation is the involved Lanczos decomposition, which is implemented by an iterative flow. We determine how to choose the iteration number in the Lanczos decomposition to balance the wavefield simulation accuracy and the computational cost with the help of numerical examples. We also analyze the stability condition of MTM numerically, and we compare its computational cost with that of the traditional Fourier-transform method.

Effective Q-compensated reverse time migration using new decoupled fractional Laplacian viscoacoustic wave equation

Seismic waves are attenuated and distorted during propagation because of the conversion of acoustic energy to heat energy. We focus on intrinsic attenuation, which is caused by [Formula: see text], which is the portion of energy lost during each cycle or wavelength. Amplitude attenuation can decrease the energy of the wavefields, and dispersion effects distort the phase of seismic waves. Attenuation and dispersion effects can reduce the resolution of image, and they can especially distort the real position of interfaces. On the basis of the viscoacoustic wave equation consisting of a single standard linear solid, we have derived a new viscoacoustic wave equation with decoupled amplitude attenuation and phase dispersion. Subsequently, we adopt a theoretical framework of viscoacoustic reverse time migration that can compensate the amplitude loss and the phase dispersion. Compared with the other variable fractional Laplacian viscoacoustic wave equations with decoupled amplitude attenuation and phase dispersion terms, 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 indicates high efficiency because only one constant fractional Laplacian operator exists in this new viscoacoustic wave equation, which can reduce the number of inverse Fourier transforms to improve the computation efficiency of forward modeling and [Formula: see text]-compensated reverse time migration ([Formula: see text]-RTM). We tested the [Formula: see text]-RTM by using Marmousi and BP gas models and compared the [Formula: see text]-RTM images with those without compensation and attenuation (the reference image). [Formula: see text]-RTM results match well with the reference images. We also compared the field data migration images with and without compensation. Results demonstrate the accuracy and efficiency of the presented new viscoacoustic wave equation.

Fractional Laplacian Viscoacoustic Wave Equation Low-Rank Temporal Extrapolation

Description of seismic wave attenuation is a hot topic in the geophysical area, and it is the basis of the attenuation-compensated seismic imaging technique, which aims to retrieve a high-resolution subsurface image for geological structure analysis and hydrocarbon reservoir prediction. The numerical simulation of viscoacoustic wave equation is an effective way to observe the seismic attenuation in lossy media. The existing study confirms that the seismic-quality-factor (Q) that is used to represent the strength of the viscous behavior of the earth is nearly independent of frequency, which is referred to as the constant-Q (CQ) model. The mathematical concept of fractional Laplacian is recently introduced to the geophysical area to form a compact CQ wave equation to describe seismic wave propagation. However, numerically solving the fractional Laplacian CQ wave equation by the traditional pseudospectral time-domain (PSTD) method suffers from a strict stability condition and great numerical dispersion due to a low-order temporal finite-difference (FD) approximation. To improve the temporal extrapolation accuracy, we derive the analytical wavenumber (k)-space domain propagators underlying the fractional Laplacian wave equation. We regard the k-space operators as mixed-domain matrices in the case of heterogeneous media and adopt a low-rank decomposition to approximate the matrices. With the low-rank approximation, an efficient time-marching formula is obtained for wavefield temporal extrapolation. We formulate the time-marching formula into a first-order equation system in terms of pressure and particle-velocity to welcome the perfectly matched layer (PML) absorbing boundary condition to eliminate the wraparound effects caused by the Fourier transform. A spatial-variable density is also incorporated to simulate more realistic amplitude variation. The numerical examples are carried out to verify the accuracy and stability of the viscoacoustic low-rank extrapolation.

A constant fractional-order viscoelastic wave equation and its numerical simulation scheme

Efficient modeling schemes currently exist to handle the spatially variable-order fractional Laplacians in the fractional Laplacian viscoacoustic wave equation. The simplest approach is to change the spatially variable-order fractional Laplacians into a linear combination of several constant fractional-order Laplacians. We generalize the constant fractional-order scheme to a spatially variable fractional-order viscoelastic wave equation and develop an almost-equivalent constant fractional-order viscoelastic wave equation. Our constant fractional-order scheme avoids the simulation error introduced by directly averaging the spatially varying fractional order; thus, our scheme simulates seismic wave propagation in viscoelastic media with sharp [Formula: see text] contrasts well. The fast Fourier transform is used in the approximation of the fractional Laplacians, which improves the spectral accuracy in space. Several simulation examples are performed to verify that the numerical solution of a homogeneous [Formula: see text] model obtained by solving our constant fractional-order viscoelastic wave equation agrees well with that obtained by solving the original viscoelastic wave equation. The numerical simulations for spatially varying [Formula: see text] models obtained by the new wave equation are more straightforward than those currently in use and match the reference solutions obtained by accurate, but inefficient, methods. This match of simulation results verifies the accuracy of our viscoelastic wave equation.

Locally solving fractional Laplacian viscoacoustic wave equation using Hermite distributed approximating functional method

Accurate seismic modeling in realistic media serves as the basis of seismic full-waveform inversion and imaging. Recently, viscoacoustic seismic modeling incorporating attenuation effects has been performed by solving a fractional Laplacian viscoacoustic wave equation. In this equation, attenuation, being spatially heterogeneous, is represented partially by the spatially varying power of the fractional Laplacian operator previously approximated by the global Fourier method. We have developed a local-spectral approach, based on the Hermite distributed approximating functional (HDAF) method, to implement the fractional Laplacian in the viscoacoustic wave equation. Our approach combines local methods’ simplicity and global methods’ accuracy. Several numerical examples are developed to evaluate the feasibility and accuracy of using the HDAF method for the frequency-independent [Formula: see text] fractional Laplacian wave equation.

Two efficient modeling schemes for fractional Laplacian viscoacoustic wave equation

Recently, a decoupled fractional Laplacian viscoacoustic wave equation has been developed based on the constant-[Formula: see text] 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 ([Formula: see text]). We also compared computational efficiency of our two modeling schemes using numerical tests.

Efficient reverse time migration based on fractional Laplacian viscoacoustic wave equation

Efficient reverse time migration based on fractional Laplacian viscoacoustic wave equation