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.

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.

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.

Nearly perfectly matched layer absorber for viscoelastic wave equations

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. T335-T345
Author(s):  
Enjiang Wang ◽  
José M. Carcione ◽  
Jing Ba ◽  
Mamdoh Alajmi ◽  
Ayman N. Qadrouh
Keyword(s):  
Wave Equations ◽  
Pseudospectral Method ◽  
Perfectly Matched Layer ◽  
Zener Model ◽  
Fourier Pseudospectral Method ◽  
Time Stepping ◽  
First Case ◽  
Stress Strain Relation ◽  
Viscoelastic Wave ◽  
Spatial Derivatives

We have applied the nearly perfectly matched layer (N-PML) absorber to the viscoelastic wave equation based on the Kelvin-Voigt and Zener constitutive equations. In the first case, the stress-strain relation has the advantage of not requiring additional physical field (memory) variables, whereas the Zener model is more adapted to describe the behavior of rocks subject to wave propagation in the whole frequency range. In both cases, eight N-PML artificial memory variables are required in the absorbing strips. The modeling simulates 2D waves by using two different approaches to compute the spatial derivatives, generating different artifacts from the boundaries, namely, 16th-order finite differences, where reflections from the boundaries are expected, and the staggered Fourier pseudospectral method, where wraparound occurs. The time stepping in both cases is a staggered second-order finite-difference scheme. Numerical experiments demonstrate that the N-PML has a similar performance as in the lossless case. Comparisons with other approaches (S-PML and C-PML) are carried out for several models, which indicate the advantages and drawbacks of the N-PML absorber in the anelastic case.

scholarly journals Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Ming-Sheng Hu ◽  
Ravi P. Agarwal ◽  
Xiao-Jun Yang
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Series Solution ◽  
Differential Operators ◽  
Fractional Wave Equation ◽  
Local Fractional Calculus ◽  
Vibrating String ◽  
Fractional Differential ◽  
Fractional Differential Operators

We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag-Leffler function.

The transversely isotropic poroelastic wave equation including the Biot and the squirt mechanisms: Theory and application

Geophysics ◽  
1997 ◽  
Vol 62 (1) ◽  
pp. 309-318 ◽  
Author(s):  
Jorge O. Parra
Keyword(s):  
Wave Equation ◽  
Fluid Flow ◽  
Analytical Solution ◽  
Seismic Waves ◽  
Transversely Isotropic ◽  
P Wave ◽  
Fluid Properties ◽  
Permeability Anisotropy ◽  
Attenuation And Dispersion ◽  
Maximum Permeability

The transversely isotropic poroelastic wave equation can be formulated to include the Biot and the squirt‐flow mechanisms to yield a new analytical solution in terms of the elements of the squirt‐flow tensor. The new model gives estimates of the vertical and the horizontal permeabilities, as well as other measurable rock and fluid properties. In particular, the model estimates phase velocity and attenuation of waves traveling at different angles of incidence with respect to the principal axis of anisotropy. The attenuation and dispersion of the fast quasi P‐wave and the quasi SV‐wave are related to the vertical and the horizontal permeabilities. Modeling suggests that the attenuation of both the quasi P‐wave and quasi SV‐wave depend on the direction of permeability. For frequencies from 500 to 4500 Hz, the quasi P‐wave attenuation will be of maximum permeability. To test the theory, interwell seismic waveforms, well logs, and hydraulic conductivity measurements (recorded in the fluvial Gypsy sandstone reservoir, Oklahoma) provide the material and fluid property parameters. For example, the analysis of petrophysical data suggests that the vertical permeability (1 md) is affected by the presence of mudstone and siltstone bodies, which are barriers to vertical fluid movement, and the horizontal permeability (1640 md) is controlled by cross‐bedded and planar‐laminated sandstones. The theoretical dispersion curves based on measurable rock and fluid properties, and the phase velocity curve obtained from seismic signatures, give the ingredients to evaluate the model. Theoretical predictions show the influence of the permeability anisotropy on the dispersion of seismic waves. These dispersion values derived from interwell seismic signatures are consistent with the theoretical model and with the direction of propagation of the seismic waves that travel parallel to the maximum permeability. This analysis with the new analytical solution is the first step toward a quantitative evaluation of the preferential directions of fluid flow in reservoir formation containing hydrocarbons. The results of the present work may lead to the development of algorithms to extract the permeability anisotropy from attenuation and dispersion data (derived from sonic logs and crosswell seismics) to map the fluid flow distribution in a reservoir.

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