Modal expansion of one‐way operators in laterally varying media

Grimbergen, J. L. T.; Dessing, F. J.; Wapenaar, Kees

doi:10.1190/1.1444410

Cited by 60 publications

(42 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is required, however, that the medium properties cðxÞ and ϱðxÞ vary only smoothly in the lateral direction at and around the redatuming boundaries. It is well-understood that discontinuity of the medium properties at these boundaries can complicate the numerical computation of the square-root operators (Grimbergen et al, 1998). The effects of such discontinuities on solving the multidimensional Marchenko equation and on redatuming by inversion (Ravasi et al, 2015a) have been described in the existing literature, too.…”

Section: The Modelmentioning

confidence: 99%

“…where H 1 is a square-root operator that obeys Grimbergen et al, 1998). In Appendix A, we show how this operator can be computed numerically.…”

Section: Reviewmentioning

confidence: 99%

“…Further, Δ is a first-order finite-difference operator. The square-root operator H 1k can be computed by eigenvalue decomposition of matrix H 2k (Grimbergen et al, 1998)…”

Section: Acknowledgmentsmentioning

confidence: 99%

See 2 more Smart Citations

Target-enclosed seismic imaging

et al. 2017

View full text Add to dashboard Cite

Seismic reflection data can be redatumed to a specified boundary in the subsurface by solving an inverse (or multidimensional deconvolution) problem. The redatumed data can be interpreted as an extended image of the subsurface at the redatuming boundary, depending on the subsurface offset and time. We retrieve target-enclosed extended images by using two redatuming boundaries, which are selected above and below a specified target volume. As input, we require the upgoing and downgoing wavefields at both redatuming boundaries due to impulsive sources at the earth’s surface. These wavefields can be obtained from actual measurements in the subsurface, they can be numerically modeled, or they can be retrieved by solving a multidimensional Marchenko equation. As output, we retrieved virtual reflection and transmission responses as if sources and receivers were located at the two target-enclosing boundaries. These data contain all orders of reflections inside the target volume but exclude all interactions with the part of the medium outside this volume. The retrieved reflection responses can be used to image the target volume from above or from below. We found that the images from above and below are similar (given that the Marchenko equation is used for wavefield retrieval). If a model with sharp boundaries in the target volume is available, the redatumed data can also be used for two-sided imaging, where the retrieved reflection and transmission responses are exploited. Because multiple reflections are used by this strategy, seismic resolution can be improved significantly. Because target-enclosed extended images are independent on the part of the medium outside the target volume, our methodology is also beneficial to reduce the computational burden of localized inversion, which can now be applied inside the target volume only, without suffering from interactions with other parts of the medium.

show abstract

Section: The Modelmentioning

confidence: 99%

“…where H 1 is a square-root operator that obeys Grimbergen et al, 1998). In Appendix A, we show how this operator can be computed numerically.…”

Section: Reviewmentioning

confidence: 99%

See 1 more Smart Citation

Target-enclosed seismic imaging

et al. 2017

View full text Add to dashboard Cite

show abstract

“…The numerical examples in this paper and by Liu and Zhang [2006] demonstrate the effectiveness of the stabilized processing. In practice, very strong laterally heterogeneous case, for which the stabilized processing may yield a significant hurt, is beyond the current dual-domain and finite-difference methods [Grimbergen et al, 1998]. …”

Section: Algorithmmentioning

confidence: 99%

Optimum split‐step Fourier one‐way operators for seismic modeling and imaging in 3D VTI media

Liu¹,

Zhang²

2006

Geophysical Research Letters

View full text Add to dashboard Cite

[1] One-way operators are proposed for modeling wideangle wave propagation in 3D heterogeneous, transversely isotropic media with a vertically symmetric axis (VTI). The operators are built in the wavenumber-space domain such that the terms in the space domain and in the wavenumber domain are separable. This means that the resulting algorithm can be implemented alternately in the space and wavenumber domains using the fast Fourier transforms. The proposed one-way operators can accommodate a wide range of anisotropy rather than the weak anisotropy. An optimization scheme is used to determine the coefficients in the operators by directly matching to spatially varying, exact anisotropic phase-shift operators over a prescribed angular range of wave propagation. We demonstrate the accuracy of the one-way operators by a comparison with the two-way elastic anisotropic finite-difference modeling in the presence of severe lateral variations. Citation: Liu, L., and J. Zhang (2006), Optimum split-step Fourier one-way operators for seismic modeling and imaging in 3D VTI media, Geophys. Res. Lett., 33, L09308,

