A K-Space-Based Temporal Compensating Scheme for a First-Order Viscoacoustic Wave Equation with Fractional Laplace Operators

Abstract

Inherent constant Q attenuation can be described using fractional Laplacian operators. Typically, the fractional Laplacian viscoacoustic or viscoelastic wave equations are addressed utilizing the staggered-grid pseudo-spectral (SGPS) method. However, this approach results in time numerical dispersion errors due to the low-order finite difference approximation. In order to address these time-stepping errors, a k-space-based temporal compensating scheme is established to solve the first-order viscoacoustic wave equation. This scheme offers the advantage of being nearly free from grid dispersion for homogeneous media and enhances simulation stability. Numerical examples indicate that the proposed k-space scheme aligns well with analytical solutions for homogeneous media. Additionally, this method demonstrates excellent applicability for complex models and is more efficient due to its ability to adopt a larger time step compared with conventional staggered-grid pseudo-spectral methods.

1. Introduction

Capturing the characteristics of wave propagation with precision is highly significant for understanding and exploring underground structures in the field of geophysics [1,2,3]. Especially in viscous media, wave propagation involves both amplitude attenuation and phase dispersion [4,5], which have a substantial impact on seismic wave imaging [6,7] and inversion [8]. Conventional methods for simulating viscoacoustic wave equations, such as the finite difference method [2,9] and pseudo-spectral method [10,11,12,13,14,15], each possess unique advantages within their respective domains. However, these methods are not without challenges when it comes to long-time simulations, including time step errors and numerical dispersion [16], which could potentially compromise the accuracy and efficiency of the simulations.

In order to reduce the time dispersion error of the viscoacoustic wave equation, scholars have proposed numerous enhancement strategies, including the Lax–Wendroff approach [16,17,18], the rapid expansion method [19,20], the one-step method [21,22], the k-space method [23,24,25,26,27], the Fourier finite difference method [28,29], and various filter-based strategies for eliminating time dispersions from synthetic data [30,31,32,33,34]. This paper specifically focuses on the k-space method, known for its stability and ease in handling heterogeneous media.

The k-space method has its roots in scattering research [23,35] and is extensively utilized in bioengineering [36] (Cox et al., 2007) as well as in geophysical problems for simulation [37,38]. There are two principal approaches for computing the compensation operator of the k-space method: the finite difference (FD) or hybrid FD–spectral methods and the pure spectral method [39,40,41]. The former still deals with the grid dispersion and stability issues inherent in FD approximations. The latter produces dispersion-free wave fields for homogeneous media and demonstrates exceptional modeling stability [42]. However, when dealing with highly heterogeneous media, the computational cost of the spectral-based compensation operator becomes excessively high due to its space-wavenumber mixed-domain nature. To address this issue, Tabei et al. (2002) [43] proposed using the maximum velocity instead of a spatially variable velocity for a specific model. However, this kind of approach may not be appropriate for subsurface geological materials that often exhibit strong heterogeneity and considerable velocity contrasts. Fomel et al. (2013) [44] put forward the low-rank approximation approach to handle the mixed-domain issue in heterogeneous circumstances. The merit of this low-rank technique lies in its capacity to directly approximate the mixed-domain operator through a separable representation [45]. Nevertheless, contrary to pseudo-spectral (PS) methods, this method demands additional Fourier transforms.

In this paper, we introduce a new k-space formulation for the first-order decoupled fractional Laplacian wave equation. Over recent years, fractional-order equations have been attracting growing attention and have been employed in the numerical simulations of viscoelastic seismic wavefields [11,14] and heat distribution in a porous medium [46]. In the case of uniform media, the suggested approach removes grid dispersion and achieves high accuracy because it is based on the precise dispersion relation of the viscoacoustic wave equation. For heterogeneous media, we apply the low-rank method [44,47] to accurately represent the mixed-domain operators that arise from variations in Q and velocity. Despite the augmented complexity in computational calculations for the correction terms, there is no diminution in the accuracy of approximation, and the computational cost scarcely increases. The freshly introduced k-space method offers enhanced temporal accuracy and a more permissive stability criterion in contrast to traditional SGPS methods. This allows for high-precision wavefield extrapolation over extended time intervals, thus enhancing overall computational efficiency.

