GNGTS 2016 - Atti del 35° Convegno Nazionale

GNGTS 2016 S essione 3.1 491 are sequentially performed starting from different parts in the model space to increase the reliability of the result. In addition, it is known that the samples accepted at the beginning of the chain (during the so called “burn-in” period) may not accurately represent the target PPD. Therefore, these samples are usually not considered in the computation of the final posterior distribution. Following the Bayesian notation and applying the chain rule, the target posterior distribution for the analyzed case can be written as: (2) To derive such posterior probability, I implement the following MCMC algorithm. The following steps are used to define the initial model at the beginning of the chain: 1a) pick a litho-fluid facies from the prior probability distribution p(F) ; 2a) define the petrophysical parameters for the initial model by drawing random numbers from the conditional probability p(R|F); 3a) apply the empirical RPM to convert the petrophysical properties into the elastic parameters; 4a) add to the derived elastic parameters the uncertainties associated to the rock physics model. The probability distribution of this uncertainty (assumed to be Gaussian) can be computed during the definition of the rock physics model by comparing the measured and the predicted elastic properties. This step is used to draw a sample from the conditional probability p(E|R,F) ; 5a) use the Zoeppritz equations to compute the likelihood p(d|E,R,F) for the considered model. The likelihood function I consider is based on a least-squares measure of misfit in which the noise is assumed to be normally distributed with a null mean value and a diagonal covariance matrix. I compute this covariance matrix by comparing the AVA responses of adjacent CMP gathers and by assuming that these responses are produced by similar petrophysical properties. Then, the differences between the AVA responses extracted from adjacent CMPs have been attributed only to noise contamination; 6a) accept the initial model as the current model. After generating this model, a candidate model must be defined. The steps advocated to this aim are the following: 1b) draw a random number p uniformly distributed over [0,1]; 2b) if p <0.2, perturb the litho-fluid facies for the current model by selecting a litho-fluid facies from the prior distribution p(F) . After this perturbation draw a random sample from p(R|F) to define the petrophysical properties for the candidate model; 3b) if p ≥0.2, define the petrophysical properties of the candidate model by perturbing the petrophysical properties of the current model. This perturbation follows a random walk that sample the distribution p(R|F). Note that in this case the litho-fluid facies for the candidate model and for the current model are the same. After step 2b) or step 3b) I compute the elastic parameters associated to the candidate model and its likelihood by repeating steps 3a), 4a) and 5a). Then the candidate model is accepted according to the Metropolis rule. If the candidate model is accepted m curr =m cand, and if the burn- in period is over m cand is collected. In all inversion tests described in the following I use 10 different randomwalks that start from different initial models. In each walk 2000 models are collected and only 1500 are considered in the computation of the final PPD, thus considering a burn-in period of 500 models. Application to field data. The available well log data and the geological knowledge about the investigated area are exploited to define the number of components of the a-priori Gaussian mixture distribution for the petrophysical properties. In this case I have considered three components each one associated with a given litho-fluid facies: shale, brine sand and gas