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.
Citations
More filters
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
Book•
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.
Abstract: Only for you today! Discover your favourite seismic wave propagation and scattering in the heterogeneous earth second edition book right here by downloading and getting the soft file of the book. This is not your time to traditionally go to the book stores to buy a book. Here, varieties of book collections are available to download. One of them is this seismic wave propagation and scattering in the heterogeneous earth second edition as your preferred book. Getting this book b on-line in this site can be realized now by visiting the link page to download. It will be easy. Why should be here?
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)....
[...]
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)....
[...]
References
More filters
TL;DR: In this paper, a precise treatment of the diffuse intensity is derived which automatically includes the effects of boundary layers, and effects such as the enhanced backscatter cone and imaging of objects in opaque media are also discussed.
Abstract: important corrections are presented. These corrections are calculated with the radiative transfer or Schwarzschild-Milne equation, which describes intensity transport at the ‘‘mesoscopic’’ level and is derived from the ‘‘microscopic’’ wave equation. A precise treatment of the diffuse intensity is derived which automatically includes the effects of boundary layers. Effects such as the enhanced backscatter cone and imaging of objects in opaque media are also discussed within this framework. This approach is extended to mesoscopic correlations between multiple scattered intensities that arise when scattering is strong. These correlations arise from the underlying wave character. The derivation of correlation functions and intensity distribution functions is given and experimental data are discussed. Although the focus is on light scattering, the theory is also applicable to microwaves, sound waves, and noninteracting electrons. [S0034-6861(99)00601-7]
615 citations
"On seismic interferometry, the gene..." refers methods in this paper
...We obtain an expansion for which the different erms account for primary and multiple scattering at the point scaterer van Rossum and Nieuwenhuizen, 1999 ....
[...]
TL;DR: In this paper, the authors formalize the classical diffraction stack by relating it to linearized seismic inversion and the generalized Radon transform, which can handle both complex velocity models and arbitrary configurations of sources and receivers.
Abstract: A new approach to seismic migration formalizes the classical diffraction (or common-tangent) stack by relating it to linearized seismic inversion and the generalized Radon transform. This approach recasts migration as the problem of reconstructing the earth’s acoustic scattering potential from its integrals over isochron surfaces. The theory rests on a solution of the wave equation with the geometrical-optics Green function and an approximate inversion formula for the generalized Radon transform. The method can handle both complex velocity models and (nearly) arbitrary configurations of sources and receivers. In this general case, the method can be implemented as a weighted diffraction stack, with the weights determined by tracing rays from image points to the experiment’s sources and receivers. When tested on a finite-difference simulation of a deviated-well vertical seismic profile (a hybrid experiment which is difficult to treat with conventional wave-equation methods), the algorithm accurately reconstructed faulted-earth models. Analytical reconstruction formulas are derived from the general formula for zero-offset and fixed-offset surface experiments in which the background velocity is constant. The zero-offset inversion formula resembles standard Kirchhoff migration. Our analysis provides a direct connection between the experimental setup (source and receiver positions, source wavelet, background velocity) and the spatial resolution of the reconstruction. Synthetic examples illustrate that the lateral resolution in seismic images is described well by the theory and is improved greatly by combining surface data and borehole data. The best resolution is obtained from a zero-offset experiment that surrounds the region to be imaged.
430 citations
TL;DR: In this paper, the authors present an overview and a detailed description of the key logic steps and mathematical-physics framework behind the development of practical algorithms for seismic exploration derived from the inverse scattering series.
Abstract: This paper presents an overview and a detailed description of the key logic steps and mathematical-physics framework behind the development of practical algorithms for seismic exploration derived from the inverse scattering series. There are both significant symmetries and critical subtle differences between the forward scattering series construction and the inverse scattering series processing of seismic events. These similarities and differences help explain the efficiency and effectiveness of different inversion objectives. The inverse series performs all of the tasks associated with inversion using the entire wavefield recorded on the measurement surface as input. However, certain terms in the series act as though only one specific task, and no other task, existed. When isolated, these terms constitute a task-specific subseries. We present both the rationale for seeking and methods of identifying uncoupled task-specific subseries that accomplish: (1) free-surface multiple removal; (2) internal multiple attenuation; (3) imaging primaries at depth; and (4) inverting for earth material properties. A combination of forward series analogues and physical intuition is employed to locate those subseries. We show that the sum of the four task-specific subseries does not correspond to the original inverse series since terms with coupled tasks are never considered or computed. Isolated tasks are accomplished sequentially and, after each is achieved, the problem is restarted as though that isolated task had never existed. This strategy avoids choosing portions of the series, at any stage, that correspond to a combination of tasks, i.e., no terms corresponding to coupled tasks are ever computed. This inversion in stages provides a tremendous practical advantage. The achievement of a task is a form of useful information exploited in the redefined and restarted problem; and the latter represents a critically important step in the logic and overall strategy. The individual subseries are analysed and their strengths, limitations and prerequisites exemplified with analytic, numerical and field data examples.
382 citations
354 citations
Book•
01 Jan 1993
TL;DR: In this paper, the reciprocity theorem was chosen as the central theme of the seismic wave theory, and the seismic experiment was formulated as terme of a geological system response to a known source function.
Abstract: Progress in seismic data processing requires the knowledge of all the theoretical aspects of the acoustic wave theory. We choose the reciprocity theorem as the central theme, because it constitutes the fundaments of the seismic wave theory (Fokkema and van den Berg, 1993). In essence, two states are distinguished in this theorem. These can be completely different, although they share the same time -invariant domgin of application and they are related via an interaction quantity. The particular choice of the two states determines the acoustic application. This makes it possible to formulate the seismic experiment in terme of a geological system response to a known source function.
334 citations