Aggregating flood damage functions: The peril of Jensen's gap

Flood risk models provide important information for disaster planning through estimating flood damage to exposed assets, such as houses. At large scales, computational constraints or data coarseness often lead modelers to aggregate asset data using a single statistic (e.g., the mean) prior to applying non-linear damage functions. This practice of aggregating inputs to nonlinear functions introduces error and is known as Jensen's inequality; however, the impact of this practice on flood risk models has so far not been investigated. With a Germany-wide approach, we isolate and compute the error resulting from aggregating four typical concave damage functions under 12 scenarios for flood magnitude and aggregation size. In line with Jensen's 1906 proof, all scenarios result in an overestimate, with the most extreme scenario of a 1 km aggregation for the 500-year flood risk map yielding a country-wide average bias of 1.19. Further, we show this bias varies across regions, with one region yielding a bias of 1.58 for this scenario. This work applies Jensen's 1906 proof in a new context to demonstrate that all flood damage models with concave functions will introduce a positive bias when aggregating and that this bias can be significant.