GNGTS 2017 - 36° Convegno Nazionale

GNGTS 2017 S essione 3.2 681 known (Major and Silic, 1981) that the simultaneous estimation of seven model parameters is a severely ill-conditioned problem in case of a frequency spectrum with a single peak as the one shown in Fig. 1d. Figs. 3a and 3b represent the comparison between true and predicted parameters by the LM and PSO algorithms, respectively. Differently from the previous example, note that due to the increased ill-conditioning of the inverse problem, the ten inversion tests converge to different solutions in the model space. The estimates of m 1 , c 1 τ 1 and show the most significant changes and most scattered trends. Differently, the estimations of and show negligible changes for the different tests. If we compare the PSO and LM results we observe that the local method outperforms the global approach as it provides more accurate and stable estimations for the , and parameters. This characteristic could indicate the difficulty of the PSO algorithm to provide stable and accurate solutions in case of a severely ill-conditioned inverse problem. The more stable performances of the LM approach with respect to the PSO algorithm, are also demonstrated by the very similar data misfit values reached by the local method in the ten inversion tests (Fig. 3c). The analysis of the eigenvectors in model space derived from the Jacobian matrix (Fig. 3d), confirms the qualitative conclusions previously drawn from Figs. 3a and 3b. Indeed, the fifth, sixth and seventh eigenvectors (that roughly speaking define the null-space) point toward , and , which are the parameters with highest variations in Figs. 3a and 3b. For this reason, these are the least resolvable parameters. Conversely, the first and second eigenvectors point toward and which resulted the best resolved model parameters. By the analysis of the singular values represented in Fig. 3e we derive in this case a condition number equal to . This value indicates a highly ill-conditioned inverse problem characterised by a wide equivalence region in the model space (Martinez et al., 2012). Conclusion. In this work, we tested and compared a local and a global optimisation method to solve the full-waveform inversion of IP data (FWI-IP). We adopted the Levenberg-Marquartd (LM) algorithm for the local optimisation and the particle swarm optimisation (PSO) for the global inversion. The synthetic tests showed that the two algorithms achieve very similar and stable results when a single dispersion model is assumed. Differently, in case of a double dispersion model, the LM approach provided more accurate and stable predictions than the PSO Fig. 2 - a) Residual function map. The white arrow indicates the global minimum. Maximum residual and minimum residual are indicated respectively in yellow and blue. b) Estimated CCP parameters by ten different LM inversions by assuming a single dispersion model. c) As b) but for the PSO algorithm. In c) and b) the 10 inversion tests are represented by different colours. d) Singular values of the Jacobian matrix in case of a single dispersion model.