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  • Bulletin of the Seismological Society of America, 89, 4, pp. 978-988, August 1999

    Multimodal Function Optimization with a Niching Genetic Algorithm:

    A Seismological Example

    by Keith D. Koper,* Michael E. Wysession, and Douglas A. Wiens

    Abstract We present a variant of a traditional genetic algorithm, known as a niching genetic algorithm (NGA), which is effective at multimodal function optimi- zation. Such an algorithm is useful for geophysical inverse problems that contain more than one distinct solution. We illustrate the utility of an NGA via a multimodal seismological inverse problem: the inversion of teleseismic body waves for the source parameters of the Mw 7.2 Kuril Islands event of 2 February 1996. We assume the source to be a pure double-couple event and so parametrize our models in terms of strike, dip, and slip, guaranteeing that two global minima exist, one of which rep- resents the fault plane and the other the auxiliary plane. We use ray theory to compute the fundamental P and S H synthetic seismograms for a given source-receiver ge- ometry; the synthetics for an arbitrary fault orientation are produced by taking linear combinations of these fundamentals, yielding a computationally fast forward prob- lem. The NGA is successful at determining that two major solutions exist and at maintaining the solutions in a steady state. Several inferior solutions representing local minima of the objective function are found as well. The two best focal solutions we find for the Kuril Islands event are very nearly conjugate planes and are consistent with the focal planes reported by the Harvard CMT project. The solutions indicate thrust movement on a moderately dipping fault--a source typical of the convergent margin near the Kuril Islands.

    Introduction

    A common goal of optimization problems is finding the global minimum of a multidimensional objective function. In geophysics this is realized through inverse problems where the objective function to be minimized is often a norm describing the error between a specific data set and model (Menke, 1989). In cases such as seismic tomography, where model parameters number in the hundreds or thousands, it- erative matrix methods are often used (Aki, 1993). In cases where models can be described by a much smaller number of parameters, and the calculation of the forward problem is computationally fast, global search methods such as simu- lated annealing (e.g., Bina, 1998) and genetic algorithms (e.g., Stoffa and Sen, 1991) have been effective. These meth- ods are not, however, designed for finding multiple, distinct solutions to mulfimodal optimization problems. Although various solutions can be found by repeatedly using such methods with different starting points, since these methods tend to be biased by the initial conditions (Vasco et aL, 1996), distinct solutions are not explicitly solved for. An

    *Present address: Department of Geosciences, University of Ar- izona, Tucson, Arizona 85721.

    efficient alternative method for explicitly producing distinct solutions to a given optimization problem is the use of a genetic algorithm (GA) variant known as a niching genetic algorithm (NGA) (e.g., Holland, 1975; Goldberg, 1989; Mahfoud, 1995).

    Niching genetic algorithms provide a formal mechanism for finding several solutions to a multimodal optimization problem. In the case of an objective function that has more than one global optimum, or several interesting local min- ima, an NGA can in theory find the regions in model space where the minima occur, and maintain the solutions indefi- nitely. The number of minima does not need to be known a priori, and in fact an NGA can be used to determine the number of distinct solutions empirically, given a user- defined criterion for model similarity. We apply an NGA to two example problems: optimization of an analytic, 1D, five- modal function on the unit interval, and the inversion of teleseismic body waves for the focal parameters of the Mw 7.2 Kuril Islands event of 7 February 1996. For the latter problem we assume a double-couple source and parametrize the models in terms of strike, dip, and slip, ensuring that the objective function has two broad, global minima represent- ing the fault plane and the auxiliary plane. The fundamental

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  • Multimodal Function Optimization with a Niching Genetic Algorithm: A Seismological Example 979

    bimodal nature of this problem is well known, thus it pro- vides an excellent venue for the seismological illustration of an NGA.

    Niching Genetic Algorithms

    Although GAs were originally developed to study evo- lutionary phenomena, their utility as optimization tools is widely recognized (Goldberg, 1989). GAs proceed stochas- tically by the repeated application of biologically inspired operators on a randomly generated population of candidate models. A wide variety of operators and model parameteri- zations have been implemented, but the most common op- erators are crossover and mutation, which use the binary model representation (genotype), and selection, which op- erates on the decimal representation of the models (pheno- type). With time the operators gradually improve the quality of the population of models, quantified by objective function evaluations, while avoiding the lure of local minima. Al- though GAs are not guaranteed to find the global minimum of an arbitrary objective function, they are explicitly de- signed to avoid local minima. Nevertheless the possibility of using variations of a genetic algorithm (NGAs) to find distinct solutions of a multimodal optimization problem (lo- cal minima of the objective function) has been recognized ever since genetic algorithms have been applied to optimi- zation problems in general (Holland, 1975; Goldberg, 1989). A thorough review of the subject is given by Mahfoud (1995), and here we only briefly comment on the back- ground of niching genetic algorithms.

    Most previous work with NGAs has been carried out using either a (1) crowding (e.g., Dejong, 1975) or (2) shar- ing scheme (e.g., Goldberg and Richardson, 1987). In the first case, individual models in a given population are com- pared with a random subpopulation of m models, where m is known as the crowding factor and is of the order 2 to 3. The similarity of two models is defined on a bit by bit basis using the genotypic representation of the models. The model in this subpopulation that is most similar to the original model is deleted from the population. The replacement, or weeding out, of similar models promotes diversity in the population as a whole and allows different species of models to evolve from the main population, with each species in- habiting a niche representing a local minima of the objective function. In the second case, that of sharing, again the goal is to induce a population of models to evolve into separate demes representing local minima of the objective function. Such implicit speciation is accomplished by scaling (reduc- ing) the fitnesses of similar models. All the models in the population have their fitnesses artificially reduced in pro- portion to their similarity to the rest of the population. The exact proportion of reductions is defined by a sharing func- tion. Thus several models within the population that are very similar to one another will have their fitnesses reduced to a much greater extent than the more distinct models in the population, and so will tend to be eliminated during the nat-

    ural course of a GA run. This mechanism inhibits the con- vergence of the model population as a whole (due to genetic drift) and, like the crowding scheme, allows for distinct so- lutions to a multimodal optimization problem to be found.

    The NGA used in this work differs from those described above in that the number of subpopulations, or demes, of models is explicitly defined at the beginning of the run. Our goal is not to have a single population of models evolve into separate demes, but to have several artificially separated sub- populations migrate to the different optimal areas in the model space--those regions representing the local minima of the objective function. This is an extreme example of the application of mating restrictions to genetic algorithm search (Goldberg, 1989). We do this to increase the temporal sta- bility of the niches. In a standard NGA it is possible that demes appear and disappear owing to the stochastic nature of GAs and the finite size of the population; however, with the demes explicitly separated we can use intrademe elitist selection to maintain the integrity of the niches.

    Although the number of demes is decided upon without explicit knowledge about the objective function and the number of local minima that exist, we can use our NGA to determine the number of local minima empirically. This is accomplished by individually analyzing the behavior of the demes for different scales of similarity. We illustrate this in a later section via the seismological example problem of teleseismic waveform inversion.

    The NGA used in this work is described as follows. Ini- tially we construct n demes of models. Each deme has the same number of members and is carried out as a typical GA run, with equivalent probabilities of mutation and crossover, with one important distinction. The first deme is allowed to run independently of the others and behaves as a regular GA. The second deme is run similarly except that after every generation the similarity of each member of this deme is calculated with respect to the best model from the first deme--the alpha solution. If this similarity is greater than a problem-specific criterion, that model

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