GNGTS 2019 - Atti del 38° Convegno Nazionale

664 GNGTS 2019 S essione 3.2 Among the different numerical approaches employed for the solution of Eq. (1) under piecewise boundary conditions, the Galerkin formulation of the finite-element method has been applied by numerous authors over the past years (e.g.Günther et al. 2006).In this case, the numerical solution of eq. (1) is found as: Φ = [ Y ] –1 I (2) which requires the stiffness matrix Y to be invertible. Whereas 2D conditions apply, Eq. (1) is generally Fourier transformed in the ( x, λ ,z ) space and then the potential back-transformed after solving Eq. (2). Once the potential has been derived, we can compute the predicted data (apparent resistivity ρ a pre ) as (De Donno and Cardarelli, 2017) (3) Being N Q the number of quadrupoles, K the geometric factor, Z q = M p,c φ c with M the measurement operator, c =1,2,…, C the current stimulations and p =1, 2,…, P the potential measurements. Inversion algorithm. In this work, the very fast simulated annealing (VFSA) algorithm (Ingber 1989; Sen and Stoffa2013) is selected for inversion, due to its robustness and the small number of inversion parameters. The algorithm draws random models from a Cauchy-like distribution that is, in turn, a function of temperature. Models that lower energy are always accepted, while models that increase energy are accepted with a finite and temperature-dependent probability. This probability-dependent acceptance criterion, together with an adequate cooling schedule and a sufficient number of random samples, assure the inversion process to converge towards the global minimum of the energy function, which is virtually independent of the initial model (Cercato 2011). In this work, the energy function E is set as the relative root mean square error (RMSE): (4) where ρ a obs is the observed apparent resistivity and m the resistivity model. In order to increase the general stability of the solution, after each simulation step the randomly altered resistivity values are smoothed by a 5-point cross median filter. The median filter is chosen since it does not destroy high-contrast boundaries as an averaging smoother would and it is not affected by isolated spikes that occasionally occur in tomographic images (Weber 2000). The random search is performed only within a closed resistivity interval, which represents the a priori information given to the inversion procedure. Results. We compare the results of VFSA inversion with those achieved by the Gauss- Newton (GN) inversion code embedded into VEMI, using an initial damping value of 0.1 halved at each iteration. Firstly, we present a simple synthetic model (SIN1, Fig. 1a), where a single ERT line of 8 electrodes spaced 1 m apart investigating a two-medium subsoil, conductive in its left part (10 Wm) and resistive in the right one (100 Ωm). The synthetic dataset was carried out using a dipole-dipole configuration with a max =2 and n max =5 (17 measurements) with 1% Gaussian noise added. Both local and global inversion incorporate inequality constraints in the range 5-150 Ωm, starting from a homogenous initial model equal to the mean observed apparent resistivity value. The 10 independent runs of the VFSA algorithm start from an initial temperature of 1 decreased towards 10 -5 in 100’000 random steps with 5 trials for each temperature value. Results are shown in Fig. 1b for GN inversion and in Fig. 1c for VFSA inversion, while the error progress is displayed in Fig. 1d. Both resistivity models converge approximately to the noise level added, even though the VFSA inversion is able to enhance the sharp interface