GNGTS 2018 - 37° Convegno Nazionale

724 GNGTS 2018 S essione 3.3 distributed, and other local minima are distant from the non-centred global minimum (located at [420.9687, …,420.9687] n ), or are even located at the opposite edge of the model space. For this function the initial ensemble of candidate solutions in each method is 100 times the model space dimension, whereas the accuracy is set at 1. Fig. 1b shows that in this case all the algorithms have a 100% convergence probability for a 2D case, whereas this percentage progressively decreases as the number of dimensions increases. The ICA is characterized by the most significant decrease of this percentage as the number of unknowns increases, whereas the FA is still able to successfully converge in 15 out of 20 tests for a 10-D model space. Note, as expected, that for a given model space dimension the number of model evaluations requested to converge in the Schwefel function is much higher than in the Rastrigin function. In this case, both ICA and QPSO require a number of model evaluations that is always one order of magnitude higher than that requested by FA. In other words, in this test the FA clearly outperforms the other two methods because it exhibits a faster convergence rate and successfully identifies the global minimum even in high dimensional model space. Residual statics corrections. In the following test we compare FA, ICA and QPSO on CMP-consistent residual statics corrections performed on a synthetic CMP gather derived from actual well log information by employing a 1D convolutional forward modelling. Sajeva et al. (2017) showed that this geophysical optimization problem is characterized by an objective function with some similarities to both the Rastrigin and Schwefel functions. To simulate residual statics in the data, we apply to each trace in the reference CMP gaussian-distributed random time shifts uniformly distributed over ±10 ms. In the subsequent optimization process we allow time shifts within the range ±15 ms. Similarly, to the analytic objective function examples we test the three methods for different model space dimensions, that is for 20-, 40- and 60-trace CMP gathers. In this example, the ensemble of solutions for each method is 20 Fig. 1 - Results for the Rastrigin and Schwefel functions (a, and b, respectively). Left: percentage of successful tests for different model space dimensions. Right: Average number of model evaluations requested to converge toward the global minimum within the selected accuracy.