Delineation of hazardous regions in the unit hypercube: A copula-based approximation for the identification of low-frequency compound occurrences

The probabilistic characterization of compound extreme events, defined as those arising from the joint action of several (generally dependent) drivers, is a central problem in engineering hydrology and water resources. The copula-based scenarios customarily used in the literature (i.e., OR, AND, Kendall, and survival Kendall) summarize multivariate risk by tracing a single boundary in the unit hypercube and assigning to it a target probability, or equivalently a return period. Except for the rectangular OR and AND cases, the prescribed probability attaches to the bounding layer rather than to the hazardous region itself, which is then determined only implicitly. Moreover, the marginal exceedance rates of the drivers are fixed by the level of the critical layer and cannot be set independently. Here we introduce a copula-based approximation that characterizes the hazardous region directly as a full-dimensional set to which the target exceedance frequency is attached. The boundary of this set consists of linear segments meeting at a vertex at the point of maximum curvature of an equiprobability surface of the copula. Crucially, the per-variable exceedance rates are predefined by the user, so that application-specific marginal rates are reconciled with the joint target within a single region, allowing asymmetric compound occurrences (e.g., high discharge under moderate coastal water level, or the reverse) to be accommodated naturally. We develop the formulation in the frequency domain for both exchangeable and non-exchangeable copulas, provide a direct extension to higher dimensions, and demonstrate it on the symmetric and asymmetric versions of the Clayton and Gumbel copulas, taken as representative models of lower- and upper-tail dependence. The resulting construction is simple to implement and particularly suited to early-warning systems, in which a hazardous combination of drivers must be recognized as it develops.