The remainder of this paper is structured as follows. We first examine the viscoacoustic wave equation [48] in relation to the second-order decoupled fractional Laplacians and its first-order equivalent. Following that, we provide a comprehensive explanation of the proposed first-order k-space formulation, along with a discussion on stability analysis and numerical implementation. Subsequently, the accuracy and computation efficiency are verified through three simulation examples. Finally, some conclusions are presented.

2. Theory

2.1. Viscoacoustic Wave Equation

Based on the frequency-independent Q model [49], the viscoacoustic wave equation consisting of fixed-power Laplacian operators [48] is expressed as

$${\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^{2}p}{\partial t^2}}={\nabla }^{2}p+\left(\gamma {\displaystyle \frac{{\omega }_0}{c}}{\left(-{\nabla }^2\right)}^{{\displaystyle \frac{1}{2}}}-\gamma {\displaystyle \frac{c}{{\omega }_0}}{\left(-{\nabla }^2\right)}^{{\displaystyle \frac{3}{2}}}\right)p+\left(-\pi \gamma {\displaystyle \frac{1}{c}}{\left(-{\nabla }^2\right)}^{{\displaystyle \frac{1}{2}}}+\pi {\gamma }^2{\displaystyle \frac{1}{{\omega }_0}}{\nabla }^2\right){\displaystyle \frac{\partial }{\partial t}}p,$$

(1)

where $p=p(\mathit{x},t)$ denotes the pressure wavefield, while $\mathit{x}=(x,z)$ represents the position-related vector; ${\nabla }^2$ corresponds to the Laplace operator, $c=c_0\cos \left({\displaystyle \frac{\pi \gamma }{2}}\right)$, where $\gamma ={\displaystyle \frac{1}{\pi }}\arctan \left({\displaystyle \frac{1}{Q}}\right)$ and $c_0$ is the wave propagation velocity defined at reference angular frequency ${\omega }_0$.

Because of the relation of Laplacian and gradient operators (${\nabla }^2=\nabla ·\nabla$ and $\nabla =\left({\displaystyle \frac{\partial }{\partial x}},{\displaystyle \frac{\partial }{\partial z}}\right)$), by defining $\mathit{v}=\left(v_x,v_z\right)=-{\displaystyle \frac{1}{\rho }}\nabla p$ and $u={\displaystyle \frac{\partial p}{\partial t}}$, Equation (1) is reformulated into the first-order equations:

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial u}{\partial t}}=\mathscr{H}\left({\displaystyle \frac{\partial v_x}{\partial x}}+{\displaystyle \frac{\partial v_z}{\partial z}}\right)\\ {\displaystyle \frac{\partial v_x}{\partial t}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial u}{\partial x}}\\ {\displaystyle \frac{\partial v_z}{\partial t}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial u}{\partial z}}\end{array}\right.,$$

(2)

where

$$\mathscr{H}=\rho \gamma c{\omega }_0{\left(-{\nabla }^2\right)}^{-{\displaystyle \frac{1}{2}}}-\rho c^2-\rho \gamma {\displaystyle \frac{c^3}{{\omega }_0}}{\left(-{\nabla }^2\right)}^{{\displaystyle \frac{1}{2}}}-\rho \pi \gamma c{\left(-{\nabla }^2\right)}^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{\partial }{\partial t}}-\rho \pi {\gamma }^2{c}^2{\displaystyle \frac{1}{{\omega }_0}}{\displaystyle \frac{\partial }{\partial t}},$$

(3)

in which $\rho$ is the density.

