GNGTS 2016 - Atti del 35° Convegno Nazionale

GNGTS 2016 S essione 3.1 513 A HYBRID METHOD TO ESTIMATE UNCERTAINTY IN 2D FWI: APPLICATION TO AN INCLUSION MODEL A. Sajeva, M. Aleardi, A. Mazzotti Dipartimento di Scienze della Terra, Università di Pisa, Italy Introduction. Full-waveform inversion (FWI) is a valuable tool to derive high-resolution models of the subsurface having available a reliable macro-model containing the correct large- wavelengths of the model. FWI is generally cast in the framework of deterministic approaches, and hence it returns a single best-fitting model in which no information is given regarding the associated uncertainties of the model parameters. However, in practice, many inverse problems are ill-posed meaning that many solutions explain observations and theory equally well. Therefore, estimating the uncertainties that affects the final result of an inverse problem provides valuable insights on the equivalence region of the solutions (Fernández et al. , 2012). Sajeva et al. (2016) proposed a workflow to determine uncertainties in two-dimensional FWI. This workflow can be divided in two parts. In the first part, a genetic algorithm combined with a Gibbs Sampler (GS) (Sambridge, 1999; Aleardi and Mazzotti, 2016) derives a low- resolution P-wave velocity (Vp) model and its uncertainties. In the second part, the PPD derived by the GS is used to perform a set of full-waveform inversions using iterative descent- based techniques, which in turn are used to perform a statistical analysis of the final high- resolution solution. In this work, we apply this method to a simple example model modified from Mora (1989), that consists in a background gradient model with a horizontal reflector and a spherical inclusion. Theory. Genetic algorithms (GAs) are a class of randomized search methods that can be applied to large-scale optimization problems. They treat models collectively, and they make evolve the ensemble of models (or population) toward new generations with lower misfit by means of selection, recombination, and mutation. Since their introduction (Holland, 1975), several implementations of GAs, both binary and real coded, have been proposed. A recent and promising version of GAs is the Breeder Genetic Algorithm (BGA), an efficient real-coded algorithm in which the optimisation is mainly guided by the selection and recombination steps (������������������ ��� ����������� ������ Schlierkamp-Voosen and M �� ühlenbein, 1993). GAs are not a Markov Chain Monte Carlo (MCMC) method, consequently, the ensemble of models explored during a GA inversion is not sampled according to the posterior probability distribution (PPD). Therefore, a biased estimation of the PPD is produced if it is directly computed from the collected models and their associated likelihoods. From a Bayesian point of view the posterior probability distribution is the solution to the inverse problem, and it contains all information available on the model. The PPD calculation depends on the data, any prior information, and the noise statistics (which is assumed known). At any point m in model space M the PPD is given by: P ( m | d 0 ) = k ρ(m ) L ( d 0 | m ), (1) where ρ(m) is the prior probability distribution, L is the likelihood function, k is a normalizing constant, d 0 is the observed data set. To convert the ensemble of GA models to a non-biased PPD, we make use of the procedure of Sambridge (1999), that resamples the model space using the Gibbs Sampler. In more details, the model space is divided into Voronoi cells, each one associated with a single GA model and its likelihood. This constructs a multi-dimensional interpolant that is resampled using the Gibbs sampler. Method . The inversion procedure that we use is composed by the following steps: 1. perform a global inversion using the BGA, and collect all the explored models and their misfit; 2. appraise the entire ensemble of BGA models using the Gibbs Sampler following the procedure of Sambridge (1999). This step returns the uncertainty affecting the low- resolution velocity model derived by GA;