Viscoelastic Full Waveform Inversion Research Articles

One-Dimensional Elastic and Viscoelastic Full-Waveform Inversion in Heterogeneous Media Using Physics-Informed Neural Networks

One-Dimensional Elastic and Viscoelastic Full-Waveform Inversion in Heterogeneous Media Using Physics-Informed Neural Networks

Multiparameter 2-D viscoelastic full-waveform inversion of Rayleigh waves: a field experiment at Krauthausen test site

SUMMARYFull-waveform inversion (FWI) has been proven to be an effective tool for high-resolution multiparameter imaging of the shallow subsurface. It has been shown that the Gauss–Newton (GN) optimization method uses the off-diagonal information contained in the Hessian matrix and can increase resolution and mitigate crosstalk in multiparameter viscoelastic FWI. In this work, we demonstrate the advantages of GN viscoelastic FWI over the conventional FWI with a conjugate gradient optimization method by using synthetic examples. We also investigate the potential of shallow seismic-wave 2-D viscoelastic FWI as a method for high-resolution hydrogeological characterization. The GN viscoelastic FWI is applied to two orthogonal profiles acquired at the Krauthausen natural laboratory (Germany). The groundwater table is located at around 2 m, which nicely agrees with an abrupt increase of P-wave velocity in the inverted results. FWI also reveals a low S-wave velocity layer at the depth of 4–6 m with high Poisson’s ratio values close to 0.5, which corresponds to a saturated sand layer known from previous studies. A K-mean cluster analysis is used to further analyse the multiparameter FWI results. By considering the derived Poisson’s ratio, P- and S-wave velocities, we convert the complex relationship between the multivariate data into a lithological meaningful zonation of the shallow subsurface. By comparing the lithological units in the alluvial aquifer with the cone penetration tests clusters, we conclude that the divided facies describe valuable characterization information about the heterogeneity and connectivity of the aquifer. This experiment indicates that the multiparameter models derived by viscoelastic FWI contain useful information for high-resolution aquifer characterization, and the potential of multiparameter FWI combined with cluster analysis in shallow subsurface characterization is encouraging.

Three-dimensional Gauss–Newton constant-Q viscoelastic full-waveform inversion of near-surface seismic wavefields

SUMMARY Full-waveform inversion (FWI) methods rely on accurate numerical simulation of wave propagation in the analysed medium. Acoustic or elastic wave equations are often used to model seismic wave propagation. These types of simulations do not account for intrinsic attenuation effects due to material anelasticity, and thus correction techniques have been utilized in practice to partially compensate the anelasticity. These techniques often only consider the waveform amplitude correction based on averaging of overall amplitude response over the entire data set, and ignore the phase correction. Viscoelastic wave equations account for the anelastic response in both waveform amplitude and phase, and are therefore a more suitable alternative. In this study, we present a novel 3-D Gauss–Newton viscoelastic FWI (3-D GN-VFWI) method. To address the main challenge of the Gauss–Newton optimization, we develop formulas to compute the Jacobian efficiently by the convolution of virtual sources and backward wavefields. The virtual sources are obtained by directly differentiating the viscoelastic wave equations with respect to model parameters. In order to resolve complex 3-D structures with reasonable computational effort, a homogeneous attenuation (Q factor) is used throughout the analysis to model the anelastic effects. Synthetic and field experiments are performed to demonstrate the utility of the method. The synthetic results clearly demonstrate the ability of the method in characterizing a challenging velocity profile, including voids and reverse velocity layers. The field experimental results show that method successfully characterizes the complex substructure with two voids and undulating limestone bedrock, which are confirmed by invasive tests. Compared to 3-D elastic FWI results, the presented viscoelastic method produces more accurate results regarding depths of the voids and bedrock. This study suggests that the improvement of imaging accuracy would warrant the widespread use of viscoelastic wave equations in FWI problems. To our best knowledge, this is the first reported study on 3-D GN-VFWI at any scale. This study provides the new theory and formulation for the use of Gauss–Newton optimization on the 3-D viscoelastic problem.

