Smooth Velocity Model Research Articles

Joint refraction traveltime tomography and migration for multilayer near-surface imaging

In near-surface velocity structure estimation, first-arrival traveltime tomography tends to produce a smooth velocity model. If the shallow structures include a weathering layer over high-velocity bedrock, first-arrival traveltime tomography may fail to recover the sharp interface. However, with the same traveltime data, refraction traveltime migration proves to be an effective tool for accurately mapping the refractor. The approach downward continues the refraction traveltime curves and produces an image (position) of the refractor for a given overburden velocity model. We first assess the validity of the refraction traveltime migration method and analyze its uncertainties with a simple model. We then develop a multilayer refraction traveltime migration method and apply the migration image to constrain traveltime tomographic inversion by imposing discontinuities at the refraction interfaces in model regularization. In each subsequent iteration, the shape of the migrated refractors and the velocity model are simultaneously updated. The synthetic tests indicate that the joint inversion method performs better than the conventional first-arrival traveltime tomography method with Tikhonov regularization and the delay-time method in reconstructing near-surface models with high-velocity contrasts. In application to field data, this method produces a more accurately resolved velocity model, which improves the quality of common midpoint stacking by making long-wavelength static corrections.

Imaging vertical interfaces using acoustic time reversal

We have developed a novel approach to image vertical structures using multiply scattered waves. This method requires only a smooth seismic velocity model and data recorded at the surface. Previous methods to image near-vertical interfaces or faults using migration methods all require prior information about small-scale details in the seismic velocity model to infer the locations of multiple-scattering wave interactions. We used waves that had their last scattering interaction with near-vertical interfaces, while their other scattering points might be anywhere in the earth, including at the free surface. Our algorithm then images the final scattering point using a time-reversed mirror-style imaging condition, so we refer to the method as time-reversed mirror imaging. Artifacts in the images produced have clear causes and can be filtered out by stacking over shots and including contributions from multiples. Our numerical examples demonstrate the successful application of the method for staircase structures and a section of the Marmousi model. They also reveal a new way to diagnose errors in the smooth or reference velocity model used. In addition, our method can be used to image point scatterers in active seismic surveys or for event location in passive surveys.

Joint least-squares reverse time migration of primary and prismatic waves

Direct imaging of the steeply dipping structures is challenging for conventional reverse time migration (RTM), especially when there are no strong reflectors in the migration velocity model. To address this issue, we have enhanced the imaging of the steeply dipping structures by incorporating the prismatic waves. We formulate the imaging problem in a nonlinear least-squares optimization framework because the prismatic waves cannot be linearly mapped from the model perturbation. Primary and prismatic waves are jointly imaged to provide a single consistent image that includes structures illuminated by both types of waves, avoiding the complexities in scaling and/or interpreting primary and prismatic images separately. A conjugate gradient algorithm is used to iteratively solve the least-squares normal equation. This inversion procedure can become unstable if directly using the recorded data for migration because it is hindered by the crosstalk caused by imaging primary waves with the prismatic imaging operator. Therefore, we isolate the prismatic waves from the recorded data and image them with the prismatic imaging operator. Our scheme only requires a kinematically accurate and smooth migration velocity model, without the need to explicitly embed the strong reflectors in the migration velocity model. Realistic 2D numerical examples demonstrate that our method can resolve the steeply dipping structures much better than conventional least-squares RTM of primary waves.

Direct and indirect inversion and a new and comprehensive perspective on the role of primaries and multiples in seismic data processing for structure determination and amplitude analysis

The removal and use of multiples have a single shared goal and objective: the imaging and inversion of primaries. There are two kinds of primaries: recorded primaries and unrecorded primaries. For imaging recorded primaries using an industry standard practice smooth velocity model, recorded multiples must be removed, to avoid false and misleading images due to the multiples. Similarly, to find an approximate image of an unrecorded primary, that is a subevent of a recorded multiple, unrecorded multiples that are subevents of the recorded multiple must be removed, for exactly the same problem and reason that recorded multiples are needed to be eliminated. Direct inverse methods are employed to derive this new comprehensive perspective on primaries and multiples. Direct inverse methods not only assure that the problem of interest is solved, but equally important, that the problem of interest is the relevant problem that we (the petroleum industry) need to be interested in.

