  Full-Waveform Inversion (FWI) has now become a widely accepted tool to obtain
high-resolution velocity models from seismic data. Typically, the velocity
model in its discrete form is represented on a rectangular grid, and we solve
for the elastic properties at these grid points. FWI is mostly solved employing
a local optimization method, where one obtains a velocity update by minimizing
the misfit between the observed and the calculated seismograms. Note also that
FWI is a highly non-linear problem which is known to be prone to
non-uniqueness. The convergence to a globally optimum solution is not
guaranteed; it depends on the choice of the starting model. Thus, a Bayesian
formulation of the inverse problem with subsequent sampling of the posterior
distribution is a preferred choice, since it enables uncertainty
quantification. However, with the increase in the dimension of a model,
sampling search space becomes computationally expensive. We employ a recently
developed trans-dimensional sampling method called Reversible Jump Hamiltonian
Monte Carlo (RJHMC), to the 2D full waveform inversion problem. We represent
our velocity model using Voronoi cells, determined from the distribution of
certain nuclei points in the model space. This method offers two advantages.
First, it solves for a variable dimensional velocity updates by using a
trans-dimensional reversible jump Markov Chain Monte Carlo (RJMCMC) step and
thus tries to achieve an optimum number of nuclei to represent the model and
minimize the misfit. A smaller number of parameters helps in an efficient
sampling of the model search space. Second, it applies the gradient-based
Hamiltonian Monte Carlo (HMC) step, which further improves the sampling by
allowing the algorithm to take a large step guided by the gradient. This
two-step algorithm proves to be a useful tool for model exploration and
uncertainty quantification in FWI.
