En un bolsillo, Mariana guarda p cantidad de dinero y en el otro, el doble

The Non-Dominated Sorting Genetic Algorithm (NSGA-II, Deb et al. 2002) is a specific version of a general genetic algorithm. In general, genetic algorithms are iterative stochastic searching mechanisms that evaluate progressively improved mathematical models using a learning mechanism to improve the efficiency and accuracy of the search mechanism (Deb et al. 2002). The learning mechanism involves at each iteration an evaluation of the quality of the current proposed velocity models and the previous velocity models. The best models are kept and the lesser quality models are discarded. In NSGA–II the solution to a multi-objective problem is a set of solutions which are better than the rest of the solutions in all dimensions of the search space. A solution is called dominant if each objective value is smaller than its counterpart. The best solutions are known as non-dominated solutions or Pareto-optimal solutions (fig. 4-4) (Deb et al. 2002). As all non- dominated solutions are good ones, any single solution from the non-dominated solutions is satisfactory and acceptable (Deb et al. 2002). The process of sorting solutions is based on their degree of non-dominance in population, referred to as ranking or fitness function. Each solution is assigned a fitness or rank equal to its non-dominated level (1 is the best level, 2 is the next best level, and so on). Rank 1 solutions are non-dominated if Rank 1 solutions are removed from the dataset, then Rank 2 solutions are the non-dominated set of the remainder. The ranking continues until all data points are ranked (fig. 4-5).

The first step in the NSGA-II procedure is to generate an initial population, in our case a group of velocity models P°. Each initial model 𝑋_𝑖(𝑡), 𝑖 = 1, … , µ is then utilized to produce synthetic waveforms. These are compared to the observed seismograms using a fitness or objective function as shown in equation 4-1 below

𝑓_𝑘(𝑋_𝑖(𝑡)) = ∑ |𝑔(𝑋_𝑖(𝑡)) − 𝑔(𝑌𝑜𝑏𝑠_{)| , 𝑘 = 1, … . 𝑛, 𝑖 = 1, … , 𝑛, ……. (4-1)} Where 𝑓_𝑘(𝑋_𝑖(𝑡)) is fitness objective, 𝑔(𝑋_𝑖(𝑡)) is the synthetic waveform, 𝑔(𝑌𝑜𝑏𝑠_{) is the observed waveform and 𝑛 is the total number of objectives, which} for this study is 4. The initial population is a set of random velocity models. Each

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iteration (or generation) within this genetic algorithm begins with a population of velocity models or generation. Each velocity model in each generation is evaluated to assess how closely it corresponds to the observed seismogram through the fitness function. Then fitness values are compared and sorted using the Pareto optimal solution approach, where the best velocity models are non-dominated.

These models are then utilized to obtain the next iteration of velocity models through the process of recombination and mutation. The recombination process involves the random selection of a proportion (𝑃𝑐) of pairs of models to produce the next generation, through weighted averages biased toward the better solutions. For instance at iteration t, consider two velocity models; 𝑋_𝑖(𝑡)𝑎𝑛𝑑 𝑋_𝑖′

(𝑡)

. If 𝑥𝑖𝑗 is a velocity at the 𝑗𝑡ℎ layer on the 𝑖𝑡ℎ model, then through recombination, two new velocity models are calculated from the models and the velocity at each layer is estimated by equations 4-2 and 4-3,

𝑥_𝑖𝑗(𝑡+1) = (𝑥𝑖𝑗 (𝑡)_+𝑥 𝑖′𝑗 (𝑡)₎ 2 + 𝛼 𝛽 _{………... (4-2)} 𝑥_𝑖′_𝑗 (𝑡+1) ₌ (𝑥𝑖𝑗(𝑡)+𝑥_𝑖′𝑗(𝑡)) 2 − 𝛼 𝛽_{……… ……….. (4-3)}

where 𝛼𝛽 is a random weighted distance from the closest range end point. The obtained result is a model in the range of the two models and is considered a weighted average of the prior models.

A mutation is a process of perturbing the new generation models to avoid becoming trapped in local minima, and at the same time helps to find new regions in the model space. A proportion of models (𝑃𝑚) are chosen to be mutated. Each layer in the selected model has a perturbed velocity. At this point all models from adjacent generations are evaluated and sorted, and the best used to initiate the next generation.

Part of the evaluation performed during the iteration is to check the models against geological constraints. We have applied constraints to improve the modelling in terms of reducing the trade-off between number of model parameters, and lessening the probability of converging to a local minima and raising the convergence rate by considering only part of the model space. To account for and

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decrease these effects, we allowed variation in model parameters within prescribed limits. Although these trade-offs can give rise to several local minima, the genetic algorithm is capable of identifying the global minimum. We constrained errors in modelled Pn arrival times to less than one second, and the average crustal velocity to values between between 6.4 and 6.8 km/s. We also only permitted models where velocity increased with depth. The algorithm is considered to have converged when all models passed to the next generation are rank 1 and meet the constraints for multiple generations.

We apply the NSGA-II to solve four objectives functions, matching the whole radial and vertical component seismograms and the Pnl wavetrain in both radial and vertical components. The four objectives functions are used because the whole waveform is dominated by the large amplitude Rayleigh wave, but important information on crustal structure is also contained in the smaller amplitude Pnl wavetrain (fig. 4-3). We ran the NSGA-II for 100 generations using a population size of 30. In addition, the probability of recombination (𝑃_𝑐) was set to be 0.9, and the probabilities of mutation (𝑃𝑚) set to 0.05. These numbers are chosen to allow convergence while preventing being caught in local minima. A range of both values were tested. The values we attained are similar to other seismological studies (Boomer and Brazier, 2009).