Huub Douma

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.

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.

Explicit expressions for prestack map time migration in isotropic and VTI media and the applicability of map depth migration in heterogeneous anisotropic media

We present 3D prestack map time migration in closed form for qP-, qSV-, and mode-converted waves in homogeneous transversely isotropic media with a vertical symmetry axis (VTI). As far as prestack time demigration is concerned, we present closed-form expressions for mapping in homogeneous isotropic media, while for homogeneous VTI media we present a system of four nonlinear equations with four unknowns to solve numerically. The expressions for prestack map time migration in VTI homogeneous media are directly applicable to the problem of anisotropic parameter estimation (i.e., the anellipticity parameter η) in the context of time-migration velocity analysis. In addition, we present closed-form expressions for both prestack map time migration and demigration in the common-offset domain for pure-mode (P-P or SS) waves in homogeneous isotropic media that use only the slope in the common-offset domain as opposed to slopes in both the common-shot and common-receiver (or equivalently the common-offset and common-midpoint) domains. All time-migration and demigration equations presented can be used in media with mild lateral and vertical velocity variations, provided the velocity is replaced with the local rms velocity. Finally, we discuss the condition for applicability of prestack map depth migration and demigration in heterogeneous anisotropic media that allows the formation of caustics and explain that this condition is satisfied if, given a velocity model and acquisition geometry, one can map-depth-migrate without ambiguity in either the migrated location or the migrated orientation of reflectors in the image.

Nonhyperbolic moveout analysis in VTI media using rational interpolation

Anisotropic velocity analysis using qP-waves in transversely isotropic media with a vertical symmetry axis (VTI) usually is done by inferring the anellipticity parameter η and the normal moveout velocity VNMO from the nonhyperbolic character of the moveout. Several approximations explicit in these parameters exist with varying degrees of accuracy. Here, we present a rational interpolation approach to nonhyperbolic moveout analysis in the t-x domain. This method has no additional computational overhead compared to using expressions explicit in η and VNMO . The lack of such overhead stems from the observation that, for fixed η and zero-offset two-way traveltime t0 , the moveout curve for different values of VNMO can be calculated by simple stretching of the offset axis. This observation is based on the assumptions that the traveltimes of qP-waves in transversely isotropic media mainly depend on η and VNMO , and that the shear-wave velocity along the symmetry axis has a negligibleinfluence on these traveltim...

Rational interpolation of qP-traveltimes for semblance-based anisotropy estimation in layered VTI media

The -p domain is the natural domain for anisotropy parameterestimationinhorizontallylayeredmedia.Theneedto transformthedatatothe-p domainortopicktraveltimesin thet-xdomainis,however,apracticaldisadvantage.Toovercomethis,wecombine-p-derivedtraveltimesandoffsetsin horizontally layered transversely isotropic media with a verticalsymmetryaxisVTIwitharationalinterpolationprocedureappliedinthet-xdomain.Thiscombinationresultsinan accurate and efficient t-x-based semblance analysis for anisotropy parameter estimation from the moveout of qPwaves in horizontally layered VTI media. The semblance analysisisappliedtothemoveouttosearchdirectlyfortheinterval values of the relevant parameters. To achieve this, the method is applied in a layer-stripping fashion. We demonstrate the method using synthetic data examples and show thatitisrobustinthepresenceofrandomnoiseandmoderate statics.

The reciprocity theorem for the scattered field is the progenitor of the generalized optical theorem

By analyzing correlation-type reciprocity theorems for wavefields in perturbed media, it is shown that the correlation-type reciprocity theorem for the scattered field is the progenitor of the generalized optical theorem. This reciprocity theorem, in contrast to the generalized optical theorem, allows for inhomogeneous background properties and does not make use of a far-field condition. This theorem specializes to the generalized optical theorem when considering a finite-size scatterer embedded in a homogeneous background medium and when utilizing the far-field condition. Moreover, it is shown that the reciprocity theorem for the scattered field is responsible for the cancellation of non-physical (spurious) arrivals in seismic interferometry, and as such provides the mathematical description of such arrivals. Even though here only acoustic waves are treated, the presented treatment is not limited to such wavefields and can be generalized to general wavefields. Therefore, this work provides the framework for deriving equivalents of the generalized optical theorem for general wavefields.

Rayleigh-wave tomography at Coronation Field, Canada: The topography effect

Within reflection seismology, surface waves or ground roll, are often considered a form of unwanted source-generated noise. Unlike body waves, surface waves propagate exclusively in the lateral direction and are virtually insensitive to structure deeper than one wavelength. For a nominal frequency of 5 Hz and phase velocity of 500 m/s, this means that a surface wave of the Rayleigh or Love type only feels the upper 100 m of the subsurface. As a result, surface waves cannot be used for imaging deep reflectors; however, they can be used to estimate near-surface properties (Xia et al., 1999; Ross et al., 2008), in particular the shear-wave velocity. Knowledge of near-surface velocity structure in turn can be used to estimate shear-wave statics in reflection seismology. Estimating statics in the presence of laterally varying structure (i.e., obtaining the long-wavelength static component) can be challenging.

A unified optical theorem for scalar and vectorial wave fields

The generalized optical theorem is an integral relation for the angle-dependent scattering amplitude of an inhomogeneous scattering object embedded in a homogeneous background. It has been derived separately for several scalar and vectorial wave phenomena. Here a unified optical theorem is derived that encompasses the separate versions for scalar and vectorial waves. Moreover, this unified theorem also holds for scattering by anisotropic elastic and piezoelectric scatterers as well as bianisotropic (non-reciprocal) EM scatterers.

Surface‐wave inversion for near‐surface shear‐wave velocity estimation at Coronation field

We study the use of surface waves to invert for a near-surface shear-wave velocity model and use this model to calculate shear-wave static corrections. We invert both group-velocity and phase-velocity measurements, each of which provide independent information on the shear-wave velocity structure. For the phase-velocity we use both slant-stacking and eikonal tomography to obtain the dispersion curves. We compare models and static solutions obtained from all different methods using field data. For the Coronation field data it appears that the phase-velocity inversion obtains a better estimate of the longwavelength static than does the group-velocity inversion.