Two-step Explicit Marching Method for Reverse Time Migration

Soubaras, Robert; Zhang, Y.

doi:10.3997/2214-4609.20147702

Cited by 22 publications

(24 citation statements)

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“…pðk; t þ ΔtÞ þpðk; t − ΔtÞ ¼ 2 cosðjkjvΔtÞpðk; tÞ: (6) Equation 6 describes an accurate time-marching scheme in the wavenumber domain (Etgen, 1989;Soubaras and Zhang, 2008). If we approximate the cosine function by Taylor's series expansion at jkjvΔt ¼ 0 and truncate it to the second order…”

Section: Methods Theorymentioning

confidence: 99%

“…Fourier methods (Soubaras and Zhang, 2008;Zhang and Zhang, 2009;Pestana and Stoffa, 2010;Song and Fomel, 2011;Tessmer, 2011;Alkhalifah, 2013;Fomel et al, 2013) have been developed to solve equation 6 effectively. To explore the possibility of achieving high temporal accuracy without FFTs, we derive an ETE scheme from equation 6 using values at a set of grid points (stencil):…”

Section: Methods Theorymentioning

confidence: 99%

“…In acoustic media, the time evolution of the wavefield can be formulated analytically by an integral of the product of the current wavefield and a cosine function in wavenumber domain, known as the Fourier integral (e.g., Soubaras and Zhang, 2008;Song and Fomel, 2011;Alkhalifah, 2013). Soubaras and Zhang (2008) first propose to approximate the cosine function in the Fourier integral by a polynomial, resulting in a two-step explicit marching scheme that allows large time steps. Zhang and Zhang (2009) further introduce a complex wavefield and suggest a one-step extrapolation by using multiple times of fast Fourier transforms (FFTs).…”

Section: Introductionmentioning

confidence: 99%

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An explicit time evolution method for acoustic wave propagation

Liu

Dai²,

Niu

et al. 2014

GEOPHYSICS

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Cost-effective waveform modeling is the key to practical reverse time migration (RTM) and full-waveform inversion (FWI) implementations. We evaluated an explicit time evolution (ETE) method to efficiently simulate wave propagation in acoustic media with high temporal accuracy. We started from the constant-density acoustic wave equation and obtained an analytical time-marching scheme in the wavenumber domain. We then formulated an ETE scheme in the time-space domain by introducing a cosine function approximation. Although the ETE operator appears to be similar to the second-order temporal finite-difference (FD) operator, the exact nature of the ETE formula ensures high accuracy in time. We further introduced a set of optimum stencils and coefficients by minimizing evolution errors in a least-squares sense. Our numerical tests indicated that ETE can achieve similar waveform accuracy as FD with four times larger time steps. Meanwhile, the compact ETE operator keeps the computation efficient. The efficiency and capability to handle complex velocity field make ETE an attractive engine in acoustic RTM and FWI.

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Section: Methods Theorymentioning

confidence: 99%

Section: Methods Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An explicit time evolution method for acoustic wave propagation

Liu

Dai²,

Niu

et al. 2014

GEOPHYSICS

View full text Add to dashboard Cite

show abstract

“…Therefore, we need to guarantee that the number of terms used in the Chebyshev expansion will give a good approximation to cos(L t) and the extrapolation of the wavefield will occur in a stable way. Moreover, based on Tal-Ezer (1986), the Chebyshev expansion will converge for M > R t. Now, if the wavefield P (n) is calculated by a second-order finite-difference scheme in time (Soubaras and Zhang 2008) and take also the first equation in (14), we have the following system of equations:…”

Section: A C K N O W L E D G E M E N T Smentioning

confidence: 99%

Perfectly matched layer boundary conditions for the second‐order acoustic wave equation solved by the rapid expansion method

Araujo

Pestana

2019

Geophysical Prospecting

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We derive a governing second‐order acoustic wave equation in the time domain with a perfectly matched layer absorbing boundary condition for general inhomogeneous media. Besides, a new scheme to solve the perfectly matched layer equation for absorbing reflections from the model boundaries based on the rapid expansion method is proposed. The suggested scheme can be easily applied to a wide class of wave equations and numerical methods for seismic modelling. The absorbing boundary condition method is formulated based on the split perfectly matched layer method and we employ the rapid expansion method to solve the derived new perfectly matched layer equation. The use of the rapid expansion method allows us to extrapolate wavefields with a time step larger than the ones commonly used by traditional finite‐difference schemes in a stable way and free of dispersion noise. Furthermore, in order to demonstrate the efficiency and applicability of the proposed perfectly matched layer scheme, numerical modelling examples are also presented. The numerical results obtained with the put forward perfectly matched layer scheme are compared with results from traditional attenuation absorbing boundary conditions and enlarged models as well. The analysis of the numerical results indicates that the proposed perfectly matched layer scheme is significantly effective and more efficient in absorbing spurious reflections from the model boundaries.

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“…Similarly, we can discretize using the second‐order time stepping scheme as follows The second‐order time stepping method corresponds to a second‐order Taylor series expansion of a formal cosine extrapolator (Fowler, Du and Fletcher 2010b; Chu and Stoffa 2011). For better accuracy, one may choose to use high‐order expansion methods (e.g., Dablain 1986; Chen 2007; Soubaras and Zhang 2008; Pestana and Stoffa 2010; Tessmer 2011). We inject sources to both and () and take the average of P and Q as the pressure field (e.g., Fowler et al 2010a).…”

Section: Numerical Examplesmentioning

confidence: 99%

Pure acoustic wave propagation in transversely isotropic media by the pseudospectral method

Chu

Macy

Anno

2012

Geophysical Prospecting

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Seismic wave propagation in transversely isotropic (TI) media is commonly described by a set of coupled partial differential equations, derived from the acoustic approximation. These equations produce pure P‐wave responses in elliptically anisotropic media but generate undesired shear‐wave components for more general TI anisotropy. Furthermore, these equations suffer from instabilities when the anisotropy parameter ε is less than δ. One solution to both problems is to use pure acoustic anisotropic wave equations, which can produce pure P‐waves without any shear‐wave contaminations in both elliptical and anelliptical TI media. In this paper, we propose a new pure acoustic transversely isotropic wave equation, which can be conveniently solved using the pseudospectral method. Like most other pure acoustic anisotropic wave equations, our equation involves complicated pseudo‐differential operators in space which are difficult to handle using the finite difference method. The advantage of our equation is that all of its model parameters are separable from the spatial differential and pseudo‐differential operators; therefore, the pseudospectral method can be directly applied. We use phase velocity analysis to show that our equation, expressed in a summation form, can be properly truncated to achieve the desired accuracy according to anisotropy strength. This flexibility allows us to save computational time by choosing the right number of summation terms for a given model. We use numerical examples to demonstrate that this new pure acoustic wave equation can produce highly accurate results, completely free from shear‐wave artefacts. This equation can be straightforwardly generalized to tilted TI media.

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Two-step Explicit Marching Method for Reverse Time Migration

Cited by 22 publications

References 0 publications

An explicit time evolution method for acoustic wave propagation

An explicit time evolution method for acoustic wave propagation

Perfectly matched layer boundary conditions for the second‐order acoustic wave equation solved by the rapid expansion method

Pure acoustic wave propagation in transversely isotropic media by the pseudospectral method

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