Visualization is an essential operation when assessing the risk of rare events such as coastal or river floodings. The goal is to display a few prototype events that best represent the probability law of the observed phenomenon, a task known as quantization. It becomes a challenge when data is expensive to generate and critical events are scarce, like extreme natural hazard. In the case of floodings, each event relies on an expensive-to-evaluate hydraulic simulator which takes as inputs offshore meteo-oceanic conditions and dyke breach parameters to compute the water level map. In this article, Lloyd's algorithm, which classically serves to quantize data, is adapted to the context of rare and costly-to-observe events. Low probability is treated through importance sampling, while Functional Principal Component Analysis combined with a Gaussian process deal with the costly hydraulic simulations. The calculated prototype maps represent the probability distribution of the flooding events in a minimal expected distance sense, and each is associated to a probability mass. The method is first validated using a 2D analytical model and then applied to a real coastal flooding scenario. The two sources of error, the metamodel and the importance sampling, are evaluated to quantify the precision of the method.

翻译：可视化是评估如沿海或河流洪水等稀有事件风险时的关键步骤。其目标是展示最能代表观测现象概率规律的少数典型事件，这一任务被称为量化。当数据生成成本高昂且关键事件（如极端自然灾害）稀缺时，量化成为一项挑战。在洪水案例中，每个事件依赖于一个计算成本高昂的水力模拟器，该模拟器以近海气象海洋条件和堤坝溃口参数为输入，计算水位图。本文对经典用于数据量化的Lloyd算法进行了改进，使其适应稀有且观测成本高昂的事件。通过重要性采样处理低概率问题，同时结合函数主成分分析与高斯过程应对高成本的水力模拟。计算出的典型地图以最小期望距离意义表征洪水事件的概率分布，每个地图关联一个概率质量。该方法首先使用二维解析模型进行验证，随后应用于一个真实沿海洪水场景。通过评估元模型和重要性采样两种误差来源，量化了方法的精确度。