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Grond adapts an optimization scheme using elements of simulated annealing (ref) and the bootstrap technique (2 refs). Rather than optimizing a single objective function, it tries to find volumes in parameter-space which satisfy low misfit values over an ensemble of N variations of the objective function. These variations are created by applying the bootstrap technique, but other methods could be used instead. The creation of one bootstrap entity is made by reweighting contributions to the global misfits (e.g. the individual trace misfits) in a way that from M such contributions M are chosen with replacement (allowing for multiple or skipped contributions). The N sets of weights defining the bootstrap entities are chosen only once before the optimization process.
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One point in parameter space defines one potential source model. During the optimization, iteratively new candidate models are generated and evaluated. For each new model, the objective functions are evaluated, leading to N misfit values, one for each bootstrap entity.
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For each bootstrap entity a fixed size set of K best performing models is maintained (high score lists). The current set of N*K models participating in the high score lists are called 'high score population'. Within the high score population, the same model may be included multiple times, because it may perform well in several bootstrap entities.
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In every optimization method there is a trade-off between number of tested models and the ability to find the global minimum without getting trapped in local minima. Usually a compromise is made depending on characteristics of the problem to be solved. A global search is explorative whereas a directed search is targeted. In our optimization, the explorativeness of the search can be influenced by the choice how new candidate models are generated (model sampling). We divide the optimization process into four successive phases differing in their sampling method: uniform, transition, explorative and non-explorative. Not all phases have to be applied and the schedule and their properties can be tuned problem-specific. In the uniform phase, the model parameters are drawn from uniform distributions within pre-defined ranges. In the explorative phase, new models are chosen based on statistical properties of the high score population models. First, either a multivariate normal distribution or parameter-wise normal distributions are determined. Then the spread of the distribution is scaled by a factor and moved to a random model of the high score population. Finally a new model is drawn from the scaled and shifted distribution. The non-explorative phase only differs from the scheme in the explorative phase in that the centroid of the distribution is not moved. The transition phase only differs from the explorative phase in that the scaling factor applied to the distribution is slowly decreased logarithmically. |