An innovating analysis of the Nataf transformation from the copula viewpoint

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Reliability-based design optimization using kriging surrogates and subset simulation (2011, background): “The reader is referred to Lebrun and Dutfoy (2009) for a recent discussion on such mappings u = T (x).”
Model Uncertainty in Finite-Element Analysis: Bayesian Finite Elements (2011, unknown): “is associated with pitfalls (Liu and Der Kiureghian 1986; Lebrun and Dutfoy 2009).”
Metamodel-based sensitivity analysis: polynomial chaos expansions and Gaussian processes (2016, unknown): “For instance, if the input vector X is defined by a set of marginal distributions and a Gaussian copula, it can be transformed into a set of independent standard normal variables using the Nataf transform [20; 35].”
An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis (2010, unknown): “[23, 24]).”
Structural reliability analysis using a copula-function-based evidence theory model (2014, background): “Lebrun and Dutfoy [45] pointed out that the commonly used Nataf transformation in traditional reliability analysis was equivalent to the Gaussian copula function.”