Paul Sava

Angle‐domain common‐image gathers by wavefield continuation methods

Migration in the angle domain creates seismic images for different reflection angles. We present a method for computing angle-domain common-image gathers from seismic images obtained by depth migration using wavefield continuation. Our method operates on prestack migrated images and produces the output as a function of the reflection angle, not as a function of offset ray parameter as in other alternative approaches. The method amounts to a radial-trace transform in the Fourier domain and is equivalent to a slant stack in the space domain. We obtain the angle gathers using a stretch technique that enables us to impose smoothness through regularization. Several examples show that our method is accurate, fast, robust, and easy to implement. The main anticipated applications of our method are in the areas of migration-velocity analysis and amplitude-versus-angle analysis.

Offset and angle-domain common image-point gathers for shot-profile migration

Prestack depth migration of shot profiles by downward continuation is a practical imaging algorithm that is especially cost-effective for sparse-shot wide-azimuth geometries. The interpretation of offset as the displacement between the downward-propagating (shot) wavefield and upward-propagating (receiver) wavefield enables us to extract offset-domain common image-point (CIP) gathers during shot-profile migration. The offset-domain gathers can then be transformed to the angle domain with a radial-trace mapping originally introduced for shot-geophone migration. The computational implications of this procedure include both the additional cost of multioffset imaging and an implicit transformation from shot-geophone to midpoint-offset coordinates. Although this algorithm provides a mechanism for imaging angle-dependent reflectivity via shot-profile migration, for sparse-shot geometries the fundamental problem of shot-aliasing may severely impact the quality of CIP gathers.

Time-shift imaging condition in seismic migration

Seismic imaging based on single-scattering approximationisintheanalysisofthematchbetweenthesourceandreceiver wavefields at every image location. Wavefields at depth are functions of space and time and are reconstructed from surface data either by integral methods Kirchhoff migration or by differential methods reverse-time or wavefieldextrapolationmigration.Differentmethodscanbeused toanalyzewavefieldmatching,ofwhichcrosscorrelationisa popular option. Implementation of a simple imaging condition requires time crosscorrelation of source and receiver wavefields, followed by extraction of the zero time lag. A generalized imaging condition operates by crosscorrelation in both space and time, followed by image extraction at zero time lag. Images at different spatial crosscorrelation lags are indicators of imaging accuracy and are also used for imageangle decomposition. In this paper, we introduce an alternative prestack imaging condition in which we preserve multiple lags of the time crosscorrelation. Prestack images are described as functions of time shifts as opposed to space shifts betweensourceandreceiverwavefields.Thisimagingcondition is applicable to migration by Kirchhoff, wavefield extrapolation, or reverse-time techniques. The transformation allows construction of common-image gathers presented as functionsofeithertimeshiftorreflectionangleateverylocationinspace.Inaccuratemigrationvelocityisrevealedbyangle-domain common-image gathers with nonflat events. Computational experiments using a synthetic data set from a complex salt model demonstrate the main features of the method.

Wave-Equation Migration Velocity Analysis

We overcome the limitations of conventional MVA in regions of high wavefield complexity (subsalt) using a wave-equation migration velocity analysis technique (Sava and Biondi, 2004a,a), and illustrate it on a realistic synthetic salt-dome dataset.

Isotropic angle-domain elastic reverse-time migration

Multicomponent data usually are not processed with specifically designed procedures but with procedures analogous to those used for single-component data. In isotropic media, the vertical and horizontal components of the data commonly are taken as proxies for the P- and S-wave modes, which are imaged independently with the acoustic wave equations.This procedure works only if the vertical and horizontal components accurately represent P- and S-wave modes, which generally is not true. Therefore, multicomponent images constructed with this procedure exhibit artifacts caused by incorrect wave-mode separation at the surface.An alternative procedure for elastic imaging uses the full vector fields for wavefield reconstruction and imaging. Thewavefieldsarereconstructedusingthemulticomponentdata as a boundary condition for a numerical solution to the elastic wave equation. The key component for wavefield migration is theimagingcondition,whichevaluatesthematchbetweenwavefields reconstructed from sources and receivers. For vector wave fields, a simple component-by-component crosscorrelation between two wavefields leads to artifacts caused by crosstalk between the unseparated wave modes. We can separate elastic wavefields after reconstruction in the subsurface and implement theimagingconditionascrosscorrelationofpurewavemodesinstead of the Cartesian components of the displacement wavefield.Thisapproachleadstoimagesthatareeasiertointerpretbecause they describe reflectivity of specified wave modes at interfaces of physical properties.As for imaging with acoustic wavefields, the elastic imaging condition can be formulated conventionally crosscorrelation with zero lag in space and time and extendedtononzerospaceandtimelags.Theelasticimagesproduced by an extended imaging condition can be used for angle decomposition of primary PP or SS and converted PS or SP reflectivity. Angle gathers constructed with this procedure have applicationsformigrationvelocityanalysisandamplitude-variation-with-angleanalysis.

