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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 shortspread 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 shortscale 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. [ABSTRACT FROM AUTHOR]

Full waveform inversion (FWI) of 3-D data sets has recently been possible thanks to the development of high performance computing. However, FWI remains a computationally intensive task when high frequencies are injected in the inversion or more complex wave physics (viscoelastic) is accounted for. The highest computational cost results from the numerical solution of the wave equation for each seismic source. To reduce the computational burden, one well-known technique is to employ a random linear combination of the sources, rather that using each source independently. This technique, known as source encoding, has shown to successfully reduce the computational cost when applied to real data. Up to now, the inversion is normally carried out using gradient descent algorithms. With the idea of achieving a fast and robust frequency-domain FWI, we assess the performance of the random source encoding method when it is interfaced with second-order optimization methods (quasi-Newton l-BFGS, truncated Newton). Because of the additional seismic modelings required to compute the Newton descent direction, it is not clear beforehand if truncated Newton methods can indeed further reduce the computational cost compared to gradient algorithms. We design precise stopping criteria of iterations to fairly assess the computational cost and the speed-up provided by the source encoding method for each optimization method. We perform experiment on synthetic and real data sets. In both cases, we confirm that combining source encoding with second-order optimization methods reduces the computational cost compared to the case where source encoding is interfaced with gradient descent algorithms. For the synthetic data set, inspired from the geology of Gulf of Mexico, we show that the quasi-Newton l-BFGS algorithm requires the lowest computational cost. For the real data set application on the Valhall data, we show that the truncated Newton methods provide the most robust direction of descent. [ABSTRACT FROM AUTHOR]

Multiparameter full waveform inversion (FWI) is a challenging quantitative seismic imaging method for lithological characterization and reservoir monitoring. The difficulties in multiparameter FWI arise from the variable influence of the different parameter classes on the phase and amplitude of the data, and the trade-off between these. In this framework, choosing a suitable parametrization of the subsurface and designing the suitable FWI workflow are two key methodological issues in non-linear waveform inversion. We assess frequency-domain visco-acoustic FWI to reconstruct the compressive velocity (VP), the density (ρ) or the impedance (IP) and the quality factor (QP), from the hydrophone component, using a synthetic data set that is representative of the Valhall oil field in the North Sea. We first assess which of the (VP, ρ) and (VP, IP) parametrizations provides the most reliable FWI results when dealing with wide-aperture data. Contrary to widely accepted ideas, we show that the (VP, ρ) parametrization allows a better reconstruction of both the VP, ρ and IP parameters, first because it favours the broad-band reconstruction of the dominant VP parameter, and secondly because the trade-off effects between velocity and density at short-to-intermediate scattering angles can be removed by multiplication, to build an impedance model. This allows for the matching of the reflection amplitudes, while the broad-band velocity model accurately describes the kinematic attributes of both the diving waves and reflections. Then, we assess different inversion strategies to recover the quality factor QP, in addition to parameters VP and ρ. A difficulty related to attenuation estimation arises because, on the one hand the values of QP are on average one order of magnitude smaller than those of VP and ρ, and on the other hands model perturbations relative to the starting models can be much higher for QP than for VP and ρ during FWI. In this framework, we show that an empirical tuning of the FWI regularization, which is adapted to each parameter class, is a key issue to correctly account for the attenuation in the inversion. We promote a hierarchical approach where the dominant parameter VP is reconstructed first from the full data set (i.e. without any data preconditioning) to build a velocity model as kinematically accurate as possible before performing the joint update of the three parameter classes during a second step. This hierarchical imaging of compressive wave speed, density and attenuation is applied to a real wide-aperture ocean-bottom-cable data set from the Valhall oil field. Several geological features, such as accumulation of gas below barriers of claystone and soft quaternary sediment are interpreted in the FWI models of density and attenuation. The models of VP, ρ and QP that have been developed by visco-acoustic FWI of the hydrophone data can be used as initial models to perform visco-elastic FWI of the geophone data for the joint update of the compressive and shear wave speeds. [ABSTRACT FROM AUTHOR]

