Dynamic mode decomposition of extreme events

Abstract. Most data-driven methods, among them Dynamic Mode Decomposition (DMD), focus on analysing and reconstructing the average behaviour of a system. However, the primary interest often lies in the anomalous behaviour, known as extreme events. This is especially the case in climate research, where extreme events have significant economic and societal costs. Therefore, we extend a DMD method to account for extreme events by adding a penalisation term. This extension allows us to not only better reconstruct the extreme events, but also extract the spatiotemporal structures related to those extreme events. DMD was originally developed by Schmid and Sesterhenn (Schmid and Sesterhenn, 2008) to enable the fluid dynamics community to identify spatiotemporal coherent structures (called modes) from high-dimensional data. In its essence DMD uses most relevant modes to filter the noise and reconstruct the original signal. We ask “Is the noise really noise”! Or can we attribute some of these dynamic modes, that result from the DMD, to extreme events? We applied this new method to the climate system, well known for its high-dimensionality. As a proof of concept, we applied the method to two well-studied European heatwaves: those of 2003 and 2010. Across both cases, our extreme DMD improves reconstruction accuracy at extreme spatiotemporal points, achieving a 0.45 %–0.85 % relative reduction in error compared with standard DMD, a difference that is small in magnitude but statistically significant. The approach also reveals coherent spatial modes that contribute specifically to the development of heat extremes. This framework represents a general extension of DMD and can be applied to other high-dimensional dynamical systems where extreme events are of interest.