Fig 1.
Forward and inverse problems and misfit function.
The FWI iterative process solves the inverse seismic problem by successively applying the direct seismic problem, fitting the model to the data, until the misfit is minimal.
Fig 2.
Uniform mesh of nx points in 1D for the FDM scheme.
The image shows the stencil and points where the wave (PDE) and the boundary conditions (BC) equations are applied.
Fig 3.
Exploration versus exploitation.
(a) The three vectors associated with updating the velocity of particle in the original PSO—inertia, cognitive and social learning terms and (b) Balancing between exploration and exploitation mechanisms.
Table 1.
The initial and final values for each PSO parameter in all application.
Value range for w, C1, C2, rrp and τ used in this research.
Fig 4.
Movements defined in the original Nelder-Mead Simplex.
(a) Reflection of X3, given by vertex Xr, (b) Expansion, represented by vertex Xe, (c) Outside contraction, given by vertex Xc, (d) Inside contraction, defined by vertex Xcc and (e) shrinkage, represented by a smaller triangle of vertices. Figure adapted from: http://www.scholarpedia.org/article/Nelder-Mead_algorithm.
Fig 5.
Schematic visualization of the work of the K-means algorithm.
The data is in black. The stars mark the centroids. The clusters are represented by the colors blue, green and red.
Fig 6.
Examples of the Hilbert curves.
(a) 3D Hilbert curve and (b) 2D Hilbert curve and particles randomly distributed in regions.
Fig 7.
Details of the optimization process of the PSO-Kmeans-ANMS algorithm in Phase 1 for the stopping criterion based on the ratio between the sizes of the clusters.
Fig 8.
Details of the optimization process of the PSO-Kmeans-ANMS algorithm in Phase 2.
Table 2.
Results of the PSO-Kmeans-ANMS algorithm for the Rosenbrock function.
The number of objective function evaluations in Phases 1 and 2 corresponds, respectively, to 1008 and 40.
Fig 9.
Details of the hybrid optimization process in phase 1 for the stopping criterion based on the relationship between the standard deviations of the objective function values.
Fig 10.
Details of the optimization process of the PSO-Kmeans-ANMS algorithm in Phase 1 for the stopping criterion based on the ratio between the sizes of the clusters.
Fig 11.
Details of the optimization process of the PSO-Kmeans-ANMS algorithm in Phase 2.
Table 3.
Results of the PSO-Kmeans-ANMS algorithm for the Rastrigin function.
The number of objective function evaluations in Phases 1 and 2 corresponds, respectively, to 1008 and 36.
Table 4.
Population versus success rate for the Rosenbrock function.
Percentage success rate sr(%) for each algorithm referring to swarm sizes with 8, 12, 20, 28 and 36 particles.
Table 5.
Population versus success rate for the Rastrigin function.
Percentage success rate sr(%) for each algorithm referring to swarm sizes with 8, 12, 20, 28 and 36 particles.
Fig 12.
Comparison between simulations for all algorithms (success rate versus number of swarm particles).
(a) Rosenbrock function and (b) Rastrigin function.
Fig 13.
Simulations details for the Rosenbrock function.
The 100 results obtained by the algorithms ANMS, PSO classic, PSO mod and PSO-Kmeans-ANMS for the swarms with 08 (upper) and 36 (lower) particles, respectively.
Fig 14.
Simulations details for the Rastrigin function.
The 100 results obtained by the algorithms ANMS, PSO classic, PSO mod and PSO-Kmeans-ANMS for the swarms with 08 (upper) and 36 (lower) particles, respectively.
Fig 15.
The solid black line is the true model, where the step indicates the position of the reflector. The dashed gray lines delimit the overall search space for the model parameters.
Table 6.
True model parameter and constraints.
The exact values of the parameters V1, V2 and hrf for true model and their lower (Llow) and upper (Lup) limits.
Fig 16.
Wavefield propagation and observed data.
(a) Evolution of wavefield propagation u(x, t) and (b) The observed data dobs or synthetic seismic trace recorded at x = 0.15.
Fig 17.
(a) The jumping moment from Phase 1 (PSO-Kmeans) for Phase 2 (ANMS). Represented by the color green (initial swarm), blue (final swarm), yellow (best solution for PSO) and red (optimal for ANMS) and (b) Details of Cluster 1 (empty blue circles), Cluster 2 (filled blue circles) and of the construction of the initial Simplex (black tetrahedron).
Fig 18.
(a) Diameters of the initial and final swarms and (b) Run time of Phases 1 and 2.
Table 7.
Results of the success rate (sr) and average execution time (rt) of each optimization algorithm.
Table 8.
Results of model parameters and objective function (misfit) for the best solution obtained in each optimization algorithm.
Table 9.
Results of the mean value and the respective standard deviation for the parameters of the successful models.
Table 10.
Results of the mean value and the respective standard deviation for the CPU time and the objective function.
Fig 19.
(a), (b), and (c) are all results obtained by the ANMS algorithm for variables V1, V2, and hrf, respectively. Successful models are highlighted in blue and unsuccessful ones are in red. (d), (e), and (f) are their respective histograms.
Fig 20.
(a), (b), and (c) are all results obtained by the PSO classic algorithm for variables V1, V2, and hrf, respectively. (d), (e), and (f) are their respective histograms.
Fig 21.
(a), (b), and (c) are all results obtained by the PSO mod algorithm for variables V1, V2, and hrf, respectively. (d), (e), and (f) are their respective histograms.
Fig 22.
(a), (b), and (c) are all results obtained by the PSO-Kmeans-ANMS algorithm for variables V1, V2, and hrf, respectively. (d), (e), and (f) are their respective histograms.
Fig 23.
Histograms of the CPU time spent by the four optimization algorithms, ANMS, PSO classic, PSO mod, and PSO-Kmeans-ANMS, respectively. Only applied to successful models.
Fig 24.
(a) Comparison between algorithms divided into: Class 1, success rates; Class 2, average CPU times and Class 3, average objective function values (ϕ(m) × 102). (b) Objective function values for successful models.
Table 11.
Results of the success rate (sr) and the average execution time (rt) for each optimization algorithm.
Table 12.
Results of model parameters and the respective objective function (misfit) for the best solutions obtained in each optimization algorithm.
Fig 25.
(a) Comparison between algorithms divided into: Class 1, success rates; Class 2, average CPU times and Class 3, average objective function values (ϕ(m) × 102). (b) Objective function values for successful models.
Table 13.
Results of the success rate (sr) and the average execution time (rt) for each optimization algorithm.
Table 14.
Results of model parameters and the respective objective function (misfit) for the best solutions obtained in each optimization algorithm.
Fig 26.
(a) Comparison between algorithms divided into: Class 1, success rates; Class 2, average CPU times and Class 3, average objective function values (ϕ(m) × 102). (b) Objective function values for successful models.
Fig 27.
Comparison between simulations for all algorithms and for all cases (1, 2 and 3).
(a) success rate, sr(%) versus population and (b) average execution time, rt(s) versus population for the ANMS, PSO classic, PSO mod and PSO-Kmeans-ANMS algorithms.