Calculation of Synthetic Seismograms with Gaussian Beams

Nowack, Robert L.

doi:10.1007/pl00012547

Cited by 35 publications

(17 citation statements)

References 61 publications

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“…A long‐standing question about the description of pure head waves by the Gaussian‐beam method (Červený 1985; White et al 1987; Weber 1988; Nowack 2003) has essentially been answered using the exact forward and inverse coherent‐state transform. However, resolving this idealized theoretical issue is not the ultimate goal here.…”

Section: Discussionmentioning

confidence: 99%

“…In fact, the Gaussian‐beam pre‐stack migration method of Hill (2001) displays essentially these attributes already. The Gaussian‐beam summation (Červený et al 1982; Červený 2001; Nowack 2003) and Maslov–Fourier methods are somewhat similar and one could say the CS method unifies them. The higher dimensional space needed to describe the CS rays is an important conceptual distinction, but the fact that in practice the real rays of the physical wave front will probably suffice brings the method into close agreement with its two predecessors.…”

Section: Introductionmentioning

confidence: 99%

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Coherent-state analysis of the seismic head wave problem: an overcomplete representation and its relationship to rays and beams

Thomson

2004

Geophysical Journal International

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S U M M A R YThe coherent-state transform (CST) is a Gaussian-windowed Fourier transform and compared with the usual plane wave expansion of seismic wavefields it provides an overcomplete basis of partial waves. These overcomplete partial waves have associated rays and the relationship of these rays to those of a physical wave front is explained here by appealing to a familiar analytic example. The exact CST of the standard Fourier plane wave summation for a point-source primary wave can be interpreted as a sum of damped plane waves forming a focussed or beamlike wavefield. This primary wave CST can also be approximated as a bundle of complex rays in a position-slowness space with higher dimension than that usually considered, a consequence of the overcompleteness. The higher-dimensional ray spreading controls the coherent-state (CS) amplitude. This complex-ray bundle in turn can be approximated as a paraxial Gaussian beam carried by a real ray associated with the physical wave front.The exact CST of the standard plane wave summation for a point-source reflection shows how the complex-ray and paraxial-beam approximations generalize to interfaces. Around a critical angle the standard plane wave summation has an asymptotic form involving the Weber function and this function also necessarily arises in the complex-ray and paraxial-beam approximations for the reflection CST. The inverse CST giving the point-source wavefield is a sum of coherent states and those contributing rays or paraxial beams that are incident around the critical angle should be given a reflection coefficient involving the Weber function. CS beams that are incident away from the critical angle collect a peripheral branch-point diffraction. In general, both types of critical-ray/branch-point signal should be included if the integrated CS response is to correctly describe the head wave.An advantage of the CST is that it combines with ray theory for gradually varying media to give a convenient solution to the caustic and pseudocaustic problems. It may be said to unify the Maslov and standard Gaussian-beam methods of seismic modelling, in part by avoiding the ray-centered coordinates conventionally employed in the latter. The general idea of such overlapping partial wave expansions, their flexibility and the smoothing they impart may have other benefits in seismic analysis and processing.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Coherent-state analysis of the seismic head wave problem: an overcomplete representation and its relationship to rays and beams

Thomson

2004

Geophysical Journal International

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“…There is a rich literature on the use of Gaussian beam summation in computing high‐frequency seismic wavefields in smoothly varying inhomogeneous media and on the successful applications of Gaussian beams to seismic depth migration (Červený, Popov, and Pšenčík ; Popov ; Nowack and Aki ; Červený ; Babich and Popov ; Hill , ; Nowack ; Gray ; Červený, Klimeš, and Pšenčík ; Zhu, Gray, and Wang ; Gray and Bleistein ; Popov et al . ).…”

