The hazard produced by natural phenomena on infrastructure and urban populations has been widely studied in the last 50 years. Researchers have recognised that the real danger posed by these phenomena depends on their extreme values. Most researchers focus on the extremes of natural phenomena considered in isolation, one variable at a time. However, what is relevant in hazard studies is coincident extremes of several climatic variables, i.e., the presence of compound extremes. The peak value of these extremes seldom coincides, but off-peak values located in the tail of the distributions are often concurrent and can lead to catastrophic events. What is essential in hazard studies is to calculate the probabilistic distribution of the extremes of coincident climatic variables. The presence of correlations between these variables complicates the problem. This paper presents a computationally efficient and robust mathematical methodology to solve the problem. The procedure is based on the convolution of the distributions of the climatic variables. Once the probabilistic distribution of the compound variables is found, it is possible to calculate the curves of the return period, which is the indicator of importance in hazard and risk studies. This compound Return Period is computed using the Statistics of Extreme Values. To illustrate the problem, the case of a cyclone landing close to a low-gradient coastal city is discussed, and its probability of flooding and recurrence period is calculated. We show that the failure to correctly model the correlation between variables can result in overestimating the Return Period curve, consequently increasing mitigation costs.
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