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- Volume 61, Issue 6, 2013
Geophysical Prospecting - 6 - Challenges of Seismic Imaging and Inversion Devoted to Goldin, 2013
6 - Challenges of Seismic Imaging and Inversion Devoted to Goldin, 2013
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Physical interpretation of degeneracies of the Christoffel equation in the theory of anisotropic elasticity†
By S.V. GoldinABSTRACTAccording to the classical approach to physical interpretation of multiple roots of the Christoffel equation in the theory of elastic wave propagation in anisotropic media, a sum of isonormal plane waves propagating in the direction of acoustic axes can have either arbitrary or circular polarization. The main question posed in this paper is as follows: can one apply conclusions drawn for plane waves to other phenomena, in particular, to waves generated by a point source? The paper proposes a new principle of physical interpretation of degeneracies stating that any assessment of the polarizations (and the group velocities) of waves propagating in anisotropic media is reasonable if there exists an experiment with a point source in which the assessment agrees with the general symmetry of the experiment. From the viewpoint of this interpretation, all degeneracies are considered on the wavefront. The inferences drawn from the performed analysis might appear surprising: in all considered cases of degeneracies (such as conic axes, tangent degeneracy on the symmetry axis of infinite order in transversely isotropic media, intersections of the slowness surfaces), ambiguity in determination of the polarization vectors either does not exist for any experiment or can be removed based on the symmetry of an experiment. A condition for convexity of the slowness surface of the fastest wave is formulated in this context.
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Full‐waveform inversion for macro velocity model reconstruction in look‐ahead offset vertical seismic profile: numerical singular value decomposition‐based analysis
Authors Ilya Silvestrov, Dmitry Neklyudov, Clement Kostov and Vladimir TcheverdaABSTRACTFull‐waveform inversion is currently considered as a potential tool for improving depth‐velocity models for areas with complex geology. It is well‐known that success of the inversion is very sensitive to the available low‐frequency content of the data. In the paper we investigate this issue considering a look‐ahead offset vertical seismic profile survey and applying singular value decomposition analysis of a linearized forward map as the main tool. We demonstrate with this technique the difference between the sequential full‐waveform inversion strategy and the original time‐domain approach proposed in the early 1980s. We emphasize the role of the lowest frequency in the data, which is necessary for reliable velocity model inversion in particular cases. Finally we show the existence of a trade‐off between the lowest frequency and a regularization parameter of the inversion procedure. The presented approach may be adapted to answer general questions regarding the quality of data and acquisition system parameters required for feasible full‐waveform inversion.
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Model‐uncertainty quantification in seismic tomography: method and applications
ABSTRACTUncertainty is inherent in every stage of the oil and gas exploration and production (E&P) business and understanding uncertainty enables mitigation of E&P risks. Therefore, quantification of uncertainty is beneficial for decision making and uncertainty should be managed along with other aspects of business. For example, decisions on well positioning should take into account the structural uncertainty related to the non‐uniqueness of a velocity model used to create a seismic depth image. Moreover, recent advances in seismic acquisition technology, such as full‐azimuth, long‐offset techniques, combined with high‐accuracy migration algorithms such as reverse‐time migration, can greatly enhance images even in highly complex structural settings, provided that an Earth velocity model with sufficient resolution is available. Modern practices often use non‐seismic observation to better constrain velocity model building. However, even with additional information, there is still ambiguity in our velocity models caused by the inherent non‐uniqueness of the seismic experiment. Many different Earth velocity models exist that match the observed seismic (and well) data and this ambiguity grows rapidly away from well controls. The result is uncertainty in the seismic velocity model and the true positions of events in our images. Tracking these uncertainties can lead to significant improvement in the quantification of exploration risk (e.g., trap failure when well‐logging data are not representative), drilling risk (e.g., dry wells and abnormal pore pressure) and volumetric uncertainties. Whilst the underlying ambiguity can never be fully eradicated, a quantified measure of these uncertainties provides a valuable tool for understanding and evaluating the risks and for development of better risk‐mitigation plans and decision‐making strategies.
