Efficient modeling of wave propagation in a vertical transversely isotropic attenuative medium based on fractional Laplacian

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. T121-T131 ◽  
Author(s):  
Tieyuan Zhu ◽  
Tong Bai
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Fractional Laplacian ◽  
Numerical Solutions ◽  
Pseudospectral Method ◽  
Transversely Isotropic ◽  
Seismic Attenuation ◽  
Fourier Pseudospectral Method ◽  
Frequency Independent ◽  
Theoretical Predictions

To efficiently simulate wave propagation in a vertical transversely isotropic (VTI) attenuative medium, we have developed a viscoelastic VTI wave equation based on fractional Laplacian operators under the assumption of weak attenuation ([Formula: see text]), where the frequency-independent [Formula: see text] model is used to mathematically represent seismic attenuation. These operators that are nonlocal in space can be efficiently computed using the Fourier pseudospectral method. We evaluated the accuracy of numerical solutions in a homogeneous transversely isotropic medium by comparing with theoretical predictions and numerical solutions by an existing viscoelastic-anisotropic wave equation based on fractional time derivatives. To accurately handle heterogeneous [Formula: see text], we select several [Formula: see text] values to compute their corresponding fractional Laplacians in the wavenumber domain and interpolate other fractional Laplacians in space. We determined its feasibility by modeling wave propagation in a 2D heterogeneous attenuative VTI medium. We concluded that the new wave equation is able to improve the efficiency of wave simulation in viscoelastic-VTI media by several orders and still maintain accuracy.

Numerical simulation of seismic wave propagation in viscoelastic-anisotropic media using frequency-independent Q wave equation

Geophysics ◽  
2017 ◽  
Vol 82 (4) ◽  
pp. WA1-WA10 ◽  
Author(s):  
Tieyuan Zhu
Keyword(s):  
Numerical Modeling ◽  
Wave Propagation ◽  
Wave Equation ◽  
Inhomogeneous Medium ◽  
Seismic Anisotropy ◽  
Numerical Solutions ◽  
Elastic Anisotropy ◽  
Transversely Isotropic ◽  
Theoretical Formula ◽  
Frequency Independent

Seismic anisotropy is the fundamental phenomenon of wave propagation in the earth’s interior. Numerical modeling of wave behavior is critical for exploration and global seismology studies. The full elastic (anisotropy) wave equation is often used to model the complexity of velocity anisotropy, but it ignores attenuation anisotropy. I have presented a time-domain displacement-stress formulation of the anisotropic-viscoelastic wave equation, which holds for arbitrarily anisotropic velocity and attenuation [Formula: see text]. The frequency-independent [Formula: see text] model is considered in the seismic frequency band; thus, anisotropic attenuation is mathematically expressed by way of fractional time derivatives, which are solved using the truncated Grünwald-Letnikov approximation. I evaluate the accuracy of numerical solutions in a homogeneous transversely isotropic (TI) medium by comparing with theoretical [Formula: see text] and [Formula: see text] values calculated from the Christoffel equation. Numerical modeling results show that the anisotropic attenuation is angle dependent and significantly different from the isotropic attenuation. In synthetic examples, I have proved its generality and feasibility by modeling wave propagation in a 2D TI inhomogeneous medium and a 3D orthorhombic inhomogeneous medium.

A generalization of the Fourier pseudospectral method

Geophysics ◽  
2010 ◽  
Vol 75 (6) ◽  
pp. A53-A56 ◽  
Author(s):  
José M. Carcione
Keyword(s):  
Wave Equation ◽  
Fractional Laplacian ◽  
Differential Operators ◽  
Density Wave ◽  
Pseudospectral Method ◽  
Uniform Density ◽  
Fourier Pseudospectral Method ◽  
Spatial Derivatives ◽  
Fourier Pseudospectral ◽  
Derivatives Of

The Fourier pseudospectral (PS) method is generalized to the case of derivatives of nonnatural order (fractional derivatives) and irrational powers of the differential operators. The generalization is straightforward because the calculation of the spatial derivatives with the fast Fourier transform is performed in the wavenumber domain, where the operator is an irrational power of the wavenumber. Modeling constant-[Formula: see text] propagation with this approach is highly efficient because it does not require memory variables or additional spatial derivatives. The classical acoustic wave equation is modified by including those with a space fractional Laplacian, which describes wave propagation with attenuation and velocity dispersion. In particular, the example considers three versions of the uniform-density wave equation, based on fractional powers of the Laplacian and fractional spatial derivatives.

