scholarly journals Nearly constant Q dissipative models and wave equations for general viscoelastic anisotropy

2021 ◽  
Vol 477 (2251) ◽  
pp. 20210170
Author(s):  
Qi Hao ◽  
Stewart Greenhalgh
Keyword(s):  
Differential Form ◽  
Wave Equations ◽  
Weighting Function ◽  
Waveform Inversion ◽  
Second Order ◽  
Common Assumption ◽  
Wave Energy Dissipation ◽  
First Derivative ◽  
Seismic Waveform ◽  
Anisotropic Case

The quality factor ( Q ) links seismic wave energy dissipation to physical properties of the Earth’s interior, such as temperature, stress and composition. Frequency independence of Q , also called constant Q for brevity, is a common assumption in practice for seismic Q inversions. Although exactly and nearly constant Q dissipative models are proposed in the literature, it is inconvenient to obtain constant Q wave equations in differential form, which explicitly involve a specified Q parameter. In our recent research paper, we proposed a novel weighting function method to build the first- and second-order nearly constant Q dissipative models. Of importance is the fact that the wave equations in differential form for these two models explicitly involve a specified Q parameter. This behaviour is beneficial for time-domain seismic waveform inversion for Q , which requires the first derivative of wavefields with respect to Q parameters. In this paper, we extend the first- and second-order nearly constant Q models to the general viscoelastic anisotropic case. We also present a few formulations of the nearly constant Q viscoelastic anisotropic wave equations in differential form.

Stable P-wave modeling for reverse-time migration in tilted TI media

Geophysics ◽  
2011 ◽  
Vol 76 (2) ◽  
pp. S65-S75 ◽  
Author(s):  
Eric Duveneck ◽  
Peter M. Bakker
Keyword(s):  
Wave Equations ◽  
Symmetry Axis ◽  
Second Order ◽  
P Wave ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
First Derivative ◽  
Time Migration ◽  
Mixed Derivatives ◽  
Spatial Derivatives

We present an approach for P-wave modeling in inhomogeneous transversely isotropic media with tilted symmetry axis (TTI media), suitable for anisotropic reverse-time migration. The proposed approach is based on wave equations derived from first principles — the equations of motion and Hooke’s law — under the acoustic TI approximation. Consequently, no assumptions are made about the spatial variation of medium parameters. A rotation of the stress and strain tensors to a local coordinate system, aligned with the TI-symmetry axis, makes it possible to benefit from the simple and sparse form of the TI-elastic tensor in that system. The resulting wave equations can be formulated either as a set of five first-order or as a set of two second-order partial differential equations. For the constant-density case, the second-order TTI wave equations involve mixed and nonmixed second-order spatial derivatives with respect to global, nonrotated coordinates. We propose a numerical implementation of these equations using high-order centered finite differences. To minimize modeling artifacts related to the use of centered first-derivative operators, we use discrete second-derivative operators for the nonmixed second-order spatial derivatives and repeated discrete first-derivative operators for the mixed derivatives. Such a combination of finite-difference operators leads to a stable wave propagator, provided that the operators are designed properly. In practice, stability is achieved by slightly weighting down terms that contain mixed derivatives. This has a minor, practically negligible, effect on the kinematics of wave propagation. The stability of the presented scheme in inhomogeneous TTI models with rapidly varying anisotropic symmetry axis direction is demonstrated with numerical examples.

Sinkhole detection with 3D full seismic waveform tomography

Geophysics ◽  
2020 ◽  
Vol 85 (5) ◽  
pp. B169-B179
Author(s):  
Majid Mirzanejad ◽  
Khiem T. Tran ◽  
Michael McVay ◽  
David Horhota ◽  
Scott J. Wasman
Keyword(s):  
Wave Equations ◽  
Waveform Inversion ◽  
Field Tests ◽  
Waveform Analysis ◽  
Full Waveform Inversion ◽  
3D Seismic ◽  
Data Set ◽  
Full Waveform ◽  
Seismic Waveform ◽  
Elastic Wave Equations

Sinkhole collapse may result in significant property damage and even loss of life. Early detection of sinkhole attributes (buried voids, raveling zones) is critical to limit the cost of remediation. One of the most promising ways to obtain subsurface imaging is 3D seismic full-waveform inversion. For demonstration, a recently developed 3D Gauss-Newton full-waveform inversion (3D GN-FWI) method is used to detect buried voids, raveling soils, and characterize variable subsurface soil/rock layering. It is based on a finite-difference solution of 3D elastic wave equations and Gauss-Newton optimization. The method is tested first on a data set constructed from the numerical simulation of a challenging synthetic model and subsequently on field data collected from two separate test sites in Florida. For the field tests, receivers and sources are placed in uniform 2D surface grids to acquire the seismic wavefields, which then are inverted to extract the 3D subsurface velocity structures. The inverted synthetic results suggest that the approach is viable for detecting voids and characterizing layering. The field seismic results reveal that the 3D waveform analysis identified a known manmade void (plastic culvert), unknown natural voids, raveling, as well as laterally variable soil/rock layering including rock pinnacles. The results are confirmed later by standard penetration tests, including depth to bedrock, two buried voids, and a raveling soil zone. Our study provides insight into the application of the 3D seismic FWI technique as a powerful tool in detecting shallow voids and other localized subsurface features.

