Quantizing Rare Random Maps: Application to Flooding Visualization

Abstract

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. Supplementary materials for this article are available online.

Keywords:

• Flood maps
• Metamodel
• Quantization
• Rare events
• Visualization

Supplementary Materials

Supplementary material description: Note describing the git repository (readme.md).

Codes related to the Campbell2D case: Git repository containing R notebooks to reproduce all the experiments related to the Campbell2D function that are described in the article. (https://github.com/charliesire/quantization_Campbell2D.git)

R package FunQuant: R package containing functions to perform the Prototype Maps Algorithm and compute the associated performance metrics. (https://github.com/charliesire/FunQuant.git)

Importance sampling performance metrics: Note providing details about the computation of the importance sampling performance metrics.(is_perf_metrics.pdf).

Complexity of the Prototype Maps Algorithm: Note explaining the computation of the complexity of the Prototype Maps Algorithm (complexity_pma.pdf).

Disclosure Statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.

Acknowledgments

This research was conducted with the support of IRSN and BRGM, through the consortium in Applied Mathematics CIROQUO (https://doi.org/10.5281/zenodo.6581217), gathering partners in technological and academia in the development of advanced methods for Computer Experiments.

Notes

1 data available on data.shom.fr

2 see for example, https://sealevel.nasa.gov/data_tools/17)

Additional information

Funding

Bureau de Recherches Géologiques et Minières; Institut de Radioprotection et de SÛreté Nucléaire; Chair Ciroquo;