GNGTS 2015 - Atti del 34° Convegno Nazionale

22 GNGTS 2015 S essione 3.1 and the weights of the mixture, is the expectation maximization algorithm (Hastie et al., 2005). This Gaussian mixture model allows us to describe the multimodality and the correlation that often characterize the distribution of the petrophysical properties in the subsurface. If we assume that ε is Gaussian with zero mean and covariance Σε , the conditional probability p(m|R) can be expressed as: (5) where: Σε can be estimated by comparing actual and predicted well-log data and is assumed independent from R and only related to ε . Note that this formulation allows us to account for uncertainties associated with the rock-physics model predictions that are expressed by ε and Σε . The joint distribution of the elastic and the petrophysical properties is again a Gaussian mixture: (6) If the rock-physics model f RPM is linear, this joint distribution can be derived analytically from the prior distribution p(R) . Conversely, if f RPM is not linear the joint distribution p(m,R) can be obtained from a semi-analytical approach that makes use of Monte Carlo samples. This last approach applies the expectation maximization algorithm to Monte Carlo samples to compute the characteristics of the joint distribution (see Grana and Della Rossa, 2010, for full details). Since the joint distribution is a Gaussian mixture, the conditional distribution p(R|m) is again a Gaussian mixture and can be written as follows: (7) in which and are analytically computed from the joint distribution p(m,R), from the prior model p(R) and from the results ( m ) of Bayesian AVA inversion: (8) and (9) The weights c k in the conditional GM distribution p(R|m) can be computed as follows: (10) To compute the final conditional probability p(R|d obs ) , which expresses the probability of petrophysical variables conditioned by seismic data, we need to propagate the uncertainties that characterize the results of the Bayesian AVA inversion into the conditional probability p(R|m). To this end the Chapman-Kolmogorov equation can be used (Papoulis, 1984): (11) This conditional probability is the final result of the petrophysical inversion conditioned by seismic data. Results. To define the number of components of the a-priori Gaussian mixture distribution of the petrophysical properties ( p(R); Eq. 4) we consider three different litho-fluid facies that are shale, brine sand and gas sands. These facies are defined on the basis of available well- log data and geological knowledge of the investigated area. This trivariate Gaussian mixture allows us to account for correlations observed between petrophysical variables in each litho- fluid class. The parameters that define this GM distribution were obtained applying the