Tensorial elastodynamics for isotropic media

Geophysics ◽  
2020 ◽  
Vol 85 (6) ◽  
pp. T359-T373
Author(s):  
Jeffrey Shragge ◽  
Tugrul Konuk
Keyword(s):  
Free Surface ◽  
Surface Topography ◽  
Numerical Solutions ◽  
Waveform Inversion ◽  
Real Life ◽  
Staggered Grid ◽  
Surface Boundary ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration

Numerical solutions of 3D isotropic elastodynamics form the key computational kernel for many isotropic elastic reverse time migration and full-waveform inversion applications. However, real-life scenarios often require computing solutions for computational domains characterized by non-Cartesian geometry (e.g., free-surface topography). One solution strategy is to compute the elastodynamic response on vertically deformed meshes designed to incorporate irregular topology. Using a tensorial formulation, we have developed and validated a novel system of semianalytic equations governing 3D elastodynamics in a stress-velocity formulation for a family of vertically deformed meshes defined by Bézier interpolation functions between two (or more) nonintersecting surfaces. The analytic coordinate definition also leads to a corresponding analytic free-surface boundary condition (FSBC) as well as expressions for wavefield injection and extraction. Theoretical examples illustrate the utility of the tensorial approach in generating analytic equations of 3D elastodynamics and the corresponding FSBCs for scenarios involving free-surface topography. Numerical examples developed using a fully staggered grid with a mimetic finite-difference formulation demonstrate the ability to model the expected full-wavefield behavior, including complex free-surface interactions.

Acoustic wave propagation in tilted transversely isotropic media: Incorporating topography

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. C265-C278 ◽  
Author(s):  
Jeffrey Shragge
Keyword(s):  
Wave Propagation ◽  
Free Surface ◽  
Surface Topography ◽  
Domain Boundary ◽  
Waveform Inversion ◽  
Transversely Isotropic ◽  
Surface Boundary ◽  
Reverse Time ◽  
Time Migration ◽  
Tti Media

Simulating two-way acoustic wavefield propagation directly from a free-surface boundary in the presence of topography remains a computational challenge for applications of reverse time migration (RTM) or full-waveform inversion (FWI). For land-seismic settings involving heavily reworked geology (e.g., fold and thrust belts), two-way wavefield propagation operators should also handle commonly observed complex anisotropy including tilted transversely isotropic (TTI) media. To address these issues, I have extended a system of coupled partial differential equations used to model 3D acoustic TTI wave propagation in Cartesian coordinates to more generalized 3D geometries, including a deformed computational mesh with a domain boundary conformal to free-surface topography. A generalized curvilinear transformation is used to specify a system of equations governing 3D acoustic TTI wave propagation in the “topographic” coordinate system. The developed finite-difference time-domain numerical solution adapts existing Cartesian TTI operators to this more generalized geometry with little additional computational overhead. Numerical evaluations illustrate that 2D and 3D impulse responses are well-matched to those simulated on Cartesian meshes and analytic traveltimes for homogeneous elliptical TTI media. Accordingly, these generalized acoustic TTI propagators and their numerical adjoints are useful for undertaking most RTM or FWI applications using computational domains conforming to free-surface topography.

Tensorial Elastodynamics for Anisotropic Media

Geophysics ◽  
2021 ◽  
pp. 1-60
Author(s):  
Tugrul Konuk ◽  
Jeffrey Shragge
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Surface Topography ◽  
Anisotropic Media ◽  
Essential Elements ◽  
Constitutive Relationship ◽  
Staggered Grid ◽  
General Strategy ◽  
Reverse Time ◽  
Time Migration

Elastic wavefield solutions computed by finite-difference (FD) methods in complex anisotropic media are essential elements of elastic reverse-time migration and full waveform inversion analyses. Cartesian formulations of such solution methods, though, face practical challenges when aiming to represent curved interfaces (including free-surface topography) with rectilinear elements. To forestall such issues, we propose a general strategy for generating solutions of tensorial elastodynamics for anisotropic media (i.e., tilted transversely isotropic (TTI) or even lower symmetry) in non-Cartesian computational domains. For the specific problem of handling free-surface topography, we define an unstretched coordinate mapping that transforms an irregular physical domain to a regular computational grid on which FD solutions of the modified equations of elastodynamics are straightforward to calculate. Our fully staggered grid with a mimetic finite-difference (FSG+MFD) approach solves the velocity-stress formulation of the tensorial elastic wave equation where we compute the stress-strain constitutive relationship in Cartesian coordinates and then transform the resulting stress tensor to generalized coordinates to solve the equations of motion. The resulting FSG+MFD numerical method has a computational complexity comparable to Cartesian scenarios using a similar FSG+MFD numerical approach. Numerical examples demonstrate that the proposed solution can simulate anisotropic elastodynamic field solutions on irregular geometries and is thus a reliable tool for anisotropic elastic modeling, imaging and inversion applications in generalized computational domains including handling free-surface topography.

