GNGTS 2019 - Atti del 38° Convegno Nazionale

726 GNGTS 2019 S essione 3.3 of available field seismic data, we validate the implemented algorithm by inverting synthetic seismic data derived on the basis of actual well log data. The approach is also benchmarked against an analytical inversion approach (see de Figueiredo et al. 2018) that assume Gaussian- mixture distributed elastic parameters. The final predictions and the convergence analysis of the implemented method prove that our approach retrieve reliable estimations and accurate uncertainty quantifications with a reasonable computational effort. The implemented MCMC algorithm. Let π and e be the facies and the elastic properties. In our case of a mixed discrete-continuous inverse problem, the posterior pdf can be written as: (1) where m = [ e , π ]. Before the MCMC inversion, we exploit the available borehole data and/or the available geological information about the investigated area to define the p ( π ) and p ( e | π ) distributions. In our implementation p ( e | π ) is a non-parametric mixture distribution that is directly derived from the available data (e.g. well log data) by means of the kernel density estimation algorithm. Then, we apply a normal score transformation to convert each non- parametric component of the prior to a Gaussian model, thus deriving the p ( z | π ) distribution where z represents the normal-score transformed elastic properties. After this transformation the conditional p ( z | π ) is a Gaussian mixture model from which we extract the mean vector and the covariance matrix of each component and the variogram model expressing the expected lateral or vertical variability of the elastic parameters. The transformation to a Gaussian mixture model allows for an easy inclusion of geostatistical constraints into the MCMC sampling in the form of a variogram model. Being m the current model and m΄ the proposed (perturbed) model, the probability for the MCMC chain to move from m to m΄ can be computed from the Metropolis-Hasting rule: (2) where q () is the proposal distribution that defines the new model m΄ as a random deviate from a probability distribution q ( m΄|m ) conditioned only on the current model m . Note that the proposal ratio term in equation (2) vanishes if symmetric proposals (for example a Gaussian proposal) are employed. If m΄ is accepted m = m΄ and another model is generated as a random perturbation of m . The ensemble of accepted models after the burn-in period is used to numerically compute the posterior pdf . To derive a reliable posterior model, we adopt multiple chains that start from different initial points defined on the basis of the a-priori information. To increase the computational efficiency of the algorithm we employ a parallel tempering strategy, in which multiple and interactive chains are simultaneously run at different temperature levels T = [ T 1 , T 2 , ..., T max ]. According to stochastic criteria, swaps of models are allowed between chains at different temperatures, and in this context the high temperature chains ensure that low-temperature chains access all the high probability regions while maintaining an efficient exploitation capability. In the case of AVA inversion, we expect the posterior pdf having different spreads along the Vp , Vs and density directions due to the different resolvability of these parameters. For this reason, we increase the efficiency of the implemented MCMC sampling by using a delayed rejection scheme: This strategy automatically adapts the characteristics of the proposal distribution to the spread of the posterior pdf associated to different model parameters. Finally, we promote the mixing of the chains by including some principles coming from the Differential Evolution Markov Chain (DEMC) algorithm into our MCMC recipe. In brief our algorithm uses iterative perturbation of the facies, and of the elastic properties of the current model to sample the posterior pdf . A normal score transformation is used to easily include geostatistical constraint on the elastic properties, whereas a vertical transition matrix is used to constraints the facies