Romain Brossier

Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion

Quantitative imaging of the elastic properties of the subsurface at depth is essential for civil engineering applications and oil- and gas-reservoir characterization. A realistic synthetic example provides for an assessment of the potential and limits of 2D elastic full-waveform inversionFWIof wide-aperture seismic data for recovering high-resolution P- and S-wave velocity models of complex onshore structures. FWI of land data is challengingbecauseoftheincreasednonlinearityintroducedbyfreesurface effects such as the propagation of surface waves in the heterogeneous near-surface. Moreover, the short wavelengths of the shear wavefield require an accurate S-wave velocity starting modeliflowfrequenciesareunavailableinthedata.Weevaluated different multiscale strategies with the aim of mitigating the nonlinearities. Massively parallel full-waveform inversion was implemented in the frequency domain. The numerical optimization relies on a limited-memory quasi-Newton algorithm that outperforms the more classic preconditioned conjugate-gradient algorithm. The forward problem is based upon a discontinuous Galerkin DG method on triangular mesh, which allows accuratemodelingoffree-surfaceeffects.Sequentialinversionsofincreasingfrequenciesdefinethemostnaturallevelofhierarchyin multiscale imaging. In the case of land data involving surface waves, the regularization introduced by hierarchical frequency inversions is not enough for adequate convergence of the inversion. A second level of hierarchy implemented with complexvalued frequencies is necessary and provides convergence of the inversion toward acceptable P- and S-wave velocity models. Amongthepossiblestrategiesforsamplingfrequenciesintheinversion, successive inversions of slightly overlapping frequency groups is the most reliable when compared to the more standard sequential inversion of single frequencies. This suggests that simultaneous inversion of multiple frequencies is critical when consideringcomplexwavephenomena.

Which data residual norm for robust elastic frequency-domain full waveform inversion?

Elastic full-waveform inversion is an ill-posed data-fitting procedure that is sensitive to noise, inaccuracies of the starting model,definitionofmultiparameterclasses,andinaccuratemodeling of wavefield amplitudes. We have investigated the performance of different minimization functionals as the least-squares norm 2, the least-absolute-values norm 1, and combinations of both the Huber and so-called hybrid criteria with reference to twonoisyoffshoreValhallmodelandonshoreoverthrustmodel synthetic data sets. The four minimization functionals were implemented in 2D elastic frequency-domain full-waveform inversion FWI, where efficient multiscale strategies were designed by successive inversions of a few increasing frequencies. For the offshore and onshore case studies, the 1-norm provided the most reliable models for P- and S-wave velocities VP and VS, even when strongly decimated data sets that correspond to fewfrequencieswereusedintheinversionandwhenoutlierspolluted the data. The 2-norm can provide reliable results in the presence of uniform white noise for VP and VS if the data redundancyisincreasedbyrefiningthefrequencysamplingintervalin the inversion at the expense of computational efficiency. The 1-norm and the Huber and hybrid criteria, unlike the 2-norm, allowforsuccessfulimagingoftheVSmodelfromnoisydataina soft-seabed environment, where the P-to-S-waves have a small footprint in the data. However, the Huber and hybrid criteria are sensitive to a threshold criterion that controls the transition between the criteria and that requires tedious trial-and-error investigations for reliable estimation. The 1-norm provides a robust alternativetothe2-normforinvertingdecimateddatasetsinthe frameworkofefficientfrequency-domainFWI.

Full waveform inversion and the truncated Newton method: quantitative imaging of complex subsurface structures

Full Waveform Inversion (FWI) is a powerful tool for quantitative seismic imaging from wide-azimuth seismic data. The method is based on the minimization of the misfit between observed and simulated data. This amounts to the resolution of a large-scale nonlinear minimization problem. The inverse Hessian operator plays a crucial role in this reconstruction process. Accounting accurately for the effect of this operator within the minimization scheme should correct for illumination deficits, restore the amplitude of the subsurface parameters, and help to remove artifacts generated by energetic multiple reflections. Conventional preconditioned gradient-based minimization methods only roughly approximate the effect of this operator. We are interested in this study to another class of minimization methods, named as truncated Newton methods. These methods are based on the computation of the model update through a matrix-free conjugate gradient resolution of the Newton linear system. The aim of this study is to present a feasible implementation of this method for the FWI problem, based on a second-order adjoint state formulation for the computation of Hessian-vector products. We compare this method with the nonlinear conjugate gradient and the l-BFGS method within the context of 2D acoustic frequency FWI for the reconstruction of P-wave velocity models. Two test cases are investigated. The first is the synthetic BP 2004 model, representative of the Gulf Of Mexico geology with high velocity contrasts associated with the presence of salt structures. The second is a 2D real data-set from the Valhall oil field in North sea. These tests emphasize the interesting properties of the truncated Newton method regarding conventional optimization methods within the context of FWI.

