On seismic interferometry, the generalized optical theorem, and the scattering matrix of a point scatterer
TL;DR: In this article, a far-field approximation of the Green's function representation for seismic interferometry is analyzed, and the generalized optical theorem is derived from the nonlinear scattering matrix of a point scatterer.
Abstract: We have analyzed the far-field approximation of the Green's function representation for seismic interferometry. By writing each of the Green's functions involved in the correlation process as a superposition of a direct wave and a scattered wave, the Green's function representation is rewritten as a superposition of four terms. When the scattered waves are modeled with the Born approximation, it appears that a three-term approximation of the Green's function representation (omitting the term containing the crosscorrelation of the scattered waves) yields a nearly exact retrieval, whereas the full four-term expression leads to a significant nonphysical event. This is because the Born approximation does not conserve energy and therefore is an insufficient model to explain all aspects of seismic interferometry. We use the full four-term expression of the Green's function representation to derive the generalized optical theorem. Unlike other recent derivations, which use stationary phase analysis, our derivation uses reciprocity theory. From the generalized optical theorem, we derive the nonlinear scattering matrix of a point scatterer. This nonlinear model accounts for primary and multiple scattering at the point scatterer and conforms with well-established scattering theory of classical waves. The model is essential to explain fully the results of seismic interferometry, even when it is applied to the response of a single point scatterer. The nonlinear scattering matrix also has implications for modeling, inversion, and migration.
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TL;DR: In this article, a trace-by-trace deconvolution process was proposed to compensate for complex source functions and the attenuation of the medium, which can also compensate for the effects of one-sided and/or irregular illumination.
Abstract: In the 1990s, the method of time-reversed acoustics was developed. This method exploits the fact that the acoustic wave equation for a lossless medium is invariant for time reversal. When ultrasonic responses recorded by piezoelectric transducers are reversed in time and fed simultaneously as source signals to the transducers, they focus at the position of the original source, even when the medium is very complex. In seismic interferometry the time-reversed responses are not physically sent into the earth, but they are convolved with other measured responses. The effect is essentially the same: The time-reversed signals focus and create a virtual source which radiates waves into the medium that are subsequently recorded by receivers. A mathematical derivation, based on reciprocity theory, formalizes this principle: The crosscorrelation of responses at two receivers, integrated over different sources, gives the Green’s function emitted by a virtual source at the position of one of the receivers and observed by the other receiver. This Green’s function representation for seismic interferometry is based on the assumption that the medium is lossless and nonmoving. Recent developments, circumventing these assumptions, include interferometric representations for attenuating and/or moving media, as well as unified representations for waves and diffusion phenomena, bending waves, quantum mechanical scattering, potential fields, elastodynamic, electromagnetic, poroelastic, and electroseismic waves. Significant improvements in the quality of the retrieved Green’s functions have been obtained with interferometry by deconvolution. A trace-by-trace deconvolution process compensates for complex source functions and the attenuation of the medium. Interferometry by multidimensional deconvolution also compensates for the effects of one-sided and/or irregular illumination.
212 citations
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01 Feb 2012
TL;DR: In this article, the authors proposed a method to discover the seismic wave propagation and scattering in the heterogeneous earth second edition book right here by downloading and getting the soft file of the book.
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205 citations
Cites background from "On seismic interferometry, the gene..."
...We note that the Green’s function retrieval for a scattering medium is strongly linked with the generalized optical theorem in the framework of scattering theory (Lu et al. 2011; Margerin and Sato 2011a,b; Snieder and Fleury 2010; Wapenaar et al. 2010)....
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TL;DR: The specific aspects of borehole radar are discussed and recent developments to become more sensitive to orientation and to exploit the supplementary information in different components in polarimetric uses of radar data are described.
