A stable constraint for pseudoelastic anisotropic reverse time migration

Liu, Hongwei; Fox, Adam; Sliz, K.; Zhang, Houzhu; Zhao, Yang

doi:10.1190/segam2018-2978628.1

Cited by 2 publications

(2 citation statements)

References 14 publications

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“…The 2D elastic wave equation in a VTI medium can also be expressed in the following form (Barucq et al, 2014;Liu et al, 2018):…”

Section: Elastic Wave Equation In Vti/tti Mediummentioning

confidence: 99%

“…The 2D elastic wave equation in a VTI medium can also be expressed in the following form (Barucq et al ., 2014; Liu et al ., 2018):

\frac{\partial u}{\partial t} = G u,

where

G = [\begin{matrix} 0 & 0 & \frac{1}{ρ} \frac{\partial}{\partial x} & 0 & \frac{1}{ρ} \frac{\partial}{\partial z} \\ 0 & 0 & 0 & \frac{1}{ρ} \frac{\partial}{\partial z} & \frac{1}{ρ} \frac{\partial}{\partial x} \\ C_{11} \frac{\partial}{\partial x} & C_{13} \frac{\partial}{\partial z} & 0 & 0 & 0 \\ C_{13} \frac{\partial}{\partial x} & C_{33} \frac{\partial}{\partial z} & 0 & 0 & 0 \\ C_{55} \frac{\partial}{\partial z} & C_{55} \frac{\partial}{\partial x} & 0 & 0 & 0 \end{matrix}],

and

u = {(V_{x}, V_{z}, τ_{xx}, τ_{zz}, τ_{xz})}^{T}

, where

τ_{x x}

τ_{z z}

and

τ_{x z}

are the components of the stress tensor and

C_{11}, C_{13}, C_{33} ...

…”

Section: Theorymentioning

confidence: 99%

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Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

Araujo

Pestana

2020

Geophysical Prospecting

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We propose a new numerical solution to the first-order linear acoustic/elastic wave equation. This numerical solution is based on the analytic solution of the linear acoustic/elastic wave equation and uses the Lie product formula, where the time evolution operator of the analytic solution is written as a product of exponential matrices where each exponential matrix term is then approximated by Taylor series expansion. Initially, we check the proposed approach numerically and then demonstrate that it is more accurate to apply a Taylor expansion for the exponential function identity rather than the exponential function itself. The numerical solution formulated employs a recursive procedure and also incorporates the split perfectly matched layer boundary condition. Thus, our scheme can be used to extrapolate wavefields in a stable manner with even larger time-steps than traditional finite-difference schemes. This new numerical solution is examined through the comparison of the solution of full acoustic wave equation using the Chebyshev expansion approach for the matrix exponential term. Moreover, to demonstrate the efficiency and applicability of our proposed solution, seismic modelling results of three geological models are presented and the processing time for each model is compared with the computing time taking by the Chebyshev expansion method. We also present the result of seismic modelling using the scheme based in Lie product formula and Taylor series expansion for the firstorder linear elastic wave equation in vertical transversely isotropic and tilted transversely isotropic media as well. Finally, a post-stack migration results are also shown using the proposed method.

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“…The 2D elastic wave equation in a VTI medium can also be expressed in the following form (Barucq et al, 2014;Liu et al, 2018):…”

Section: Elastic Wave Equation In Vti/tti Mediummentioning

confidence: 99%

“…The 2D elastic wave equation in a VTI medium can also be expressed in the following form (Barucq et al ., 2014; Liu et al ., 2018):

\frac{\partial u}{\partial t} = G u,

where

G = [\begin{matrix} 0 & 0 & \frac{1}{ρ} \frac{\partial}{\partial x} & 0 & \frac{1}{ρ} \frac{\partial}{\partial z} \\ 0 & 0 & 0 & \frac{1}{ρ} \frac{\partial}{\partial z} & \frac{1}{ρ} \frac{\partial}{\partial x} \\ C_{11} \frac{\partial}{\partial x} & C_{13} \frac{\partial}{\partial z} & 0 & 0 & 0 \\ C_{13} \frac{\partial}{\partial x} & C_{33} \frac{\partial}{\partial z} & 0 & 0 & 0 \\ C_{55} \frac{\partial}{\partial z} & C_{55} \frac{\partial}{\partial x} & 0 & 0 & 0 \end{matrix}],

and

u = {(V_{x}, V_{z}, τ_{xx}, τ_{zz}, τ_{xz})}^{T}

, where

τ_{x x}

τ_{z z}

and

τ_{x z}

are the components of the stress tensor and

C_{11}, C_{13}, C_{33} ...

…”

Section: Theorymentioning

confidence: 99%

Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

Araujo

Pestana

2020

Geophysical Prospecting

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Reverse time adjoint migration

2019

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From a mathematical perspective, it is desirable to apply the adjoint operator for back propagating the receiver wavefield for migration of data residuals for inversion. For frequency-domain direct solvers, it is straightforward to apply the adjoint operator, whereas in most real applications of time-domain finite-difference (TDFD) stencils, the forward-propagation kernel function is reused for backward propagation for simplicity. However, when applying the exact adjoint operator, the migration result will be different, especially when strong variations exist in the velocity model. Actually, these three operators (forward, backward, and adjoint of the forward) can be expressed in matrix forms under the Born approximation for acoustic wave equations. These expressions linearly relate model perturbations to recorded seismic data. Every element in the matrices is well-defined, and all involved operations, such as the imaging condition and wavefield sampling, are included in the closed-form matrix expressions. These matrix expressions provide a platform for analyzing seismic modeling and inversion via mature linear algebra methodologies and provide clear strategies for developing computer algorithms. By analyzing the similarity of the matrix expressions, one can find that the time stepping approaches for all three operators are essentially the same. Based on this observation, a new time-marching stencil can be designed to realize the TDFD adjoint operator. Compared with traditional reverse time migration, the new method using the adjoint operator can provide better image quality, especially at sharp velocity boundaries.

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A stable constraint for pseudoelastic anisotropic reverse time migration

Cited by 2 publications

References 14 publications

Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

Reverse time adjoint migration

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