Derivation and application of extended parabolic wave theories. II. Path integral representations

Fishman, Louis

doi:10.1063/1.526150

Cited by 52 publications

(20 citation statements)

References 18 publications

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“…Large errors may exist around sharp boundaries. The expansion of the symbol in terms of h-f series can be found in FISHMAN and MCCOY (1984). Retaining only the leading term leads to a simple form of c HF ð0Þ ¼ fa 2 ðX T Þ À a 2 T g 1=2 which is the principal part of the symbol , where c is the vertical slowness, aðX T Þ is the local slowness (inverse velocity) at a transverse position X T and a T is the horizontal slowness.…”

Section: Examples Of Wide-angle 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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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: Examples Of Wide-angle 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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“…Thus we arrive at a ‘time’‐ordered product integral representation (De Witte‐Morette et al 1979) of the one‐sided propagators ( cf. associated with the one‐way wave equations (Fishman & McCoy 1984a,b; De Hoop 1996; Fishman et al 1997) where ‘time’ here refers to the vertical coordinate x 3 , where H denotes the Heaviside function. In this expression, the operator ordering is initiated by acting on followed by applying to the result, successively for increasing ζ.…”

Section: Trotter‐product One‐way Wave Propagatormentioning

confidence: 99%

Symplectic structure of wave-equation imaging: a path-integral approach based on the double-square-root equation

Hoop

Rousseau

Biondi

2003

Geophysical Journal International

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SUMMARY We carry out high‐frequency analyses of Claerbout's double‐square‐root equation and its (numerical) solution procedures in heterogeneous media. We show that the double‐square‐root equation generates the adjoint of the single‐scattering modelling operator upon substituting the leading term of the generalized Bremmer series for the background Green function. This adjoint operator yields the process of ‘wave‐equation’ imaging. We finally decompose the wave‐equation imaging process into common image point gathers in accordance with the characteristic strips in the wavefront set of the data.

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“…Numerical solutions for operations on functions can be obtained by an appropriate approximation and discretization of the operator symbol through the operator kernel. Employing the pseudo‐differential operator calculus, the relation between a pseudo‐differential D‐t‐N operator and its symbol can be obtained as a function of the transverse space coordinates and the Fourier dual transverse wavenumber coordinates, together constituting the transverse phase space (Fishman & McCoy 1984a,b; Grubb 1996; Fishman et al 2000). This calculus generalizes previous methods where medium parameters are assumed to be locally constant, enabling a reduction to an ordinary differential equation by Fourier transform over the transverse coordinates (Grubb 1996).…”

Section: Introductionmentioning

confidence: 99%

Recovering acoustic reflectivity using Dirichlet-to-Neumann maps and left- and right-operating adjoint propagators

Dillen

Fokkema

Wapenaar

2005

Geophysical Journal International

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S U M M A R YConstructing an image of the Earth subsurface from acoustic wave reflections has previously been described as a recursive downward redatuming of sources and receivers. Most of the methods that have been presented involve reflectivity and propagators associated with oneway wavefield components. In this paper, we consider the reflectivity relation between twoway wavefield components, each a solution of a Helmholtz equation. To construct forward and inverse propagators, and a reflection operator, the invariant-embedding technique is followed, using Dirichlet-to-Neumann maps. Employing bilinear and sesquilinear forms, the forwardand inverse-scattering problems, respectively, are treated analogously. Through these mathematical constructs, the relationship between a causality radiation condition and symmetry, with respect to a bilinear form, is associated with the requirement of an anticausality radiation condition with respect to a sesquilinear form. Using reciprocity, sources and receivers are redatumed recursively to the reflector, employing left-and right-operating adjoint propagators. The exposition of the proposed method is formal, that is numerical applications are not derived.The key to applications lies in the explicit representation, characterization and approximation of the relevant operators (symbols) and fundamental solutions (path integrals). Existing constructive work which could be applied to the proposed method are referred to in the text.In Berkhout (1985) acoustic wave propagation and reflection in discretized space is represented by matrix operations. According to this model the reflected wavefield from an acoustic contrast is obtained by multiplying a multisource wavefields matrix by a product of three matrices. Evaluating this product from the right to the left, modelling a surface seismic experiment, the first matrix propagates the source wavefields downward to the reflector, the second matrix reflects these and the third matrix propagates the reflected wavefields upward to the measurement surface. Because the wavefields are considered in the temporal frequency domain, forward wavefields propagation is recursive with respect to depth. This means that propagation can be handled incrementlly through the medium by ordered matrix multiplications. Inversion for the reflection matrix is also represented by a three-matrix product. The measured wavefield matrix, in which each column represents a different common source gather, is then multiplied from the right and the left with an approximate inverse-propagation matrix, which is the Hermitian of the forward-propagation matrix.In Wapenaar (1996a) Berkhout's model is generalized to R 3 for one-way wavefields employing operators for matrices. Wavefield decomposition into one-way wavefields (Weston 1988;de Hoop 1996;Wapenaar & Grimbergen 1996) follows from a factorization of the two-way Helmholtz wave equation. Propagation in the marching direction, determined by the square-root Helmholtz operator, is then separated from scattering from medium v...

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Derivation and application of extended parabolic wave theories. II. Path integral representations

Cited by 52 publications

References 18 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

Symplectic structure of wave-equation imaging: a path-integral approach based on the double-square-root equation

Recovering acoustic reflectivity using Dirichlet-to-Neumann maps and left- and right-operating adjoint propagators

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