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, the authors define the seismic image as a locally scattered wavefield and introduce a new imaging condition that is suitable and practical for nonlinear imaging, which is similar to the one in this paper.
Abstract: Imaging highly complex subsurface structures is a challenging problem because it must ultimately deal with nonlinear multiple-scattering effects (e.g., migration of multiples, wavefield extrapolation with nonlinear model interactions, amplitude-preserving migration) to overcome the limitations of linear imaging. Most of the current migration techniques rely on the linear single-scattering assumption, and therefore, fail to handle these complex scattering effects. Recently, seismic imaging has been related to scattering-based image-domain interferometry to address the fully nonlinear imaging problem. Building on this connection between imaging and interferometry, we define the seismic image as a locally scattered wavefield and introduce a new imaging condition that is suitable and practical for nonlinear imaging. A previous formulation of nonlinear scattering-based imaging requires the evaluation of volume integrals that cannot easily be incorporated in current imaging algorithms. Our method consisted of a...
42 citations
Cites background or methods from "On seismic interferometry, the gene..."
...…is responsible for an erroneous estimate of the scattering amplitude of scattering objects such as reflectors and diffractors, for missing high order scattering events in the image, and for spurious scattering events mapped in the image (Curtis and Halliday, 2010; Wapenaar et al., 2010)....
[...]
...The Born approximation does not satisfy the generalized optical theorem (Wapenaar et al., 2010) in the same way that it violates energy conservation (Rodberg and Thaler, 1967; Wolf and Born, 1965)....
[...]
...The conventional imaging condition is only accurate in the Born sense and does not conserve power (Wolf and Born, 1965; Rodberg and Thaler, 1967; Wapenaar et al., 2010)....
[...]
...Handling such waves incorrectly may lead to spurious arrivals (Snieder et al., 2008; Fleury et al., 2010; Wapenaar et al., 2010)....
[...]
TL;DR: In this article, a multichannel analysis of surface waves (MASW) is used to estimate shallow shear-wave velocity (V S ) structure often needed to estimate ground motions using recent ground motion prediction relations.
Abstract: Refraction microtremor (ReMi) and multichannel analysis of surface waves (MASW) are effective approaches to estimate shallow shear-wave velocity ( V S ) structure often needed to estimate ground motions using recent ground motion prediction relations. Interferometric MASW (IMASW) uses slowness-frequency slant-stack analyses combined with interferometric time-domain dispersion analyses to improve resolution of lower-frequency Rayleigh-wave dispersion to better constrain V S . Cross-correlation interferometry is used to obtain deterministic correlation Green’s function (CGF) IMASW seismograms from ambient-noise and/or active-source wave fields contained in ReMi and/or MASW data. The CGFs are processed using the multiple-filter technique to estimate phase and group dispersion. In the IMASW approach, active seismic sources ensure that the stationary-phase contributions to cross correlations dominate CGF responses. In a single IMASW profile, each geophone represents a virtual source, and the IMASW approach stacks CGF common-offset data from all virtual sources to obtain a single averaged forward- and reverse-record section. CGF time-domain and slowness-frequency phase-slowness estimates are combined with CGF time-domain group slowness estimates for a consistency check on dispersion picks. A multistate Monte Carlo approach is used to estimate mean slowness depth and slowness uncertainties. IMASW is evaluated with passive ReMi data from two sites and active-source IMASW at six sites with independent downhole velocity–depth logs. Comparison of six P-S suspension log–IMASW profile pairs across the Van Norman Complex in northern San Fernando Valley shows that, on average, 30-m-depth shear-wave velocity estimates between the two methods differed by P-S suspension log measurements of V S were made at the IMASW profile midpoint, the IMASW V S depth inversions resolve 3-m thickness V S variations accurately to the bottom of one borehole at 40-m depth and to 100-m depth at a >200-m-deep borehole site.
34 citations
TL;DR: In this article, a correlation-type representation theorem for perturbed elastic media, commonly used in seismic interferometry to explain how a scattered wave response between two receivers/ sources may be predicted given a boundary of sources/receivers, can be considered as a starting point for the derivation.
