GNGTS 2017 - 36° Convegno Nazionale

680 GNGTS 2017 S essione 3.2 where k is a geometrical coefficient dependent on the acquisition geometry. Although the filtering procedure significantly contributes to reduce the noise, the amplitude and phase spectra are still characterised by a poor signal to noise ratio at high frequencies. For this reason, the resampling of the linearly-spaced spectra to a logarithmic-spaced one (necessary for the inversion algorithm), was performed by using a sliding Parzen window with variable- width, weighted by the normalised inverse function of the power spectrum of noise. The final amplitude and phase spectra (Figs. 1d and 1e) show a significant improvement of the S/N ratio over the entire frequency range. Inversion method. The full-waveform inversion of IP data (FWI-IP) is a non-linear and ill- conditioned inverse problem that aims to reproduce both the amplitude and the phase spectra of observed IP data. Obviously, the ill-conditioning and the non-linearity of this inverse problem severely increase with the number of model parameters (i.e. passing from a single to a double dispersion model). The FWI-IP is typically performed by applying gradient-based methods in which a regularisation strategy is used to stabilise the problem. In this work, we use Levenberg- Marquardt algorithm as a local optimisation algorithm in which a finite difference approach is used to compute the Jacobian matrix, whereas the standard L-curve (Aster et al., 2005) served us to set the optimal value for the damping parameter. Similarly to other local optimisation methods, the performances of the LM approach are highly dependent on the starting model. If needed the starting model can be also defined by analysing the observed data. Global optimisation methods have demonstrated to be powerful tools to solve non-linear geophysical inverse problems with complex (i.e multiminima, ill-conditioned) misfit functions (Sajeva et al. , 2017). These methods perform a wide exploration of the model space, thus resulting less affected by the choice of the starting model and less prone to get trapped in local minima. The downside of global methods is that their computational cost exponentially increases with the number of unknown model parameters. In this work, we assess the applicability of a global optimisation to solve the FWI-IP. Among the many global methods, we choose the particle swarm optimisation (PSO) which is inspired by the social behaviour of bird flocking or fish schooling (Sajeva et al., 2017). For both the LM and PSO algorithms the objective function is the L2 norm difference between observed and predicted data. To better analyse the ill-conditioning of the inverse problem, residual function maps and SVD decomposition of the Jacobian matrix (Aleardi et al., 2016) are used. In particular, residual function maps provide projections of the objective function onto a 1D, 2D, or 3D space. Inversion tests. We start by describing the residual function map computed on the reference model previously described (Fig. 1a). To make a 3D representation of the 7D residual function map possible, we keep fixed the parameters fixed to their true values, while the other parameters are varied within plausible physical ranges. Note that the objective function of Fig. 2a is characterised by an elongated valley along the and parameters and a single, well-defined, global minimum. For this reason, Fig. 2a proves that the objective function is mainly effected by ill-posedness problems rather than by the presence of multiple minima. In the first test, we consider a single-dispersion model, that is we have four model parameters to determine. To produce statistically significant results, we perform 10 inversion tests for each algorithm. In each test, the starting model for the LM algorithm and the initial generation of birds for the PSO, are randomly generated with uniform distribution within plausible physical ranges of the inverted parameters. Figs. 2b and 2c show the results achieved by the two algorithms for each inversion test. In particular, note that both algorithms provide equivalent results for each inversion test, thus confirming the lack of several local minima in the objective function. From the analysis of the singular values represented in Fig. 2d we can derive a condition number equal approximately to 6 for this single dispersion model, a value that indicates a not-severely ill-conditioned inverse problem. In the second test, we invert for all the seven model parameters (double dispersion model). Similarly to the previous example, we run 10 different inversion tests for each algorithm. It is