Reducing computation cost by Lax-Wendroff methods with fourth-order temporal accuracy

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. T109-T119 ◽  
Author(s):  
Hongwei Liu ◽  
Houzhu Zhang
Keyword(s):  
Wave Equations ◽  
Numerical Solutions ◽  
Waveform Inversion ◽  
Computational Cost ◽  
Second Order ◽  
Numerical Dispersion ◽  
Reverse Time ◽  
Time Migration ◽  
L2 Norm ◽  
Time Marching

Explicit time-marching finite-difference stencils have been extensively used for simulating seismic wave propagation, and they are the most computationally intensive part of seismic forward modeling, reverse time migration, and full-waveform inversion. The time-marching step, determined by both the stability condition and numerical dispersion, is a key factor in the computational cost. In contrast with the widely used second-order temporal stencil, the Lax-Wendroff stencil is more cost effective because the time-marching step can be much larger. It can be proved, using theory and numerical tests, that the Lax-Wendroff stencil does enable larger time steps. In terms of numerical dispersion, the time steps for second-order and Lax-Wendroff stencils are functions of the number of shortest wavelengths away from the source location. These functions are derived by evaluating the relative L2-norm of the differences between the analytical and numerical solutions. The method for determining the time-marching step is model adaptive and easy to implement. We use the pseudo spectral method for the computation of spatial derivatives, and the wave equations that we solved are for isotropic media only, but the described principles can be easily implemented for more complicated types of media.

scholarly journals Method for Measuring the Second-Order Moment of Atmospheric Turbulence

Atmosphere ◽  
2021 ◽  
Vol 12 (5) ◽  
pp. 564
Author(s):  
Hong Shen ◽  
Longkun Yu ◽  
Xu Jing ◽  
Fengfu Tan
Keyword(s):  
Atmospheric Turbulence ◽  
Weighting Function ◽  
Structure Constant ◽  
Turbulence Models ◽  
Second Order ◽  
Index Structure ◽  
Order Moment ◽  
Site Testing ◽  
Optical Effects ◽  
Novel Method

The turbulence moment of order m (μm) is defined as the refractive index structure constant Cn2 integrated over the whole path z with path-weighting function zm. Optical effects of atmospheric turbulence are directly related to turbulence moments. To evaluate the optical effects of atmospheric turbulence, it is necessary to measure the turbulence moment. It is well known that zero-order moments of turbulence (μ0) and five-thirds-order moments of turbulence (μ5/3), which correspond to the seeing and the isoplanatic angles, respectively, have been monitored as routine parameters in astronomical site testing. However, the direct measurement of second-order moments of turbulence (μ2) of the whole layer atmosphere has not been reported. Using a star as the light source, it has been found that μ2 can be measured through the covariance of the irradiance in two receiver apertures with suitable aperture size and aperture separation. Numerical results show that the theoretical error of this novel method is negligible in all the typical turbulence models. This method enabled us to monitor μ2 as a routine parameter in astronomical site testing, which is helpful to understand the characteristics of atmospheric turbulence better combined with μ0 and μ5/3.

Periodic solutions of second-order wave equations. IV

1989 ◽  
Vol 40 (6) ◽  
pp. 639-644
Author(s):  
Yu. A. Mitropol'skii ◽  
G. P. Khoma
Keyword(s):  
Periodic Solutions ◽  
Wave Equations ◽  
Second Order

Time dispersion correction for arbitrarily even-order Lax-Wendroff methods and the application on full waveform inversion

Geophysics ◽  
2021 ◽  
pp. 1-89
Author(s):  
Zhiming Ren ◽  
Qianzong Bao ◽  
Bingluo Gu
Keyword(s):  
Waveform Inversion ◽  
Second Order ◽  
High Order ◽  
Full Waveform Inversion ◽  
Dispersion Correction ◽  
Equal Time ◽  
Full Waveform ◽  
Time Dispersion ◽  
Correction Scheme ◽  
Even Order

A second-order accurate finite-difference (FD) approximation is commonly used to approximate the second-order time derivative of wave equation. The second-order accurate FD scheme may introduce time dispersion in wavefield extrapolation. Lax-Wendroff methods can suppress such dispersion by replacing the high-order time FD error-terms with space FD error correcting terms. However, the time dispersion cannot be completely eliminated and the computation cost dramatically increases with increasing order of (temporal) accuracy. To mitigate the problem, we extend the existing time dispersion correction scheme for second- or fourth-order Lax-Wendroff method to a scheme for arbitrary even-order methods, which uses the forward and inverse time dispersion transform (FTDT and ITDT) to add and remove the time dispersion from synthetic data. We test the correction scheme using a homogeneous model and the Sigsbee2A model. Modeling examples suggest that the use of derived FTDT and ITDT pairs on high-order Lax-Wendroff methods can effectively remove time dispersion errors from high-frequency waves while using longer time steps than allowed in low-order Lax-Wendroff methods. We investigate the influence of the time dispersion on full waveform inversion (FWI) and show an anti-dispersion workflow. We apply the FTDT to source terms and recorded traces before inversion, resulting in that the source and adjoint wavefields contain equal time dispersion from source-side wave propagation, and the modeled and observed traces accumulate equal time dispersion from source- and receiver-side wave propagation. Inversion results reveal that the anti-dispersion workflow is capable of increasing the accuracy of FWI for arbitrary even-order Lax-Wendroff methods. Additionally, the high-order method can obtain better inversion results compared to the second-order method with the same anti-dispersion workflow.

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