• Bayesian inversion
• acoustic full waveform inversion
• uncertainty estimation

Full Waveform Inversion (FWI) is a geophysical technique that aims to estimate a high – resolution velocity model exploiting all the seismic waves that are recorded at the receivers. FWI is an inverse problem, and as such it can be solved in a deterministic or probabilistic framework. The deterministic approach offers a fast convergence, but it is incapable to estimate a complete uncertainty on the inversion result. While the Bayesian approach can propagate uncertainty from the data, the prior information, and the forward modelling up to the recovered result. The Bayesian solution is the Posterior Probability Density (PPD) function in the model space. For non – linear problems a possibility to compute the statistical property of the PPD is to sample the posterior density by means of sampling methods such us the Markov chain Monte Carlo (MCMC) algorithm. These are very computationally demanding algorithms that work well in reduced model spaces. We tested a Gradient – Based Markov chain Monte Carlo (GB - MCMC) algorithm that exploits the local gradient and Hessian information of the misfit function to rapidly converge to a stable posterior uncertainty. We applied the GB – MCMC FWI algorithm in the acoustic case to synthetic and field seismic data combined with a compression of model and data spaces. As a possible compression strategy for both the data and the model spaces we used the 2D Discrete Cosine Transform (DCT) reparameterization. The compression allows to make more feasible the manipulation and computation of large dimensions matrices during the inversion run.

I want to underline that the aim of the thesis was not the implementation of the GB - MCMC algorithm, which was provided to us complete, but we limit to test the algorithm on synthetic and field data. The algorithm was first tested on a synthetic case and then on the real data set. For what concerns the synthetic case, we performed two inversion tests each composed of two chains. The use of two chains per test allows to demonstrate that even if starting from different initial velocity models the chain was able to reach the sampling of high posterior probability areas avoiding being affected by the cycle skipping problem. The idea of more tests was to understand the degree of cycle skipping the algorithm was able to handle. Therefore, we used increasingly simpler initial models and with different noise contamination on the seismic data. The aim of the others synthetic tests was to assess the amount of amplitude, frequency variation and phase errors in the estimation of the source wavelet the algorithm can tolerate. Then we compared the GB – MCMC with a local inversion to test advantages and the drawbacks of each method. We found that the local inversion is very sensitive to a good starting velocity model. In the second part of the thesis project, we moved to the real data set. We used the Mobil Avo Viking Graben Line 12 is a marine data set acquired in the North Sea offshore Norway. The data were limited in terms of maximum offset and low signal content at low frequency that make the application of FWI challenging. Despite these conditions we were able by means of several processing steps to make the data more feasible for that application. In any case we did not achieve a good data matching, however we should point out that, differently from the predicted data the observed data contain the superposition of the direct waves with the reflection, whereas in the predicted data we subtracted the direct waves from the seismograms to have a similar kinematic of the events with the observed case.