Regularization tunnelling for full waveform inversion

SUMMARY Prior knowledge can be a powerful source of information in seismic inversion, leading to substantially more accurate models than those derived from seismic data alone. Prior knowledge about clustering of rock physics properties, supplied by appropriate rock physics models, or nearby well-logs, may be especially informative. However, incorporating general forms of this type of information into local optimization methods, such as those used in full waveform inversion (FWI), is problematic, since it will often call for model updates involving a departure from the basin of a local minimum. Here, we propose an optimization strategy for FWI, in which global regularization information is partially accounted for, and in which a model can, under the right circumstances, ‘tunnel’ between basins, to honour prior information. The process is based on standard FWI updates and begins with an initial model in which every model grid cell is in one of a range of pre-defined clusters. At each iteration, a standard update is computed, but it is then followed by a second update, in which each grid cell tunnels to a new cluster if the move is warranted by the update history of the model (its ‘momentum’) and the model regularization penalty (its ‘potential’). To limit the ability of the tunnelling procedure from introducing spatial variations on scales smaller than those contained in the seismic data, these conditions are considered in combination with a small mode filter. We implement the procedure by building on an existing 2-D viscoelastic FWI algorithm, limited such that only the P-wave velocity and density are updated, and evaluate it by subjecting it to a variety of basic tests involving simulated data. We conclude that the basic approach, and this specific implementation, using a simple initial model, provides a combination of well-resolved structures and grid cells occupying appropriate clusters that cannot be produced by standard means.

Full Waveform Inversion of Viscoelastic Media Based on P–S Separation

In viscoelastic full waveform inversion, there is interference between the gradients of the P- and S- wave velocities, leading to the degradation of the inversion accuracy. Therefore, the interference problem in viscoelastic media needs to be solved beforehand. Even though the P-S separation method has been demonstrated to be effective in reducing the crosstalk in elastic media, little attention has been paid to its efficacy in viscoelastic media. We incorporate the P-S separation method in the full waveform inversion of viscoelastic media (QEFWI) and derive a gradient formula for the QEFWI. Numerical tests on the QEFWI demonstrate that the proposed method can successfully reduce the interference problem between the gradients of P- and S- wave velocities, resulting in significant improvement of the viscoelastic full waveform inversion.

3-D multicomponent full waveform inversion for shallow-seismic target: Ettlingen Line case study

SUMMARY In this work, we investigate methodological development and application of viscoelastic full waveform inversion to a multicomponent source and receiver data set at the near-surface scale. The target is located in Germany and lies in an ancient war trench from the War of the Spanish Succession that was buried at approximately 1 m depth. We present the pre-processing of the data, including a matching filter correction to compensate for different source and receiver coupling conditions during the acquisition, as well as a dedicated multistep workflow for the reconstruction of both P-wave and S-wave velocities. Our implementation is based on viscoelastic modelling using a spectral element discretization to accurately account for the wave propagation complexity in this shallow region. We illustrate the inversion stability by starting from different initial models, either based on dispersion curve analysis or homogeneous models consistent with first arrivals. We recover similar results in both cases. We also illustrate the importance of accounting for the attenuation by comparing elastic and viscoelastic results. The 3-D results make it possible to recover and locate the trench precisely in terms of interpretation. They also exhibit another trench structure, in a direction forming an angle at 45° with the direction of the targeted trench. This new structure had been previously interpreted as an artifact in former 2-D inversion results. The archaeological interpretation of this new structure is still a matter of discussion.

Multiparameter viscoelastic full-waveform inversion of shallow-seismic surface waves with a pre-conditioned truncated Newton method

SUMMARY 2-D full-waveform inversion (FWI) of shallow-seismic Rayleigh waves has become a powerful method for reconstructing viscoelastic multiparameter models of shallow subsurface with high resolution. The multiparameter reconstruction in FWI is challenging due to the potential presence of cross-talk between different parameters and the unbalanced sensitivity of Rayleigh-wave data with respect to different parameter classes. Accounting for the inverse Hessian using truncated Newton methods based on second-order adjoint methods provides an effective tool to mitigate cross-talk caused by the coupling between different parameters. In this study, we apply a pre-conditioned truncated Newton (PTN) method to shallow-seismic FWI to simultaneously invert for multiparameter near-surface models (P- and S-wave velocities, attenuation of P and S waves, and density). We first investigate scattered wavefields caused by these parameters to evaluate the coupling between them. Then we investigate the performance of the PTN method on shallow-seismic FWI of Rayleigh wave for reconstructing all five parameters simultaneously. The application to spatially correlated and uncorrelated models demonstrates that the PTN method helps to mitigate the cross-talk and improves the resolution of the multiparameter reconstructions, especially for the weak parameters with small sensitivity such as attenuation and density parameters. The attenuation of Pwaves cannot be inverted reliably due to its negligible sensitivity on the Rayleigh wave. The comparison with the classical pre-conditioned conjugate gradient method highlights the improved performance of the PTN method and thus the benefit of accounting for the information included in the Hessian.