Wavelength-dependent Fresnel beam propagator and migration in VTI media

Ray-based wave propagators are widely applied in seismic migration due to implementation flexibility and computational efficiency. The classic ray theory that under the high-frequency assumption requires sufficiently smooth velocity models, which limits the applications of ray-based methods because seismic waves are band-limited. Besides, it is desirable to extend seismic propagation and migration to general anisotropic case since the reality of subsurface. We adapt a wavelength-dependent smoothing (WDS) operator for transversely isotropic media with a vertical symmetry axis (VTI), which considers both characteristics of band-limited wave propagation and local anisotropic heterogeneity. Frequency-dependent traveltimes are computed with the WDS models by using an anisotropic dynamic programming approach. Then, a wavelength-dependent Fresnel beam propagator is constructed based on the frequency-dependent traveltimes. Analysis of traveltime fields demonstrates that wavelength-dependent Fresnel beam propagator can provide accurate wave propagating directions and traveltimes. We develop a wavelength-dependent Fresnel beam migration (WDFBM) method based on the wavelength-dependent Fresnel beam propagator for VTI media, which generates angle domain common imaging gathers (ADCIGs) efficiently. Numerical examples are included to demonstrate the effectiveness of our approach, which shows its potential application for subsequent velocity estimation via ADCIGs.

Automated seismic waveform location using multichannel coherency migration (MCM)–I: theory

With the proliferation of dense seismic networks sampling the full seismic wavefield, recorded seismic data volumes are getting bigger and automated analysis tools to locate seismic events are essential. Here, we propose a novel Multichannel Coherency Migration (MCM) method to locate earthquakes in continuous seismic data and reveal the location and origin time of seismic events directly from recorded waveforms. By continuously calculating the coherency between waveforms from different receiver pairs, MCM greatly expands the available information which can be used for event location. MCM does not require phase picking or phase identification, which allows fully automated waveform analysis. By migrating the coherency between waveforms, MCM leads to improved source energy focusing. We have tested and compared MCM to other migration-based methods in noise-free and noisy synthetic data. The tests and analysis show that MCM is noise resistant and can achieve more accurate results compared with other migration-based methods. MCM is able to suppress strong interference from other seismic sources occurring at a similar time and location. It can be used with arbitrary 3D velocity models and is able to obtain reasonable location results with smooth but inaccurate velocity models. MCM exhibits excellent location performance and can be easily parallelized giving it large potential to be developed as a real-time location method for very large datasets.

A Nonlinear Method for Imaging with Acoustic Waves Via Reduced Order Model Backprojection

We introduce a novel nonlinear imaging method for the acoustic wave equation based on data-driven model order reduction. The objective is to image the discontinuities of the acoustic velocity, a coefficient of the scalar wave equation from the discretely sampled time domain data measured at an array of transducers that can act as both sources and receivers. We treat the wave equation along with transducer functionals as a dynamical system. A reduced order model (ROM) for the propagator of such system can be computed so that it interpolates exactly the measured time domain data. The resulting ROM is an orthogonal projection of the propagator on the subspace of the snapshots of solutions of the acoustic wave equation. While the wavefield snapshots are unknown, the projection ROM can be computed entirely from the measured data, thus we refer to such ROM as data-driven. The image is obtained by backprojecting the ROM. Since the basis functions for the projection subspace are not known, we replace them with the ones computed for a known smooth kinematic velocity model. A crucial step of ROM construction is an implicit orthogonalization of solution snapshots. It is a nonlinear procedure that differentiates our approach from the conventional linear imaging methods (Kirchhoff migration and reverse time migration - RTM). It resolves all dynamical behavior captured by the data, so the error from the imperfect knowledge of the velocity model is purely kinematic. This allows for almost complete removal of multiple reflection artifacts, while simultaneously improving the resolution in the range direction compared to conventional RTM.

