134 Seismic Imaging For simplicity, we have used acoustic modelling. With elasticity, the presence of surface waves largely complicates the imaging strategy (see below in the discussion part) (Brossier et al., 2009). The modelling is based on a staggered grid strategy with a velocity-stress formulation (Virieux, 1986). The particle velocity is converted at the surface into a pressure wave field. The acquisition geometry consists of regular shots every 1 m along the profile and receivers at ±10 m around the source position, knowing that the maximum depth to be investigated is at around 10 m. It means that in this configuration, the interesting information is contained in the reflected waves and not in the transmitted waves for which longer offsets would be needed (see Chapters 2 and 3 on “refraction surface” and “seismic tomography”). Inversion results For the definition of the initial model (here the vp and density models), we used a simple gradient model, where the parameters linearly increase with depth (Figure 5.4). This model is consistent with a smooth version of the exact model on the left or right parts, but differs in the central part (Figure 5.3). The only modification brought to this gradient model was the introduction of the shallow layer (first 50 cm) with a dipping interface around x = 12 and 53 m (Figure 5.4). Figure 5.4 Initial vp (top, in m/s) and density (bottom, in kg/m3) models. In the first strategy (A), the minimum and maximum data frequencies were 30 and 300 Hz, respectively. All frequencies were inverted simultaneously. Note that the central frequency was about 120 Hz. The non-linear minimization was obtained with a standard quasi-Newton scheme (L-BFGS, (Nocedal, 1980)), for which the last 5 iterations are used to build the inverse of the Hessian to speed up convergence
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