B uilding high-resolution models of several physical properties of the subsurface by multiparameter full waveform inversion (FWI) of multicomponent data will be a challenge for seismic imaging for the next decade. The physical properties, which govern propagation of seismic waves in visco-elastic media, are the velocities of the P-and S-waves, density, attenuation, and anisotropic parameters. Updating each property is challenging because several parameters of a different nature can have a coupled effect on the seismic response for a particular propagation regime (from transmission to reflection). This is generally referred to as trade-off or crosstalk between parameters. Moreover, different parameter classes can have different orders of magnitude or physical units and footprints of different strength in the wavefield, which can make the inversion poorly conditioned if it is not properly scaled. These difficulties raise the issue of a suitable parameterization for multiparameter FWI, where the term parameterization must be understood as a set of independent parameter classes that fully describe the subsurface properties. Many combinations of parameters can be viewed and this choice is not neutral as the parameterization controls the trade-off between parameters and the local resolution with which they can be reconstructed. Once this parameterization is selected, the subset of parameter classes in the parameterization, that can be reliably updated during the inversion, must be identified to avoid overparameterization of the optimization problem. The purpose of this tutorial is to provide a comprehensive overview of the promise, pitfalls, and open questions underlying multiparameter FWI. We first review the main FWI ingredient that controls the tradeoff between parameters, namely the radiation pattern of the so-called virtual sources, and some tools for analyzing these trade-offs. Then, we present some illustrative examples of multiparameter FWI, which should provide some guidelines to choose a suitable parameterization for FWI in visco-acoustic anisotropic media. We conclude by proposing a data-driven and model-driven workflow for visco-elastic anisotropic FWI of multicomponent marine data, which has been inspired by a real data case study from Valhall.

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Abstract. Full Waveform Inversion (FWI) is a powerful method for reconstructing subsurface parameters from local measurements of the seismic wavefield. This method consists in minimizing a distance between predicted and recorded data. The predicted data is computed as the solution of a wave propagation problem. Conventional numerical methods for the resolution of FWI problems are gradient-based methods, such as the preconditioned steepest-descent, or more recently the l-BFGS quasi-Newton algorithm. In this study, we investigate the interest of applying a truncated Newton method to FWI. The inverse Hessian operator plays a crucial role in the parameter reconstruction. The truncated Newton method allows one to better account for this operator. This method is based on the computation of the Newton descent direction by solving the corresponding linear system through an iterative procedure such as the conjugate gradient method. The large-scale nature of FWI problems requires however to carefully implement this method to avoid prohibitive computational costs. First, this requires to work in a matrix-free formalism, and the capability of computing efficiently Hessian-vector products. To this purpose, we propose general second-order adjoint state formulas. Second, special attention must be payed to define the stopping criterion for the inner linear iterations associated with the computation of the Newton descent direction. We propose several possibilities and establish a theoretical link between the Steihaug-Toint method, based on trust-regions, and the Eisenstat stopping criterion, designed for method globalized by linesearch. We investigate the application of the truncated Newton method to two test cases: the first is a standard test case in seismic imaging based on the Marmousi II model. The second one is inspired by a nearsurface imaging problem for the reconstruction of high velocity structures. In the latter case, we demonstrate that the presence of large amplitude multi-scattered waves prevents standard methods from converging while the truncated Newton method provides more reliable results.

In the recent years, the use of optimal transport distance has yielded significant progress in image processing for pattern recognition, shape identification, and histograms matching. In this study, the use of this distance is investigated for a seismic tomography problem exploiting the complete waveform; the full waveform inversion. In its conventional formulation, this high resolution seismic imaging method is based on the minimization of the L 2 distance between predicted and observed data. Application of this method is generally hampered by the local minima of the associated L 2 misfit function, which correspond to velocity models matching the data up to one or several phase shifts. Conversely, the optimal transport distance appears as a more suitable tool to compare the misfit between oscillatory signals, for its ability to detect shifted patterns. However, its application to full waveform inversion is not straightforward, as the mass conservation between the compared data can not be guaranteed, a crucial assumption for optimal transport. In this study, the use of a distance based on the Kantorovich-Rubinstein norm is introduced to overcome this difficulty. Its mathematical link with the optimal transport distance is made clear. An efficient numerical strategy for its computation, based on a proximal splitting technique, is introduced. Each iteration of the corresponding algorithm requires the solution of a linear system which is demonstrated to be a second-order finite-difference discretization of the Poisson equation, for which fast solvers can be used, relying either on the fast Fourier transform or multigrid techniques. The development of this numerical method make possible applications to industrial scale data, involving tenths of millions of discrete unknowns. The results we obtain on such large scale synthetic data illustrate the potentialities of the optimal transport for seismic imaging. Starting from crude initial velocity models, optimal transport based inversion yields significantly better velocity reconstructions than those based on the L 2 distance, both in 2D and 3D contexts.

Computationally efficient 3-D frequency-domain full waveform inversion (FWI) is applied to ocean-bottom cable data from the Valhall oil field in the visco-acoustic vertical transverse isotropic (VTI) approximation. Frequency-domain seismic modelling is performed with a parallel sparse direct solver on a limited number of computer nodes. A multiscale imaging is performed by successive inversions of single frequencies in the 3.5-10 Hz frequency band. The vertical wave speed is updated during FWI while density, quality factor Q P and anisotropic Thomsen's parameters δ and are kept fixed to their initial values. The final FWI model shows the resolution improvement that was achieved compared to the initial model that was built by reflection traveltime tomography. This FWI model shows a glacial channel system at 175 m depth, the footprint of drifting icebergs on the palaeo-seafloor at 500 m depth, a detailed view of a gas cloud at 1 km depth and the base cretaceous reflector at 3.5 km depth. The relevance of the FWI model is assessed by frequency-domain and time-domain seismic modelling and source wavelet estimation. The agreement between the modelled and recorded data in the frequency domain is excellent up to 10 Hz although amplitudes of modelled wavefields propagating across the gas cloud are overestimated. This might highlight the footprint of attenuation, whose absorption effects are underestimated by the homogeneous background Q P model (Q P = 200). The match between recorded and modelled time-domain seismograms suggests that the inversion was not significantly hampered by cycle skipping. However, late arrivals in the synthetic seismograms, computed without attenuation and with a source wavelet estimated from short-offset early arrivals, arrive 40 ms earlier than the recorded seismograms. This might result from dispersion effects related to attenuation. The repeatability of the source wavelets inferred from data that are weighted by a linear gain with offset is dramatically improved when they are estimated in the FWI model rather than in the smooth initial model. The two source wavelets, estimated in the FWI model from data with and without offset gain, show a 40 ms time-shift, which is consistent with the previous analysis of the time-domain seismograms. The computational efficiency of our frequency-domain approach is assessed against a recent time-domain FWI case study performed in a similar geological environment. This analysis highlights the efficiency of the frequency-domain approach to process a large number of sources and receivers with limited computational resources, thanks to the efficiency of the substitution step performed by the direct solver. This efficiency can be further improved by using a block-low rank version of the multifrontal solver and by exploiting the sparsity of the source vectors during the substitution step. Future work will aim to update attenuation and density at the same time of the vertical wave speed.

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