A generalization of the Nataf transformation to distributions with elliptical copula

In the first article [Lebrun R, Dutfoy A. An innovating viewpoint of the isoprobabilistic Nataf transformation with the copula theory within exceedance threshold uncertainty analysis. Probabilistic Structure Engineering Structural Reliability. 2008 [in press]], we showed that the Nataf transformation is a way to model the dependence structure of a random vector by a normal copula, parameterized by its correlation matrix. Following this analysis, we propose an extension of this transformation to any elliptical copula, and give the equivalent of the First Order Reliability Method (FORM) and the Second Order Reliability Method (SORM) for this generalization. In particular, we derive the Breitung asymptotic approximation in this new context.