Ivan Vasconcelos

Interferometry by deconvolution: Part 1 — Theory for acoustic waves and numerical examples

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 (1) excitation is a complicated source-time function and/or (2) 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.

Interferometry by deconvolution: Part 2 — Theory for elastic waves and application to drill-bit seismic imaging

Deconvolution interferometry successfully recovers the impulse response between two receivers without the need for an independent estimate of the source function. Here we extend the method of interferometry by deconvolution to multicomponent data in elastic media. As in the acoustic case, elastic deconvolution interferometry retrieves only causal scattered waves that propagate between two receivers as if one acts as a pseudosource of the point-force type. Interferometry by deconvolution in elastic media also generates artifacts because of a clamped-point boundary condition imposed by the deconvolution process. In seismic-while-drilling (SWD) practice, the goal is to determine the subsurface impulse response from drill-bit noise records. Most SWD technologies rely on pilot sensors and/or models to predict the drill-bit source function, whose imprint is then removed from the data. Interferometry by deconvolution is of most use to SWD applications in which pilot records are absent or provide unreliable estimates of bit excitation. With a numerical SWD subsalt example, we show that deconvolution interferometry provides an image of the subsurface that cannot be obtained by correlations without an estimate of the source autocorrelation. Finally, we test the use of deconvolution interferometry in processing SWD field data acquired at the San Andreas Fault Observatory at Depth (SAFOD). Because no pilot records were available for these data, deconvolution outperforms correlation in obtaining an interferometric image of the San Andreas fault zone at depth.

Nonlinear extended images via image-domain interferometry

Wave-equation, finite-frequency imaging and inversion still face many challenges in addressing the inversion of highly complexvelocitymodelsaswellasindealingwithnonlinearimaging e.g., migration of multiples, amplitude-preserving migration. Extended images EIs 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 practiceofseismicinterferometry.Inlightofthisconnection,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.,migrationofmultiplesandamplitudecorrections.In this case, the practice of two-way imaging departs considerably from the one-way approach.We have studied the differences betweentheseapproachesinthecontextofnonlinearimaging,analyzingthedifferencesinthewavefieldextrapolationstepsaswell as in imposing the extended imaging conditions.When invoking single-scatteringeffectsandignoringamplitudeeffectsingenerating EIs, the one- and two-way approaches become essentially the same as those used in today’s migration practice, with the straightforwardadditionofspaceandtimelagsinthecorrelationbased imaging condition. Our formal description of the EIs and theinsightthattheyarescatteredfieldsintheimagedomainmay beusefulinfurtherdevelopmentofimagingandinversionmethodsinthecontextoflinear,migration-basedvelocityinversionor inmoresophisticatedimage-domainnonlinearinversescattering approaches.

Imaging internal multiples from subsalt VSP data — Examples of target-oriented interferometry

Seismic interferometry has become a technology of growing interest for imaging borehole seismic data. We demonstrate that interferometry of internal multiples can be used to image targets above a borehole receiver array. By internal multiples, we refer to all types of waves that scatter multiple times inside the model. These include, for instance, interbed, intrasalt, and water-bottom multiples as well as conversions among them. We use an interferometry technique that is based on representation theorems for perturbed media and targets the reconstruction of specific primary reflections from multiply reflected waves. In this interferometry approach, we rely on shot-domain wavenumber separation to select the directions of waves arriving at a given receiver. Using a numerical walkaway (WAW) VSP experiment recorded by a subsalt borehole receiver array in the Sigsbee salt model, we use the interference of internal multiples to image the salt structure from below. In this numerical example, the interferometric i...

The influence of crack shape on the effective elasticity of fractured rocks

A circle is the basic fracture shape adopted by conventional effective media theories to describe the overall elasticity of cracked solids. Fractures in rocks do not resemble circles, so it is important to find out to what extent the available theoretical results apply to realistic fracture shapes. To address this issue, we conduct 3D numerical experiments on the so-called digital rocks containing irregular cracks that might be partially closed and might intersect each other. Despite profound deviations of our fracture geometries from circles, we find that basic theoretical results originally developed for penny-shaped cracks remain valid for arbitrary fracture shapes. Based on a series of finite-element computations, we show that as far as the effective elasticity is concerned, flat fractures with random in-plane irregularities are represented accurately by circular ones. We also show that approximate effective elliptical orthotropy established for multiple sets of dry, penny-shaped cracks embedded in is...