Staggered mesh for the anisotropic and viscoelastic wave equation

Geophysics ◽  
1999 ◽  
Vol 64 (6) ◽  
pp. 1863-1866 ◽  
Author(s):  
José M. Carcione
Keyword(s):  
Wave Equation ◽  
Differential Operators ◽  
Dynamic Range ◽  
Pseudospectral Method ◽  
Transversely Isotropic ◽  
Staggered Grid ◽  
Physical Situation ◽  
Fourier Pseudospectral Method ◽  
Regular Grids ◽  
Grid Solution

Computation of the spatial derivatives with nonlocal differential operators, such as the Fourier pseudospectral method, may cause strong numerical artifacts in the form of noncausal ringing. This situation happens when regular grids are used. The problem is attacked by using a staggered pseudospectral technique, with a different scheme for each rheological relation. The nature and causes of acausal ringing in regular grid methods and the reasons why staggered‐grid methods eliminate this problem are explained in papers by Fornberg (1990) and Özdenvar and McMechan (1996). Thus, the objective here is not to propose a new method but to develop the algorithm for the viscoelastic and transversely isotropic (VTI) wave equation, for which the technique can be implemented without interpolation. The algorithm is illustrated for one physical situation that requires very high accuracy, such as a fluid‐solid interface, where very large contrasts in material properties occur. The staggered‐grid solution is noise free in the dynamic range where regular grids generate artifacts that may have amplitudes similar to those of physical arrivals.

Chebyshev multidomain pseudospectral method to solve cardiac wave equations with rotational anisotropy

2018 ◽  
Vol 09 (04) ◽  
pp. 1850025 ◽  
Author(s):  
Jairo Rodríguez-Padilla ◽  
Daniel Olmos-Liceaga
Keyword(s):  
Wave Propagation ◽  
Numerical Methods ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Pseudospectral Method ◽  
Finite Difference Schemes ◽  
Cardiac Models ◽  
Computational Resources ◽  
Realistic Geometries

The implementation of numerical methods to solve and study equations for cardiac wave propagation in realistic geometries is very costly, in terms of computational resources. The aim of this work is to show the improvement that can be obtained with Chebyshev polynomials-based methods over the classical finite difference schemes to obtain numerical solutions of cardiac models. To this end, we present a Chebyshev multidomain (CMD) Pseudospectral method to solve a simple two variable cardiac models on three-dimensional anisotropic media and we show the usefulness of the method over the traditional finite differences scheme widely used in the literature.

Modeling viscoacoustic wave propagation using a new spatial variable-order fractional Laplacian wave equation

Geophysics ◽  
2021 ◽  
pp. 1-74
Author(s):  
Xinru Mu ◽  
Jianping Huang ◽  
Lei Wen ◽  
Subin Zhuang
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Fractional Order ◽  
Fractional Laplacian ◽  
Spatial Variable ◽  
Variable Order ◽  
New Wave ◽  
Dissipation Term ◽  
Laplacian Operators ◽  
Dispersion Term

