In offshore engineering design, nonlinear wave models are often used to propagate stochastic waves from an input boundary to the location of an offshore structure. Each wave realization is typically characterized by a high-dimensional input time series, and a reliable determination of the extreme events is associated with substantial computational effort. As the sea depth decreases, extreme events become more difficult to evaluate. We here construct a low-dimensional characterization of the candidate input time series to circumvent the search for extreme wave events in a high-dimensional input probability space. Each wave input is represented by a unique low-dimensional set of parameters for which standard surrogate approximations, such as Gaussian processes, can estimate the short-term exceedance probability efficiently and accurately. We demonstrate the advantages of the new approach with a simple shallow-water wave model based on the Korteweg-de Vries equation for which we can provide an accurate reference solution based on the simple Monte Carlo method. We furthermore apply the method to a fully nonlinear wave model for wave propagation over a sloping seabed. The results demonstrate that the Gaussian process can learn accurately the tail of the heavy-tailed distribution of the maximum wave crest elevation based on only $1.7%$ of the required Monte Carlo evaluations.
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