Autotuning Hamiltonian Monte Carlo for efficient generalized nullspace exploration

SUMMARYWe propose methods to efficiently explore the generalized nullspace of (non-linear) inverse problems, defined as the set of plausible models that explain observations within some misfit tolerance. Owing to the random nature of observational errors, the generalized nullspace is an inherently probabilistic entity, described by a joint probability density of tolerance values and model parameters. Our exploration methods rest on the construction of artificial Hamiltonian systems, where models are treated as high-dimensional particles moving along a trajectory through model space. In the special case where the distribution of misfit tolerances is Gaussian, the methods are identical to standard Hamiltonian Monte Carlo, revealing that its apparently meaningless momentum variable plays the intuitive role of a directional tolerance. Its direction points from the current towards a new acceptable model, and its magnitude is the corresponding misfit increase. We address the fundamental problem of producing independent plausible models within a high-dimensional generalized nullspace by autotuning the mass matrix of the Hamiltonian system. The approach rests on a factorized and sequentially preconditioned version of the L-BFGS method, which produces local Hessian approximations for use as a near-optimal mass matrix. An adaptive time stepping algorithm for the numerical solution of Hamilton’s equations ensures both stability and reasonable acceptance rates of the generalized nullspace sampler. In addition to the basic method, we propose variations of it, where autotuning focuses either on the diagonal elements of the mass matrix or on the macroscopic (long-range) properties of the generalized nullspace distribution. We quantify the performance of our methods in a series of numerical experiments, involving analytical, high-dimensional, multimodal test functions. These are designed to mimic realistic inverse problems, where sensitivity to different model parameters varies widely, and where parameters tend to be correlated. The tests indicate that the effective sample size may increase by orders of magnitude when autotuning is used. Finally, we present a proof of principle of generalized nullspace exploration in viscoelastic full-waveform inversion. In this context, we demonstrate (1) the quantification of inter- and intraparameter trade-offs, (2) the flexibility to change model parametrization a posteriori, for instance, to adapt averaging length scales, (3) the ability to perform dehomogenization to retrieve plausible subwavelength models and (4) the extraction of a manageable number of alternative models, potentially located in distinct local minima of the misfit functional.

SeisElastic2D: An open-source package for multiparameter full-waveform inversion in isotropic-, anisotropic- and visco-elastic media

Full-waveform inversion (FWI) has emerged as a powerful technique to obtain high-resolution subsurface elastic properties. However, several complicating features, including interparameter trade-offs, cycle-skipping, and high computational cost, motivate a careful assessment and validation of candidate versions of FWI prior to use. Field data application of elastic FWI remains a challenging task, and widely available tools for inversion and appraisal do not exist. SeisElastic2D is an open-source package for multiparameter FWI in isotropic-, anisotropic- and visco-elastic media consisting of a set of FORTRAN 90 routines and SHELL scripts. This package has a modular structure built upon several existing open-source packages including upgraded SPECFEM2D, seisDD, and Seismic Unix. Various model parameterizations for isotropic- and anisotropic-elastic FWI, designed to reduce interparameter trade-offs, are available, as are different misfit functions designed to mitigate cycle-skipping and source-structure trade-offs. New misfit functions, measuring amplitude variation and central-frequency shift, are provided to estimate attenuation models in visco-elastic FWI. This package can be installed and used on high-performance computing cluster and local workstation supporting massively parallel interface, in order to satisfy the high computational requirements of elastic FWI. These features of SeisElastic2D make it a powerful tool for elastic FWI to bridge the gap between academic studies and industrial applications.