We propose a new time-domain viscoacoustic wave equation for simulating wave propagation in anelastic media. The new wave equation is derived by inserting the complex-valued phase velocity described by the Kjartansson attenuation model into the frequency-wavenumber domain acoustic wave equation. Our wave equation includes one second-order temporal derivative and two spatial variable-order fractional Laplacian operators. The two fractional Laplacian operators describe the phase dispersion and amplitude attenuation effects, respectively. To facilitate the numerical solution for the proposed wave equation, we use the arbitrary-order Taylor series expansion (TSE) to approximate the mixed domain fractional Laplacians and achieve the decoupling of the wavenumber and the fractional order. Then the proposed viscoacoustic wave equation can be directly solved using the pseudospectral method (PSM). We adopt a hybrid pseudospectral/finite-difference method (HPSFDM) to stably simulate wave propagation in arbitrarily complex media. We validate the high accuracy of the proposed approximate dispersion term and approximate dissipation term in comparison with the accurate dispersion term and accurate dissipation term. The accuracy of numerical solutions is evaluated by comparison with the analytical solutions in homogeneous media. Theory analysis and simulation results show that our viscoacoustic wave equation has higher precision than the traditional fractional viscoacoustic wave equation in describing constant- Q attenuation. For a model with Q < 10, the calculation cost for solving the new wave equation with TSE HPSFDM is lower than that for solving the traditional fractional-order wave equation with TSE HPSFDM under the high numerical simulation precision. Furthermore, we demonstrate the accuracy of HPSFDM in heterogeneous media by several numerical examples.

A spectral scheme for wave propagation simulation in 3-D elastic‐anisotropic media

Geophysics ◽  
1992 ◽  
Vol 57 (12) ◽  
pp. 1593-1607 ◽  
Author(s):  
José M. Carcione ◽  
Dan Kosloff ◽  
Alfred Behle ◽  
Geza Seriani
Keyword(s):  
Wave Propagation ◽  
Fourier Method ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Computational Effort ◽  
Second Order ◽  
Rapid Expansion ◽  
Integration Algorithm ◽  
Fourier Pseudospectral Method ◽  
Band Limited

This work presents a new scheme for wave propagation simulation in three‐dimensional (3-D) elastic-anisotropic media. The algorithm is based on the rapid expansion method (REM) as a time integration algorithm, and the Fourier pseudospectral method for computation of the spatial derivatives. The REM expands the evolution operator of the second‐order wave equation in terms of Chebychev polynomials, constituting an optimal series expansion with exponential convergence. The modeling allows arbitrary elastic coefficients and density in lateral and vertical directions. Numerical methods which are based on finite‐difference techniques (in time and space) are not efficient when applied to realistic 3-D models, since they require considerable computer memory and time to obtain accurate results. On the other hand, the Fourier method permits a significant reduction of the working space, and the REM algorithm gives machine accuracy with half the computational effort as the usual second-order temporal differencing scheme. The new algorithm provides spectral accuracy for band limited wave propagation with no temporal or spatial dispersion. Hence, the combination REM/Fourier method could be considered at present the fastest and the most accurate of all the algorithms based on spectral methods in terms of efficiency of computer time. The technique is successfully tested with the analytical solution in the symmetry axis of a 3-D homogeneous transversely isotropic solid. Computed radiation patterns in clay shale and sandstone show the characteristics predicted by the theory. A final example computes synthetic seismograms showing the effects of shear‐wave splitting of a model where an azimuthally anisotropic cracked shale layer is inside a transversely isotropic sandstone.

A separable strong-anisotropy approximation for pure qP-wave propagation in transversely isotropic media

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. C337-C354 ◽  
Author(s):  
Jörg Schleicher ◽  
Jessé C. Costa
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Dispersion Relation ◽  
Transversely Isotropic ◽  
Wave Dispersion ◽  
Low Rank ◽  
S Wave ◽  
Finite Difference Approximations ◽  
Transversely Isotropic Media ◽  
Isotropic Media

The wave equation can be tailored to describe wave propagation in vertical-symmetry axis transversely isotropic (VTI) media. The qP- and qS-wave eikonal equations derived from the VTI wave equation indicate that in the pseudoacoustic approximation, their dispersion relations degenerate into a single one. Therefore, when using this dispersion relation for wave simulation, for instance, by means of finite-difference approximations, both events are generated. To avoid the occurrence of the pseudo-S-wave, the qP-wave dispersion relation alone needs to be approximated. This can be done with or without the pseudoacoustic approximation. A Padé expansion of the exact qP-wave dispersion relation leads to a very good approximation. Our implementation of a separable version of this equation in the mixed space-wavenumber domain permits it to be compared with a low-rank solution of the exact qP-wave dispersion relation. Our numerical experiments showed that this approximation can provide highly accurate wavefields, even in strongly anisotropic inhomogeneous media.

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