2.2. New First-Order Viscoacoustic Wave Equation Based on the K-Space Method

To tackle the problem of poor simulation time accuracy, we propose a compensation method based on the k-space operator. This method can eliminate the time dispersion caused by grid dispersion. In the domain, Equation (1) is expressed as follows:

$${\left(i\omega \right)}^2+a\left(i\omega \right)+b=0,$$

(4)

where $a=\pi \gamma ck+\pi {\gamma }^2{c}^2{\displaystyle \frac{1}{{\omega }_0}}{k}^2$ and $b=-\gamma c{\omega }_{0}k+c^2{k}^2+\gamma {\displaystyle \frac{c^3}{{\omega }_0}}{k}^3$. The variable $i\omega$ has the following solution:

$$i\omega ={\displaystyle \frac{-a\pm i\sqrt{4b-a^2}}{2}}.$$

(5)

Equation (5) is multiplied by wavefield $\tilde{p}=\tilde{p}\left(k,\omega \right)$ on both sides, and then, according to the Fourier relations of $i\omega \to {\displaystyle \frac{\partial }{\partial t}}$, the following time domain equation can be obtained:

$${\displaystyle \frac{\partial \tilde{p}}{\partial t}}-g\cdot \tilde{p}=0,$$

(6)

where $g={\displaystyle \frac{-a\pm i\sqrt{4b-a^2}}{2}}$. The analytical solution of Equation (6) [50] (Boyce et al., 2021) is

$${\tilde{p}}^t={\tilde{p}}^0{e}^{g\cdot t}.$$

(7)

The wavefields ${\tilde{p}}^{t+\Delta t}$ and ${\tilde{p}}^{t-\Delta t}$ can be evaluated by Equation (7). We can further determine that

$${\displaystyle \frac{{\tilde{p}}^{t+\Delta t}-2{\tilde{p}}^t+{\tilde{p}}^{t-\Delta t}}{\Delta t^2}}={\displaystyle \frac{2{\tilde{p}}^t\left(e^{-\frac{a}{2}\Delta t}\cos \left(\frac{\sqrt{4b-a^2}}{2}\Delta t\right)-1\right)}{\Delta t^2}}+{\displaystyle \frac{{\tilde{p}}^{t-\Delta t}\left(1-e^{-a\Delta t}\right)}{\Delta t^2}}.$$

(8)

The first-order equations equivalent to Equation (8) have the following form:

$$\left\{\begin{array}{l}{\displaystyle \frac{{\tilde{u}}^{t+\raisebox{1ex}{$\Delta t$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\tilde{u}}^{t-\raisebox{1ex}{$\Delta t$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\Delta t}}=-\rho c^2\left(i{k}_x{\tilde{v}}_x^t+i{k}_z{\tilde{v}}_z^t\right)+\varphi (k){\tilde{u}}^{t-\raisebox{1ex}{$\Delta t$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\\ {\displaystyle \frac{{\tilde{v}}_x^{t+\Delta t}-{\tilde{v}}_x^t}{\Delta t}}=-{\displaystyle \frac{1}{\rho }}i{k}_x\phi (k){\tilde{u}}^{t+\raisebox{1ex}{$\Delta t$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\\ {\displaystyle \frac{{\tilde{v}}_z^{t+\Delta t}-{\tilde{v}}_z^t}{\Delta t}}=-{\displaystyle \frac{1}{\rho }}i{k}_z\phi (k){\tilde{u}}^{t+\raisebox{1ex}{$\Delta t$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right.,$$

(9)

where ${\tilde{u}}^t={\displaystyle \frac{{\tilde{p}}^{t+\raisebox{1ex}{$\Delta t$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\tilde{p}}^{t-\raisebox{1ex}{$\Delta t$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\Delta t}}$, ${\tilde{v}}_x^t=-{\displaystyle \frac{1}{\rho }}i{k}_x\phi (k){\tilde{p}}^t$, ${\tilde{v}}_z^t=-{\displaystyle \frac{1}{\rho }}i{k}_z\phi (k){\tilde{p}}^t$, $\varphi (k)={\displaystyle \frac{e^{-a\Delta t}-1}{\Delta t}}$, and $\phi (k)={\displaystyle \frac{2e^{-\frac{a}{2}\Delta t}\cos \left(\frac{\sqrt{4b-a^2}}{2}\Delta t\right)-e^{-a\Delta t}-1}{-k^2{c}^2\Delta t^2}}$.