First-order particle velocity equations of decoupled P- and S-wavefields and their application in elastic reverse time migration

Geophysics ◽  
2021 ◽  
pp. 1-78
Author(s):  
Zhiyuan Li ◽  
Youshan Liu ◽  
Guanghe Liang ◽  
Guoqiang Xue ◽  
Runjie Wang
Keyword(s):  
Particle Velocity ◽  
Computational Cost ◽  
Staggered Grid ◽  
Additional Stress ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Computational Accuracy ◽  
First Order ◽  
Time Migration ◽  
Stress Variable

The separation of P- and S-wavefields is considered to be an effective approach for eliminating wave-mode cross-talk in elastic reverse-time migration. At present, the Helmholtz decomposition method is widely used for isotropic media. However, it tends to change the amplitudes and phases of the separated wavefields compared with the original wavefields. Other methods used to obtain pure P- and S-wavefields include the application of the elastic wave equations of the decoupled wavefields. To achieve a high computational accuracy, staggered-grid finite-difference (FD) schemes are usually used to numerically solve the equations by introducing an additional stress variable. However, the computational cost of this method is high because a conventional hybrid wavefield (P- and S-wavefields are mixed together) simulation must be created before the P- and S-wavefields can be calculated. We developed the first-order particle velocity equations to reduce the computational cost. The equations can describe four types of particle velocity wavefields: the vector P-wavefield, the scalar P-wavefield, the vector S-wavefield, and the vector S-wavefield rotated in the direction of the curl factor. Without introducing the stress variable, only the four types of particle velocity variables are used to construct the staggered-grid FD schemes, so the computational cost is reduced. We also present an algorithm to calculate the P and S propagation vectors using the four particle velocities, which is simpler than the Poynting vector. Finally, we applied the velocity equations and propagation vectors to elastic reverse-time migration and angle-domain common-image gather computations. These numerical examples illustrate the efficiency of the proposed methods.

Q least-squares reverse time migration based on the first-order viscoacoustic quasidifferential equations

Geophysics ◽  
2021 ◽  
pp. 1-65
Author(s):  
Yingming Qu ◽  
Yixin Wang ◽  
Zhenchun Li ◽  
Chang Liu
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Surface Topography ◽  
Density Perturbation ◽  
Second Order ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
First Order ◽  
Time Migration ◽  
Grid Scheme

Seismic wave attenuation caused by subsurface viscoelasticity reduces the quality of migration and the reliability of interpretation. A variety of Q-compensated migration methods have been developed based on the second-order viscoacoustic quasidifferential equations. However, these second-order wave-equation-based methods are difficult to handle with density perturbation and surface topography. In addition, the staggered grid scheme, which has an advantage over the collocated grid scheme because of its reduced numerical dispersion and enhanced stability, works in first-order wave-equation-based methods. We have developed a Q least-squares reverse time migration method based on the first-order viscoacoustic quasidifferential equations by deriving Q-compensated forward-propagated operators, Q-compensated adjoint operators, and Q-attenuated Born modeling operators. Besides, our method using curvilinear grids is available even when the attenuating medium has surface topography and can conduct Q-compensated migration with density perturbation. The results of numerical tests on two synthetic and a field data sets indicate that our method improves the imaging quality with iterations and produces better imaging results with clearer structures, higher signal-to-noise ratio, higher resolution, and more balanced amplitude by correcting the energy loss and phase distortion caused by Q attenuation. It also suppresses the scattering and diffracted noise caused by the surface topography.

Least-squares reverse time migration via linearized waveform inversion using a Wasserstein metric

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. S411-S423
Author(s):  
Peng Yong ◽  
Jianping Huang ◽  
Zhenchun Li ◽  
Wenyuan Liao ◽  
Luping Qu
Keyword(s):  
Least Squares ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Gradient Algorithm ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Wasserstein Metric ◽  
Time Migration ◽  
Imaging Results ◽  
Primal Dual

Least-squares reverse time migration (LSRTM), an effective tool for imaging the structures of the earth from seismograms, can be characterized as a linearized waveform inversion problem. We have investigated the performance of three minimization functionals as the [Formula: see text] norm, the hybrid [Formula: see text] norm, and the Wasserstein metric ([Formula: see text] metric) for LSRTM. The [Formula: see text] metric used in this study is based on the dynamic formulation of transport problems, and a primal-dual hybrid gradient algorithm is introduced to efficiently compute the [Formula: see text] metric between two seismograms. One-dimensional signal analysis has demonstrated that the [Formula: see text] metric behaves like the [Formula: see text] norm for two amplitude-varied signals. Unlike the [Formula: see text] norm, the [Formula: see text] metric does not suffer from the differentiability issue for null residuals. Numerical examples of the application of three misfit functions to LSRTM on synthetic data have demonstrated that, compared to the [Formula: see text] norm, the hybrid [Formula: see text] norm and [Formula: see text] metric can accelerate LSRTM and are less sensitive to non-Gaussian noise. For the field data application, the [Formula: see text] metric produces the most reliable imaging results. The hybrid [Formula: see text] norm requires tedious trial-and-error tests for the judicious threshold parameter selection. Hence, the more automatic [Formula: see text] metric is recommended as a robust alternative to the customary [Formula: see text] norm for time-domain LSRTM.

scholarly journals Reverse-time migration-based reflection tomography using teleseismic free surface multiples

2013 ◽  
Vol 196 (2) ◽  
pp. 996-1017 ◽  
Author(s):  
S. Burdick ◽  
M. V. de Hoop ◽  
S. Wang ◽  
R. D. van der Hilst
Keyword(s):  
Free Surface ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Reflection Tomography
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