GNGTS 2016 - Atti del 35° Convegno Nazionale

514 GNGTS 2016 S essione 3.1 3. use a MCMC algorithm to extract a sufficiently large set of models from the PPD estimated at the previous step; 4. apply local FWI to each model obtained at step 3; 5. apply a non-parametric method (e.g. the kernel density estimation) to the entire set of final models derived at the previous step. This step gives the final PPD that represents the uncertainties associated with the final Vp models. Note that the second step yields a non-biased PPD which expresses the uncertainties affecting the best-fitting BGAmodel. This method returns the 1D marginal PPD for each model parameter. To mitigate the so-called curse of dimensionality in the GAoptimization, we reduced the number of model parameters by resampling the prior model onto an irregular grid with cell sizes chosen according to seismic resolution criteria, that is, proportional to a quarter of the dominant wavelength for the vertical resolution and proportional to the first Fresnel zone for the horizontal resolution. See Sajeva et al. (2016) for more details. Synthetic example. We apply the method to an acoustic inclusion model similar to the one introduced by Mora (1989). This model is constituted by a spherical homogenous inclusion in a background velocity model characterized by a constant gradient with depth and a deep reflection (see Fig. 1a). For the forward modelling we use the finite-difference method, with accuracy of second order in time and fourth order in space, a vertical and horizontal space step of 48 m, a time step of 4 ms, and a 6 Hz Ricker wavelet as the source signature. The acquisition geometry consists of 31 sources equally spaced at the surface, which illuminate all the evenly spaced 127 receivers at the surface. To evaluate the misfit, we use the L2 norm applied to low- pass filtered (0-3 Hz) and trace-by-trace normalized data. As prior information, we use a simple 1D Vp model with velocity linearly increasing with depth from 1500 to 3000 m/s. This model is used to centre the GA inversion ranges and also to build the irregular GA. The grid and the 1D model are shown in Fig. 1b. This grid has 26 nodes. These nodes are bilinearly interpolated to the finite-difference grid for the forward-modelling. In the inversion, we performed 16k model evaluations and the final best-fitting model is shown in Fig. 2a. This result may be considered a good macro model, since it contains the long-wavelengths of the true model. In addition, we desire to appraise the entire ensemble of GA models to quantify the uncertainty affecting the BGA solution of Fig. 1c. To this end, we employ the GS. Fig. 1d shows some of the resulting 1D marginal PPDs (first row: from left to Fig. 1 – a) The true inclusion model; b) the “starting model” superimposed with the grid nodes; c) the final model after GA; d) the uncertainties associated with some model parameters (first row: at surface from left to right; second row, central line from top to bottom).