Recursive integral time extrapolation methods for scalar waves

Fowler, Paul J.; Du, Xiuli; Fletcher, Richard

doi:10.1190/1.3513514

Cited by 17 publications

(14 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Equation 17 corresponds to the one-step method Fowler, 2010b). Fomel et al (2013) adopt instead a two-step implementation, which uses only the ϕ 1 term in equation 20 to cancel out the imaginary part of the wave extrapolation operator (Etgen and Brandsberg-Dahl, 2009): pðx; t þ ΔtÞ þ pðx; t − ΔtÞ ≈ 2 Z Pðk; tÞe ik·x cos½Vðx; kÞjkjΔtdk:…”

Section: Variable Velocity and Anisotropymentioning

confidence: 99%

“…Recently, alternative strategies have been proposed to propagate waves by mixed-domain space-wavenumber operators (Soubaras and Zhang, 2008;Wards et al, 2008;Etgen and BrandsbergDahl, 2009;Liu et al, 2009;Zhang and Zhang, 2009;Du et al, 2010;Pestana and Stoffa, 2010;Chu and Stoffa, 2011;Fomel et al, 2013;Wu and Alkhalifah, 2014). Fowler (2010b) and Du et al (2014) refer to these methods as recursive integral time extrapolation (RITE) methods.…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Low-rank one-step wave extrapolation for reverse time migration

2016

View full text Add to dashboard Cite

Reverse time migration (RTM) relies on accurate wave extrapolation engines to image complex subsurface structures. To construct such operators with high efficiency and numerical stability, we have developed a one-step wave extrapolation approach using complex-valued low-rank decomposition to approximate the mixed-domain space-wavenumber wave extrapolation symbol. The low-rank one-step method involves a complex-valued phase function, which is more flexible than a real-valued phase function of two-step schemes, and thus it is capable of modeling a wider variety of dispersion relations. Two novel designs of the phase function leads to the desired properties in wave extrapolation. First, for wave propagation in inhomogeneous media, including a velocity gradient term assures a more accurate phase behavior, particularly when the velocity variations are large. Second, an absorbing boundary condition, which is propagation-direction-dependent, can be incorporated into the phase function as an anisotropic attenuation term. This term allows waves to travel parallel to the boundary without absorption, thus reducing artificial reflections at wide incident angles. Using numerical experiments, we revealed the stability improvement of a one-step scheme in comparison with two-step schemes. We observed the low-rank one-step operator to be remarkably stable and capable of propagating waves using large time step sizes, even beyond the Nyquist limit. The stability property can help to minimize the computational cost of seismic modeling or RTM. The low-rank one-step wave extrapolation also handles anisotropic wave propagation accurately and efficiently. When applied to RTM in anisotropic media, the proposed method generated high-quality images.

show abstract

Section: Variable Velocity and Anisotropymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Low-rank one-step wave extrapolation for reverse time migration

2016

View full text Add to dashboard Cite

show abstract

“…If we use the second‐order finite difference method for the temporal derivatives and the pseudospectral method for the spatial derivatives, equations () and () become where , , Δ t is the temporal sampling rate and n is the time stepping index. 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).…”

Section: Numerical Examplesmentioning

confidence: 99%

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

Chu

Macy

Anno

2012

Geophysical Prospecting

View full text Add to dashboard Cite

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.

show abstract

“…Increasing wavefield approximation accuracy in time has become a popular topic in recent years (e.g., Soubaras and Zhang, 2008;Etgen and Brandsberg-Dahl, 2009;Fowler et al, 2010;Song and Fomel, 2011;Alkhalifah, 2013;Fomel et al, 2013;Tan and Huang, 2014a). For the second-order acoustic wave equation with a constant density, the wavefield approximation in time can be formulated analytically by an integral of the product of the current wavefield and a cosine function in the wavenumber domain, known as the Fourier integral (e.g., Soubaras and Zhang, 2008;Song and Fomel, 2011).…”

Section: Introductionmentioning

confidence: 99%

A k-space operator-based least-squares staggered-grid finite-difference method for modeling scalar wave propagation

et al. 2016

View full text Add to dashboard Cite

Two staggered-grid finite-difference (SGFD) schemes with fourth-and sixth-order accuracies in time have been developed recently based on new SGFD stencils. The SGFD coefficients of the two schemes are determined by a Taylorseries expansion (TE) approach, which is accurate only near a zero wavenumber. We have adopted the same SGFD stencils and determined the SGFD coefficients by minimizing the errors between the wavenumber responses of the SGFD operators and the first-order k (wavenumber)-space operator in a least-squares (LS) sense. We solved the LS problems by performing a weighted pseudoinverse of nonsquare matrices to obtain the SGFD coefficients and to yield LSbased SGFD methods. We have developed an efficient LS implementation by using a small number of representative wavenumbers, which makes the computational time for estimating the coefficients negligible. Dispersion analysis and numerical examples demonstrate that our LS-based SGFD methods can preserve the original temporal accuracy and achieve better spatial accuracy than the existing TE-based SGFD methods. Extensive numerical tests proved that our LS-based methods do not damage the stability of the corresponding TE-based methods significantly.

show abstract

Recursive integral time extrapolation methods for scalar waves

Cited by 17 publications

References 24 publications

Low-rank one-step wave extrapolation for reverse time migration

Low-rank one-step wave extrapolation for reverse time migration

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

A k-space operator-based least-squares staggered-grid finite-difference method for modeling scalar wave propagation

Contact Info

Product

Resources

About