show abstract

“…One-way approximation for wave propagation has been introduced and widely used as propagators in forward and inverse problems of scalar, acoustic and elastic waves (e.g., Claerbout, 1970Claerbout, , 1976Landers and Claerbout, 1972;Flatté and Tappert, 1975;Corones, 1975;Tappert, 1977;McCoy, 1977;Hudson, 1980;Ma, 1982;Wales and McCoy, 1983;McCoy, 1984, 1985;Wales, 1986;McCoy and Frazer, 1986;Collins, 1989Collins, , 1993Collins and Westwood, 1991;Stoffa et al, 1990;Fisk and McCartor, 1991;Wu and Huang, 1992;Ristow and Ruhl, 1994;Wu, 1994Wu, , 1996Wu, , 2003Wu and Xie, 1994;Huang and Wu, 1996;Wu and Jin, 1997;Grimbergen et al, 1998;Van Stralen et al, 1998;Wild and Hudson, 1998;Huang et al, 1999aHuang et al, , 1999bThomson, 1999Thomson, , 2005De Hoop et al, 2000;Lee et al, 2000;Wild et al, 2000;Wu et al, 2000aWu et al, , 2000bLe Rousseau and De Hoop, 2001;Wu, 2001, 2006;Wu, 2001, 2005;…”

Section: Introductionmentioning

confidence: 99%

One-way and one-return approximations (de wolf approximation) for fast elastic wave modeling in complex media

Wu¹,

Xie²,

Wu³

2007

Advances in Wave Propagation in Heterogenous Earth

View full text Add to dashboard Cite

The De Wolf approximation has been introduced to overcome the limitation of the Born and Rytov approximations in long range forward propagation and backscattering calculations. The De Wolf approximation is a multiple-forescattering-singlebackscattering (MFSB) approximation, which can be implemented by using an iterative marching algorithm with a single backscattering calculation for each marching step (a thin-slab). Therefore, it is also called a one-return approximation. The marching algorithm not only updates the incident field step-by-step, in the forward direction, but also the Green's function when propagating the backscattered waves to the receivers. This distinguishes it from the first order approximation of the asymptotic multiple scattering series, such as the generalized Bremmer series, where the Green's function is approximated by an asymptotic solution. The De Wolf approximation neglects the reverberations (internal multiples) inside thin-slabs, but can model all the forward scattering phenomena, such as focusing/defocusing, diffraction, refraction, interference, as well as the primary reflections.In this chapter, renormalized MFSB (multiple-forescattering-single-backscattering) equations and the dual-domain expressions for scalar, acoustic and elastic waves are derived by using a unified approach. Two versions of the one-return method (using MFSB approximation) are given: one is the wide-angle, dual-domain formulation (thin-slab approximation) (compared to the screen approximation, no small-angle approximation is made in the derivation); the other is the screen approximation. In the screen approximation, which involves a small-angle approximation for the wave-medium interaction, it can be clearly seen that the forward scattered, or transmitted waves are mainly controlled by velocity perturbations; while the backscattered or reflected waves, are mainly controlled by impedance perturbations. Later in this chapter the validity of the thin-slab and screen methods, and the wide-angle capability of the dual-domain implementation are demonstrated by numerical examples. Reflection coefficients of a plane interface, derived from numerical simulations by the wide-angle method, are shown to match the theoretical curves well up to critical angles. The methods are applied to the fast calculation of synthetic seismograms. The results are compared with finite difference (FD good agreement between the two methods verifies the validity of the one-return approach. However, the one-return approach is about 2-3 orders of magnitude faster than the elastic FD algorithm. The other example of application is the modeling of amplitude variation with angle (AVA) responses for a complex reservoir with heterogeneous overburdens. In addition to its fast computation speed, the one return method (thin-slab and complex-screen propagators) has some special advantages when applied to the thin-bed and random layer responses.

show abstract

Modal expansion of one‐way operators in laterally varying media

Cited by 60 publications

References 19 publications

Target-enclosed seismic imaging

Target-enclosed seismic imaging

Optimum split‐step Fourier one‐way operators for seismic modeling and imaging in 3D VTI media

One-way and one-return approximations (de wolf approximation) for fast elastic wave modeling in complex media

Contact Info

Product

Resources

About