GNGTS 2024 - Atti del 42° Convegno Nazionale

Session 3.3 GNGTS 2024 This kind of approach makes the inversion highly demanding from a computatonal point of view, so it is necessary to adopt some strategies to alleviate this efort: here we employ the Discrete Cosine Transform (DCT) to compress both data and model space. This technique reduces the number of unknown in the inversion and also the dimensions of the matrices and vectors involved in the ES-MDA approach. The DCT is a Fourier-related transform through which a signal can be expressed as sum of cosine functons. Since the DCT concentrates most of the energy of the signal in low order coefcients, it is possible to get an approximaton of the original signal by discarding those that are very close to zero and retaining only the low order ones. Other informaton can be found in Britanak et al. (2010) and Aleardi et al. (2021a). In this work we restrict the applicaton of EB-FWI to a synthetc case, but its utlizaton to feld data is being prepared. In fact, we processed a 2D seismic line from the FORGE (Fronter Observatory for Research in Geothermal Energy) geothermal experiment located in Utah, USA (Miller at al., 2018). Precisely for the purpose of applying a FWI to this dataset, we performed a dedicated processing, comprehensive of Migraton Velocity Analysis (MVA), for improving the velocity feld estmaton. Method In this work we use an ensemble-based approach implementng the ES-MDA algorithm to cast the FWI in a Bayesian inference framework. The ensemble-based method represents a data assimilaton algorithm in which the posterior distributon consists of a set, also called ensemble, of model realizatons. It can be demonstrated that ES corresponds to a single Gauss-Newton step, but it usually requires many iteratons to ensure a good data predicton when compared to MDA, which speeds up the convergence performing multple assimilatons (correctons) of the data. The steps of MDA algorithm are the following: choice of the number of data assimilatons (iteratons); generaton of the startng ensemble of models drawn from a Gaussian prior distributon; for each iteraton and for each model of the ensemble, computaton of the data associated to each member of the ensemble, perturbaton of each data and update of the models. A schematc representaton of the algorithm is shown in Fig.1. The perturbaton of each data vector is made according to , where: is the observed data, is a random perturbaton of the observed data, is called infaton coefcient, is the data covariance matrix and , with representng a Gaussian distributon and the identty matrix. The update of each model of the ensemble is defned as follows: , with , where is the number of models in the ensemble, the superscripts and refers to variable computed at the current iteraton (updated) and to the previous one, d ~ k = d + α i C 1 2 d ∙ n d d ~ k α C d n = N (0, Id ) N ( d , C d ) Id m u k = m p k + K ~ ( d ~ k − d p k ) k = 1,…, N N u p