By Fourier transform, the first-order compensated viscoacoustic equations in the space-time domain are derived:

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial u^t}{\partial t}}=-\rho c^2\left({\displaystyle \frac{\partial v_x^t}{\partial x}}+{\displaystyle \frac{\partial v_z^t}{\partial z}}\right)+{\mathcal{F}}^{-1}\left\{\varphi (k){\tilde{u}}^{t-\raisebox{1ex}{$\Delta t$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right\}\\ {\displaystyle \frac{\partial v_x^{t+\raisebox{1ex}{$\Delta t$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\partial t}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{{\partial }^{\ast }{u}^{t+\raisebox{1ex}{$\Delta t$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\partial x}}\\ {\displaystyle \frac{\partial v_z^{t+\raisebox{1ex}{$\Delta t$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\partial t}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{{\partial }^{\ast }{u}^{t+\raisebox{1ex}{$\Delta t$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\partial z}}\end{array}\right.,$$

(10)

where ${\partial }_x^{\ast }={\mathcal{F}}^{-1}\left\{i{k}_x\phi (k)\mathcal{F}\left[·\right]\right\}$, and ${\partial }_z^{\ast }={\mathcal{F}}^{-1}\left\{i{k}_z\phi (k)\mathcal{F}\left[·\right]\right\}$.

2.3. Stability Analysis

To facilitate the analysis of the stability condition, Equation (8) is rewritten as the matrix recursive form

$$\left(\begin{array}{c}{\tilde{p}}^{t+\Delta t}\\ {\tilde{p}}^t\end{array}\right)=\left(\begin{array}{cc}2e^{-\frac{a}{2}\Delta t}\cos \left(\frac{\sqrt{4b-a^2}}{2}\Delta t\right)& -e^{-a\Delta t}\\ 1& 0\end{array}\right)\left(\begin{array}{c}{\tilde{p}}^t\\ {\tilde{p}}^{t-\Delta t}\end{array}\right)=\mathit{A}\left(\begin{array}{c}{\tilde{p}}^t\\ {\tilde{p}}^{t-\Delta t}\end{array}\right),$$

(11)

According to the eigenvalue method [51], the eigenvalue $\lambda$ of matrix A in Equation (11) must satisfy the condition $\left|\lambda \right|\le 1$ to ensure stability. In accordance with the characteristic polynomial $\left|A-\lambda E\right|=0$ of matrix A, its eigenvalues are

$$\lambda =e^{-\frac{a}{2}\Delta t}\cos \left(\frac{\sqrt{4b-a^2}}{2}\Delta t\right)\pm \sqrt{{\left[e^{-\frac{a}{2}\Delta t}\cos \left(\frac{\sqrt{4b-a^2}}{2}\Delta t\right)\right]}^2-e^{-a\Delta t}}.$$

(12)

In the event that $\left|\lambda \right|\le 1$, it is imperative to ensure that the following inequality is satisfied:

$${\left[e^{-\frac{a}{2}\Delta t}\cos \left(\frac{\sqrt{4b-a^2}}{2}\Delta t\right)\right]}^2-e^{-a\Delta t}\le 0,$$

(13)

which can be simplified as

$${\cos }^2\left(\frac{\sqrt{4b-a^2}}{2}\Delta t\right)\le 1.$$

(14)

This inequality is consistently satisfied, thus guaranteeing the simulation’s perpetual stability.

2.4. Numerical Implementation

Assuming a uniform medium, it becomes straightforward to extend wavefields using Equation (10) in conjunction with the SGPS method. The particle velocity and pressure wavefield are then modified in the following manner:

$$\left\{\begin{array}{l}{u}^{+}=u^{-}+\Delta t\left\{-\rho c^2\left({\displaystyle \frac{{\partial }^{-}{v}_x^{+}}{\partial x}}+{\displaystyle \frac{{\partial }^{-}{v}_z^{+}}{\partial z}}\right)+{\mathcal{F}}^{-1}\left[\varphi (k)\mathcal{F}\left(u^{-}\right)\right]\right\}\\ v_x^{+}=v_x^{-}+\left(-{\displaystyle \frac{\Delta t}{\rho }}{\displaystyle \frac{{\partial }^{\ast +}{u}^{-}}{\partial x}}\right)\\ v_z^{+}=v_z^{-}+\left(-{\displaystyle \frac{\Delta t}{\rho }}{\displaystyle \frac{{\partial }^{\ast +}{u}^{-}}{\partial z}}\right)\end{array}\right..$$

(15)

