S U M M A R YIterative 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.
A B S T R A C TWavefield-based migration velocity analysis using the semblance principle requires computation of images in an extended space in which we can evaluate the imaging consistency as a function of overlapping experiments. Usual industry practice is to assemble those seismic images in common-image gathers that represent reflectivity as a function of depth and extensions, e.g., reflection angles. We introduce extended common-image point (CIP) gathers constructed only as a function of the spaceand time-lag extensions at sparse and irregularly distributed points in the image. Semblance analysis using CIP's constructed by this procedure is advantageous because we do not need to compute gathers at regular surface locations and we do not need to compute extensions at all depth levels. The CIP's also give us the flexibility to distribute them in the image at irregular locations aligned with the geologic structure. Furthermore, the CIP's remove the depth bias of common-image gathers constructed as a function of the depth axis. An interpretation of the CIP's using the scattering theory shows that they are scattered wavefields associated with sources and receivers inside the subsurface. Thus, when the surface wavefields are correctly reconstructed, the extended CIP's are characterized by focused energy at the origin of the spaceand time-lag axes. Otherwise, the energy defocuses from the origin of the lag axes proportionally with the cumulative velocity error in the overburden. This information can be used for wavefield-based tomographic updates of the velocity model, and if the velocity used for imaging is correct, the coordinate-independent CIP's can be a decomposed as a function of the angles of incidence.
A B S T R A C TThe azimuthally varying non-hyperbolic moveout of P-waves in orthorhombic media can provide valuable information for characterization of fractured reservoirs and seismic processing. Here, we present a technique to invert long-spread, wide-azimuth P-wave data for the orientation of the vertical symmetry planes and five key moveout parameters: the symmetry-plane NMO velocities, V (1) nmo and V (2) nmo , and the anellipticity parameters, η (1) , η (2) and η (3) . The inversion algorithm is based on a coherence operator that computes the semblance for the full range of offsets and azimuths using a generalized version of the Alkhalifah-Tsvankin non-hyperbolic moveout equation.The moveout equation provides a close approximation to the reflection traveltimes in layered anisotropic media with a uniform orientation of the vertical symmetry planes. Numerical tests on noise-contaminated data for a single orthorhombic layer show that the best-constrained parameters are the azimuth ϕ of one of the symmetry planes and the velocities V (1) nmo and V (2) nmo , while the resolution in η (1) and η (2) is somewhat compromised by the trade-off between the quadratic and quartic moveout terms. The largest uncertainty is observed in the parameter η (3) , which influences only long-spread moveout in off-symmetry directions. For stratified orthorhombic models with depth-dependent symmetry-plane azimuths, the moveout equation has to be modified by allowing the orientation of the effective NMO ellipse to differ from the principal azimuthal direction of the effective quartic moveout term.The algorithm was successfully tested on wide-azimuth P-wave reflections recorded at the Weyburn Field in Canada. Taking azimuthal anisotropy into account increased the semblance values for most long-offset reflection events in the overburden, which indicates that fracturing is not limited to the reservoir level. The inverted symmetryplane directions are close to the azimuths of the off-trend fracture sets determined from borehole data and shear-wave splitting analysis. The effective moveout parameters estimated by our algorithm provide input for P-wave time imaging and geometricalspreading correction in layered orthorhombic media.
Seismic imaging provides much of our information about the Earth's crustal structure. The principal source of imaging errors derives from simplistic modelled predictions of the complex, scattered wavefields that interact with each subsurface point to be imaged. A new method of wavefield extrapolation based on inverse scattering theory in mathematical physics produces accurate estimates of these subsurface scattered wavefields, while still using relatively little information about the Earth's properties. We use it for the first time to create real target-oriented seismic images of a North Sea field. We synthesise underside illumination from surface reflection data, and use it to reveal subsurface features that are not present in an image from conventional migration of surface data. To reconstruct underside reflections, we rely on the so-called downgoing focusing function, whose coda consists entirely of transmission-born multiple scattering. As such, with the method presented here, we provide the first field data example of reconstructing underside reflections with contributions from transmitted multiples, without the need to first locate or image any reflectors in order to reconstruct multiple scattering effects.
