An optimal PE-type wave equation

Berman, David H.; Wright, E. B.; Baer, Ralph N.

doi:10.1121/1.398338

Cited by 18 publications

(6 citation statements)

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“…The accuracy of this improved one-way propagator has been analyzed in those papers and more recently by HUANG and FEHLER (1998). The other approach to improve the phasescreen propagator was to match its travel time with the ray equation (TOLSTOY et al, 1985;BERMAN et al, 1989). BERMAN et al (1989) changed the phase correction term of the screen into log n, where n is the refraction index of the medium.…”

Section: Classical Scalar-wave Dual-domain Propagatorsmentioning

confidence: 99%

“…The other approach to improve the phasescreen propagator was to match its travel time with the ray equation (TOLSTOY et al, 1985;BERMAN et al, 1989). BERMAN et al (1989) changed the phase correction term of the screen into log n, where n is the refraction index of the medium. However, all the improvement is kept in the realm of classical phase-screen correction.…”

Section: Classical Scalar-wave Dual-domain Propagatorsmentioning

confidence: 99%

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Wave Propagation, Scattering and Imaging Using Dual-domain One-way and One-return Propagators

Wu¹

2003

Pure appl. geophys.

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Dual-domain one-way propagators implement wave propagation in heterogeneous media in mixed domains (space-wavenumber domains). One-way propagators neglect wave reverberations between heterogeneities but correctly handle the forward multiple-scattering including focusing/defocusing, diffraction, refraction and interference of waves. The algorithm shuttles between space-domain and wavenumber-domain using FFT, and the operations in the two domains are self-adaptive to the complexity of the media. The method makes the best use of the operations in each domain, resulting in efficient and accurate propagators. Due to recent progress, new versions of dual-domain methods overcame some limitations of the classical dual-domain methods (phase-screen or split-step Fourier methods) and can propagate large-angle waves quite accurately in media with strong velocity contrasts. These methods can deliver superior image quality (high resolution/high fidelity) for complex subsurface structures. One-way and one-return (De Wolf approximation) propagators can be also applied to wave-field modeling and simulations for some geophysical problems. In the article, a historical review and theoretical analysis of the Born, Rytov, and De Wolf approximations are given. A review on classical phase-screen or split-step Fourier methods is also given, followed by a summary and analysis of the new dual-domain propagators. The applications of the new propagators to seismic imaging and modeling are reviewed with several examples. For seismic imaging, the advantages and limitations of the traditional Kirchhoff migration and time-space domain finite-difference migration, when applied to 3-D complicated structures, are first analyzed. Then the special features, and applications of the new dual-domain methods are presented. Three versions of GSP (generalized screen propagators), the hybrid pseudo-screen, the wide-angle Pade´-screen, and the higherorder generalized screen propagators are discussed. Recent progress also makes it possible to use the dualdomain propagators for modeling elastic reflections for complex structures and long-range propagations of crustal guided waves. Examples of 2-D and 3-D imaging and modeling using GSP methods are given.

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Section: Classical Scalar-wave Dual-domain Propagatorsmentioning

confidence: 99%

Section: Classical Scalar-wave Dual-domain Propagatorsmentioning

confidence: 99%

Wave Propagation, Scattering and Imaging Using Dual-domain One-way and One-return Propagators

Wu¹

2003

Pure appl. geophys.

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“…PSi (Tolstoy, 1985;Berman, 1989). For the case of thin-slab, we will omit the z position z -zz/2 of the symbol.…”

Section: Wideang1e Pseudo-screen and Phase-screen Approximationsmentioning

confidence: 97%

<title>Accuracy analysis and numerical tests of screen propagators for wave extrapolation</title>

Hoop

1996

SPIE Proceedings

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One-way wave extrapolators, such as the split-step Fourier method, generalized screen methods including the phase-screen, complex screen and wide-angle pseudo-screen methods, have been proposed recently as part of the solution to lessen the CPU and memory size requirement of wave equation based 3D subsurface imaging methods. The phase space path-integral formulation and the vertical slowness symbol analysis provide a general and convenient background for accuracy estimation and improvement of screen propagators. In this paper we review and elucidate the relevant theory and formulation in paper I (de Hoop and Wu, 1996), and present the results of accuracy analysis and numerical tests for screen propagators. By comparing the dispersion relations of screen propagators in the high-frequency limit with the leading term high-frequency asymptotics to the vertical slowness symbol, and through numerical experiments of thin-slab transmission, it is seen that for weak perturbations both the phase-screen and the wide-angle pseudo-screen propagators perform well; While for the case of strong medium contrasts, the modified wide-angle pseudo-screen propagator has much better accuracy for large-angle waves than the phase screen propagator. Unlike the traditional phase screen propagator which is purely a space domain operator, the wide-angle pseudo-screen propagator is a dual-damain operator. The screen propagators have great potential in the application to 3D prestack depth migration/inversion as extrapolators.