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Extended structure tensors for multiple directionality estimation
Authors Fredrik Andersson and Anton A. DuchkovABSTRACTStandard structure tensors provide a robust way of directionality estimation of waves (or edges) but only for the case when they do not intersect. In this work, a structure tensor extension using a one‐way wave equation is proposed as a tool for estimating directionality in seismic data and images in the presence of conflicting dips. Detection of two intersecting waves is possible in a two‐dimensional case. In three dimensions both two and three intersecting waves can be detected. Moreover, a method for directionality filtering using the estimated directions is proposed. This method makes use of the ideas of a one‐way wave equation but can be applied to generic images not related to wave propagation.
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Imaging improvement by prestack signal enhancement
Authors Vadim Buzlukov and Evgeny LandaABSTRACTThe quality of recorded seismic data depends on many factors and a low signal‐to‐noise ratio leads to a low quality of processing and imaging. The zero‐offset common‐reflection‐surface stack and multifocusing methods have been successfully applied to improve the prestack signal‐to‐noise ratio by the partial summation of coherent seismic events. However, in the case of non‐hyperbolic traveltime behaviour of seismic events these approaches can result in non‐optimal partial summation.
We develop a local common‐offset approximation for the traveltime stacking surface. It allows us to adequately approximate the traveltimes of reflection events in the vicinity of an arbitrary offset. Here, the stacking operator is interpreted as a local second‐order traveltime approximation of the seismic event and it is used for the purpose of partial summation. An algorithm and a numerical implementation scheme are discussed. The proposed signal enhancement procedure was applied to synthetic and real 3D data. Imaging results of the enhanced data show a high potential for reliable imaging in complex subsurface environments.
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Inversion of microseismic data for triclinic velocity models
Authors Vladimir Grechka and Sergey YaskevichABSTRACTModern downhole microseismic surveys often employ geometries in which ray trajectories generated by a collection of locatable events provide full polar and azimuthal coverage, making it possible to estimate the in situ seismic anisotropy. We show that traveltimes and particle motions of the direct P‐ and shear‐waves acquired in such geometries can constrain stiffness tensors of triclinic media. While obtaining all 21 stiffness coefficients of a homogeneous triclinic space simultaneously with locating pertinent microseismic events from data recorded in a single vertical well is relatively straightforward, the same methodology does not necessarily succeed in layered formations because the combination of their vertical heterogeneity and azimuthal anisotropy might invalidate the commonly adopted approximation of the event azimuths by those of the P‐wave polarization vectors. When the event azimuths cannot be derived from the particle motions, traveltimes observed in two or more wells are required to locate the events and build layered triclinic or higher‐symmetry azimuthally anisotropic velocity models. As our numerical tests indicate, the multi‐well event‐location methods are expected to perform better than their single‐well counterparts because they rely solely on triangulation and eliminate the usually pronounced azimuthal uncertainties in the event locations that stem from noises adversely affecting hodogram analysis.
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Mapping of moveout in tilted transversely isotropic media
Authors Alexey Stovas and Tariq AlkhalifahABSTRACTThe computation of traveltimes in a transverse isotropic medium with a tilted symmetry axis tilted transversely isotropic is very important both for modelling and inversion. We develop a simple analytical procedure to map the traveltime function from a transverse isotropic medium with a vertical symmetry axis (vertical transversely isotropic) to a tilted transversely isotropic medium by applying point‐by‐point mapping of the traveltime function. This approach can be used for kinematic modelling and inversion in layered tilted transversely isotropic media.
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2D common‐offset traveltime based diffraction enhancement and imaging
Authors Endrias G. Asgedom, Leiv ‐J. Gelius and Martin TygelABSTRACTThe diffracted energy in a seismic recording contains key information about small‐scale inhomogeneities or discontinuities of the subsurface. Diffractions can therefore lead to high‐resolution imaging of subsurface structures associated with hydrocarbon traps. However, seismic diffracted signals are often much weaker than specular reflections and consequently require enhancement before they can be utilized. In this paper diffractions are enhanced relative to reflections based on two traveltime techniques, namely the modified common‐reflection‐surface approach, which uses the common‐reflection‐surface technique with some modification to tailor it for diffractions and the replacement‐media approach derived here for the purpose of a simplified parametrization of diffraction traveltimes. Both approaches are implemented in the common‐offset domain with the use of a finite‐offset central ray unlike the zero‐ or small‐offset diffraction enhancement techniques that use a zero‐offset central ray. The validity of the two moveout expressions is tested using velocity data taken from a smooth isotropic Marmousi model. A feasibility test was also carried out with respect to the new replacement‐media traveltime approximation addressing the various effects of signal‐to‐noise ratio, depth and lateral displacement of the diffractor location. Finally, diffraction enhancement and imaging was performed on 2D seismic data from the Jequitinhonha basin offshore Brazil. Diffractions were significantly enhanced and a high‐resolution image of the discontinuities of the subsurface was obtained.
