Possibilistic response surfaces: incorporating fuzzy thresholds into bottom-up flood vulnerability analysis

Abstract. Several alternatives have been proposed to shift the paradigms of water management under uncertainty from predictive to decision-centric. An often-mentioned tool is the stress-test response surface, mapping system performance to a large sample of future hydro-climatic conditions. Dividing this exposure space between acceptable and unacceptable states requires a criterion of acceptable performance defined by a threshold. In practice, however, stakeholders and decision-makers may be confronted with ambiguous objectives for which the the acceptability threshold is not clearly defined (crisp). To accommodate such situations, this paper integrates fuzzy thresholds to the response surface tool. Such integration is not straightforward when response surfaces also have their own irreducible uncertainty, from the limited number of descriptors and the stochasticity of hydro-climatic conditions. Incorporating fuzzy thresholds therefore requires articulating uncertainties that are different in nature: the irreducible uncertainty of the response itself relative to the variables that describe change, and the ambiguity of the acceptability threshold. We thus propose possibilistic surfaces to assess flood vulnerability with fuzzy acceptability thresholds. An adaptation of the logistic regression for fuzzy set theory combines the probability of acceptable outcome and the ambiguity of the acceptability criterion within a single possibility measure. We use the flood-prone reservoir system of the Upper Saint-Francois River Basin in Canada as a case study to illustrate the proposed approach. Results show how a fuzzy threshold can be quantitatively integrated when generating a response surface, and how ignoring it might lead to different decisions. This study suggests that further theoretical development should link the decision-making under deep uncertainty framework with the existing experience of fuzzy set theory, notably for hydro-climatic vulnerability analysis.