The operators ${\partial }_m^{\pm }$ and ${{\partial }_m^{\ast }}^{\pm }$ are expressed as

$${\partial }_m^{\pm }\mathit{u}={\mathcal{F}}_m^{-1}\left(i{k}_m{e}^{\pm i{k}_m\Delta m/2}{\mathcal{F}}_m\mathit{u}\right),{{\partial }_m^{\ast }}^{\pm }\mathit{u}={\mathcal{F}}^{-1}\left(i{k}_m{e}^{\pm i{k}_m\Delta m/2}\phi (k)\mathcal{F}\mathit{u}\right),m=xorz.$$

(16)

where $\mathit{u}={\left(u^{+},v_x^{-},v_z^{-}\right)}^T$, and the symbols ${\mathcal{F}}_m$ and ${\mathcal{F}}_m^{-1}$ represent the 1D Fourier and corresponding inverse transform, respectively. Similarly, the symbols $\mathcal{F}$ and ${\mathcal{F}}^{-1}$ are the 2D Fourier and its corresponding inverse transformation, respectively. In the e-exponential, the + and − indicate a shift of a half spatial interval to the left (up) and right (down), respectively. In vector $\mathit{u}$, the + and − refer to the current and previous time. To simulate wave propagation in heterogeneous media, it is essential to deal with mixed-domain operators $\phi (\mathit{x},k)$ and $\varphi (\mathit{x},k)$ in Equation (10). To approximate the mixed-domain operators, we use the low-rank decomposition method [44]. Take operator $\phi (\mathit{x},k)$ as an example, it has a matrix decomposition form as follows:

$$\phi (\mathit{x},k)={\mathit{A}}_{p\times l}{\mathit{C}}_{l\times n}{\mathit{B}}_{n\times q},$$

(17)

where the matrices ${\mathit{A}}_{p\times l}={\mathit{A}}_{p\times l}\left(\mathit{x},k_l\right)$ and ${\mathit{B}}_{n\times q}={\mathit{B}}_{n\times q}\left({\mathit{x}}_n,k\right)$ represent the l columns and n rows sub-matrices of $\phi (\mathit{x},k)$, while ${\mathit{C}}_{l\times n}$ represents the weighting matrix. With the matrix decomposition, the operator ${{\partial }_m^{\ast }}^{\pm }$ in Equation (15) can be denoted as

$${{\partial }_m^{\ast }}^{\pm }\mathit{u}\left(\mathit{x},t\right)={\mathcal{F}}^{-1}\left\{i{k}_m{e}^{\pm i{k}_m\Delta m/2}\phi (k)\mathcal{F}\left[\mathit{u}\left(\mathit{x},t\right)\right]\right\}\approx {\mathit{A}}_{p\times l}{\mathit{C}}_{l\times n}{\mathcal{F}}^{-1}\left\{i{k}_m{e}^{\pm i{k}_m\Delta m/2}{\mathit{B}}_{n\times q}\mathit{u}\left(k,t\right)\right\}.$$

(18)

The numerical implementation of homogeneous and heterogeneous media is described in detail in Algorithms 1 and 2.

Algorithm 1 The numerical implementation of homogeneous medium

Require: Simulation parameters

Output: Wavefields (u) and common-gather (r)

For t = 1: nt   Calculate velocity component vx using the SGPS method, where the symbol Lx represents the derivative with respect to the variable x;   vxnew = vxold − Lx (uold);   Calculate velocity component vz using the SGPS method, where the symbol Lz represents the derivative with respect to the variable z;   vzt+1 = vzold − Lz(uold);   Compute the spatial derivative using the SGPS method (L) and compensation term calculated by Fourier pseudo-spectral method (P);   unew = uold − L(vxnew, vznew) + P(uold);   Load the source (s);   unew = unew + s;   save common-gather (r)

end

Algorithm 2 The numerical implementation of heterogeneous medium

Require: Simulation parameters

Output: Wavefields (u) and common-gather (r)

For t = 1: nt

Calculate velocity component vx and vz using the low-rank method (LR);

vxnew = vxold − LR (uold);

vznew = vzt − LR(uold);

Compute the spatial derivative using the SGPS method (L) and compensation term calculated by low-rank method (LR);

unew = uold − L(vxnew, vznew) + LR(uold);