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“…The normal pro-cedure of neglecting the higher order derivatives in the propagation direction leads to the standard PE (parabolic equation)type phase-screen [Tappert, 1977;Flatte'and Tappert, 1975;Martin and Flatte• 1988], which can be also derived by the truncated Taylor expansion of the square-root operator. The other approach for improving the accuracy is to match the ray equation of the phase-screen method which is a one-way equation, with the ray equation of the two-way Helmholtz equation [Tolstoy et al, 1985;Berman et al, 1989]. The propagation term is a wide-angle free propagator, while the interaction term is a multiplication by (n-l), where n is the refractive index.…”

Section: Limiting Case Of Scalar Wavesmentioning

confidence: 99%

Wide‐angle elastic wave one‐way propagation in heterogeneous media and an elastic wave complex‐screen method

Wu¹

1994

J. Geophys. Res.

179

134

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In this paper a system of equations for wide‐angle one‐way elastic wave propagation in arbitrarily heterogeneous media is formulated in both the space and wavenumber domains using elastic Rayleigh integrals and local elastic Born scattering theory. The wavenumber domain formulation leads to compact solutions to one‐way propagation and scattering problems. It is shown that wide‐angle scattering in heterogeneous elastic media cannot be formulated as passage through regular phase‐screens, since the interaction between the incident wavefield and the heterogeneities is not local in both the space domain and the wavenumber domain. Our more generally valid formulation is called the “thin‐slab” formulation. After applying the small‐angle approximation, the thin‐slab effect degenerates to that of an elastic complex‐screen (or “generalized phase‐screen”). Compared with scalar phase‐screen, the elastic complex‐screen has the following features. (1) For P‐P scattering and S‐S in‐plane scattering, the elastic complex‐screen acts as two separate scalar phase‐screens for P and S waves respectively. The phase distortions are determined by the P and S wave velocity perturbations respectively. (2) For P‐S and S‐P conversions, the screen is no longer a pure phase‐screen and becomes complex (with both phase and amplitude terms); both conversions are determined by the shear wave velocity perturbation and the shear modulus perturbation. For Poisson solids the S wave velocity perturbation plays a major role. In the special case of α0 = 2β0, S wave velocity perturbation becomes the only factor for both conversions. (3) For the cross‐coupling between in‐plane S waves and off‐plane S waves, only the shear modulus perturbation δμ has influence in the thin‐slab formulation. For the complex‐screen method the cross‐coupling term is neglected because it is a higher order small quantity for small‐angle scattering. Relative to prior derivations of vector phase‐screen method, our method can correctly treat the conversion between P and S waves and the cross‐coupling between differently polarized S waves. A comparison with solutions from three‐dimensional finite difference and exact solutions using eigenfunction expansion is made for two special cases. One is for a solid sphere with only P velocity perturbation; the other is with only S velocity perturbation. The Elastic complex‐screen method generally agrees well with the three‐dimensional finite difference method and the exact solutions. In the limiting case of scalar waves, the derivation in this paper leads to a more generally valid new method, namely, a scalar thin‐slab method. When making the small‐angle approximation to the interaction term while keeping the propagation term unchanged, the thin‐slab method approaches the currently available scalar wide‐angle phase‐screen method.

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An optimal PE-type wave equation

Cited by 18 publications

References 0 publications

Wave Propagation, Scattering and Imaging Using Dual-domain One-way and One-return Propagators

Wave Propagation, Scattering and Imaging Using Dual-domain One-way and One-return Propagators

<title>Accuracy analysis and numerical tests of screen propagators for wave extrapolation</title>

Wide‐angle elastic wave one‐way propagation in heterogeneous media and an elastic wave complex‐screen method

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