GNGTS 2023 - Atti del 41° Convegno Nazionale

Session 3.3 ______ ___ GNGTS 2023 We have observed that less than 10 coefficients along the two DCT spatial dimensions explain more than 96% of variability of the original model. The Vp and Vs models were treated as separate images to which the 2D DCT is applied. This means that the compression allows for a reduction of the 44 x 280 x 2 = 24640-D elastic space to a 6 x 8 x 2 = 96-D domain (we are considering the same number of DCT coefficients for both the Vp and Vs models). A similar analysis has been carried out on our seismic data, and in this case 70 x 70 = 4900 retained coefficients in the data space explain almost the total variability of the uncompressed seismogram. Therefore, the full 901 x 101 = 91001-D data space has been reduced to a 4900-D domain. We can see that all the matrices that we are using in our inversion procedure are a lot more tractable in the compressed domain, for example the 24640 x 24640 Hessian in the full domain has been reduced to a 96 x 96 matrix in the compressed space. Notice that, although the MCMC inversions run in the reduced DCT space, the sampled models must be projected back into the full domain just before the forward modelling phase performed with Devito, generates the predicted data needed to compute the likelihood value. We have used six cores on an Intel® Core ™ i7-8700 CPU @ 3.20GHz to run the numerical tests. Each iteration, including computing the Jacobian, the gradient, the Hessian matrix and drawing a sample, takes approximately 70 s wall clock time. We have run one single chain with 15.000 iterations, and it finished running within 40 h . The chain starts from constant velocity models (Vs=300 m/s and Vp=600 m/s ), used also as a prior information, and we can observe in Figure 2 that the posterior mean models (predicted models) exactly reproduce the vertical velocity variations of the true models and the lateral variations are a bit smoother, due to the effect of the DCT compression. The predicted Vs model is better with respect to the predicted Vp model, as expected, considering the weaker influence of the Vp values to the surface waves compared to the Vs values. The standard deviation for both velocity models ( Figure 2 ) has very low values, and the higher values are at the bottom and at the edges of the models, where we have lower illumination. Analyzing the evolution of the negative log-likelihood ( Figure 3 ) we can observe that after a certain number of iterations we reach the stationary regime, where the likelihood still oscillates but around the same values. We have decided to consider the first 3000 iterations as the burn-in period and we have discarded the corresponding samples, so only the following 12000 samples for each chain were used to generate the mean and standard deviation models. The predicted data ( Figure 3 ), which was calculated on the posterior mean model, is very close to the observed data. CONCLUSIONS In this work we have presented a GB-MCMC sampling method based on the Bayesian inference framework to solve an ill-posed inverse problem in high dimensions. We have shown that, with the help of the local gradient and Hessian information, a proposal distribution which is a good approximation of the posterior distribution, is easy to construct and samples can be drawn efficiently from the proposal. We have shown the results of the GB-MCMC elastic FWI applied to a