International audienceNonlinear elastic waveform inversion has advanced to the point where it is now possible to invert real multiple‐shot seismic data. The iterative gradient algorithm that we employ can readily accommodate robust minimization criteria which tend to handle many types of seismic noise (noise bursts, missing traces, etc.) better than the commonly used least‐squares minimization criteria. Although there are many robust criteria from which to choose, we have tested only a few. In particular, the Cauchy criterion and the hyperbolic secant criterion perform very well in both noise‐free and noise‐added inversions of numerical data. Although the real data set, which we invert using the sech criterion, is marine (pressure sources and receivers) and is very much dominated by unconverted P waves, we can, for the most part, resolve the short wavelengths of both P impedance and S impedance. The long wavelengths of velocity (the background) are assumed known. Because we are deriving nearly all impedance information from unconverted P waves in this inversion, data acquisition geometry must have sufficient multiplicity in subsurface coverage and a sufficient range of offsets, just as in amplitude‐versus‐offset (AVO) inversion. However, AVO analysis is implicitly contained in elastic waveform inversion algorithms as part of the elastic wave equation upon which the algorithms are based. Because the real‐data inversion is so large—over 230,000 unknowns (340,000 when density is included) and over 600,000 data values—most statistical analyses of parameter resolution are not feasible. We qualitatively verify the resolution of our results by inverting a numerical data set which has the same acquisition geometry and corresponding long wavelengths of velocity as the real data, but has semirandom perturbations in the short wavelengths of P and S impedance
International audienceComplex velocity models characterized by strong lateral variations are certainly a great motivation, but also a great challenge, for depth imaging. In this context, some unexpected results can occur when using depth imaging algorithms. In general, after a common shot or common offset migration, the resulting depth images are sorted into common‐image gathers (CIG), for further processing such as migration‐based velocity analysis or amplitude‐variation‐with‐offset analysis. In this paper, we show that CIGs calculated by common‐shot or common‐offset migration can be strongly affected by artifacts, even when a correct velocity model is used for the migration. The CIGs are simply not flat, due to unexpected curved events (kinematic artifacts) and strong lateral variations of the amplitude (dynamic artifacts). Kinematic artifacts do not depend on the migration algorithm provided it can take into account lateral variations of the velocity model. This can be observed when migrating the 2‐D Marmousi dataset either with a wave‐equation migration or with a multivalued Kirchhoff migration/inversion. On the contrary, dynamic artifacts are specific to multi‐arrival ray‐based migration/inversion. This approach, which should provide a quantitative estimation of the reflectivity of the model, provides in this context dramatic results. In this paper, we propose an analysis of these artifacts through the study of the ray‐based migration/inversion operator. The artifacts appear when migrating a single‐fold subdata set with multivalued ray fields. They are due to the ambiguous focusing of individual reflected events at different locations in the image. No information is a priori available in the single‐fold data set for selecting the focusing position, while migration of multifold data would provide this information and remove the artifacts by the stack of the CIGs. Analysis of the migration/inversion operator provides a physical condition, the imaging condition, for insuring artifact free CIGs. The specific cases of common‐shot and common‐offset single‐fold gathers are studied. It appears clearly that the imaging condition generally breaks down in complex velocity models for both these configurations. For artifact free CIGs, we propose a novel strategy: compute CIGs versus the diffracting/reflecting angle. Working in the angle domain seems the natural way for unfolding multivalued ray fields, and it can be demonstrated theoretically and practically that common‐angle imaging satisfies the imaging condition in the great majority of cases. Practically, the sorting into angle gathers can not be done a priori over the data set, but is done in the inner depth migration loop. Depth‐migrated images are obtained for each angle range. A canonical example is used for illustrating the theoretical derivations. Finally, an application to the Marmousi model is presented, demonstrating the relevance of the approach
International audienceThe aim of inverting seismic waveforms is to obtain the “best” earth model. The best model is defined as the one producing seismograms that best match (usually under a least‐squares criterion) those recorded. Our approach is nonlinear in the sense that we synthesize seismograms without using any linearization of the elastic wave equation. Since we use rather complete data sets without any spatial aliasing, we do not have the problem of secondary minima (Tarantola, 1986). Nevertheless, our gradient methods fail to converge if the starting earth model is far from the true earth (Mora, 1987; Kolb et al., 1986; Pica et al., 1989)
Classical algorithms used for traveltime tomography are not necessarily well suited for handling very large seismic data sets or for taking advantage of current supercomputers. The classical approach of first-arrival traveltime tomography was revisited with the proposal of a simple gradient-based approach that avoids ray tracing and estimation of the Fréchet derivative matrix. The key point becomes the derivation of the gradient of the misfit function obtained by the adjointstate technique. The adjoint-state method is very attractive from a numerical point of view because the associated cost is equivalent to the solution of the forward-modeling problem, whatever the size of the input data and the number of unknown velocity parameters. An application on a 2D synthetic data set demonstrated the ability of the algorithm to image near-surface velocities with strong vertical and lateral variations and revealed the potential of the method.
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