Parameter crosstalk and leakage between spatially separated unknowns in viscoelastic full-waveform inversion

Elastic and attenuative effects play a major role in the determination of wave amplitudes and phases observed at seismic sensors. Viscoelastic full-waveform inversion (FWI) has the potential to recover much of the information content of measured seismic data by simultaneously accounting for these effects. However, the frequency variations and phase information present in viscoelastic FWI introduce new challenges to the inversion, especially through their impact on interparameter crosstalk. Crosstalk is typically characterized through analysis of the radiation patterns of point scatterers; however, the point scatterer model is not well suited to viscoelastic FWI because (1) attenuation introduces a significant potential for crosstalk between variables distant from one another in space and (2) interpreting the effect of frequency and phase dependence on the radiation patterns of point scatterers is not straightforward. We have introduced and examined a numerical approach for assessing the viscoelastic crosstalk modes expected for a given parameterization, optimization strategy, and acquisition geometry based on differencing various synthetic inversion results. With this approach, we have characterized the viscoelastic crosstalk for a typical parameterization for several possible acquisition geometries. Of particular note is the strong tendency for [Formula: see text] variables to leak into elastic variables from which they are spatially separated.

2-D multiparameter viscoelastic shallow-seismic full-waveform inversion: reconstruction tests and first field-data application

SUMMARY 2-D full-waveform inversion (FWI) of shallow-seismic wavefields has recently become a novel way to reconstruct S-wave velocity models of the shallow subsurface with high vertical and lateral resolution. In most applications, seismic wave attenuation is ignored or considered as a passive modelling parameter only. In this study, we explore the feasibility and performance of multiparameter viscoelastic 2-D FWI in which seismic velocities and attenuation of P and S waves, respectively, and mass density are inverted simultaneously. Synthetic reconstruction experiments reveal that multiple crosstalks between all viscoelastic material parameters may occur. The reconstruction of S-wave velocity is always robust and of high quality. The parameters P-wave velocity and density exhibit weaker sensitivity and can be reconstructed more reliably by multiparameter viscoelastic FWI. Anomalies in S-wave attenuation can be recovered but with limited resolution. In a field-data application, a small-scale refilled trench is nicely delineated as a low P- and S-wave velocity anomaly. The reconstruction of P-wave velocity is improved by the simultaneous inversion of attenuation. The reconstructed S-wave attenuation reveals higher attenuation in the shallow weathering zone and weaker attenuation below. The variations in the reconstructed P- and S-wave velocity models are consistent with the reflectivity observed in a ground penetrating radar (GPR) profile.

On the influence of different misfit functions for attenuation estimation in viscoelastic full-waveform inversion: synthetic study

SUMMARY Estimating subsurface attenuation distribution is essential to compensate the amplitude and phase distortions in seismic imaging and characterize attenuative reservoirs. Full-waveform inversion (FWI) methods represent promising techniques to invert for both velocity and attenuation models with arbitrary spatial distributions. However, simultaneously determining velocity and attenuation properties introduces the problem of interparameter trade-off in viscoelastic FWI. Ignoring attenuation effects can result in inaccurate velocity estimations. Velocity errors may produce significant parameter crosstalk artefacts in the inverted attenuation models. An appropriate misfit function measuring specific seismic attribute is essential to capture the influence of attenuation on the seismic data and thus is expected to reduce the influences of velocity errors for attenuation estimation. In this study, we evaluate the performances of different misfit functions for attenuation estimation in viscoelastic FWI accompanied with a two-stage sequential inversion strategy. Synthetic examples with different acquisition surveys are given to show that in the presence of strong velocity errors, the amplitude-based misfit functions, including envelope-difference, root-mean-square amplitude-ratio and spectral amplitude-ratio, can invert for the attenuation models more reliably, compared to the waveform-difference and instantaneous phase misfit functions. With the two-stage inversion approach, more reliable velocity and attenuation models can be obtained using viscoelastic FWI.

