Fitness functions, genetic algorithms and hybrid optimization in seismic waveform inversion

Due to the nonlinear nature of the seismic waveform inversion problem, global optimization methods such as simulated annealing (SA) and genetic algorithm (GA) have been applied to these problems. Here we evaluate some fundamental issues related to the application of global optimization methods to seismic waveform inversion with the aim of achieving greater accuracy and reducing computational cost. They are: a generalized form of an error or correlation function and a hybrid scheme that efficiently combines a genetic algorithm with a gradient descent scheme. We redefine the two commonly used correlation functions in terms of a geometric and a harmonic measure of misfit and generalize them to have a general order of exponent. That is, this generalized error function is allowed to have any power of data misfit residual which may even take values that are less than unity. The effect of changing this power is to accentuate or de-emphasize the differences between the observed and the synthetic data. A fractional harmonic measure of error seems to help improve the diversity of the population in the GA and prevents and reduces the influence of model parameters that would unduly bias the fitness function as the optimization procedure converges. In order to improve the search efficiency of a GA, we develop a hybrid scheme that incorporates a local gradient sear ch at each step of GA. At each generation of a genetic search, the best fit model is perturbed by one step of a local search algorithm. By this process we substantially improve the performance of the GA. The new method takes advantage of the convergence properties of the local search approach while the global search is carried out using GA. The two methods working together improve the directivity of the model ensemble increasing the fitness and accelerating the convergence to near the global minimum.