An analytic signal based accurate time-domain visco-acoustic wave equation from the Constant-Q theory

Geophysics ◽  
2021 ◽  
pp. 1-58
Author(s):  
Hongwei Liu ◽  
Yi Luo
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Time Domain ◽  
Seismic Waves ◽  
Heterogeneous Media ◽  
Relaxation Property ◽  
Analytic Signal ◽  
Acoustic Wave Equation ◽  
Real Component ◽  
Frequency Components

We present a concise time-domain wave equation to accurately simulate wave propagation in visco-acoustic media. The central idea behind this work is to dismiss the negative frequency components from a time-domain signal by converting the signal to its analytic format. The negative frequency components of any analytic signal are always zero, meaning we can construct the visco-acoustic wave equation to honor the relaxation property of the media for positive frequencies only. The newly proposed complex-valued wave equation (CWE) represents the wavefield with its analytic signal, whose real part is the desired physical wavefield, while the imaginary part is the Hilbert transform of the real component. Specifically, this CWE is accurate for both weak and strong attenuating media in terms of both dissipation and dispersion and the attenuation is precisely linear with respect to the frequencies. Besides, the CWE is easy and flexible to model dispersion-only, dissipation-only or dispersion-plus-dissipation seismic waves. We have verified these CWEs by comparing the results with analytical solutions, and achieved nearly perfect matching. Except for the homogeneous Q media, we have also extended the CWEs to heterogeneous media. The results of the CWEs for heterogeneous Q media are consistent with those computed from the nonstationary operator based Fourier Integral method and from the Standard Linear Solid (SLS) equations.

Combining wave‐equation imaging with traveltime tomography to form high‐resolution images from crosshole data

Geophysics ◽  
1991 ◽  
Vol 56 (2) ◽  
pp. 208-224 ◽  
Author(s):  
R. Gerhard Pratt ◽  
Neil R. Goulty
Keyword(s):  
Wave Equation ◽  
High Resolution ◽  
Acoustic Wave ◽  
Equation Method ◽  
Single Frequency ◽  
Acoustic Wave Equation ◽  
Elastic Wave Equation ◽  
Traveltime Tomography ◽  
Combined Use ◽  
Frequency Components

Traveltime tomography is an appropriate method for estimating seismic velocity structure from arrival times. However, tomography fails to resolve discontinuities in the velocities. Wave‐equation techniques provide images using the full wave field that complement the results of traveltime tomography. We use the velocity estimates from tomography as a reference model for a numerical propagation of the time reversed data. These “backpropagated” wave fields are used to provide images of the discontinuities in the velocity field. The combined use of traveltime tomography and wave‐equation imaging is particularly suitable for forming high‐resolution geologic images from multiple‐source/multiple‐receiver data acquired in borehole‐to‐borehole seismic surveying. In the context of crosshole imaging, an effective implementation of wave‐equation imaging is obtained by transforming the data and the algorithms into the frequency domain. This transformation allows the use of efficient frequency‐domain numerical propagation methods. Experiments with computer‐generated data demonstrate the quality of the images that can be obtained from only a single frequency component of the data. Images of both compressional [Formula: see text] and shear wave [Formula: see text] velocity anomalies can be obtained by applying acoustic wave‐equation imaging in two passes. An imaging technique derived from the full elastic wave‐equation method yields superior images of both anomalies in a single pass. To demonstrate the combined use of traveltime tomography and wave‐equation imaging, a scale model experiment was carried out to simulate a crosshole seismic survey in the presence of strong velocity contrasts. Following the application of traveltime tomography, wave‐equation methods were used to form images from single frequency components of the data. The images were further enhanced by summing the results from several frequency components. The elastic wave‐equation method provided slightly better images of the [Formula: see text] discontinuities than the acoustic wave‐equation method. Errors in picking shear‐wave arrivals and uncertainties in the source radiation pattern prevented us from obtaining satisfactory images of the [Formula: see text] discontinuities.

Error Analysis of Chebyshev Spectral Element Methods for the Acoustic Wave Equation in Heterogeneous Media

2018 ◽  
Vol 26 (03) ◽  
pp. 1850035 ◽  
Author(s):  
Saulo Pomponet Oliveira
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Error Analysis ◽  
Heterogeneous Media ◽  
Spectral Element ◽  
Variable Density ◽  
Dirichlet Boundary Conditions ◽  
Acoustic Wave Equation ◽  
Polynomial Degree ◽  
Mesh Parameter

This work concerns the error analysis of the spectral element method with Gauss–Lobatto–Chebyshev collocation points with the implicit Newmark average acceleration scheme for the two-dimensional acoustic wave equation. The analysis is restricted to homogeneous Dirichlet boundary conditions, constant compressibility and variable density. The proposed error estimates are optimal with respect to the mesh parameter although suboptimal on the polynomial degree. Numerical examples illustrate the theoretical results.

Solving the 3D acoustic wave equation on generalized structured meshes: A finite-difference time-domain approach

Geophysics ◽  
2014 ◽  
Vol 79 (6) ◽  
pp. T363-T378 ◽  
Author(s):  
Jeffrey Shragge
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Time Domain ◽  
Finite Difference Time Domain ◽  
3D Seismic ◽  
Acoustic Wave Equation ◽  
Structured Meshes ◽  
Cartesian Geometry ◽  
And Inversion ◽  
Difference Time

The key computational kernel of most advanced 3D seismic imaging and inversion algorithms used in exploration seismology involves calculating solutions of the 3D acoustic wave equation, most commonly with a finite-difference time-domain (FDTD) methodology. Although well suited for regularly sampled rectilinear computational domains, FDTD methods seemingly have limited applicability in scenarios involving irregular 3D domain boundary surfaces and mesh interiors best described by non-Cartesian geometry (e.g., surface topography). Using coordinate mapping relationships and differential geometry, an FDTD approach can be developed for generating solutions to the 3D acoustic wave equation that is applicable to generalized 3D coordinate systems and (quadrilateral-faced hexahedral) structured meshes. The developed numerical implementation is similar to the established Cartesian approaches, save for a necessary introduction of weighted first- and mixed second-order partial-derivative operators that account for spatially varying geometry. The approach was validated on three different types of computational meshes: (1) an “internal boundary” mesh conforming to a dipping water bottom layer, (2) analytic “semiorthogonal cylindrical” coordinates, and (3) analytic semiorthogonal and numerically specified “topographic” coordinate meshes. Impulse response tests and numerical analysis demonstrated the viability of the approach for kernel computations for 3D seismic imaging and inversion experiments for non-Cartesian geometry scenarios.

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