A gradient-based Markov chain Monte Carlo algorithm for elastic pre-stack inversion with data and model space reduction

The main challenge of Markov chain Monte Carlo sampling is to define a proposal distribution that simultaneously is a good approximation of the posterior probability while being inexpensive to manipulate. We present a gradient-based Markov chain Monte Carlo inversion of pre-stack seismic data in which the posterior sampling is accelerated by defining a proposal that is a local, Gaussian approximation of the posterior model, while a non-parametric prior is assumed for the distribution of the elastic properties. The proposal is constructed from the local Hessian and gradient information of the log posterior, whereas the non-linear, exact Zoeppritz equations constitute the forward modelling engine for the inversion procedure. Hessian and gradient information is made computationally tractable by a reduction of data and model spaces through a discrete cosine transform reparameterization. This reparameterization acts as a regularization operator in the model space, while also preserving the spatial and temporal continuity of the elastic properties in the sampled models. We test the implemented algorithm on synthetic pre-stack inversions under different signal-to-noise ratios in the observed data. We also compare the results provided by the presented method when a computationally expensive (but accurate) finite-difference scheme is used for the Jacobian computation, with those obtained when the Jacobian is derived from the linearization of the exact Zoeppritz equations. The outcomes of the proposed approach are also compared against those yielded by a gradient-free Monte Carlo sampling and by a deterministic least-squares inversion. Our tests demonstrate that the gradient-based sampling reaches accurate uncertainty estimations with a much lower computational effort than the gradient-free approach.