Limitations of correlation-based redatuming methods

Redatuming aims to correct seismic data for the consequences of an acquisition far from the target. That includes the effects of an irregular acquisition surface and of complex geological structures in the overburden such as strong lateral heterogeneities or layers with low or very high velocity. Interferometric techniques can be used to relocate sources to positions where only receivers are available and have been used to move acquisition geometries to the ocean bottom or transform data between surface–seismic and vertical seismic profiles. Even if no receivers are available at the new datum, the acquisition system can be relocated to any datum in the subsurface to which the propagation of waves can be modeled with sufficient accuracy. By correlating the modeled wavefield with seismic surface data, one can carry the seismic acquisition geometry from the surface closer to geologic horizons of interest. Specifically, we show the derivation and approximation of the one-sided seismic interferometry equation for surface-data redatuming, conveniently using Green's theorem for the Helmholtz equation with density variation. Our numerical examples demonstrate that correlation-based single-boundary redatuming works perfectly in a homogeneous overburden. If the overburden is inhomogeneous, primary reflections from deeper interfaces are still repositioned with satisfactory accuracy. However, in this case artifacts are generated as a consequence of incorrectly redatumed overburden multiples. These artifacts get even worse if the complete wavefield is used instead of the direct wavefield. Therefore, we conclude that correlation-based interferometric redatuming of surface–seismic data should always be applied using direct waves only, which can be approximated with sufficient quality if a smooth velocity model for the overburden is available.

Interrelation between Laplace constants and the gradient distortion effect in Laplace-domain waveform inversion

Laplace-domain waveform inversion (WI) is generally used to generate smooth initial velocity models for frequency- or time-domain full-waveform inversion. However, in the inversion results of Laplace-domain WI, anomalies such as salt domes are sometimes shifted. We evaluate the “gradient-distortion effect” that causes undesirable changes in parameter updates and found that this is caused by the relationship between the partial derivatives of Laplace wavefields with respect to two different parameters. By analyzing the gradient of the Laplace-domain misfit function, we found that the gradient distortion effect increases as the Laplace constants used in the Laplace-domain WI decrease. The velocity model inverted in the Laplace domain is generally blurred from shallower parameters to deeper parameters because the partial derivatives of the Laplace wavefields with respect to shallower parameters tend to be larger than those of deeper parameters. We found two solutions for suppressing the gradient distortion effect. The first one is the sequentially ordered Laplace constant approach with multiple Laplace constants. We discover that a dense, broad set of Laplace constants should be sequentially used in this approach. The second solution is the Gauss-Newton method, in which the Hessian matrix is considered. Numerical tests performed using a four-layer model and the BP benchmark model show the gradient distortion effect appeared in the inversion results and the effectiveness of the sequentially ordered Laplace constant approach. In addition, tests using an inverted BP benchmark model determine that the inversion results can be improved by applying a broad, dense set of Laplace constants to synthetic data. Finally, we verify the effectiveness of the Gauss-Newton method at suppressing the gradient distortion effect using the BP benchmark model.

Utilizing diffractions in wavefront tomography

Wavefront tomography is known to be an efficient and stable approach for velocity inversion that does not require accurate starting models and does not interact directly with the prestack data. Instead, the original data are transformed to physically meaningful wavefront attribute fields. These can be automatically estimated using local-coherence analysis by means of the common-reflection-surface (CRS) stack, which has been shown to be a powerful tool for data analysis and enhancement. In addition, the zero-offset wavefront attributes acquired during the CRS stack can be used for sophisticated subsequent processes such as wavefield characterization and separation. Whereas in previous works, wavefront tomography has been applied mainly to reflection data, resulting in smooth velocity models suitable for migration of targets with moderately complex overburden, we have emphasized using the diffracted contributions in the data for velocity inversion. By means of simple synthetic examples, we reveal the potential of diffractions for velocity inversion. On industrial field data, we suggest a joint inversion based on reflected and diffracted contributions of the measured wavefield, which confirms the general finding that diffraction-based wavefront tomography can help to increase the resolution of the velocity models. Concluding our work, we compare the quality of a reverse time migrated result using the estimated velocity model with the result based on the inversion of reflections, which reveals an improved imaging potential for a complex salt geometry.

