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1. Massively parallel frequency-domain FWI (FFWI code)

Figures 1-4 illustrate some results of 3D visco-acoustic VTI frequency-domain FWI on Ocean Bottom Cable (OBC) data collected in the North Sea. Both seismic modelling and inversion are performed in the frequency domain. The linear system resulting from the discretization of the time-harmonic wave equation has been solved with the sparse multifrontal direct solver MUMPS using quite limited computational resources provided by the computer center SIGAMM hosted by Observatory of Côte d'Azur. This may sound counter intuitive as sparse direct solvers have been considered as untractable for years to tackle 3D problems due to the memory demand of LU factorization and its limited scalability for large scale problems. In contrast, we have shown with the MUMPS team the efficiency of this approach to tackle problems involving a few tens millions of unknowns, hence validating our feasibility analysis published in 2007 ( Operto et al., 2007). See Operto et al., 2014; Operto et al., 2015; Amestoy et al., 2016; Operto & Miniussi, 2018 for more details.

*Figure 1:* Seismic imaging of an oil field in the North Sea by frequency-domain FWI. (Left): P-wave velocity model of the subsurface across a low-velocity gas cloud. (Right): A 10Hz monochromatic wavefield computed by solving the Helmholtz equation with the sparse multifrontal direct solver MUMPS is superimposed ( Operto et al., 2015; Amestoy et al., 2016; Operto & Miniussi, 2018).

Figure 2: On the resolution power of FWI. The above figure shows some horizontal and vertical sections of P-wave velocity models of the oil field. On the left, the velocity model was obtained by reflection traveltime tomography. On the right, by frequency-domain FWI. From top to bottom, the first three slices cross-cut sand channel deposits at 175m depth, scarves left by drifting icebergs on the paleo sea bed at 500m depth, and a gas cloud at 1km depth. The two bottom panels show vertical sections across the gas cloud and its periphery ( Operto et al., 2015; Amestoy et al., 2016; Operto & Miniussi, 2018).

*Figure 3:* (a) Velocity model of the oil reservoir. (b) Q model of the oil reservoir reconstructed by frequency-domain FWI. The imprint of attenuation in the seismic data fit is highlighted in Figure 4. Overall, the values of Q are consistent with the expected geology with a positive correlation between low velocities in the soft sediments and the gas cloud and high attenuating zones ( Operto & Miniussi, 2018).

*Figure 4:* Comparison between recorded and simulated data. The recorded data are plotted with a red/white/blue color scale. The simulated data are superimposed with black variable area. The two sets of synthetic are in phase if the black overprint the red. The simulated data are computed in FWI models when attenuation has not be taken into account during FWI (a) and when attenuation is updated during FWI (b). We show that dispersive waves generated by a shallow wave guide are better matched when attenuation is taken into account (yellow arrow) ( Operto & Miniussi, 2018).