  Various methods have been proposed for defining an environmental contour,
based on different concepts of exceedance probability. In the inverse
first-order reliability method (IFORM) and the direct sampling (DS) method,
contours are defined in terms of exceedances within a region bounded by a
hyperplane in either standard normal space or the original parameter space,
corresponding to marginal exceedance probabilities under rotations of the
coordinate system. In contrast, the more recent inverse second-order
reliability method (ISORM) and highest density (HD) contours are defined in
terms of an isodensity contour of the joint density function in either standard
normal space or the original parameter space, where an exceedance is defined to
be anywhere outside the contour. Contours defined in terms of the total
probability outside the contour are significantly more conservative than
contours defined in terms of marginal exceedance probabilities. In this work we
study the relationship between the marginal exceedance probability of the
maximum value of each variable along an environmental contour and the total
probability outside the contour. The marginal exceedance probability of the
contour maximum can be orders of magnitude lower than the total exceedance
probability of the contour, with the differences increasing with the number of
variables. The full abstract is longer than arxiv's requirement of 1,920
characters (see PDF).
