Hamiltonian Monte Carlo algorithms for target-oriented and interval-oriented AVA inversions

A reliable assessment of the posterior uncertainties is a crucial aspect of any Amplitude Versus Angle (AVA) inversion due to the severe ill-conditioning of this inverse problem. To accomplish this task numerical Markov chain Monte Carlo algorithms are usually employed when the forward operator is non-linear. The downside of these algorithms is the considerable number of samples needed to attain stable posterior estimations especially in high-dimensional spaces. To overcome this issue, we assess the suitability of Hamiltonian Monte Carlo (HMC) algorithm for non-linear target-oriented and interval-oriented AVA inversions for the estimation of elastic properties and associated uncertainties from pre-stack seismic data. The target-oriented approach inverts the AVA responses of the target reflection by adopting the non-linear Zoeppritz equations, whereas the interval-oriented method inverts the seismic amplitudes along a time interval using a 1D convolutional forward model still based on the Zoeppritz equations. HMC uses an artificial Hamiltonian system in which a model is viewed as a particle moving along a trajectory in an extended space. In this context, the inclusion of the derivatives information of the misfit function makes it possible long-distance moves with a high probability of acceptance from the current position towards a new independent model. In our application we adopt a simple Gaussian a-priori distribution that allows for an analytical inclusion of geostatistical constraints into the inversion framework and we also propose a strategy that replaces the numerical computation of the Jacobian with a matrix operator analytically derived from a linearization of the Zoeppritz equations. Synthetic and field data inversions demonstrate that the HMC is a very promising approach for Bayesian AVA inversion that guarantees an efficient sampling of the model space and retrieves reliable estimations and accurate uncertainty quantifications with an affordable computational cost.