Shallow-structure characterization by 2D elastic full-waveform inversion

Assessing the effectiveness of elastic full-waveform-inversion (FWI) algorithms when applied to shallow 2D structures in the presence of a complex topography is critically important. By using FWI, we overcome inherent limitations of conventional seismic methods used for near-surface prospecting (acoustic tomography and multichannel spectral analysis of surface waves). The elastic forward problem, formulated in the frequency domain, is based on a mixed finite-element P0P1 discontinuous Galerkin method to ensure accurate modeling of complex topography effects at a reasonable computing cost. The inversion problem uses an FWI algorithm to minimize the misfit between observed and calculated data. Based on results from a numerical experiment performed on a realistic landslide model inspired from the morphostructure of the Super-Sauze earthflow, we analyzed the effect of using a hierarchical preconditioning strategy, based on a simultaneous multifrequency inversion of damped data, to mitigate the strong nonlinearities coming from the surface waves. This strategy is a key point in alleviating the strong near-surface effects and avoiding convergence toward a local minimum. Using a limited-memory quasi-Newton method improved the convergence level. These findings are analogous to recent applications on large-scale domains, although limited sourcereceiver offset ranges, low-frequency content of the source, and domination of surface waves on the signal led to some difficulties. Regarding the impact of data decimation on the inversion results, we have learned that an inversion restricted to the vertical data component can be successful without significant loss in terms of parameter imagery resolution. In our investigations of the effect of increased source spacing, we found that a sampling of 4 m (less than three times the theoretical maximum of one half-wavelength) led to severe aliasing.

Seismic wave modeling for seismic imaging

Many scientific applications require accurate modeling of seismic wave propagation in complex media. These objectives can include fundamental understanding of seismic wave propagation of the Earth on a global scale, including fluid envelopes, mitigation of seismic risk with better quantitative estimates of seismic hazard, and improved exploitation of the natural resources in the crust of the Earth. Accurate quantification is a continual quest, and benchmark protocols have been designed for model definitions and comparison of solutions. Through this quest, an impressive number of numerical tools have been developed, ranging from efficient finite-difference methods to more sophisticated finite-element methods, and including the so-called pseudospectral methods (see Wu and Maupin for a review). The main motivation behind these permanent developments has been to improve the efficiency and accuracy of forward modeling. To achieve this, one systematic choice for both finite-difference and finite-element methods has involved explicit time-stepping integration to avoid matrix inversion...

Velocity model building from seismic reflection data by full-waveform inversion

Full-waveform inversion is re-emerging as a powerful data-fitting procedure for quantitative seismic imaging of the subsurface from wide-azimuth seismic data. This method is suitable to build high-resolution velocity models provided that the targeted area is sampled by both diving waves and reflected waves. However, the conventional formulation of full-waveform inversion prevents the reconstruction of the small wavenumber components of the velocity model when the subsurface is sampled by reflected waves only. This typically occurs as the depth becomes significant with respect to the length of the receiver array. This study first aims to highlight the limits of the conventional form of full-waveform inversion when applied to seismic reflection data, through a simple canonical example of seismic imaging and to propose a new inversion workflow that overcomes these limitations. The governing idea is to decompose the subsurface model as a background part, which we seek to update and a singular part that corresponds to some prior knowledge of the reflectivity. Forcing this scale uncoupling in the full-waveform inversion formalism brings out the transmitted wavepaths that connect the sources and receivers to the reflectors in the sensitivity kernel of the full-waveform inversion, which is otherwise dominated by the migration impulse responses formed by the correlation of the downgoing direct wavefields coming from the shot and receiver positions. This transmission regime makes full-waveform inversion amenable to the update of the long-to-intermediate wavelengths of the background model from the wide scattering-angle information. However, we show that this prior knowledge of the reflectivity does not prevent the use of a suitable misfit measurement based on cross-correlation, to avoid cycleskipping issues as well as a suitable inversion domain as the pseudo-depth domain that allows us to preserve the invariant property of the zero-offset time. This latter feature is useful to avoid updating the reflectivity information at each non-linear iteration of the full-waveform inversion, hence considerably reducing the computational cost of the entire workflow. Prior information of the reflectivity in the full-waveform inversion formalism, a robust misfit function that prevents cycle-skipping issues and a suitable inversion domain that preserves the seismic invariant are the three key ingredients that should ensure well-posedness and computational efficiency of fullwaveform inversion algorithms for seismic reflection data.

Two-dimensional Seismic Imaging of the Valhall Model From Synthetic OBC Data By Frequency Domain Elastic Full-waveform Inversion

Quantitative imaging of the elastic properties of the subsurface is essential for reservoir characterization. We apply twodimensional frequency-domain full-waveform inversion (FWI) to image shallow-water synthetic VP and VS models of the Valhall oil and gas field from 3-C ocean-bottom-cable data. In soft-seabed environment where a small amount of P-to-S mode conversion occurs at the sea bottom, the seismic wavefield is dominated by the P waves whereas the S waves have a weaker signature. We first show that acoustic FWI of elastic data provides accurateVP model in such environment. Elastic FWI, that is desired for reservoir characterization, is a more difficult task. Hierarchical processing of the different parameter classes and data components is required, in addition to low-frequency data and robust multi-scale algorithm, to converge toward acceptable models.

