Square-root variable metric based elastic full-waveform inversion-Part 2: Uncertainty estimation

In our first paper (Part 1) about the square-root variable metric (SRVM) method we presented the basic theory and validation of the inverse algorithm applicable to large-scale seismic data inversions. In this second paper (Part 2) about the SRVM method, the objective is to estimate the resolution and uncertainty of the inverted resulting geophysical model. Bayesian inference allows estimating the posterior model distribution from its prior distribution and likelihood function. These distributions, when being linear and Gaussian, can be mathematically characterized by their covariance matrices. However, it is prohibitive to explicitly construct and store the covariance in large-scale practical problems. In Part 1, we applied the SRVM method to elastic full-waveform inversion in a matrix-free vector version. This new algorithm allows accessing the posterior covariance by reconstructing the inverseHessian with memory-Affordable vector series. The focus of this paper is on extracting quantitative and statistical information from the inverse Hessian for quality assessment of the inverted seismic model by FWI. To operate on the inverse Hessian more efficiently, we compute its eigenvalues and eigenvectors with randomized singular value decomposition. Furthermore, we collect point-spread functions from the Hessian in an efficient way. The posterior standard deviation quantitatively measures the uncertainties of the posterior model. 2-D Gaussian random samplers will help to visually compare both the prior and posterior distributions. We highlight our method on several numerical examples and demonstrate an uncertainty estimation analysis applicable to large-scale inversions.