GNGTS 2013 - Atti del 32° Convegno Nazionale

The shallowest part of the model is better resolved in either the inversions. This is due to the better illumination of the near surface part of the model with respect to the deeper elements. Figs. 3d and 3e graph the misfit evolution for the NA and GA, respectively. Note the slow rate of convergence of the NA, and a better performance of GA. In the GA inversion, the misfit evolution is different for each subpopulation that explores a different part of the model space, and jumps can be seen in the plot in correspondence of the generations at which the migration algorithm takes place. In particular NA does not show a significant decrease of the misfit with the iterations and this is likely due to an entrapment into a local minimum. Therefore the GA method seems to be characterized by a more efficient exploration of the entire model space. In conclusion, also this synthetic seismic inversion test confirms the previously obtained results by means of the analytical objective functions. Conclusions. In this work we compared two different global optimization methods: Genetic Algorithm and the Neighborhood algorithm. The tests on the convex analytical misfit function highlighted the better performance of the NA technique in case of a low dimensional model space, and, vice versa, a better performance of the GA for higher dimension model spaces. The cross-point at which the number of models needed for convergence for the GA is less than that needed for the NA occurs approximately at a dimension of 16 unknowns (that is for a dimension 16 of the model space). Analogously, the computational time at which GA starts converging faster than NA takes place at dimensions greater than 9. The tests on the multi-minima analytic function showed a more problematic reliability of the NA method, in fact, it resulted that the cross-point moved toward smaller dimensions of the model space (2-3) and that generally GA outperformed NA. Furthermore NA seemed to be subjected to be trapped in local minima if the model space is not adequately sampled by the previous set of models. In order to avoid it, the number of models computed for each iteration, must be set to increase at least linearly with the number of dimensions of the model space. Focusing on the computational time, the inversions on the analytical misfit functionals showed a steeper curve for NA than for GA against the parameter dimensions. The high computational costs of the NA is likely due to the time needed to evaluate the Voronoi cells, which is proportional to the square of the number of evaluated models. Thus, the computational cost of the NA increases dramatically along with the number of dimensions. In comparison, the GA computational time is significantly lower for high dimension model spaces. The synthetic seismic inversions performed with model space of dimension 64, confirmed the results obtained with the analytic functionals, that is, the better performance of GA in case of higher dimensional parameter spaces. The bad performance of NA in terms of convergence toward a good model is likely due to the small values chosen for the free parameters in the test, values that we took from the literature, but that must be probably set to higher values when moving towards higher dimension parameter spaces ( nd =30-60). In conclusion, the global optimization method must be carefully chosen depending on the expected dimension of the model space. Acknowledgements. These results were obtained within a research project with ENI. We thank ENI for the permission to publish this paper. References Fichtner A.; 2011: Full Seismic Waveform Modelling and Inversion. Springer. Geman S. and Geman D.; 1984: Stochastic Relaxation, Gibbs distributions and the Bayesian Restoration of images. IEEE Trans. Patt. Analysis Mach. Int., 6, 721-741. Goldberg D.E.; 1989: Genetic algorithm in search, optimization and machine learning , Kluwer Academic Publishers. Gould S. J. and Eldredge N.; 1977: Punctuated equilibria: the tempo and mode of evolution reconsidered, Paleobiology. 3 (2), 115-151. Holland J.H.; 1975: Adaptation in natural and artificial systems . University of Michigan Press. Horn J.; 1993: Finite Markov Chain Analysis of genetic Algorithms with Niching. Proceedings of the 5 th international Conference on genetic algorithms, 110-117. 65 GNGTS 2013 S essione 3.1