Amplitude-based misfit functions in viscoelastic full-waveform inversion applied to walk-away vertical seismic profile data

Viscoelastic full-waveform inversion is applied to walk-away vertical seismic profile data acquired at a producing heavy-oil field in Western Canada for the determination of subsurface velocity models (P-wave velocity [Formula: see text] and S-wave velocity [Formula: see text]) and attenuation models (P-wave quality factor [Formula: see text] and S-wave quality factor [Formula: see text]). To mitigate strong velocity-attenuation trade-offs, a two-stage approach is adopted. In Stage I, [Formula: see text] and [Formula: see text] models are first inverted using a standard waveform-difference (WD) misfit function. Following this, in Stage II, different amplitude-based misfit functions are used to estimate the [Formula: see text] and [Formula: see text] models. Compared to the traditional WD misfit function, the amplitude-based misfit functions exhibit stronger sensitivity to attenuation anomalies and appear to be able to invert [Formula: see text] and [Formula: see text] models more reliably in the presence of velocity errors. Overall, the root-mean-square amplitude-ratio and spectral amplitude-ratio misfit functions outperform other misfit function choices. In the final outputs of our inversion, significant drops in the [Formula: see text] to [Formula: see text] ratio (~1.6) and Poisson’s ratio (~0.23) are apparent within the Clearwater Formation (depth ~0.45–0.50 km) of the Mannville Group in the Western Canada Sedimentary Basin. Strong [Formula: see text] (~20) and [Formula: see text] (~15) anomalies are also evident in this zone. These observations provide information to help identify the target attenuative reservoir saturated with heavy-oil resources.

3D viscoelastic full waveform inversion of seismic waves for geotechnical site investigation

Full waveform inversion (FWI) methods are becoming efficient geophysical tools for geotechnical site investigation. They rely on a mathematical description of wave propagation phenomena to generate synthetic wave-fields, which are then matched with measured wave-fields for extraction of material properties. Due to the inherent computational challenges in implementing a viscoelastic treatment of the wave propagation phenomena, most of the existing 3D FWI algorithms are based on an elastic treatment, and hence ignore the anelastic behavior of materials. The focus of this study is to account for the anelastic effect of materials in the inversion analysis to improve inverted results. Therefore, a 3D viscoelastic FWI method is developed and tested on synthetic and field datasets. The method is based on an existing solution of 3D viscoelastic wave equations (finite-difference solver SOFI3D) and the adjoint-state inversion approach. The results show that the viscoelastic waveform analysis was able to characterize low- and high-velocity synthetic layers, and real variable soil layers. The S-wave velocity (Vs) profiles from field experiment generally agree with invasive standard penetration test (SPT) N-values, including identification of a low-velocity zone. Vs profiles obtained from a 3D elastic FWI are also included for comparison, and results from the viscoelastic inversion are more consistent with the SPT N-values in both trend and magnitudes.

Efficient time-domain 3D elastic and viscoelastic full-waveform inversion using a spectral-element method on flexible Cartesian-based mesh

Viscoelastic full-waveform inversion is recognized as a challenging task for current acquisition deployment at the crustal scale. We have developed an efficient formulation based on a time-domain spectral-element method on a flexible Cartesian-based mesh. We consider anisotropic elastic coefficients and isotropic attenuation. Complete gradient expressions including the attenuation contribution spread into those of elastic components are given in a consistent way. The influence of attenuation on the P-wave velocity reconstruction is illustrated through a toy configuration. The numerical implementation of the forward problem includes efficient matrix-vector products for solving second-order elastodynamic equations for 3D geometries: An original high-order integration for topography effects is performed at nearly no extra cost. Combined adjoint and forward field recomputation from the final state and previously saved boundary values allows the estimation of misfit gradients for density, elastic parameters, and attenuation factors with no I/O efforts. Two-level parallelism is implemented over the sources and domain decomposition, which is necessary for a realistic 3D configuration. The misfit gradient preconditioning is performed by a so-called Bessel filter using an efficient differential implementation based on finite-element ingredients on the forward mesh instead of the often-used, costly convolution approach. A 3D synthetic illustration is provided on a subset ([Formula: see text]) of the SEG Advanced Modeling (SEAM) Phase II Foothills model with 4 lines of 20 sources. The structurally based Bessel filter and a simple data hierarchy strategy considering early body waves before all waves including surface waves allow a precise reconstruction of the P- and S-wavespeeds while keeping a smooth density description.

