GNGTS 2022 - Atti del 40° Convegno Nazionale

478 GNGTS 2022 Sessione 3.3 A BAYESIAN APPROACH TO FULL-WAVEFORM INVERSION: PRELIMINARY RESULTS S. Berti 1,2 , M. Aleardi 1 , E. Stucchi 1 1 Earth Sciences Department, University of Pisa, Italy 2 Earth Sciences Department, University of Florence, Italy Introduction. An important application of the geophysical inverse theory is seismic full- waveform inversion (FWI), in which seismic data recorded at the surface are used to estimate compressional and shear wave velocities. In FWI, seismic waveforms are exploited to update subsurface model parameters by trying to match the recorded data with the estimated data (Virieux and Operto, 2009). There are two critical issues often related to the traditional FWI problem. First of all, most optimization-based FWI methods are designed to only find the model that best fits the observations according to a specific error function. Nevertheless, the FWI problem is also ill- posed, observed data can be noisy, modelling methods can be inaccurate and prior knowledge can be insufficient. All of these factors introduce uncertainties into the inversion results. In this context, focusing on the best-fitting model only (i.e. deterministic inversion) impedes a complete assessment of the uncertainty affecting the recovered solution (i.e. probabilistic framework). Additionally, in traditional local optimization-based FWI methods, an objective function, which measures the misfit between observed and estimated data, is minimized with respect to the model parameters. The commonly used L2 -norm objective function is non-convex because of the highly nonlinear forward mapping and hence the inversion is prone to get trapped into local minima. We seek the solution for FWI in a Bayesian inference framework to address those two issues (Mosegaard and Tarantola, 2002). Bayesian inference provides a systematic framework for incorporating and propagating the uncertainties in observed data, prior knowledge and forward operator into the uncertainties affecting the recovered model. The final solution of a Bayesian inversion is the so-called posterior probability density (PPD) function in the model space which fully quantifies the uncertainties in the recovered solution. To efficiently sample the posterior distribution, we introduce a sampling algorithm in which the proposal distribution is constructed by the local gradient and the Hessian of the negative log posterior. For non-linear problems the Bayesian inversion is often solved through a Markov Chain Monte Carlo (MCMC) sampling. Monte Carlo is a technique for randomly sampling a probability distribution. Markov chain is a systematic method for generating a sequence of random variables where the current value is probabilistically dependent only on the previous state of the chain. Combining these two methods, allows random sampling of high dimensional probability distributions that honors the probabilistic dependence between samples by constructing a Markov Chain that comprise the Monte Carlo sample. Our algorithm is called gradient-based Markov chain Monte Carlo (GB-MCMC). The GB- MCMC FWI method can quantify inversion uncertainties with estimated posterior distributions given sufficiently long Markov chains. MCMC sampling methods provide the global view of the model space, so the inversion avoids the entrapment in a local region. Theoretically speaking, GB-MCMC method can accurately estimate the posterior distribution given sufficiently long Markov chains with arbitrary starting points. However, expensive forward model operators and high-dimensional parameter spaces make the application of MCMC algorithms computationally unfeasible. A suitable strategy to reduce the computational complexity of this type of inverse problem is to compress the model space through appropriate reparameterization techniques, in order