GNGTS 2024 - Atti del 42° Convegno Nazionale

Session 3.2 GNGTS 2024 where |t CG | 68 is the sixty-eighth percentle (a standard deviaton threshold) of cross-gradient’s absolute value distributon and the previous rato is calculated for each cell of the mesh. This term, ploted between 0 and 10 compressing the variaton between 1 and 10 showed the ability to well represent any case, overcoming the botleneck of the unchanged cross-gradient visualizaton. As an auxiliary tool we applied a fuzzy c-means (FCM) algorithm (Bezdek, 1981) to both individually and jointly inverted models yielding quanttatve integrated cross-sectons which resemble both models. FCM minimizes within-cluster variances (squared Euclidean distances) through an iteratve process that assigns points to clusters in a probabilistc way (Paasche et al., 2010) reducing the objectve functon: In Eq.4 n is the number of data points, c the number of clusters, u ij the degree of membership of data point d j to cluster i defned by its center v i and f is the weightng exponent, set as 2. Finally, the output of fuzzy clustering is arranged in a unique plot developed by modifying the approach afer Paasche et al. (2010), where primary colors denote diferent clusters while color saturaton is proportonal to the degree of membership to the assigned cluster. Synthetc example The synthetc example (Fig.1a) simulates a road embankment, in which a surface layer (ρ = 10 Ωm, v P = 600 m/s) overlays an harder stabilising high-resistvity base (ρ = 1000 Ωm, v P = 1800 m/s), both lying over a lower-resistvity and stfer bedrock (ρ = 100 Ωm, v P = 3000 m/s). Synthetc ERT dataset was generated by using 48 electrodes spaced 0.5 m apart in a dipole-dipole array with a maximum dipole length a max =5 and a maximum dipole separaton n max =6. The resultng 945 apparent resistvity data points were contaminated with a zero-mean 3% Gaussian error. Analogously, for SRT geophones were located at the same electrode positons, simulatng a shot every two geophones. A zero-mean Gaussian noise having a 0.1 ms standard deviaton (median travel-tme is about 5 ms) was added to the 1128 traveltme measurements. Results from separate inversions are shown in Fig.1b. The surface between superfcial and intermediate layers, as well as the lateral extent of the later are well defned in resistvity secton, while the transiton from the base to the bedrock is not well reproduced. Structural joint inversion (Fig.1c) highly improves the resistvity model, so that the base is properly placed where modelled with a signifcant upgrading in vertcal reconstructon; jointly inverted velocity model exhibits an enhancement in a sharper transiton from the superfcial layer to the deeper ones. The NCG values (Fig.1d) from individual inversions are generally higher compared to those from the joint inversion, partcularly in the area below the middle layer and in the transiton between surface layer and bedrock. Finally, cluster analysis (Fig.1e) from jointly inverted models shows a beter reconstructon, especially for the middle cluster (cluster no. 3) which is properly reconstructed within its real boundaries, and the botom (4) J = c ∑ i =1 n ∑ j =1 ( u ij ) f d j − v i 2