Load the source (s);

unew = unew + s;

save common-gather (r)

end

3. Numerical Examples

3.1. Homogeneous Model

The initial application of a homogeneous model is intended to verify the accuracy of the proposed scheme through the comparison of snapshots and traces. Simulations are performed using a two-dimensional model that has a reference velocity of 2000 m/s. The model consists of 200 × 200 cells featuring a uniform grid spacing of 20 m. The dominant frequency is set to 20 Hz. Figure 1a demonstrates that temporal dispersion influences the snapshots of the SGPS method, particularly as the time step increases. In contrast, Figure 1b indicates that the proposed k-space approach shows minimal dispersion effects even when employing a large time step.

In order to confirm the effectiveness, a comparison is made between the numerical outcome and the analytical solution acquired using the Green’s function method [52]. The receiver is positioned at a distance of 1800 m from the 20 Hz Ricker wavelet source. The time interval is 3 ms. To quantitatively evaluate the accuracy of the proposed k-space scheme, the following formula is used:

$$\mathrm{err}=\left[{\displaystyle \sum _{i=1}^N{\left(x_{ri}-x_{ni}\right)}^2}/{\displaystyle \sum _{i=1}^{nt}{x}_{ri}^2}\right]\times 100\%.$$

(19)

where $x_r$ and $x_n$ represent the reference trace and the calculated trace, respectively. N denotes the overall count of temporal sampling points. The results (Figure 2) demonstrate that the numerical results (red dash line) of the proposed k-space formulation are in close agreement with the analytical solutions (black line) across a broad range of Q, with relative errors remaining below 2% even in the presence of significant attenuation (Q = 10).

3.2. Two-Layer Model

A two-layer model is utilized to evaluate the applicability and efficiency of the proposed k-space method. The velocity model has dimensions of 400 × 400, with a grid spacing of 20 m. The P-wave velocity is determined at 20 Hz. The upper layer exhibits a velocity of 2000 m/s, whereas the lower layer has a velocity of 4000 m/s. The density and Q models are generated from the velocity model utilizing empirical functions of $\rho =310c^{0.25}$ (Castagna et al., 1993) and $Q=3.516{\left({\displaystyle \frac{c}{1000}}\right)}^{2.2}$ [53]. The model interface is located at a depth of 3 km. A Ricker wavelet featuring a dominant frequency of 25 Hz is introduced at (2.4 km, 4 km). Figure 3 presents snapshots of the wavefield at 1 s. A reference snapshot is depicted in Figure 3a, while Figure 3b,c illustrate snapshots generated by the conventional SGPS method with time intervals of 1 ms and 2 ms, respectively. Figure 3d,e exhibit snapshots computed by the suggested approach with time steps of 2 ms and 3 ms, respectively. All snapshots are presented within the identical range of amplitudes. In Figure 3c, discernible disturbances can be witnessed due to discretization errors when a time step of 2 ms is employed. Reducing the time step to 1 ms in the SGPS method markedly reduces waveform distortion (as shown in Figure 3b). Subsequently, increasing the time step to 3 ms reveals that the snapshots from the k-space method remain precise (as seen in Figure 3e). The SGPS method fails as it does not fulfill the stability conditions when such a time step is utilized. Figure 4 shows seismic traces recorded at (1.6 km, 2.36 km) within a time frame of 1.0–1.6 s. Similar to the observations from wavefield snapshots, considerable numerical dispersion can be noted in traces obtained by the SGPS method with a time step of 2 ms (Figure 4b), whereas reducing this to 1 ms leads to a substantial decrease in the relative error from 25.95% to 1.90% (Figure 4a). Seismic traces generated by the k-space method with time steps of 2 ms and 3 ms are all highly accurate, and the seismic trace generated with 2 ms is even more accurate than the trace generated by the SGPS method with 1 ms.