Wavefront picking for 3D tomography and full-waveform inversion

We have developed an efficient approach for picking first-break wavefronts on coarsely sampled time slices of 3D shot gathers. Our objective was to compute a smooth initial velocity model for multiscale full-waveform inversion (FWI). Using interactive software, first-break wavefronts were geometrically modeled on time slices with a minimal number of picks. We picked sparse time slices, performed traveltime tomography, and then compared the predicted traveltimes with the data in-between the picked slices. The picking interval was refined with iterations until the errors in traveltime predictions fell within the limits necessary to avoid cycle skipping in early arrivals FWI. This approach was applied to a 3D ocean-bottom-station data set. Our results indicate that wavefront picking has 28% fewer data slices to pick compared with picking traveltimes in shot gathers. In addition, by using sparse time samples for picking, data storage is reduced by 88%, and therefore allows for a faster visualization and quality control of the picks. Our final traveltime tomogram is sufficient as a starting model for early arrival FWI.

Multiples: Signal or noise?

Migration and migration inversion are the seismic processing methods for structural determination and subsequent amplitude analysis, respectively. To date, the most well-founded and physically interpretable migration method is based on predicting a coincident source and receiver experiment at depth at time equals zero. We have extended that migration method for heterogeneous media and to accommodate two-way propagation in a local sense at every point from the source to the target reflector and back from the reflector to the receiver and in a global sense, separately for each of the two legs from the source to the reflector and from the reflector to the receiver. That provides the first migration method that avoids high-frequency assumptions in the imaging principle and how it is implemented, and hence, it is equally effective at all frequencies at the target or reservoir. This advance for two-way wave propagation migration then provides a tool to quantitatively, unambiguously, and definitively define the role of primaries and multiples in migration. Our conclusion was that with data consisting of primaries and multiples, for an accurate discontinuous velocity model, only primaries contribute to migration with the same image and inversion results independent of whether multiples are kept or removed. However, for a smooth and continuous velocity model (i.e., generally assumed in practice), every multiple will result in a false, misleading, and potentially injurious subsurface image and hence must be removed before migration. In practice, we migrate with a smooth velocity model, and hence multiples must be removed. When the collection of primaries is incomplete, a multiple can be used to provide an approximate image of an unrecorded primary. However, it is always the migration of primaries that provides subsurface structure and amplitude information.

Full-waveform inversion as a game changer: Are we there yet?

Full-waveform inversion (FWI) has the potential to be a game changer for the seismic industry since it produces accurate velocity models at a resolution that cannot be matched by conventional traveltime tomography, with the additional advantage that it runs on basically raw data. Today, these two features should be better exploited to maximize the business value that FWI can deliver to an oil and gas operator. Provided that we acquire adequate data sets, FWI can be run starting from a smooth velocity model so that traveltime tomography is completely bypassed. As very little processing is required, FWI can be executed in parallel with the main preprocessing efforts, with the results of shortening considerably the overall project turnaround. For imaging purposes, FWI can be limited to the low-frequency range (∼ 10 Hz) but if run in the high-frequency range (∼ 30 Hz), FWI velocities have such a resolution that they can be used directly for interpretation. In particular, applications in the field of pore-pressure prediction and shallow hazard have great potential. In the current implementation, the main limitations are linked to data quality and computational power, but both these issues can be adequately addressed.