Multiparameter elastic full waveform inversion (FWI) is a promising technology that allows inferences to be made on rock and fluid properties, which thus narrows the gap between seismic imaging and reservoir characterization. Here, we assess the feasibility of 2-D vertical transverse isotropic visco-elastic FWI of wide-aperture multicomponent ocean-bottom-cable data from the Valhall oil field. A key issue is to design a suitable hierarchical data-driven and model-driven FWI workflow, the aim of which is to reduce the nonlinearity of the FWI. This nonlinearity partly arises because the shear (S) wavespeed can have a limited influence on seismic data in marine environments. In a preliminary stage, visco-acoustic FWI of the hydrophone component is performed to build a compressional (P)-wave velocity model, a density model and a quality-factor model, which provide the necessary background models for the subsequent elastic inversion. During the elastic FWI, the P and S wavespeeds are jointly updated in two steps. First, the hydrophone data are inverted to mainly update the long-to-intermediate wavelengths of the S wavespeeds from the amplitude-versus-offset variations of the P–P reflections. This S-wave velocity model is used as an improved starting model for the subsequent inversion of the better-resolving data recorded by the geophones. During these two steps, the P-wave velocity model is marginally updated, which supports the relevance of the visco-acoustic FWI results. Through seismic modelling, we show that the resulting visco-elastic model allows several P-to-S converted phases recorded on the horizontal-geophone component to be matched. Several elastic quantities, such as the Poisson ratio, and the ratio and product between the P and S wavespeeds, are inferred from the P-wave and S-wave velocity models. These attributes provide hints for the interpretation of an accumulation of gas below lithological barriers. [ABSTRACT FROM AUTHOR]

The development of dense networks of broad-band seismographs makes teleseismic data amenable to full-waveform inversion (FWI) methods for high-resolution lithospheric imaging. Compared to scattered-field migration, FWI seeks to involve the full seismic wavefield in the inversion. We present a parametric analysis of 2-D frequency-domain FWI in the framework of lithospheric imaging from teleseismic data to identify the main factors that impact on the quality of the reconstructed compressional (P)-wave and shear (S)-wave speed models. Compared to controlled-source seismology, the main adaptation of FWI to teleseismic configuration consists of the implementation with a scattered-filed formulation of plane-wave sources that impinge on the base of the lithospheric target located below the receiver network at an arbitrary incidence angle. Seismic modelling is performed with a hp-adaptive discontinuous Galerkin method on unstructured triangular mesh. A quasi-Newton inversion algorithm provides an approximate accounting for the Hessian operator, which contributes to reduce the footprint of the coarse acquisition geometry in the imaging. A versatile algorithm to compute the gradient of the misfit function with the adjoint-state method allows for abstraction between the forward-problem operators and the meshes that are during seismic modelling and inversion, respectively. An approximate correction for obliquity is derived for future application to real teleseismic data under the two-dimension approximation. Comparisons between the characteristic scales involved in exploration geophysics and in teleseismic seismology suggest that the resolution gain provided by full waveform technologies should be of the same order of magnitude for both applications. We first show the importance of the surface-reflected wavefield to dramatically improve the resolving power of FWI by combining tomography-like and migration-like imaging through the incorporation of the forward-scattered and the backscattered wavefields in the inversion. The resolution of FWI is assessed through checkerboard tests and confirms a resolution of the order of the wavelength for both the P and S speeds, when the full wavefield is incorporated in the inversion. Secondly, we show that computationally efficient strategies, which consist of decimating the number of frequency components involved in the inversion, do not apply to teleseismic acquisitions, because the scattering-angle bandwidth sampled by plane-wave sources can be narrow and coarsely sampled, compared to that provided by dense profiles of point sources in exploration seismology. The waveform inversion is less sensitive to the band of incidence angles spanned by the plane-wave sources and to the sampling of this band. However, the deficit of vertically propagating plane waves hampers the vertical resolution of planar layers. Aliasing artefacts created by coarse arrays of receivers are illustrated. We show how taking into account the Hessian in the inversion and the suitable management of frequencies in the inversion help to mitigate these artefacts. Acceptable reconstructions are shown for both the P- and S-wave speeds for a receiver spacing of up to 20 km in the 0.1–0.4 Hz frequency range. Building a reliable initial model for FWI is a highly non-linear problem in exploration seismology. We show how the low-frequency content of the teleseismic sources allow us to build accurate P- and S-wave speed models starting from simple vertical-gradient velocity models. All of these results are derived using a 2-D realistic synthetic experiment that was performed with noise-free data. Most of conclusions of this parametric study should apply in three dimensions. The impact of noise and the footprint of other experimental parameters ... [ABSTRACT FROM AUTHOR]