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Band‐limited ray tracing
Authors Can Evren Yarman, Xin Cheng, Konstantin Osypov, Dave Nichols and Maxim ProtasovABSTRACTWe develop a new ray‐tracing method, namely, band‐limited ray tracing, which aims to overcome some of the limitations of standard high‐frequency ray tracing in complex velocity models that contain complex interfaces. Band‐limited ray tracing is based on the Kirchhoff integral representation of the transmitted wavefield. In our discussion we consider the Kirchhoff integral for the acoustic wave equation, however, our method can be extended to the Kirchhoff integral for the elastic wave equation. When compared to finite‐difference wave propagation, the new method captures the wave kinematics more accurately than conventional ray tracing while staying within the ray‐tracing framework, without requiring processing or alteration to the original model. We present the theory of band‐limited ray tracing and demonstrate its capability. We compare the Kirchhoff migration examples obtained using conventional ray‐tracing and band‐limited ray‐tracing methods. Our synthetic example demonstrates that band‐limited ray tracing provides better results than conventional ray tracing, especially in estimating wavefronts. In the case of the real data example, the results are comparable to each other while presenting different focusing characteristics. The different focusing characteristics indicate the necessity of performing a full model building workflow for an extended evaluation on real data application.
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3D diffraction imaging of linear features and its application to seismic monitoring
Authors Faisal Alonaizi, Roman Pevzner, Andrej Bóna, Valeriya Shulakova and Boris GurevichABSTRACTMany subsurface features, such as faults, fractures, cracks, or fluid content terminations are defined by geological discontinuities. The seismic response from such features is encoded in diffractions. We develop an algorithm for imaging such discontinuities by detecting edge diffractions. The algorithm exploits phase‐reversal phenomena of edge diffractions and uses them as a criterion to separate these diffractions from specular reflections and diffractions produced by a leaner object. The performance of the method is demonstrated on both synthetic and real 3D seismic data. The output image focuses the diffracted energy back to its origin and shows high semblance values at the edge of the object. The method is applied on conventionally stacked data producing an image that contains only diffraction events called the D‐volume. We also reveal the potential of diffractions to image and track the changes of a CO2 plume using time‐lapse analysis and detect any possible CO2 seepage from its primary containment.
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Joint inversion of seismic traveltimes and magnetotelluric data with a directed structural constraint
Authors Dmitry M. Molodtsov, Vladimir N. Troyan, Yuri V. Roslov and Andrea ZerilliABSTRACTWe introduce a new structural constraint for joint inversion with an application to regional scale seismic traveltimes and magnetotelluric data. We call the constraint ‘directed’ as it takes into account a priori information on the sign of cross‐correlation between the gradients of reconstructed parameters. With special treatment of singularities, arising from vanishing gradients and linearization, this constraint demonstrates some properties of an edge‐preserving stabilizer – it provides blocky models with coincident discontinuities. We develop an algorithm for 2D pixel‐based joint inversion, including the proposed structural constraint as a penalty term of the objective function; additional stabilizing terms are total variation of seismic slowness and Levenberg–Marquardt damping. The resulting regularized Gauss–Newton scheme is numerically stable and demonstrates relatively fast convergence both in data misfits and in the structural similarity measure. This is shown by a numerical study of the algorithm on a simplified regional model of the Earth's crust. The considered model has a blocky structure with a positive correlation between P‐wave velocity and electrical resistivity; it represents a faulted basement overlain by sediments embedding an allochthonous salt dome. The developed joint inversion improves both velocity and resistivity reconstruction relative to separate inversions. In comparison with the cross‐gradients constraint the considered structural constraint firstly fixes the sign of correlation between the gradients of the parameters, thus reducing uncertainty in the model recovery and secondly due to its stabilizing properties it limits the amount of additional regularization, which results in sharper reconstructed models.
