Transdimensional seismic inversion using the Hamiltonian Monte-Carlo approach

Abstract:

In an inverse problem, the number of model parameters is often a choice dictated by computational convenience. In a transdimensional inverse problem, the number of model parameters is treated as an additional variable that we solve for. The Reversible jump Markov Chain Monte Carlo (RJMCMC) is generally employed for model exploration and uncertainty quantification in such problems. A typical RJMCMC is computationally expensive and therefore, its application has so far been limited to problems with a small number of model parameters, where the forward modeling can be done rapidly. Here we report a practical transdimesional seismic inversion algorithm where the model perturbations are generated according to the birth-death approach. However, we determine the model acceptance by a Hamiltonian approach that introduces a new momentum variable. This results in an update rule that makes use of the gradient information together with the Metropolis criterion. The algorithm can either be used for model exploration (sampling at a constant temperature) or model exploitation using annealing. We apply this technique to 1D waveform inversion problem of seismic reflection data where at each location we make use of several hundred model parameters. Forward modelng is carried out using reflectivity layer matrices in the frequency wavenumber domain and the objective function is evaluated in delaytime-ray parameter domain. Only at a few selected sufrace locations (CMP gathers) we carry out detailed uncertainty analysis ; at all other places we determine the best fit model. We are able to estimate geologically meaningful results that are correlated very well with collocated well logs at a few selected locations. We also explore several proposal distributions diifferent from the standard birth-date appproach for trial model generation that can be computationally fast.