GNGTS 2017 - 36° Convegno Nazionale

710 GNGTS 2017 S essione 3.3 The final goal of the optimization is the minimization of the objective or fitness function, that is, the achievement of the best fitness. To implement the algorithm in the 2D problem, we adopted the fitness function defined in Eq. (3), modified from Everett (1993) in order to control the model roughness in both directions, hence following the Occam-like approach (deGroot- Hedlin and Constable, 1990): (3) where: and are observed (or synthetic) and computed apparent resistivity, respectively; and are observed (or synthetic) and computed impedance phases; the quantities and are the standard errors in apparent resistivity and phase, respectively; M is the number of degree of freedom, i.e. the number of variables; is the Lagrange multiplier in the x- and z-direction, set as a tradeoff between the model and data misfit in order to minimize the model roughness, and is the first derivative of model in the x- and z-direction. The PSO algorithm ran until the maximum number of iterations was reached. This number is problem dependent (Engelbrecht, 2005) and was chosen with the intention of achieving a robust minimization of the fitness function. The size of the swarm (i.e. the number of particles) was assumed to be between 8 and 10 times the number of unknowns. The boundary constraints of the problem were fixed equal to the lower and upper values of resistivity in the search space the swarm can explore. The initial position of particles was maintained randomly distributed within the search space, so that the starting model was in accordance with the stochastic nature of the PSO method. However, it is typical to use a priori information in inversion problems. In the present work, this approach was applied only to the 2D case, and, in order to preserve the swarming nature of the algorithm, the a priori was given only to a small amount of particles (1% or 5% of the total) so that the initial position of the rest of the swarm was selected randomly. The a priori information can be derived from geological (well-log) data or geophysical solutions (e.g. PSO 1D). Fig. 1 - a) Observed data and predicted model response after PSO 1D for ρapp and phase. b) Final PSO models from the 11 trials of optimization and the comparison with the benchmark solution by Jones and Hutton.