GNGTS 2017 - 36° Convegno Nazionale

GNGTS 2017 S essione 3.3 721 Fig. 2a shows the 2-D Schwefel function computed within [500, 500] n . Note that this function is characterized by a much more complex topology than the Rastrigin function. In particular, in the Schwefel function the local minima are more irregularly distributed, and important local minima are distant from the non-centred global minimum (located at [420.9687, …,420.9687]), or are even located at the opposite edge of the model space. In the following tests, we consider a 4-D space, a population of 100 individuals divided in 4 subpopulations, and a maximum number of generations equal to 800. The final results are represented in Fig. 2b. Note that none of the MCA tests satisfies the convergence criterion, and that for the NGA, only 22 out of 50 tests converge. Differently, the implemented DAGA approach converges for all the tests. From the one hand, these results confirm that the GA approach is strongly affected by genetic drift in case of irregularly distributed minima. From the other hand, Fig. 2b proves that the DAGA approach effectively attenuates the genetic drift and is able to find the global minimum even in case of objective functions with very complex topology. For this test, the average convergence time for SGA and DAGA is 1.76 s and 3.86 s, respectively. Residual statics correction. The residual statics correction is a highly non-linear optimization problem that is often solved by applying global optimization methods (Rothman, 1985). In the following test, we compare the NGA and DAGA methods on CMP-consistent residual statics correction performed on a synthetic CMP gather. Sajeva et al. (2017) showed that this geophysical optimization problem is characterized by an objective function that shows some similarities to both the Rastrigin and Schwefel functions. For this reason, the GA encounters some problems in finding the global minimum in case of high-dimensional model spaces. In this case we use actual well log information and a convolutional forward modelling to generate the reference CMP gather (without residual statics; Fig. 3a). To simulate residual statics in the data, we apply to each trace in the reference CMP random time shifts uniformly distributed over the range - 15/+15 ms (Fig. 3b). In the subsequent optimization process we allow time shifts within the range -25/+25 ms. In this case the model space dimensionality is equal to 40 (i.e. we consider 40 seismic traces), while the total number of individual is 200 divided into two subpopulations. Figs. 3c and 3d show the inversions results for the NGA and DAGA approaches, respectively. We note that the DAGA method returns a final corrected CMP gather (Fig. 3c) very similar to the reference one, whereas the final CMP yielded by the NGA approach shows many misalignments of the reflections and cycle-skipped traces (Fig. 3d). In Fig. 3e, we also observe that the Fig. 3 - a) The synthetic reference CMP gather, b) the trace-shifted CMP, c) the final DAGA CMP, d) the final NGA CMP. e) Energy curves representing the evolution of the energy of the stack trace over iterations.