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High‐accuracy two‐interval approximation for normal‐moveout function in a multi‐layered anisotropic model
By Emil BliasABSTRACTThis paper discusses reducing computation costs for traveltime calculations in multi‐layered anisotropic models. Fomel and Stovas (2010) suggested a two‐ray five‐parameter approximation that they named ‘generalized’ because it reduces to several known three‐parameter forms. Model tests, demonstrated by the authors, showed that this generalized approximation provided very high accuracy, implying it can be used in place of the exact moveout function in modelling, migration and traveltime inversion. However, detailed model studies show that for some models, with a high‐velocity layer, this approximation leads to significant errors. I develop a new three‐ray eight‐parameter approximation that provides higher accuracy and can replace the exact traveltime function that requires numerical ray calculations for each receiver. I call it a ‘two‐interval approximation’ because it uses two different equations for two offset intervals. Model tests show that this two‐interval approximation can bring much higher accuracy compared to the generalized approximation due to the use of an additional reference ray. The two‐interval new approximation can be used instead of exact traveltimes for many practical purposes.
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3D elastic full‐waveform inversion of small‐scale heterogeneities in transmission geometry
Authors S. Butzer, A. Kurzmann and T. BohlenABSTRACTThree‐dimensional elastic full‐waveform inversion aims to reconstruct elastic material properties of 3D structures in the subsurface with high resolution. Here we present an implementation of 3D elastic full‐waveform inversion based on the adjoint‐state method. The code is optimized regarding runtime and storage costs by using a time‐frequency approach. The gradient is computed from monochromatic frequency‐domain particle‐velocity wavefields calculated with a time‐domain velocity‐stress finite‐difference scheme. The 3D full‐waveform inversion was applied to data of a complex random medium model, which resembles a realistic crystalline rock environment. We show synthetic inversion results of P‐wave and S‐wave velocities for two transmission geometries: (1) a 3D acquisition geometry with planes of sources and receivers and (2) a 2D geometry with two lines of sources and receivers, resembling a realistic two‐borehole geometry. The 3D inversion of data acquired with 3D source‐receiver geometry is capable to reconstruct differently sized 3D structures of shear and compressional velocities with resolution of about a wavelength. The 3D random medium data recorded with 2D acquisition geometry were inverted using 3D inversion and 2D full‐waveform inversion for comparison. The 2D inversion suffers from strong artefacts that are caused by 3D scattering. The multiparameter 3D inversion, by contrast, is capable to invert the 3D scattered waves and to reconstruct 3D structures up to about 1–2 wavelengths adjacent to the plane between sources and receivers. The resolution is lower compared to the 3D acquisition geometry result. Still, a 3D inversion of cross‐hole data can be beneficial compared to a 2D inversion in the presence of complex 3D small‐scale heterogeneities, as it is capable to resolve 3D structures next to the source‐receiver plane.
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On the role of reflections, refractions and diving waves in full‐waveform inversion
Authors V.V. Kazei, V.N. Troyan, B.M. Kashtan and W.A. MulderABSTRACTFull‐waveform inversion suffers from local minima, due to a lack of low frequencies in data. A reflector below the zone of interest may, however, help in recovering the long‐wavelength components of a velocity perturbation, as demonstrated in a paper by Mora. With the Born approximation for a perturbation in a reference model consisting in two homogeneous isotropic acoustic half‐spaces and the assumption of infinitely large apertures available in the data, analytic expressions can be found that describe the spatial spectrum of the recorded seismic signal as a function of the spatial spectrum of the inhomogeneity. Diving waves can be included if the deeper part of the homogeneous model is replaced by one that has a vertical velocity gradient. We study this spectrum in more detail by separately considering scattering of direct, reflected and head waves, as well as singly and multiply reflected diving waves for a gradient model. Taking the reflection coefficient of the deeper reflector into account, we obtain sensitivity estimates for each wavetype. Although the head waves have a relatively small contribution to the reconstruction of the velocity perturbation, compared to the other waves, they contain reliable long‐wavelength information that can be beneficial for full‐waveform inversion. If the deeper part has a constant positive velocity gradient with depth, all the energy eventually returns to the source‐receiver line, given a sufficiently large acquisition aperture. This will improve the sensitivity of the scattered reflected and refracted wavefields to perturbations in the background model. The same happens for a zero velocity gradient but with a very high impedance contrast between the two half‐spaces, which results in a large reflection coefficient.
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