3.3. Marmousi Model

To evaluate the applicability of the proposed approach for a more realistic model featuring a complex structure, a highly heterogeneous Marmousi model is utilized. The velocity model is shown in Figure 5. Q-model and density are transformed from velocity. This model encompasses 663 × 234 grid points arranged with a consistent spacing of 20 m. A Ricker wavelet, characterized by a dominant frequency of 20 Hz, is introduced at the coordinates (6.64 km, 0.04 km). We present the wavefield snapshots at 1.5 s (Figure 6). The first row displays a reference snapshot, and the second row shows the snapshots generated by the traditional SGPS method (time step of 0.5 ms) and the proposed k-space approach (time step of 1 ms), respectively. The third row presents the residuals between the reference and SGPS method/k-space methods. The smaller residuals in Figure 6f suggest that the proposed method maintains a higher simulation accuracy compared with the traditional method when the time step is larger. The common shot gather in Figure 7 also corroborates the same conclusion. To further substantiate the superiority of the proposed k-space method, the traces at x = 7 km are extracted and shown in Figure 8. The relative error values demonstrate that the time step of the k-space approach is twice that of the traditional one, yet still attains an enhancement in accuracy approaching 10 times. This undoubtedly substantiates the superiority of the simulation accuracy of the proposed scheme.

3.4. Field Model

We finally demonstrate the applicability of our proposed method to the field model. The velocity and Q models are shown in Figure 9a and Figure 9b, respectively. The model is divided into 1000 × 300 cells, featuring a spatial interval of 20 m. A Ricker wavelet with a dominant frequency of 20 Hz is introduced at the coordinates (12.04 km, 0.04 km), and a total of 480 receivers are placed on the free surface at intervals of 20 m. Figure 10 illustrates the seismic records and seismic traces, where Figure 10a and Figure 10b are obtained through the proposed method with a 2 ms time step and SGPS method with a 1 ms time step, respectively. We observe that the discrepancy between the seismogram derived from the proposed method and the reference seismogram is less pronounced than the discrepancy observed between the seismogram obtained through the SGPS method and the reference seismogram. To provide a quantitative assessment of the accuracy, seismic traces are extracted at an offset of 2 km. We find that the seismic traces calculated using the proposed method exhibit a relative error of 2.04% in comparison to the reference traces, while the seismic traces computed using the SGPS method differ significantly from the reference traces, with a relative error of 4.31%.

4. Discussion

For heterogeneous models, we discuss the computation time. In this case, the simulation time is slightly increased because a low-rank decomposition is required to process the compensated mixed-domain operators. A detailed comparison of the computation times for the Marmousi and field models is presented in

Table 1. Computation time for Marmousi model.

Numerical Scheme	Time Step (ms)	Relative Error (%)	Computation Time (s)
SGPS	0.5	2.66	905.19
k-space	1	0.23	852.45
k-space	1.5	2.43	568.37

and

Table 2. Computation time for field model.

Numerical Scheme	Time Step (ms)	Relative Error (%)	Computation Time (s)
SGPS	0.75	2.18	6.92
SGPS	1	4.31	5.19
k-space	2	2.04	4.50

, respectively. The former was simulated on a Windows operating system using MATLAB R2023b. The latter was implemented using CUDA programs on an Intel workstation. As illustrated in Table 1, the simulation accuracy of the SGPS method and the proposed method is equivalent; for instance, the trace calculated by the SGPS method at the 0.5 ms time step is identical to that obtained by the proposed method at the 1.5 ms time step. In this instance, the calculation time of the proposed method is merely 0.6 times that of the SGPS method. Similarly, in Table 2, when the simulation accuracy of the two methods is the same for the field model, the SGPS method requires a relatively small time step of 0.75 ms, whereas the proposed method employs a considerably larger time step of 2 ms. This results in a 35% reduction in computation time compared to the SGPS method.

5. Conclusions

A k-space temporal high-precision scheme has been developed for the solution of the first-order decoupled fractional Laplacian viscoacoustic wave equation. The merit of the presented scheme lies in the complete elimination of the numerical dispersion error for homogeneous media. In contrast to the conventional PS method, the suggested k-space method demonstrates greater accuracy and stability, as the relaxed stability condition allows for larger time steps in simulations. Two numerical examples conducted on heterogeneous media further confirm that the proposed approach is able to improve computational efficiency by roughly 2.0 times for 2D modeling. As a consequence, the suggested approach for efficient viscoacoustic modeling holds the potential to facilitate the advancement of practical migration and full wave inversion techniques.