A Bayesian Approach To Elastic Full-waveform inversion: Application to a synthetic dataset

Seismic full-waveform inversion (FWI) is being increasingly applied at both exploration and global scales for the determination of high-resolution subsurface models (Tarantola 1984; Virieux and Operto, 2009). In FWI, seismic waveforms are exploited to update subsurface model parameters by trying to match the recorded data with the estimated data. There are some well-known issues often related to the traditional FWI problem. Since the seismic data-model relationship is strongly nonlinear, FWI model updates are often trapped in local minima of the objective function (that usually measures the misfit between observed and estimated data), which are caused by the lack of low frequencies in the data and by inaccurate starting models. Also, focusing on the best-fitting model only (i.e., deterministic inversion) impeeds a complete assessment of the uncertainty affecting the recovered solution (i.e., probabilistic framework). Multiparameter elastic FWI is concerned with the simultaneous determination of two or more subsurface elastic properties (i.e., P-wave velocity, S-wave velocity and density) and this adds a further set of serious challenges to practical FWI application. In addition, elastic FWI generally requires extremely long computing time to achieve convergence (also the elastic forward modelling is more computationally expensive than the acoustic one). We seek the solution for elastic FWI in a Bayesian inference framework to address these issues (Mosegaard and Tarantola, 2002; Gebraad et al, 2020). 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 elastic 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 this type of inverse problem is to compress the model space through appropriate reparameterization techniques, in order to reduce the number of data points and model parameters and hence the dimensions of Ha and g. In this work, we propose a GB-MCMC elastic FWI method combined with the compression of data and model space through a discrete cosine transform (DCT) and we apply this strategy to a 2D synthetic model with one strong vertical and some lateral velocity variations.