Interferometry allows for synthesis of data recorded at any two receivers into waves that propagate between these receivers as if one of them behaves as a source. This is accomplished typically by crosscorrelations. Based on perturbation theory and representation theorems, we show that interferometry also can be done by deconvolutions for arbitrary media and multidimensional experiments. This is important for interferometry applications in which ͑1͒ excitation is a complicated source-time function and/or ͑2͒ when wavefield separation methods are used along with interferometry to retrieve specific arrivals. Unlike using crosscorrelations, this method yields only causal scattered waves that propagate between the receivers. We offer a physical interpretation of deconvolution interferometry based on scattering theory. Here we show that deconvolution interferometry in acoustic media imposes an extra boundary condition, which we refer to as the free-point or clamped-point boundary condition, depending on the measured field quantity. This boundary condition generates so-called free-point scattering interactions, which are described in detail. The extra boundary condition and its associated artifacts can be circumvented by separating the reference waves from scattered wavefields prior to interferometry. Three wavefield-separation methods that can be used in interferometry are direct-wave interferometry, dual-field interferometry, and shot-domain separation. Each has different objectives and requirements.
Reciprocity theorems for perturbed acoustic media are provided in the form of convolution- and correlation-type theorems. These reciprocity relations are particularly useful in the general treatment of both forward and inverse-scattering problems. Using Green's functions to describe perturbed and unperturbed waves in two distinct wave states, representation theorems for scattered waves are derived from the reciprocity relations. While the convolution-type theorems can be manipulated to obtain scattering integrals that are analogous to the Lippmann-Schwinger equation, the correlation-type theorems can be used to retrieve the scattering response of the medium by cross correlations. Unlike previous formulations of Green's function retrieval, the extraction of scattered-wave responses by cross correlations does not require energy equipartitioning. Allowing for uneven energy radiation brings experimental advantages to the retrieval of fields scattered by remote lossless and/or attenuative scatterers. These concepts are illustrated with a number of examples, including analytic solutions to a one-dimensional scattering problem, and a numerical example in the context of seismic waves recorded on the ocean bottom.
Wave-equation, finite-frequency imaging and inversion still face many challenges in addressing the inversion of highly complex velocity models as well as in dealing with nonlinear imaging ͑e.g., migration of multiples, amplitude-preserving migration͒. Extended images ͑EIs͒ are particularly important for designing image-domain objective functions aimed at addressing standing issues in seismic imaging, such as two-way migration velocity inversion or imaging/inversion using multiples. General oneand two-way representations for scattered wavefields can describe and analyze EIs obtained in wave-equation imaging. We have developed a formulation that explicitly connects the wavefield correlations done in seismic imaging with the theory and practice of seismic interferometry. In light of this connection, we define EIs as locally scattered fields reconstructed by model-dependent, image-domain interferometry. Because they incorporate the same one-and two-way scattering representations used for seismic interferometry, the reciprocity-based EIs can in principle account for all possible nonlinear effects in the imaging process, i.e., migration of multiples and amplitude corrections. In this case, the practice of two-way imaging departs considerably from the one-way approach. We have studied the differences between these approaches in the context of nonlinear imaging, analyzing the differences in the wavefield extrapolation steps as well as in imposing the extended imaging conditions. When invoking single-scattering effects and ignoring amplitude effects in generating EIs, the one-and two-way approaches become essentially the same as those used in today's migration practice, with the straightforward addition of space and time lags in the correlationbased imaging condition. Our formal description of the EIs and the insight that they are scattered fields in the image domain may be useful in further development of imaging and inversion methods in the context of linear, migration-based velocity inversion or in more sophisticated image-domain nonlinear inverse scattering approaches.
Imaging highly complex subsurface structures is a challenging problem because it ultimately necessitates dealing with nonlinear multiple-scattering effects (e.g., migration of multiples, amplitude corrections for transmission effects) to overcome the liminations 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 in order to address the fully nonlinear imaging problem. Building on this connection between imaging and interferometry, we define the seimic image as a locally scattered wavefield and introduce a new imaging condition that is both suitable and practical for nonlinear imaging. A previous formulation of nonlinear scatteringbased imaging requires the evaluation of volume integrals that cannot easily be incorporated in current imaging algorithms. Our method consists of adapting the conventional crosscorrelation imaging condition to account for the interference mechanisms that ensure power conservation in the scattering of wavefields. To do so, we add the zero-lag autocorrelation of scattered wavefields to the zero-lag crosscorrelation of reference and scattered wavefields. In our development, we show that this autocorrelation of scattered fields fully replaces the volume scattering term required by the previous formulation. We also show that this replacement follows from the application of the generalized optical theorem. The resulting imaging condition accounts for nonlinear multiple-scattering effects, reduces imaging artifacts and improves both amplitude preservation and illumination in the images. We address the principles of our nonlinear imaging condition and demonstrate its importance in ideal nonlinear imaging experiments, i.e., we present synthetic data examples assuming ideal scattered wavefield extrapolation and study the influence of different scattering regimes and aperture limitation.
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