Hierarchical Strategy for Full Waveform Inversion in the Frequency Domain

Full waveform inversion (FWI) of seismic traces recorded at the free surface allows the reconstruction of the physical parameters structure on the underlying medium. Our two main objectives are the reconstruction of multiple classes of parameters on one side and the formulation of both the acoustic and elastic FWI for 3D geometries. A quasi-Newtonian method with a preconditioned L-BFGS algorithm provides scaled gradients of the misfit function for each class of parameter. For onshore applications where body waves and surface waves are jointly inverted, P- and S-wave velocities (VP and VS) must be reconstructed simultaneously using a hierarchical inversion algorithm with two nested levels of data preconditioning. The first one concerns the frequency sampling from low to high frequencies during the inversion procedure. The second one performs a data preconditioning by an exponential decay after the first arrival time. Simultaneous inversion of multiple frequencies rather than successive inversions of single frequencies significantly increases the S/N ratio of the models. For offshore applications where VS can have a minor footprint in the data, a hierarchical approach which first reconstructs VP in the acoustic approximation from the hydrophone component followed by the joint reconstruction of VP and VS from the geophone components can be the approach of choice. Among all the possible minimization criteria, we found that the L1 norm provides the most robust and easy-to-tune criterion as expected for this norm. In particular, successfull reconstruction of VP and VS on a realistic synthetic offshore case study is possible when white noise with outliers has been added to the data. The feasibility of 3D FWI is highly dependent on the efficiency of the seismic modelling. Frequency domain modelling based on direct solver allows one to tackle small-scale problems involving few millions of unknowns at low frequencies. If the seismic modelling engine embeds expensive sourcedependent tasks, source encoding can be used to mitigate the computational burden of multiple-source modelling. However, the source encoding is very sensitive to noise in the framework of efficient frequency-domain FWI where a limited number of frequencies is inverted sequentially. Time-domain modelling for the estimation of harmonic components of the solution is an alternative of choice even for 3D frequency-domain FWI because it allows one to extract an arbitrary number of frequencies at a minimum extra cost, a procedure useful when multiple frequencies are inverted together.