Multi-Scale and Multi-Resolution Stochastic Modeling of Subsurface Heterogeneity by Tree-Indexed Markov Chains

A new methodology is proposed to handle multi-scale heterogeneous structures. It can be of importance in the field of hydrogeology and for petroleum engineers who are interested in characterizing subsurface heterogeneity at various scales. The framework of this methodology is based on a coarse to fine scale representation of the heterogeneous structures on trees. Different depths in the tree correspond to different spatial scales in representing the heterogeneous structures on trees. On these trees a Markov chain is used to describe scale to scale transitions and to account for the uncertainty in the stochastically generated images.