Abstract: During the past 80 years, ground-penetrating radar (GPR) has evolved from a skeptically received glacier sounder to a full multicomponent 3D volume-imaging and characterization device. The tool can be calibrated to allow for quantitative estimates of physical properties such as water content. Because of its high resolution, GPR is a valuable tool for quantifying subsurface heterogeneity, and its ability to see nonmetallic and metallic objects makes it a useful mapping tool to detect, localize, and characterize buried objects. No tool solves all problems; so to determine whether GPR is appropriate for a given problem, studying the reasons for failure can provide an understanding of the basics, which in turn can help determine whether GPR is appropriate for a given problem. We discuss the specific aspects of borehole radar and describe recent developments to become more sensitiveto orientation and to exploit the supplementary information in different components in polarimetric uses of radar data. Multicomponent GPR data contain more diverse geometric information than single-channel data, and this is exploited in developed dedicated imaging algorithms. The evolution of these imaging schemes is discussed for ground-coupled and air-coupled antennas. For air-coupled antennas, the measured radiated wavefield can be used as the basis for the wavefield extrapolator in linear-inversion schemes with an imaging condition, which eliminates the source-time function and corrects for the measured radiation pattern. A handheld GPR system coupled with a metal detector is ready for routine use in mine fields. Recent advances in modeling, tomography, and full-waveform inversion, as well as Green's function extraction through correlation and deconvolution, show much promise in this field.
165 citations
TL;DR: In this paper, the authors present a new subsurface angle-domain seismic imaging system for generating and extracting high-resolution information about the surface angle-dependent reflectivity, which enables geophysicists to use all recorded seismic data in a continuous fashion.
Abstract: We present a new subsurface angle-domain seismic imaging systemforgeneratingandextractinghigh-resolutioninformation about subsurface angle-dependent reflectivity. The system enables geophysicists to use all recorded seismic data in a continuousfashiondirectlyinthesubsurfacelocalangledomainLAD, resulting in two complementary, full-azimuth, common-imageangle gather systems: directional and reflection. The complete setofinformationfrombothtypesofanglegathersleadstoaccurate, high-resolution, reliable velocity model determination and reservoir characterization. The directional angle decomposition enables the implementation of specular and diffraction imaging in real 3D isotropic/anisotropic geological models, leading to simultaneous emphasis on continuous structural surfaces and discontinuous objects such as faults and small-scale fractures. Structural attributes at each subsurface point, e.g., dip, azimuth andcontinuity,canbederiveddirectlyfromthedirectionalangle gathers. The reflection-angle gathers display reflectivity as a function of the opening angle and opening azimuth. These gathers are most meaningful in the vicinity of actual local reflecting surfaces,wherethereflectionanglesaremeasuredwithrespectto the derived background specular direction. The reflection-angle gathers are used for automatic picking of full-azimuth angle-domainresidualmoveoutsRMOwhich,togetherwiththederived background orientations of the subsurface reflection horizons, provide a complete set of input data to isotropic/anisotropic tomography. The full-azimuth, angle-dependent amplitude variations are used for reliable and accurate amplitude versus angle andazimuthAVAZanalysisandreservoircharacterization.The proposed system is most effective for imaging and analysis below complex structures, such as subsalt and subbasalt, high-velocity carbonate rocks, shallow low-velocity gas pockets, and others. In addition, it enables accurate azimuthal anisotropic imaging and analysis, providing optimal solutions for fracture detectionandreservoircharacterization.
164 citations
TL;DR: In this article, the authors propose an iterative substitution of the coupled Marchenko equations to retrieve the Green's functions from a source or receiver array at an acquisition surface to an arbitrary location in an acoustic medium.
Abstract: Iterative substitution of the coupled Marchenko equations is a novel methodology to retrieve the Green's functions from a source or receiver array at an acquisition surface to an arbitrary location in an acoustic medium. The methodology requires as input the single-sided reflection response at the acquisition surface and an initial focusing function, being the time-reversed direct wavefield from the acquisition surface to a specified location in the subsurface. We express the iterative scheme that is applied by this methodology explicitly as the successive actions of various linear operators, acting on an initial focusing function. These operators involve multidimensional crosscorrelations with the reflection data and truncations in time. We offer physical interpretations of the multidimensional crosscorrelations by subtracting traveltimes along common ray paths at the stationary points of the underlying integrals. This provides a clear understanding of how individual events are retrieved by the scheme. Our interpretation also exposes some of the scheme's limitations in terms of what can be retrieved in case of a finite recording aperture. Green's function retrieval is only successful if the relevant stationary points are sampled. As a consequence, internal multiples can only be retrieved at a subsurface location with a particular ray parameter if this location is illuminated by the direct wavefield with this specific ray parameter. Several assumptions are required to solve the Marchenko equations. We show that these assumptions are not always satisfied in arbitrary heterogeneous media, which can result in incomplete Green's function retrieval and the emergence of artefacts. Despite these limitations, accurate Green's functions can often be retrieved by the iterative scheme, which is highly relevant for seismic imaging and inversion of internal multiple reflections.
