Massively parallel frequencydomain FWI (FFWI code)
We implemented a
3D massively parallel frequencydomain FWI code (called
FFWI ) with the goal to perform the first application of frequencydomain FWI on a real industrial OBC dataset when the forward problem is solved with
sparse direct solver . Frequencydomain FWI is more specifically suitable for stationaryrecording acquisitions such as OBC or OBN because they provide a broad angular illumination of the subsurface which allows us to limit FWI to a limited number of frequencies (see the resolution analysis of FWI in the tutorial page). In the frequency domain, wavefield simulation is a boundary value problem which requires to solve a large and sparse linear system with multiple righthand sides. Two categories of linear algebra methods can be used:
direct or
iterative methods.The advantage of direct methods is to give the solution in a finite number of operations (this is particularly beneficial for illconditioned Helmholtz problems for which iterative solvers need to be equipped with efficient preconditioner) and the efficient processing of multiple righthand sides (RHSs) by forward/backward substitution. The drawback is to involve a RHSindependent "preprocessing" step (the LU decomposition) which is memory demanding and computationally expensive. However, the time complexity of one LU decomposition \((O(N^6))\) (namely, for one frequency) is the same as that of an explicit time stepping method for a dense surface seismic acquisition, where \(N\) stands for the number of points along one dimension of a \(N^3\) grid and the number of sources scales to \(O(N^2)\). Moreover, the memory demand of one LU decomposition \((O(N^4))\) is one order of magnitude smaller than the memory storage of a timedomain dataset \((O(N^5))\). Accordingly, there was no obvious theoretical reason to prevent the assessment of this technology for 3D stationaryrecording acquisition such as OBC or OBN, in particular considering recent advances carried out by the community developing sparse direct solver to decrease the complexity of the problem with lowrank compression strategies (e.g.,
Amestoy et al., 2016).
Check the
documentation of the FFWI code (PDF) .
Figures 14 illustrate some results of 3D viscoacoustic VTI frequencydomain FWI on Ocean Bottom Cable (OBC) data collected in the North Sea. Both seismic modelling and inversion have been performed in the frequency domain with
FFWI . The linear system resulting from the discretization of the timeharmonic wave equation has been solved with the finitedifference method of
Operto et al., 2014 and 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 frugal use of computational resources may sound counter intuitive as sparse direct solvers are considered as being computionally expensive 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 frequencydomain FWI. (Left): Pwave velocity model of the subsurface across a lowvelocity gas cloud. (Right): A 10Hz monochromatic wavefield computed by solving the Helmholtz equation with the sparse multifrontal direct solver MUMPS is superimposed. A total of 685 computer cores are typically used to perform such simulations and FWI at 10Hz. ( 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 Pwave velocity models of the oil field. (Left) Velocity model obtained by reflection traveltime tomography. This model was used as starting model for FWI. (Right) Velocity model obtained by frequencydomain FWI. (ac) the slices crosscut (a) sand channel deposits at 175m depth, (b) scarves left by drifting icebergs on the paleo sea bed at 500m depth, and (c) a gas cloud at 1km depth. The two bottom panels show vertical sections across (d) the gas cloud and (e) 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 frequencydomain 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 synthetic seismograms have been simulated with the same frequencydomain modeling engine as that used to perform FWI before inverse Fourier transform. 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 (a) attenuation has not be taken into account during FWI and (b) attenuation is updated during FWI. 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).
We recently applied the FFWI code to an OBN dataset from the Gorgon field, NW continental shelf, Australia. Some preliminary results are shown in Figure XXX.
References:
 S. Operto, P. Amestoy, H. Aghamiry, S. Beller, A. Buttari, L. Combe, V. Dolean, M. Gerest, G. Guo, P. Jolivet, J.Y. L'Excellent, F. Mamfoumbi, T. Mary, C. Puglisi, A. Ribodetti, and P. H. Tournier, Is 3D frequencydomain FWI of fullazimuth/longoffset OBN data feasible? The Gorgondata FWI case study. The Leading Edge (special section on Full Waveform Inversion), March 2023.
 S. Operto and A. Miniussi, On the role of density and attenuation in 3D multiparameter viscoacoustic VTI frequencydomain FWI: an OBC case study from the North Sea, Geophysical Journal International, 213(3), 20372059, 2018, https://doi.org/10.1093/gji/ggy103.
 P. Amestoy, R. Brossier, A. Buttari, J.Y. L'Excellent, T. Mary, L. Métivier, A. Miniussi and S. Operto. Fast 3D frequencydomain full waveform inversion with a parallel Block LowRank multifrontal direct solver: application to OBC data from the North Sea, Geophysics, 81(6), pages R363  R383, 2016, https://doi.org/10.1190/geo20160052.1.
 S. Operto, A. Miniussi, R. Brossier, L. Combe, L. Métivier, V. Monteiller, A. Ribodetti, and J. Virieux. Efficient 3D frequencydomain monoparameter fullwaveform inversion of oceanbottom cable data: application to Valhall in the viscoacoustic vertical transverse isotropic approximation. Geophysical Journal International, 202(2):13621391, 2015, https://doi.org/10.1093/gji/ggv226.

Is 3D frequencydomain FWI of fullazimuth/longoffset OBN data feasible? The Gorgon data FWI case study
S. Operto, P. Amestoy, H. Aghamiry, S. Beller, A. Buttari, L. Combe, V. Dolean, M. Gerest, G. Guo, P. Jolivet, J.Y. L'Excellent, F. Mamfoumbi, T. Mary, C. Puglisi, A. Ribodetti, and P.H. Tournier
The Leading Edge 2023 42:3, 173183