Full‐wavefield migration: using surface and internal multiples in imaging

ABSTRACTMultiple scattering is usually ignored in migration algorithms, although it is a genuine part of the physical reflection response. When properly included, multiples can add to the illumination of the subsurface, although their crosstalk effects are removed. Therefore, we introduce full‐wavefield migration. It includes all multiples and transmission effects in deriving an image via an inversion approach. Since it tries to minimize the misfit between modeled and observed data, it may be considered a full waveform inversion process. However, full‐wavefield migration involves a forward modelling process that uses the estimated seismic image (i.e., the reflectivities) to generate the modelled full wavefield response, whereas a smooth migration velocity model can be used to describe the propagation effects. This separation of modelling in terms of scattering and propagation is not easily achievable when finite‐difference or finite‐element modelling is used. By this separation, a more linear inversion problem is obtained. Moreover, during the forward modelling, the wavefields are computed separately in the incident and scattered directions, which allows the implementation of various imaging conditions, such as imaging reflectors from below, and avoids low‐frequency image artefacts, such as typically observed during reverse‐time migration. The full wavefield modelling process also has the flexibility to image directly the total data (i.e., primaries and multiples together) or the primaries and the multiples separately. Based on various numerical data examples for the 2D and 3D cases, the advantages of this methodology are demonstrated.

Full waveform inversion of diving & reflected waves for velocity model building with impedance inversion based on scale separation

Full waveform inversion (FWI) aims to reconstruct high-resolution subsurface models from the full wavefield, which includes diving waves, post-critical reflections and short-spread reflections. Most successful applications of FWI are driven by the information carried by diving waves and post-critical reflections to build the long-to-intermediate wavelengths of the velocity structure. Alternative approaches, referred to as reflection waveform inversion (RWI), have been recently revisited to retrieve these long-to-intermediate wavelengths from short-spread reflections by using some prior knowledge of the reflectivity and a scale separation between the velocity macromodel and the reflectivity. This study presents a unified formalism of FWI, named as Joint FWI, whose aim is to efficiently combine the diving and reflected waves for velocity model building. The two key ingredients of Joint FWI are, on the data side, the explicit separation between the short-spread reflections and the wide-angle arrivals and, on the model side, the scale separation between the velocity macromodel and the short-scale impedance model. The velocity model and the impedance model are updated in an alternate way by Joint FWI and waveform inversion of the reflection data (least-squares migration), respectively. Starting from a crude velocity model, Joint FWI is applied to the streamer seismic data computed in the synthetic Valhall model. While the conventional FWI is stuck into a local minimum due to cycle skipping, Joint FWI succeeds in building a reliable velocity macromodel. Compared with RWI, the use of diving waves in Joint FWI improves the reconstruction of shallow velocities, which translates into an improved imaging at deeper depths. The smooth velocity model built by Joint FWI can be subsequently used as a reliable initial model for conventional FWI to increase the high-wavenumber content of the velocity model.

Image registration guided wavefield tomography for shear-wave velocity model building

Multicomponent acquisitions offer the opportunity to form elastic migration images and to estimate elastic parameters of the subsurface. Compared with better constrained P-wave velocity inversions, it is more difficult to estimate the S-wave velocity due to strong nonlinearities introduced by converted S-waves. We have developed an iterative S-wave velocity inversion method guided by image registration. Given an accurate P-wave velocity and a simple initial S-wave model, we form P-P and P-S images using elastic reverse time migration. We use image registration to find the shifts between the P-S and P-P images. The S-wave velocity model could be updated iteratively by minimizing the differences between the original and the fractionally warped P-S images. A simple layered model and a modified Marmousi model are used to demonstrate the viability of the new method. In both examples, high-quality P-S images, as well as smooth S-wave velocity models, are inverted efficiently with a homogeneous S-wave initial model.

Applications of boundary-preserving seismic tomography for delineating reservoir boundaries and zones of CO2 saturation

Delineating reservoir units is still a challenge for seismic approaches. Even high-resolution crosswell tomographic approaches that produce smooth velocity models to match traveltime data usually provide limited information about the boundaries of subsurface targets. A recent development of seismic traveltime tomography incorporated with a boundary-preserving regularization constraint promisingly helps to resolve ambiguities in reservoir boundaries, while allowing lateral variations. We applied a kind of boundary-preserving traveltime tomography to delineate boundaries of the reservoir and [Formula: see text]-saturated zones. We chose the minimum gradient support as the regularization to preserve boundaries of the geologic target by penalizing smaller model gradients and smoothing small model variations caused by noise. We evaluated several synthetic and real data applications. Synthetic examples demonstrated that the boundary-preserving algorithm produced improved recovery of the profile shape and velocity values of the blocky targets. Two real applications were for delineating the top and base of the reservoir in the King Mountain field, and for delineating a [Formula: see text]-saturated zone in the McElroy field. These inversion results suggested that the boundary-preserving inversion is able to provide better delineation of the top and base boundaries of the reservoir and boundaries of the [Formula: see text]-saturated zone than the conventional smooth-constrained inversion.

