GNGTS 2013 - Atti del 32° Convegno Nazionale

Differently, in the NA inversion the small set of free-parameters ( ns and nr ) must be adjusted depending on the dimensions of the model space in order to guarantee convergence. Roughly they must increase along with the dimension of model space proportionally to the number of minima of the misfit functional. As a rule of thumb, we multiplied each value ( ns, nr ) by 6 when adding a new dimension in the inversion, because, in this particular case, the number of minima increases by a factor of 6 when a new dimension is added. Notwithstanding the effort, we noted that the convergence is not assured, even setting ns and nr to high values (e.g., ns =10 3 , nr =10 2 for nd =4), and the method is subjected to fall on local minima. Fig. 2a shows the number of model evaluations needed to ensure convergence for the two methods. The red and blue crosses represent the NA and GA respectively, while the red and blue dotted lines represent a five degree polynomial fit for NA and GA respectively. Note that the same behavior of the first test occurs, i.e., NA is better performing for smaller dimensions while GA is more effective in case of higher dimension model spaces. However, in this case the GA results to be better performing than NA already for very small model space dimensions (around 2 in Fig. 2a) so that the GA implementation results to be the best for almost every dimensions. We stopped the NA inversion at nd =5 due to the high computational cost of the inversion procedure. Fig. 2b shows the computational time of the two different optimization methods, which have been tested on the same hardware of the first test. Note a better performance of the NA method below nd =3, where the cross-point takes place, and then a better behavior of GA for higher dimensions. As we already pointed out, NA inversions appear to need high values for the free- parameters to ensure convergence in case of high dimensional spaces (e.g. ns >10 3 for nd =5). Furthermore, based on the results in these tests, it seems that it is best to choose a nr value, the number of selected models per iteration, at least as big as the number of local minima of the functional. Obviously this information would not be available when performing an actual inversion. Differently, GA seemed to be less sensitive to the specific values of the user defined parameters, and to better escape local minima even with a relatively small population (100 models) per generation. Synthetic seismic inversion. Full-waveform inversion is a relatively novel variant of seismic tomography characterized by the numerical solution of the equations of motions (Fichtner, 2011). We performed our test simulating very low seismic frequencies (<1 Hz) and assuming Fig. 2 – Multi-minima misfit functional: (a) comparison of number of models computed to reach convergence for GA (blue) and NA (red); (b) comparison of runtime occurred to reach convergence for GA (blue) and NA (red). Each experimental dataset in the graphs was fitted with a polynomial of 5 th degree. 63 GNGTS 2013 S essione 3.1