135 5. Full waveform inversion and more importantly to equalize the contributions with depth (at the first iteration, only the very shallow part is updated). Note that only the P-wave model was updated: the density model remained unchanged, even if the exact density model differs from the initial model (Figures 5.1 and 5.2, bottom). After 30 non-linear iterations, the final model displays some interesting structures, for example around x = 27 m (Figure 5.5). We superimposed the main elements in a dashed black line extracted from the exact model (Figure 5.3). But this final model does not display homogeneous velocities within the cemented structures. The bedrock is not horizontal and its depth is slightly underestimated in the central part. In fact, the FWI algorithm converges to a local minimum as the initial model (Figure 5.4) is too far from the correct one for the frequency range of the data. Figure 5.5 Inverted vp model (m/s) after 30 iterations, with fmin = 30 Hz and fmax = 300 Hz. The density model is not updated. We thus developed the second strategy (B), consisting of two steps: (1) 15 non-linear inversions with frequencies between 30 and 60 Hz, and then (2) 30 non-linear inversions with frequencies between 30 and 300 Hz as before. The intermediate model after 15 iterations is smooth and contains higher velocities in the cemented structure, even if the limit is not clear (Figure 5.6, top). From that result, we increased the frequency range. The final model (Figure 5.6, bottom) provides a very satisfactory result. The velocity model is much more homogeneous and the main interfaces are correctly positioned. Also, the low (blue) velocity anomaly around x = 45 m and z = 5 m is well retrieved. Note that some oscillations were created to compensate for density contrasts as the density model remains unchanged. This is clearly visible on vertical sections extracted from the inverted and exact models (Figure 5.7). There is a good agreement between the inverted (blue) and exact (red) velocity models (Figure 5.7, left) and a quantitative match between the impedance models ρvp (Figure 5.7, right), where vp is the inverted model and ρ the initial model. For example, for x = 30 m, the velocity jump at 1 m depth is overestimated in vp and correctly determined in ρvp. Here, the image mainly comes from the analysis of reflected waves; this is why the impedance section is preferred compared to the velocity section.
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