Abstract: With the more widespread introduction of multicomponent recording devices in land and marine ocean-bottom seismic acquisition, elastic imaging may become mainstream in coming years. We have derived new, nonlinear, elastic imaging conditions. A correlation-type representation theorem for perturbed elastic media, commonly used in seismic interferometry to explain how a scattered wave response between two receivers/ sources may be predicted given a boundary of sources/receivers, can be considered as a starting point for the derivation. Here, we use this theorem to derive and interpret imaging conditions for elastic migration by wavefield extrapolation (e.g., elastic reverse-time migration). Some approximations lead to a known, heuristically derived imaging condition that crosscorrelates P- and S-wave potentials that are separated in the subsurface after full-wavefield extrapolation. This formal connection reveals that the nonapproximated correlation-type representation theorem can be interpreted as a nonlinear imaging condition, that accounts also for multiply scattered and multiply converted waves, properly focusing such energy at each image point. We present a synthetic data example using either an ideal (acquisition on a full, closed boundary) or a real (partial boundary) seismic exploration survey, and we demonstrate the importance of nonlinearities in pure- and converted-mode imaging. In PP imaging, they result in better illumination and artifact reduction, whereas in PS imaging they show how zero time-lag and zero space-lag crosscorrelation imaging conditions are not ideal for imaging of converted-mode waves because no conversion arises from zero-offset experiments.
29 citations
TL;DR: In this paper, a generalized internal multiple imaging (GIMI) application includes logic that generates a higher order internal multiple image using a background Green's function and logic that renders the higher-order internal multiple images for display on a display device.
Abstract: Various examples are provided for generalized internal multiple imaging (GIMI). In one example, among others, a method includes generating a higher order internal multiple image using a background Green's function and rendering the higher order internal multiple image for presentation. In another example, a system includes a computing device and a generalized internal multiple imaging (GIMI) application executable in the computing device. The GIMI application includes logic that generates a higher order internal multiple image using a background Green's function and logic that renders the higher order internal multiple image for display on a display device. In another example, a non-transitory computer readable medium has a program executable by processing circuitry that generates a higher order internal multiple image using a background Green's function and renders the higher order internal multiple image for display on a display device.
27 citations
References
More filters
Book•
[...]
01 Jan 1959
TL;DR: In this paper, the authors discuss various topics about optics, such as geometrical theories, image forming instruments, and optics of metals and crystals, including interference, interferometers, and diffraction.
Abstract: The book is comprised of 15 chapters that discuss various topics about optics, such as geometrical theories, image forming instruments, and optics of metals and crystals. The text covers the elements of the theories of interference, interferometers, and diffraction. The book tackles several behaviors of light, including its diffraction when exposed to ultrasonic waves.
19,815 citations
[...]
01 Oct 1999
TL;DR: In this article, the authors discuss various topics about optics, such as geometrical theories, image forming instruments, and optics of metals and crystals, including interference, interferometers, and diffraction.
Abstract: The book is comprised of 15 chapters that discuss various topics about optics, such as geometrical theories, image forming instruments, and optics of metals and crystals. The text covers the elements of the theories of interference, interferometers, and diffraction. The book tackles several behaviors of light, including its diffraction when exposed to ultrasonic waves.
19,503 citations
9,185 citations
TL;DR: In this article, it was shown that the acoustic Green's function between any two points in the medium can be represented by an integral of crosscorrelations of wavefield observations at those two points.
Abstract: The term seismic interferometry refers to the principle of generating new seismic responses by crosscorrelating seismic observations at different receiver locations. The first version of this principle was derived by Claerbout (1968), who showed that the reflection response of a horizontally layered medium can be synthesized from the autocorrelation of its transmission response. For an arbitrary 3D inhomogeneous lossless medium it follows from Rayleigh's reciprocity theorem and the principle of time-reversal invariance that the acoustic Green's function between any two points in the medium can be represented by an integral of crosscorrelations of wavefield observations at those two points. The integral is along sources on an arbitrarily shaped surface enclosing these points. No assumptions are made with respect to the diffusivity of the wavefield. The Rayleigh-Betti reciprocity theorem leads to a similar representation of the elastodynamic Green's function. When a part of the enclosing surface is the earth's free surface, the integral needs only to be evaluated over the remaining part of the closed surface. In practice, not all sources are equally important: The main contributions to the reconstructed Green's function come from sources at stationary points. When the sources emit transient signals, a shaping filter can be applied to correct for the differences in source wavelets. When the sources are uncorrelated noise sources, the representation simplifies to a direct crosscorrelation of wavefield observations at two points, similar as in methods that retrieve Green's functions from diffuse wavefields in disordered media or in finite media with an irregular bounding surface.
700 citations