Full-waveform inversion FWI is a challenging data-fitting procedure based on full-wavefield modeling to extract quantitative information from seismograms. High-resolution imaging at half the propagated wavelength is expected. Recent advances in high-performance computing and multifold/multicomponent wide-aperture and wide-azimuth acquisitions make 3D acoustic FWI feasible today. Key ingredients of FWI are an efficient forward-modeling engine and a local differential approach, in which the gradient and the Hessian operators are efficiently estimated. Local optimization does not, however, prevent convergence of the misfit function toward local minima because of the limited accuracy of the starting model, the lack of low frequencies, the presence of noise, and the approximate modeling of the wave-physics complexity. Different hierarchical multiscale strategiesaredesignedtomitigatethenonlinearityandill-posedness of FWI by incorporating progressively shorter wavelengths in the parameter space. Synthetic and real-data case studies address reconstructing various parameters, from VP and VS velocities to density, anisotropy, and attenuation. This review attempts to illuminate the state of the art of FWI. Crucial jumps, however, remain necessary to make it as popular as migration techniques. The challenges can be categorized as 1 building accurate starting models with automatic procedures and/or recording low frequencies, 2 defining new minimization criteria to mitigate the sensitivity of FWI to amplitude errors and increasing the robustness of FWI when multiple parameter classes are estimated, and 3 improving computational efficiency by data-compression techniquestomake3DelasticFWIfeasible.

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An overview of full-waveform inversion in exploration geophysics

SUMMARY We present an introduction to the spectral element method, which provides an innovative numerical approach to the calculation of synthetic seismograms in 3-D earth models. The method combines the £exibility of a ¢nite element method with the accuracy of a spectral method. One uses a weak formulation of the equations of motion, which are solved on a mesh of hexahedral elements that is adapted to the free surface and to the main internal discontinuities of the model. The wave¢eld on the elements is discretized using high-degree Lagrange interpolants, and integration over an element is accomplished based upon the Gauss^Lobatto^Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix, which greatly simpli¢es the algorithm. We illustrate the great potential of the method by comparing it to a discrete wavenumber/re£ectivity method for layer-cake models. Both body and surface waves are accurately represented, and the method can handle point force as well as moment tensor sources. For a model with very steep surface topography we successfully benchmark the method against an approximate boundary technique. For a homogeneous medium with strong attenuation we obtain excellent agreement with the analytical solution for a point force.

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Introduction to the spectral element method for three-dimensional seismic wave propagation

SUMMARY We use a spectral-element method to simulate seismic wave propagation throughout the entire globe. The method is based upon a weak formulation of the equations of motion and combines the flexibility of a finite-element method with the accuracy of a global pseudospectral method. The finite-element mesh honours all first- and second-order discontinuities in the earth model. To maintain a relatively constant resolution throughout the model in terms of the number of grid points per wavelength, the size of the elements is increased with depth in a conforming fashion, thus retaining a diagonal mass matrix. In the Earth’s mantle and inner core we solve the wave equation in terms of displacement, whereas in the liquid outer core we use a formulation based upon a scalar potential. The three domains are matched at the inner core and core‐ mantle boundaries, honouring the continuity of traction and the normal component of velocity. The effects of attenuation and anisotropy are fully incorporated. The method is implemented on a parallel computer using a message passing technique. We benchmark spectral-element synthetic seismograms against normal-mode synthetics for a spherically symmetric reference model. The two methods are in excellent agreement for all body- and surface-wave arrivals with periods greater than about 20 s.

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Spectral-element simulations of global seismic wave propagation—I. Validation

The perfectly matched layer (PML) absorbing boundary condition has proven to be very efficient from a numerical point of view for the elastic wave equation to absorb both body waves with nongrazing incidence and surface waves. However, at grazing incidence the classical discrete PML method suffers from large spurious reflections that make it less efficient for instance in the case of very thin mesh slices, in the case of sources located close to the edge of the mesh, and/or in the case of receivers located at very large offset. We demonstrate how to improve the PML at grazing incidence for the differential seismic wave equation based on an unsplit convolution technique. The improved PML has a cost that is similar in terms of memory storage to that of the classical PML. We illustrate the efficiency of this improved convolutional PML based on numerical benchmarks using a finite-difference method on a thin mesh slice for an isotropic material and show that results are significantly improved compared with the classical PML technique. We also show that, as the classical PML, the convolutional technique is intrinsically unstable in the case of some anisotropic materials.

An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation

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.