Elastic wave-mode separation for VTI media

Elastic wave propagation in anisotropic media is well represented by elastic wave equations. Modeling based on elasticwaveequationscharacterizesbothkinematicsanddynamics correctly. However, because P- and S-modes are both propagated using elastic wave equations, there is a need to separate P- and S-modes to efficiently apply single-mode processing tools. In isotropic media, wave modes are usually separated using Helmholtz decomposition. However, Helmholtz decomposition using conventional divergence and curl operators in anisotropic media does not give satisfactory resultsandleavesthedifferentwavemodesonlypartiallyseparated.Theseparationofanisotropicwavefieldsrequiresmore sophisticated operators that depend on local material parameters. Anisotropic wavefield-separation operators are constructedusingthepolarizationvectorsevaluatedateachpoint of the medium by solving the Christoffel equation for local mediumparameters.Thesepolarizationvectorscanberepresented in the space domain as localized filtering operators, which resemble conventional derivative operators. The spatially variable pseudo-derivative operators perform well in heterogeneous VTI media even at places of rapid velocity/ densityvariation.Syntheticresultsindicatethattheoperators can be used to separate wavefields for VTI media with an arbitrarydegreeofanisotropy.

Wave-equation migration velocity analysis by focusing diffractions and reflections

We propose a method for estimating interval velocity using the kinematic information in defocused diffractions and reflections. We extract velocity information from defocused migrated events by analyzing their residual focusing in physical space (depth and midpoint) using prestack residual migration. The results of this residual-focusing analysis are fed to a linearized inversion procedure that produces interval velocity updates. Our inversion procedure uses a wavefield-continuation operator linking perturbations of interval velocities to perturbations of migrated images, based on the principles of wave-equation migration velocity analysis introduced in recent years. We measure the accuracy of the migration velocity using a diffraction-focusing criterion instead of the criterion of flatness of migrated common-image gathers that is commonly used in migration velocity analysis. This new criterion enables us to extract velocity information from events that would be challenging to use with conventional veloci...

Riemannian wavefield extrapolation

Riemannian spaces are described by nonorthogonal curvilinear coordinates. We generalize one-way wavefield extrapolation to semiorthogonal Riemannian coordinate systems that include, but are not limited to, ray coordinate systems. We obtain a one-way wavefield extrapolation method that can be used for waves propagating in arbitrary directions, in contrast to downward continuation, which is used for waves propagating mainly in the vertical direction. Ray coordinate systems can be initiated in many different ways; for example, from point sources or from plane waves incident at various angles. Since wavefield propagation happens mostly along the extrapolation direction, we can use inexpensive finite-difference or mixed-domain extrapolators to achieve high angle accuracy. The main applications of our method include imaging of steeply dipping or overturning reflections.

Multiple attenuation in the image space

Multiples can be suppressed in the angle-domain image space after migration. For a given velocity model, primaries and multiples have different angle-domain moveout and, therefore, can be separated using techniques similar to the ones employed in the data space prior to migration. We use Radon transforms in the image space to discriminate between primaries and multiples and employ accurate functions describing angledomain moveouts. Since every individual angle-domain common-image gather incorporates complex 3D propagation effects, our method has the advantage of working with 3D data and complicated geology. Therefore, our method offers an alternative to the more expensive surface-related multiple-elimination (SRME) approach operating in the data space. Radon transforms are cheap but not necessarily ideal for separating primaries and multiples, particularly at small angles where the moveout discrepancy between the two kinds of events are not large. Better techniques involving signal/noise separation using prediction-error filters can be employed as well, although such approaches fall outside the scope of this paper. We demonstrate, using synthetic and real data examples, the power of our method in discriminating between primaries and multiples after migration by wavefield extrapolation, followed by transformation to the angle domain.

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.