Attenuation compensation for least-squares reverse time migration using the viscoacoustic-wave equation

Strong subsurface attenuation leads to distortion of amplitudes and phases of seismic waves propagating inside the earth. Conventional acoustic reverse time migration (RTM) and least-squares reverse time migration (LSRTM) do not account for this distortion, which can lead to defocusing of migration images in highly attenuative geologic environments. To correct for this distortion, we used a linearized inversion method, denoted as [Formula: see text]-LSRTM. During the least-squares iterations, we used a linearized viscoacoustic modeling operator for forward modeling. The adjoint equations were derived using the adjoint-state method for back propagating the residual wavefields. The merit of this approach compared with conventional RTM and LSRTM was that [Formula: see text]-LSRTM compensated for the amplitude loss due to attenuation and could produce images with better balanced amplitudes and more resolution below highly attenuative layers. Numerical tests on synthetic and field data illustrated the advantages of [Formula: see text]-LSRTM over RTM and LSRTM when the recorded data had strong attenuation effects. Similar to standard LSRTM, the sensitivity tests for background velocity and [Formula: see text] errors revealed that the liability of this method is the requirement for smooth and accurate migration velocity and attenuation models.

Generalized internal multiple imaging

Internal multiples deteriorate the image when the imaging procedure assumes only single scattering, especially if the velocity model does not have sharp contrasts to reproduce such scattering in the Green’s function through forward modeling. If properly imaged, internal multiples (internally scattered energy) can enhance the seismic image. Conventionally, to image internal multiples, accurate, sharp contrasts in the velocity model are required to construct a Green’s function with all the scattered energy. As an alternative, we have developed a generalized internal multiple imaging procedure that images any order internal scattering using the background Green’s function (from the surface to each image point), constructed from a smooth velocity model, usually used for conventional imaging. For the first-order internal multiples, the approach consisted of three steps, in which we first back propagated the recorded surface seismic data using the background Green’s function, then crosscorrelated the back-propagated data with the recorded data, and finally crosscorrelated the result with the original background Green’s function. This procedure images the contribution of the recorded first-order internal multiples, and it is almost free of the single-scattering recorded energy. The cost includes one additional crosscorrelation over the conventional single-scattering imaging application. We generalized this method to image internal multiples of any order separately. The resulting images can be added to the conventional single-scattering image, obtained, e.g., from Kirchhoff or reverse-time migration, to enhance the image. Application to synthetic data with reflectors illuminated by multiple scattering (double scattering) demonstrated the effectiveness of the approach.

Kirchhoff migration using eikonal-based computation of traveltime source derivatives

The computational efficiency of Kirchhoff-type migration can be enhanced by using accurate traveltime interpolation algorithms. We addressed the problem of interpolating between a sparse source sampling by using the derivative of traveltime with respect to the source location. We adopted a first-order partial differential equation that originates from differentiating the eikonal equation to compute the traveltime source derivatives efficiently and conveniently. Unlike methods that rely on finite-difference estimations, the accuracy of the eikonal-based derivative did not depend on input source sampling. For smooth velocity models, the first-order traveltime source derivatives enabled a cubic Hermite traveltime interpolation that took into consideration the curvatures of local wavefronts and can be straightforwardly incorporated into Kirchhoff antialiasing schemes. We provided an implementation of the proposed method to first-arrival traveltimes by modifying the fast-marching eikonal solver. Several simple synthetic models and a semirecursive Kirchhoff migration of the Marmousi model demonstrated the applicability of the proposed method.