Section: Introductionmentioning

confidence: 99%

Gabor‐frame‐based Gaussian packet migration

Geng

Gao

2014

Geophysical Prospecting

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We present a Gaussian packet migration method based on Gabor frame decomposition and asymptotic propagation of Gaussian packets. A Gaussian packet has both Gaussian‐shaped time–frequency localization and space–direction localization. Its evolution can be obtained by ray tracing and dynamic ray tracing. In this paper, we first briefly review the concept of Gaussian packets. After discussing how initial parameters affect the shape of a Gaussian packet, we then propose two Gabor‐frame‐based Gaussian packet decomposition methods that can sparsely and accurately represent seismic data. One method is the dreamlet–Gaussian packet method. Dreamlets are physical wavelets defined on an observation plane and can represent seismic data efficiently in the local time–frequency space–wavenumber domain. After decomposition, dreamlet coefficients can be easily converted to the corresponding Gaussian packet coefficients. The other method is the Gabor‐frame Gaussian beam method. In this method, a local slant stack, which is widely used in Gaussian beam migration, is combined with the Gabor frame decomposition to obtain uniform sampled horizontal slowness for each local frequency. Based on these decomposition methods, we derive a poststack depth migration method through the summation of the backpropagated Gaussian packets and the application of the imaging condition. To demonstrate the Gaussian packet evolution and migration/imaging in complex models, we show several numerical examples. We first use the evolution of a single Gaussian packet in media with different complexities to show the accuracy of Gaussian packet propagation. Then we test the point source responses in smoothed varying velocity models to show the accuracy of Gaussian packet summation. Finally, using poststack synthetic data sets of a four‐layer model and the two‐dimensional SEG/EAGE model, we demonstrate the validity and accuracy of the migration method. Compared with the more accurate but more time‐consuming one‐way wave‐equation‐based migration, such as beamlet migration, the Gaussian packet method proposed in this paper can correctly image the major structures of the complex model, especially in subsalt areas, with much higher efficiency. This shows the application potential of Gaussian packet migration in complicated areas.

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“…Between these end members are the so-called hybrid methods. Extensions to basic ray theory are the Kirchhoff [e.g., 9; 10], Maslov [e.g., 11; 12], Gaussian beam [e.g., 13] and coherent-state [e.g., 14] methods, which describe additional diffraction effects to various degrees. Transform methods involve separating the partial differential wave equation into an ordinary differential wave equation with appropriate boundary conditions [see 11, for a review] and are suitable for homogeneous [e.g., 15] and weakly heterogeneous [e.g., 11] layered media.…”

Section: Introductionmentioning

confidence: 99%

The One-Way Wave Equation: A Full-Waveform Tool for Modeling Seismic Body Wave Phenomena

Angus

2013

Surv Geophys

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The study of seismic body waves is an integral aspect in global, exploration and engineering scale seismology, where the forward modeling of waves is an essential component in seismic interpretation. Forward modeling represents the kernel of both migration and inversion algorithms as the Green's function for wavefield propagation, and is also an important diagnostic tool that provides insight into the physics of wave propagation and a means of testing hypotheses inferred from observational data. This paper introduces the one-way wave equation method for modeling seismic wave phenomena and specifically focuses on the so-called operatorroot one-way wave equations. To provide some motivation for this approach, this review first summarizes the various approaches in deriving one-way approximations and subsequently discusses several alternative matrix narrow-angle and wide-angle formulations. To demonstrate the key strengths of the one-way approach, results from waveform simulation for global scale shear-wave splitting modeling, reservoir-scale frequency dependent shear-wave splitting modeling, and acoustic waveform modeling in random heterogeneous media are shown. These results highlight the main feature of the one-way wave equation approach in terms of its ability to model gradual vector (for the elastic case) and scalar (for the acoustic case) waveform evolution along the underlying wavefront. Although not strictly an exact solution, the one-way wave equation shows significant advantages (e.g., computational efficiency) for a range of transmitted wave three-dimensional global, exploration and engineering scale applications.

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Calculation of Synthetic Seismograms with Gaussian Beams

Cited by 35 publications

References 61 publications

Coherent-state analysis of the seismic head wave problem: an overcomplete representation and its relationship to rays and beams

Coherent-state analysis of the seismic head wave problem: an overcomplete representation and its relationship to rays and beams

Gabor‐frame‐based Gaussian packet migration

The One-Way Wave Equation: A Full-Waveform Tool for Modeling Seismic Body Wave Phenomena

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