133 citations
Cites background from "On seismic interferometry, the gene..."
...This effect is also observed in Green’s function retrieval by seismic interferometry (Wapenaar et al. 2010)....
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References
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TL;DR: The history of the so-called optical theorem in scattering theory is traced from its beginning over 100 years ago to recent applications of its generalizations as discussed by the authors, which is the basis for this paper.
Abstract: The history of the so‐called optical theorem in scattering theory is traced from its beginning over 100 years ago to recent applications of its generalizations.
273 citations
TL;DR: It is shown that the multiple-scattering formalism developed in condensed matter physics provides a rigorous basis to analyze the field correlations in disordered media and establishes a fruitful mapping between time reversal and correlation.
Abstract: This paper presents an interdisciplinary review of the correlation properties of random wavefields. We expose several important theoretical results of various fields, ranging from time reversal in acoustics to transport theory in condensed matter physics. Using numerical simulations, we introduce the correlation process in an intuitive manner. We establish a fruitful mapping between time reversal and correlation, which enables us to transpose many known results from acoustics to seismology. We show that the multiple-scattering formalism developed in condensed matter physics provides a rigorous basis to analyze the field correlations in disordered media. We discuss extensively the various factors controlling and affecting the retrieval of the Green's function of a complex medium from the correlation of either noise or coda. Acoustic imaging of complex samples in the laboratory and seismic tomography of geologic structures give a glimpse of the promising wide range of applications of the correlation method.
225 citations
TL;DR: In this article, the authors proposed a method to estimate the thickness of the reflected/diffracted energy at each considered location in depth to improve the expected quality of PSDM images.
Abstract: Prestack depth migration (PSDM) should be the ultimate goal of seismic processing, producing angle-dependant depth images of the subsurface reflectivity. But the expected quality of PSDM images is constrained by many factors. Understanding all of these factors is necessary to improve depth imaging of geologic structures. In all PSDM approaches, e.g., Kirchhoff or wave-equation, migration always includes compensating for wave propagation in the overburden (back propagation, downward continuation, etc.), before focusing back the reflected/diffracted energy at each considered location in depth (imaging). Ideally, we would like to retrieve the reflectivity of the ground as detailed as possible to invert for the elastic parameters. But the waves perceive the reflectivity through “thick glasses,” seeing blurred structures, and not necessarily all of them, depending on the illumination. Only a filtered version of the true reflectivity is therefore retrieved. Being able to estimate these filters, the so-called re...
201 citations
TL;DR: This work presents a methodology providing a new perspective on modeling and inversion of wave propagation satisfying time-reversal invariance and reciprocity in generally inhomogeneous media using a representation theorem of the wave equation.
Abstract: We present a methodology providing a new perspective on modeling and inversion of wave propagation satisfying time-reversal invariance and reciprocity in generally inhomogeneous media. The approach relies on a representation theorem of the wave equation to express the Green function between points in the interior as an integral over the response in those points due to sources on a surface surrounding the medium. Following a predictable initial computational effort, Green's functions between arbitrary points in the medium can be computed as needed using a simple cross-correlation algorithm.
169 citations
"On seismic interferometry, the gene..." refers background in this paper
...This exact Green’s function representation is the basis for seismic nterferometry van Manen et al., 2005; Wapenaar et al., 2005 ....
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TL;DR: In this paper, the authors compare two approaches for deriving the fact that the Green's function in an arbitrary inhomogeneous open system can be obtained by cross correlating recordings of the wave field at two positions.
Abstract: We compare two approaches for deriving the fact that the Green’s function in an arbitrary inhomogeneous open system can be obtained by cross correlating recordings of the wave field at two positions. One approach is based on physical arguments, exploiting the principle of time-reversal invariance of the acoustic wave equation. The other approach is based on Rayleigh’s reciprocity theorem. Using a unified notation, we show that the result of the time-reversal approach can be obtained as an approximation of the result of the reciprocity approach.
166 citations