GNGTS 2018 - 37° Convegno Nazionale

556 GNGTS 2018 S essione 3.1 (2) where K is the total number of facies considered: in the following shale, brine sand and gas sand. The key aspect of this inversion approach is the proper choice of the joint distribution p ( m , d  | f ). To this end many assumptions can be made, for example one can simply neglect the facies dependency of m and d , thus using a simple unimodal Gaussian distribution. However, note that in this case it is no more possible performing a facies classification. More generally, the assumed statistical model should honour the multimodality of the p ( m , d  | f ) distribution, and among the many multimodal distributions, the Gaussian-mixture is often adopted. Another possible, but less common approach, is to directly approximate the joint distribution using a non-parametric technique, such as the kernel density estimation (KDE). The numerical inversion method previously described can be applied to both logged impedance values or Ip values inferred from a post-stack seismic inversion. In the following, both these cases are analysed: first, I use Fig. 1 - Non-parametric, Gaussian-mixture, and Gaussian joint p ( m , d  | f ) distributions (a, b, and c respectively).