139 5. Full waveform inversion In practice, a pre-processing step should be applied to only select reflected waves from the observed data. We ran LSM in the same initial model as for the second FWI scheme (Figure 5.6, top). Here, the velocity perturbation is updated, not the initial macro-model. It means that the Green’s functions are always computed in the same model. They cannot generally be saved in memory as their size depends on the number of sources, the spatial size of the model and the number of time samples. The final result is an oscillatory signal (Figure 5.11), here expressed in terms of velocity perturbations. We used a quantitative LSM approach, but migration algorithms are often qualitative, for a structural interpretation. Here, with the quantitative approach, it is possible to sum the macro-model (initial model, Figure 5.6, top) and the velocity perturbation, yielding a very similar result as the one provided by FWI. The reason for this is that most of the information is contained in the reflected energy, and not in the transmitted waves already explained by the initial model. Sensitivities We tested the sensitivity of FWI with respect to the • acquisition geometry; • choice of the model parameter to be inverted. We repeated the same process as in Figure 5.6, except that we selected a source every 5 m, instead of every 1 m. The final FWI is similar (Figure 5.12) and the differences are localized in the shallow part, around 1 to 2 m depth, every 5 m along the x axis. These zones with higher (white) velocities are the imprint of the acquisition design: for datasets that are too sparse, the model cannot be properly reconstructed because of aliasing effects (Gray, 2013). These effects would have been even stronger if a mask had not been applied to prevent any updates in the first meter. Figure 5.12 Same as for Figure 5.4, bottom, but for sources every 5 m instead of every 1 m as in all other examples. The differences are mainly visible in the shallow part. In the previous example, the velocity was updated, while the density model remained fixed, even if the exact Earth (Figure 5.3) contained both velocity and density variations. We discussed the fact that the best reconstruction was obtained by the impedance (product of velocity by density). Here, we ran FWI for a fixed
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