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

We introduce a numerical approach for modeling elastic wave propagation in 2-D and 3-D fully anisotropic media based upon a spectral element method. The technique solves a weak formulation of the wave equation, which is discretized using a high-order polynomial representation on a finite element mesh. For isotropic media, the spectral element method is known for its high degree of accuracy, its ability to handle complex model geometries, and its low computational cost. We show that the method can be extended to fully anisotropic media. The mass matrix obtained is diagonal by construction, which leads to a very efficient fully explicit solver. We demonstrate the accuracy of the method by comparing it against a known analytical solution for a 2-D transversely isotropic test case, and by comparing its predictions against those based upon a finite difference method for a 2-D heterogeneous, anisotropic medium. We show its generality and its flexibility by modeling wave propagation in a 3-D transversely isotropic medium with a symmetry axis tilted relative to the axes of the grid.

Simulation of anisotropic wave propagation based upon a spectral element method

We introduce a spectral-element method for modeling wave propagation in media with both fluid (acoustic) and solid (elastic) regions, as for instance in offshore seismic experiments. The problem is formulated in terms of displacement in elastic regions and a velocity potential in acoustic regions. Matching between domains is implemented based upon an interface integral in the framework of an explicit prediction-multicorrection staggered time scheme. The formulation results in a mass matrix that is diagonal by construction. The scheme exhibits high accuracy for a 2-D test case with known analytical solution. The method is robust in the case of strong topography at the fluid-solid interface and is a good alternative to classical techniques, such as finite differencing.

Wave propagation near a fluid-solid interface : A spectral-element approach

Marine reflection seismic data inversion is a compute-intensive process, especially in three dimensions. Approximations often are made to limit the number of physical parameters we invert for, or to speed up the forward modeling. Because the data often are dominated by unconverted P-waves, one popular approximation is to consider the earth as purely acoustic, i.e., no shear modulus. The material density sometimes is taken as a constant. Nonlinear waveform seismic inversion consists of iteratively minimizing the misfit between the amplitudes of the measured and the modeled data. Approximations, such as assuming an acoustic medium, lead to incorrect modeling of the amplitudes of the seismic waves, especially with respect to amplitude variation with offset (AVO), and therefore have a direct impact on the inversion results. For evaluation purposes, we have performed a series of inversions with different approximations and different constraints whereby the synthetic data set to recover is computed for a 1D elastic medium. A series of numerical experiments, although simple, help to define the applicability domain of the acoustic assumption. Acoustic full-wave inversion is applicable only when the S-wave velocity and the density fields are smooth enough to reduce the AVO effect, or when the near-offset seismograms are inverted with a good starting model. However, in many realistic cases, acoustic approximation penalizes the full-wave inversion of marine reflection seismic data in retrieving the acoustic parameters.

The domain of applicability of acoustic full-waveform inversion for marine seismic data

Anisotropy is often observed due to the thin layering or aligned micro-structures, like small fractures. At the scale of cross-well tomography, the anisotropic effects cannot be neglected. In this paper, we propose a method of full-wave inversion for transversely isotropic media and we test its robustness against structured noisy data.



Optimization inversion techniques based on a least-square formalism are used. In this framework, analytical expressions of the misfit function gradient, based on the adjoint technique in the time domain, allow one to solve the inverse problem with a high number of parameters and for a completely heterogeneous medium.



The wave propagation equation for transversely isotropic media with vertical symmetry axis is solved using the finite difference method on the cylindrical system of coordinates. This system allows one to model the 3D propagation in a 2D medium with a revolution symmetry. In case of approximately horizontal layering, this approximation is sufficient.



The full-wave inversion method is applied to a crosswell synthetic 2-component (radial and vertical) dataset generated using a 2D model with three different anisotropic regions. Complex noise has been added to these synthetic observed data. This noise is Gaussian and has the same amplitude f−k spectrum as the data. Part of the noise is localized as a coda of arrivals, the other part is not localized. Five parameter fields are estimated, (vertical) P-wave velocity, (vertical) S-wave velocity, volumetric mass and the Thomsen anisotropic parameters epsilon and delta. Horizontal exponential correlations have been used. The results show that the full-wave inversion of cross-well data is relatively robust for high-level noise even for second-order parameters such as Thomsen epsilon and delta anisotropic parameters.

Feasibility study for an anisotropic full waveform inversion of cross-well seismic data

A linear viscoelastic media with a constant quality factor provide a reasonably realistic model of Earth for many seismic application. A linear viscoelastic relation, based on the superposition of relaxation mechanisms, produced formulations that allow the computation of synthetic seismograms in time-domain for arbitrary spatial distributions of quality factors. By using this constitutive relation and the adjoint state technique, we propose a new formulation for the viscoelastic seismic inverse problem. This approach reduces the computer memory requirements to the point where computations are practical. For illustration, this full viscoelastic waveform inversion has been applied for a numerical experiment.

Christophe Barnes

Papers

Simulation of anisotropic wave propagation based upon a spectral element method

Wave propagation near a fluid-solid interface : A spectral-element approach

The domain of applicability of acoustic full-waveform inversion for marine seismic data

Feasibility study for an anisotropic full waveform inversion of cross-well seismic data

Full waveform inversion of seismic data for a viscoelastic medium