On the connection between artifact filtering in reverse-time migration and adjoint tomography

Finite-frequency sensitivity kernels in seismic tomographydefinethevolumesinsidetheearththatinfluenceseismic wavesastheytraversethroughit.Ithasrecentlybeennumerically observed that an image obtained using the impedance kernel is much less contaminated by low-frequency artifacts due to the presence of sharp wave-speed contrasts in the background model, than is an image obtained using reversetime migration. In practical reverse-time migration, these artifacts are routinely heuristically dampened by Laplacian filtering of the image. Here we show analytically that, for an isotropic acoustic medium with constant density, away from sources and receivers and in a smooth background medium, Laplacian imagefiltering is identical to imaging with the impedance kernel. Therefore, when imaging is pushed toward using background models with sharp wave-speed contrasts, the impedance kernel image is less prone to develop low-frequency artifacts than is the reverse-time migration image, due to the implicit action of the Laplacian that amplifies the higher-frequency reflectors relative to the low-frequency artifacts.Thus,theheuristicLaplacianfilteringcommonlyused in practical reverse-time migration is fundamentally rooted inadjointtomographyand,inparticular,closelyconnectedto theimpedancekernel.

Wave‐equation extended images via image‐domain interferometry

Using general twoand one-way representations for scatter ed wavefields, we analyze the nature of extended images obtaine d in wave-equation imaging. The presented formulation expli citly connects the wavefield correlations done in seismic ima ging with the theory and practice of seismic interferometry. We show that extended images actually behave as locally scatte red fields in the image domain. The wavefield behavior of twoand one-way extended images is illustrated using numerical examples. The general description of the extended images pr esented here and the derived insight that they actually are sc attered fields in the image-domain, may prove to be useful in further development of imaging and inversion methods.

Seismic characterization of multiple fracture sets at Rulison Field, Colorado

Conventional fracture-characterization methods assume the presence of a single set of aligned, vertical cracks in the subsur-face. We relax this assumption and demonstrate the feasibility of seismic characterization of multiple fracture sets. Our technique relies on recent numerical findings indicating that multiple, differently oriented, possibly intersecting planar cracks embedded in an otherwise isotropic host rock result in a nearly orthorhombic (or orthotropic) effective medium. Here, the governing parameters of crack-induced orthotropy are estimated from 3D, wide-azimuth, multicomponent seismic reflection data acquired over the tight-gas Rulison Field in Colorado. We translate strong azimuthal variations of the normal-moveout velocities intointerval crack densities, fracture orientations, type of fluid infill, and velocities of P- and S-waves in an unfractured rock. Our inversion procedure identifies a set of cracks aligned in approximately west northwest-east southeast direction in the western part...

Generalized Representations of Perturbed Fields - Applications In Seismic Interferometry And Migration

To extract only the scattered fields between two receivers as if one of them acts as a source, I present a perturbation-based interferometry formulation that applies to general physical systems. These include lossy acoustic, elastic and electromagnetic phenomena, mass or heat transport, and seismo-electrical problems. When the receivers lie between the physical sources and the imaging target, this perturbationbased theorem can be adapted for interferometry of both dual(e.g., pressure and particle velocity) and single-field measurements. In the case of oceanbottom cable (OBC) data, dual-field perturbationbased interferometry retrieves the response between the receivers without the free-surface multiples. Here the reference field is taken as the direct arrival along with water bottom multiples. In the context of seismic imaging, I show that the representation theorem in perturbed media can lead to the commonly used cross-correlation imaging condition in migration by wavefield extrapolation. Thus the exact form of the perturbed representation theorem can in principle be used for the migration of vector fields, multiplyscattered waves, as well as for corrections for migration amplitudes and artifacts.

Estimation of Azimuthally Varying Attenuation From Wide-azimuth P-wave Data

By assuming that Q is frequency independent, and the medium at each particular azimuth the medium is laterally homogeneous, we use the spectral ratio method and a regularized linear inversion scheme to estimate the quality factor in azimuth-sectored data. The regularization parameters are chosen by a χ criterion that is based on estimates of the variance in the data. Tests on synthetic data show that this regularized inversion provides robust estimates of Q for signal-to-noise ratios lower than those observed in the data.