Adjoint Hamiltonian Monte Carlo algorithm for the estimation of elastic modulus through the inversion of elastic wave propagation data

Efficient inversion of noisy seismic waveform data produced due to elastic wave propagation for the estimation of a high-dimensional elastic modulus vector is achieved. Estimation is carried out in a Bayesian framework using Hamiltonian Monte Carlo (HMC) that enables efficient statistical estimation over high-dimensional parameters. The truncated Karhunen-Loève (K-L) expansion is introduced to reduce the dimensionality of the elastic modulus vector. Expensive computations of the gradient of the state vector with respect to the parameter vector at every step are also eliminated through the adjoint method, which is developed from a general one-step discretization of the governing second-order ordinary differential equations (ODEs). An Adjoint HMC algorithm that employs a truncated K-L expansion of the elastic modulus vector is presented. The efficacy of the algorithm is investigated with respect to two representative problems with varying geometric complexity. Adjoint HMC offers a significant speed up in gradient calculation time over the direct differentiation counterpart as the number of terms in the K-L expansion increases. The algorithm is able to estimate the true elastic modulus within the credible intervals for both cases.