Time-lapse seismic imaging using regularized full-waveform inversion with a prior model: which strategy?

Full-waveform inversion is an appealing technique for time-lapse imaging, especially when prior model information is included into the inversion workflow. Once the baseline reconstruction is achieved, several strategies can be used to assess the physical parameter changes, such as parallel difference (two separate inversions of baseline and monitor data sets), sequential difference (inversion of the monitor data set starting from the recovered baseline model) and double-difference (inversion of the difference data starting from the recovered baseline model) strategies. Using syntheticMarmousi data sets, we investigate which strategy should be adopted to obtain more robust and more accurate time-lapse velocity changes in noise-free and noisy environments. This synthetic application demonstrates that the double-difference strategy provides the more robust time-lapse result. In addition, we propose a target-oriented time-lapse imaging using regularized full-waveform inversion including a prior model and model weighting, if the prior information exists on the location of expected variations. This scheme applies strong prior model constraints outside of the expected areas of timelapse changes and relatively less prior constraints in the time-lapse target zones. In application of this process to the Marmousi model data set, the local resolution analysis performed with spike tests shows that the target-oriented inversion prevents the occurrence of artefacts outside the target areas, which could contaminate and compromise the reconstruction of the effective time-lapse changes, especially when using the sequential difference strategy. In a strongly noisy case, the target-oriented prior model weighting ensures the same behaviour for both time-lapse strategies, the double-difference and the sequential difference strategies and leads to a more robust reconstruction of the weak time-lapse changes. The double-difference strategy can deliver more accurate time-lapse variation since it can focus to invert the difference data. However, the double-difference strategy requires a preprocessing step on data sets such as time-lapse binning to have a similar source/receiver location between two surveys, while the sequential difference needs less this requirement. If we have prior information about the area of changes, the target-oriented sequential difference strategy can be an alternative and can provide the same robust result as the doubledifference strategy.

2D elastic full-waveform imaging of the near-surface: application to synthetic and physical modelling data sets

Standard seismic methods are generally not well adapted to provide sharp quantitative images of the first few metres of underground. A two-dimensional full-waveform inversion of land seismic data, based on frequency-domain viscoelastic modelling, offers a promising approach to take advantage of the full complexity of seismograms and to simultaneously build 2D images of P and VS parameters. In order to understand the behaviour of this method in a near-surface context and anticipate the corresponding field applications, we perform this investigation by applying waveform inversion on a simple layered medium. We first use synthetic data obtained from numerical modelling and then we employ laboratory data obtained by small-scale physical modelling. We demonstrate that such a near-surface 2D model can be quantitatively determined even in a realistic situation where the data are dominated by high-amplitude surface waves. A comparison of results derived for the same medium from ideal synthetic data and noisy experimental data allows detecting anomalies in the reconstruction of velocity models due to the experimental nature of the data used.

Frequency-Domain Numerical Modelling of Visco-Acoustic Waves Based on Finite-Difference and Finite-Element Discontinuous Galerkin Methods

Seismic exploration is one of the main geophysical methods to extract quantitative inferences about the Earth’s interior at different scales from the recording of seismic waves near the surface. Main applications are civil engineering for cavity detection and landslide characterization, site effect modelling for seismic hazard, CO2 sequestration and nuclearwaste storage, oil and gas exploration, and fundamental understanding of geodynamical processes. Acoustic or elastic waves are emitted either by controlled sources or natural sources (i.e., earthquakes). Interactions of seismic waves with the heterogeneities of the subsurface provide indirect measurements of the physical properties of the subsurface which govern the propagation of elastic waves (compressional and shear wave speeds, density, attenuation, anisotropy). Quantitative inference of the physical properties of the subsurface from the recordings of seismic waves at receiver positions is the so-called seismic inverse problem that can be recast in the framework of local numerical optimization. The most complete seismic inversion method, the so-called full waveform inversion (Virieux & Operto (2009) for a review), aims to exploit the full information content of seismic data by minimization of the misfit between the full seismic wavefield and the modelled one. The theoretical resolution of full waveform inversion is half the propagated wavelength. In full waveform inversion, the full seismic wavefield is generally modelled with volumetric methods that rely on the discretization of the wave equation (finite difference, finite element, finite volume methods). In the regime of small deformations associated with seismic wave propagation, the subsurface can be represented by a linear elastic solid parameterized by twenty-one elastic constants and the density in the framework of the constitutive Hooke’s law. If the subsurface is assumed isotropic, the elastic constants reduce to two independent parameters, the Lame parameters, which depend on the compressional (P) and the shear (S) wave speeds. In marine environment, the P wave speed has most of the time a dominant footprint in the seismic wavefield, in particular, on the hydrophone component which records the pressure wavefield. The dominant footprint of the P wave speed on the seismic