Time domain viscoelastic full waveform inversion

Viscous attenuation can have a strong impact on seismic wave propagation, but it is rarely taken into account in full waveform inversion (FWI). When viscoelasticity is considered in time domain FWI, the displacement formulation of the wave equation is usually used instead of the popular velocity–stress formulation. However, inversion schemes rely on the adjoint equations, which are quite different for the velocity–stress formulation than for the displacement formulation. In this paper, we apply the adjoint state method to the isotropic viscoelastic wave equation in the velocity–stress formulation based on the generalized standard linear solid rheology. By applying linear transformations to the wave equation before deriving the adjoint state equations, we obtain two symmetric sets of partial differential equations for the forward and adjoint variables. The resulting sets of equations only differ by a sign change and can be solved by the same numerical implementation. We also investigate the crosstalk between parameter classes (velocity and attenuation) of the viscoelastic equation. More specifically, we show that the attenuation levels can be used to recover the quality factors of P and S waves, but that they are very sensitive to velocity errors. Finally, we present a synthetic example of viscoelastic FWI in the context of monitoring CO₂ geological sequestration. We show that FWI based on our formulation can indeed recover P- and S-wave velocities and their attenuation levels when attenuation is high enough. Both changes in velocity and attenuation levels recovered with FWI can be used to track the CO₂ plume during and after injection. Further studies are required to evaluate the performance of viscoelastic FWI on real data.

Time-domain seismic modeling in viscoelastic media for full waveform inversion on heterogeneous computing platforms with OpenCL

Full Waveform Inversion (FWI) aims at recovering the elastic parameters of the Earth by matching recordings of the ground motion with the direct solution of the wave equation. Modeling the wave propagation for realistic scenarios is computationally intensive, which limits the applicability of FWI. The current hardware evolution brings increasing parallel computing power that can speed up the computations in FWI. However, to take advantage of the diversity of parallel architectures presently available, new programming approaches are required. In this work, we explore the use of OpenCL to develop a portable code that can take advantage of the many parallel processor architectures now available. We present a program called SeisCL for 2D and 3D viscoelastic FWI in the time domain. The code computes the forward and adjoint wavefields using finite-difference and outputs the gradient of the misfit function given by the adjoint state method. To demonstrate the code portability on different architectures, the performance of SeisCL is tested on three different devices: Intel CPUs, NVidia GPUs and Intel Xeon PHI. Results show that the use of GPUs with OpenCL can speed up the computations by nearly two orders of magnitudes over a single threaded application on the CPU. Although OpenCL allows code portability, we show that some device-specific optimization is still required to get the best performance out of a specific architecture. Using OpenCL in conjunction with MPI allows the domain decomposition of large models on several devices located on different nodes of a cluster. For large enough models, the speedup of the domain decomposition varies quasi-linearly with the number of devices. Finally, we investigate two different approaches to compute the gradient by the adjoint state method and show the significant advantages of using OpenCL for FWI.

A review on the systematic formulation of 3-D multiparameter full waveform inversion in viscoelastic medium

In this paper, we study 3-D multiparameter full waveform inversion (FWI) in viscoelastic media based on the generalized Maxwell/Zener body including arbitrary number of attenuation mechanisms. We present a frequency-domain energy analysis to establish the stability condition of a full anisotropic viscoelastic system, according to zero-valued boundary condition and the elastic–viscoelastic correspondence principle: the real-valued stiffness matrix becomes a complex-valued one in Fourier domain when seismic attenuation is taken into account. We develop a least-squares optimization approach to linearly relate the quality factor with the anelastic coefficients by estimating a set of constants which are independent of the spatial coordinates, which supplies an explicit incorporation of the parameter Q in the general viscoelastic wave equation. By introducing the Lagrangian multipliers into the matrix expression of the wave equation with implicit time integration, we build a systematic formulation of multiparameter FWI for full anisotropic viscoelastic wave equation, while the equivalent form of the state and adjoint equation with explicit time integration is available to be resolved efficiently. In particular, this formulation lays the foundation for the inversion of the parameter Q in the time domain with full anisotropic viscoelastic properties. In the 3-D isotropic viscoelastic settings, the anelastic coefficients and the quality factors using bulk and shear moduli parametrization can be related to the counterparts using P and S velocity. Gradients with respect to any other parameter of interest can be found by chain rule. Pioneering numerical validations as well as the real applications of this most generic framework will be carried out to disclose the potential of viscoelastic FWI when adequate high-performance computing resources and the field data are available.

Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 2: imaging compressive-wave and shear-wave velocities

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.

Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 1: imaging compressional wave speed, density and attenuation

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.