Is Hamiltonian Monte Carlo (HMC) really worth it? An alternative exploration of hyper-parameter tuning in a time-lapse seismic scenario

Determining if uncertainty quantification is worth it or not is closely related to how that uncertainty is computed and the associated computational cost. For seismic imaging, it is typically done using Markov chain Monte Carlo algorithms (McMC). Solving an inverse problem using McMC means exploring and characterizing the ensemble of all plausible models through more or less point-wise random walk in the data misfit landscape. This is typically done using Bayes’ theorem via the computation of a posterior probability density function. Even though this can sound naively simple, it can come with a significant computational burden given the dimension of the problem to be solved and the expense of the forward solver. This is because as the number of dimensions grow, there are exponentially more possible guesses the algorithm can make, while only a few of these models will be accepted as plausible. More advanced uncertainty quantification methods such as Hamiltonian Monte Carlo (HMC) could be beneficial because they can handle higher dimensions because efficient sampling of the model space through pseudo-mechanical trajectories in the data misfit landscape is expected. In order for an HMC algorithm to efficiently sample the model space of interest and provide meaningful uncertainty estimates, three hyper-parameters need to be tuned for trajectory design: the Leapfrog steps L, the Leapfrog stepsize ε, and the Mass Matrix M. There has been already work showing how one can choose L and ε; however designing the appropriate M is far more challenging. We consider a time-lapse seismic scenario and use a local acoustic solver for fast forward solutions. We then use Singular value decomposition, in the vicinity of the true model, to transform our time-lapse optimal model to a system of normal coordinates and use only a few of the eigenvalues and eigenvectors of the Hessian as oscillators. By doing so, we can efficiently understand the impact of the initial conditions and the choice of M and gain insight on how to design M in the standard system. This gives us an intuitive way to understand the mass matrix, allowing us to determine whether gains from the HMC algorithm are worth the cost of determining the parameters.