GNGTS 2017 - 36° Convegno Nazionale

GNGTS 2017 S essione 3.3 713 conducted on a mesh with the layer thickness increasing logarithmically with depth, so that the problem unknowns were the model resistivities to be predicted for each layer. The PSO algorithm was iterated 200 times and the procedure was repeated for 11 trials in order to check the solution variability. The swarm was composed of 200 particles and the number of layers, i.e. of unknowns, was 25. The Lagrange multiplier λ was set to 0.001. Cognitive and social accelerations were constant for the whole optimization and, respectively, equal to 1 and 1.5. Fig. 1a shows how observed data and the PSO response are largely comparable for both apparent resistivity and phase. The validity of the PSO solution agrees with the inversion model in Jones and Hutton (1979) and reported in Fig. 1b as a benchmark result. PSO application to 2D MT data. The synthetic model tested for the 2D inversion was composed of a conductive anomaly of 10 Ωm, buried within a host medium of 100 Ωm, below an array of 15 MT stations, as shown in Fig. 2a. The forward modelling was run in a frequency range between 0.01 and 1000 Hz and on a big resistivity matrix in order to minimize any conductivity influence on the boundaries. Synthetic data were contaminated with uncorrelated noise of 10% drawn from a normal distribution. If a priori information derived from PSO 1D is given to the 5% of particles as their initial position, after 400 iterations in a 3000-particle swarm, we obtained a result that is quite consistent with the synthetic model (see Fig. 2b). The Lagrange multipliers ( λ x,z ) were fixed at 0.01. The fitting between synthetic and calculated data, depicted in Fig. 2c, was remarkable both for TE and TM polarizations. Final RMS errors were: RMS ρTE = 6.3; RMS phaseTE = 0.6; RMS ρTM = 6.9; RMS phaseTM = 1.4. The total runtime was 4.4 hours. When there was no initial conditioning of the swarm and the number of unknowns increased (957), after 400 iterations in a 10000-particle swarm, the result was slightly dissimilar from the true model, but it is evident from Fig. 3a that the anomaly was correctly located. The Lagrange multipliers ( λ x,z ) were fixed at 0.1. Fig. 3b shows that the synthetic and predicted data are in good agreement, which is a positive outcome considering the high number of variables and the problem of non-uniqueness of solutions in inverse problems. Final RMS errors were: RMS ρTE = 8.8; RMS phaseTE = 2.4; RMS ρTM = 10.7; RMS phaseTM = 2.8. The total runtime was 13 hours. Conclusion. The PSO algorithm has proven to be a valid method to solve MT inversion problems, both for 1D and 2D. The stochastic nature of PSO and the combination of exploration and exploitation behaviors to find the optimum solution allowed significant outcomes to be reached. The results presented in this work demonstrate that, if some a priori information is available, a small portion of the swarm can be initialized giving robust models. In any case, initial conditioning is not necessarily required for the achievement of appreciable results. A further innovation of this study was the introduction of parallel computing. This yielded a twofold advantage: computation time savings and efficiency in managing a large amount of data. Future developments will be the PSO application to 2D field data and the comparison and validation of these results with results from well-established algorithms (e.g. conjugate gradient). Acknowledgements Computational resources provided by hpc@polito (http://hpc.polito.it) . References Candansayar, M. E.; 2008: Two-dimensional inversion of magnetotelluric data with consecutive use of conjugate gradient and least-squares solution with singular value decomposition algorithms. Geophysical Prospecting, 56 (1), 141-157. deGroot-Hedlin, C. and Constable, S.; 1990: Occam’s inversion to generate smooth, two-dimensional models from magnetotelluric data. Geophysics, 55 (12),1613-1624. Ebbesen, S., Kiwitz, P. and Guzzella, L.; 2012: A generic Particle Swarm Optimization Matlab function. Proceedings of the American Control Conference, Montreal, 1519-1524. Engelbrecht, A. P.; 2005: Fundamentals of computational swarm intelligence. John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester , England. Everett, M.E. and Schultz, A.; 1993: Two-dimensional nonlinear magnetotelluric inversion using a genetic algorithm. Journal of Geomagnetism and Geoelectricity, 45 (9), 1013-1026.