SUMMARY The validity of isotropic approximation to perform acoustic full waveform inversion (FWI) of real wide-aperture anisotropic data can be questioned due to the intrinsic kinematic inconsistency between short- and large-aperture components of the data. This inconsistency is mainly related to the differences between the vertical and horizontal velocities in vertical-transverse isotropic (VTI) media. The footprint of VTI anisotropy on 2-D acoustic isotropic FWI is illustrated on a hydrophone data set of an ocean-bottom cable that was collected over the Valhall field in the North Sea. Multiscale FWI is implemented in the frequency domain by hierarchical inversions of increasing frequencies and decreasing aperture angles. The FWI models are appraised by local comparison with well information, seismic modelling, reverse-time migration (RTM) and source-wavelet estimation. A smooth initial VTI model parameterized by the vertical velocity V0 and the Thomsen parameters δ and ε were previously developed by anisotropic reflection traveltime tomography. The normal moveout () and horizontal () velocity models were inferred from the anisotropic models to perform isotropic FWI. The VNMO models allows for an accurate match of short-spread reflection traveltimes, whereas the Vh model, after updating by first-arrival traveltime tomography (FATT), allows for an accurate match of first-arrival traveltimes. Ray tracing in the velocity models shows that the first 1.5 km of the medium are sampled by both diving waves and reflections, whereas the deeper structure at the reservoir level is mainly controlled by short-spread reflections. Starting from the initial anisotropic model and keeping fixed δ and ε models, anisotropic FWI allows us to build a vertical velocity model that matches reasonably well the well-log velocities. Isotropic FWI is performed using either the NMO model or the FATT model as initial model. In both cases, horizontal velocities are mainly reconstructed in the first 1.5 km of the medium. This suggests that the wide-aperture components of the data have a dominant control on the velocity estimation at these depths. These high velocities in the upper structure lead to low values of velocity in the underlying gas layers (either equal or lower than vertical velocities of the well log), and/or a vertical stretching of the structure at the reservoir level below the gas. This bias in the gas velocities and the mispositioning in depth of the deep reflectors, also shown in the RTM images, are required to match the deep reflections in the isotropic approximation and highlight the footprint of anisotropy in the isotropic FWI of long-offset data. Despite the significant differences between the anisotropic and isotropic FWI models, each of these models produce a nearly-equivalent match of the data, which highlights the ill-posedness of acoustic anisotropic FWI. Hence, we conclude with the importance of considering anisotropy in FWI of wide-aperture data to avoid bias in the velocity reconstructions and mispositioning in depth of reflectors. Designing a suitable parameterization of the VTI acoustic FWI is a central issue to manage the ill-posedness of the FWI. [ABSTRACT FROM AUTHOR]

We compare different finite-difference schemes for two-dimensional (2-D) acoustic frequency-domain forward modelling. The schemes are based on staggered-grid stencils of various accuracy and grid rotation strategies to discretize the derivatives of the wave equation. A combination of two staggered-grid stencils on the classical Cartesian coordinate system and the 45° rotated grid is the basis of the so-called mixed-grid stencil. This method is compared with a parsimonious staggered-grid method based on a fourth-order approximation of the first derivative operator. Averaging of the mass acceleration can be incorporated in the two stencils. Sponge-like perfectly matched layer absorbing boundary conditions are also examined for each stencil and shown to be effective. The deduced numerical stencils are examined for both the wavelength content and azimuthal variation. The accuracy of the fourth-order staggered-grid stencil is slightly superior in terms of phase velocity dispersion to that of the mixed-grid stencil when averaging of the mass acceleration term is applied to the staggered-grid stencil. For fourth-order derivative approximations, the classical staggered-grid geometry leads to a stencil that incorporates 13 grid nodes. The mixed-grid approach combines only nine grid nodes. In both cases, wavefield solutions are computed using a direct matrix solver based on an optimized multifrontal method. For this 2-D geometry, the staggered-grid strategy is significantly less efficient in terms of memory and CPU time requirements because of the enlarged bandwidth of the impedance matrix and increased number of coefficients in the discrete stencil. Therefore, the mixed-grid approach should be suggested as the routine scheme for 2-D acoustic wave propagation modelling in the frequency domain. [ABSTRACT FROM AUTHOR]

We present a new numerical modelling approach for frequency-domain finite-difference (FDFD) wave simulations. The new approach is developed as an extension to standard FDFD modelling schemes, when wave propagation simulations are performed in large-scale 2-D or 3-D models with complex heterogeneous rheology. Partial differential equations are presented in matrix-type form. Wavefield solutions are computed on different coarse- and fine-discretized numerical grids by a combination of a direct solver with an iterative solver. Two different connection strategies are designed. Both compute a coarse-grid wavefield solution using a direct matrix solver. The obtained solution is projected on a fine-discretized grid, which is used as an initial solution for an iterative solver to compute the desired fine-grid solution. The wavefield projection that combines coarse and fine grids, is either based on a space interpolation scheme, called the direct iterative space solver (DISS) or on a multi-scale wavelet expansion, called the direct iterative wavelet solver (DIWS). The DISS scheme mimics a nested iteration scheme of a full multi-grid method, since numerical grids are prolonged by a simple bilinear interpolation scheme. The simple grid combination leads to wavefield solutions that are affected by spatial phase-shift artefacts (aliasing), which may be suppressed by a large number of iteration steps or a standard V- and W-cycles sequence between grids. The actual DIWS matrix construction implementation is computationally more expensive, though the wavelet iteration scheme guarantees fast and stable iterative convergence. Coarse-grid wavefield solutions are combined with fine-grid solutions through the multi-resolution scaling property of a standard orthogonal wavelet expansion. Since the wavelet transformation accounts for grid interactions, phase-shift artefacts are greatly reduced and significantly fewer iteration steps are required for convergence. We demonstrate the performance and accuracy of the DISS and DIWS strategies for two complex 2-D heterogeneous wave simulation examples. [ABSTRACT FROM AUTHOR]

The Crozet Bank, the easternmost region of the Crozet Plateau (austral Indian Ocean), is capped by two groups of islands which form the Crozet Archipelago (Terres Australes and Antarctiques Françaises). A N-S-trending 2 km deep basin, the Indivat Basin, extends between the two groups of islands and bisects the Crozet Bank into two separate domains. The interpretation of the KeOBS8 seismic refraction profile shot during the KeOBS/MD66 cruise (January-February 1991) over the eastern Crozet Plateau was carried out by ray tracing and the computation of synthetic seismograms. This interpretation leads to a rather complex seismic structure and outlines a pronounced eastward crustal thinning from 16.5 to 10 km at the expense of layer 2. The thinning of the crust is abrupt east of the Indivat Basin. Unlike below the Hawaiian Islands and the Marquesas Islands, no underplated mantle material underlies the crust of the Crozet Bank. Moreover, this deep seismic sounding has further revealed that a high-velocity feature outcropping the seabed 30 km east of Ile aux Cochons could be a major structural feature, deeply rooted within the lower crust. The occurrence of this feature (a submarine volcano?) is associated with a mantle dyke causing a steep shallowing of the lower crustal interfaces. Gravity modelling was performed along line KeOBS8, with a density structure deduced from the seismic section, to model free-air anomalies derived from altimetry data. This modelling confirms that the Indivat Basin, underlined by a gravity low running roughly N-S between the two groups of islands, is a major structural boundary. As the model generates medium-wavelength anomalies of adequate amplitude, it also confirms that the volcano, located west of the Indivat Basin, is a deeply rooted feature. The Crozet Bank clearly appears as a plume-affected structure, which may have originated from a deep thermal anomaly within the lithosphere. More recent volcanic episodes, related to a... [ABSTRACT FROM AUTHOR]