Coherent design-life risk under climate nonstationarity: trend-structure selection and fusion of observations, climate-model projections, and scenario uncertainty

The Design Life Level (DLL) proposed by Rootzén and Katz (2013) — the level exceeded with a specified probability over a specified design life period — replaces the stationary return period as the risk quantity for engineering design under a changing climate, but the paper explicitly leaves open how such a quantity should actually be estimated — which nonstationary trend structure to adopt when competing specifications fit the observational record equally well yet extrapolate to very different future hazards; how to use climate-model output whose projections of extremes are biased and coarse relative to engineering scales; and how to fold in scenario uncertainty about future human activity — and, at the design layer, how the probability of failure to be specified should be determined through costs and benefits over the design life. A decade of follow-up work has refined the risk concepts, exposed structural uncertainty as dominant, and produced case-study fusion frameworks, yet reviews through 2026 still report no consensus methodology. The problem is the paper's open estimation-and-integration question: how should these uncertainties be handled when the Design Life Level is estimated in practice? The paper states that there are no general rules for doing so.

Background

Engineering design against hydrologic and other environmental loads — dikes, dams, sewers, bridges — has traditionally used the return period and return level. If $F$ is the distribution of, say, the annual maximum rainfall in a stationary climate (one whose probability distributions do not change over time), the $T$-year return level $u_T$ is the $(1-1/T)$-quantile of $F$, so that on average one year in $T$ experiences an exceedance; the probability of at least one exceedance of $u_T$ over an $N$-year service life is then $1-(1-1/T)^{N}$. Extreme value theory (EVT), whose block-maxima limit result gives the generalized extreme value (GEV) family as the asymptotic model for annual maxima, supplies the statistical machinery for estimating such quantiles from data. These concepts presuppose stationarity, and climate change — driven by both global warming and local land-use change — breaks that assumption: a level with a 100-year return period in today's climate may be exceeded far more often in a future climate, and a structure's risk depends on exactly which years it is in service. Rootzén and Katz (2013) therefore proposed the Design Life Level, written $\mathrm{DLL}_{n}(p)$: for a design life period of $n$ years and a specified failure probability $p$, it is the level whose probability of being exceeded at least once during those $n$ years equals $p$; a Minimax variant instead caps the exceedance probability of any single year in the period. The definition fixes what should be communicated — the pair (design life period, probability of failure) — but not how to compute it in practice. Estimation requires a nonstationary extreme value model, in which a GEV parameter such as the location or log-scale is expressed as a function of time or of a physical covariate (an explanatory variable such as surface air temperature). Rootzén and Katz flag three estimation obstacles plus one design-layer obstacle. First, trend-structure (model) choice: a linear trend, a quadratic trend, or a different functional form can be almost indistinguishable over the observational record yet imply very different future hazards, and the paper states there are no general rules for handling this. Second, climate-model output: global climate models (GCMs; the coupled atmosphere–ocean simulations coordinated internationally through the Coupled Model Intercomparison Project, CMIP, whose recent phases are labeled CMIP5 and CMIP6) project future climate, but their extremes are biased and too coarse at the spatial and temporal scales engineering design needs, and projections are conventionally conditioned on scenarios of future human activity (emission pathways such as the older Representative Concentration Pathways, RCPs, or the current Shared Socioeconomic Pathways, SSPs) whose likelihoods are themselves unknown. Third, coherent combination: observations, climate-model ensembles, and scenario uncertainty must be merged into one risk estimate — the paper suggests, without developing, a Bayesian approach that combines historical trends with ensemble projections. Fourth, the design layer: choosing the probability of failure to specify in the DLL requires integrating expected costs and benefits over the design life, which the paper leaves as an open design problem.

Problem Statement

As posed by Rootzén and Katz (2013, Sections 6–7), left open after their proposal of the Design Life Level: in a changing climate, how should one actually obtain a coherent, quantitatively defensible risk estimate for engineering design — a Design-Life-Level-type level $\mathrm{DLL}_{n}(p)$, i.e., the level exceeded with specified probability $p$ over a specified design life period of $n$ years, where $n$ and $p$ are inputs specified by the decision-maker — given the linked sources of model uncertainty that its estimation involves and that the paper leaves open, stating that there are no general rules for handling them? (i) Trend-structure selection: a linear, quadratic, or other functional form (or covariate link) for the nonstationary extreme value model (an extreme value model whose parameters are made to vary with time or with a physical covariate, i.e., an explanatory variable such as surface air temperature) can fit the observational record almost equally well yet imply very different future hazards; what principled rule selects, weights, or otherwise propagates this structural uncertainty into the risk estimate? (ii) Climate-model information: how should direct climate-model output be used to estimate nonstationary extremes at the spatial and temporal scales engineering design requires, given that model projections of extremes are biased and coarse at those scales and are conditioned on scenarios of future human activity whose likelihoods are themselves unknown? (iii) Coherent combination: how should at-site observations, climate-model ensembles, and scenario uncertainty — the unknown future course of human activity that drives the scenarios — be merged into one risk estimate? These questions are interrelated: the source poses them together as the estimation problem for which it states there are no general rules.

Separately, the source notes a design-layer open problem standing outside this statistical estimation task: determining the optimal probability of failure to be specified in the DLL requires integrating expected costs and benefits over the design life period, within design processes that are also constrained by political and legal systems.

References

1. Rootzén, H., & Katz, R. W. (2013). Design Life Level: Quantifying risk in a changing climate. Water Resources Research, 49(9), 5964–5972. doi:10.1002/wrcr.20425, https://doi.org/10.1002/wrcr.20425
2. Read, L. K., & Vogel, R. M. (2015). Reliability, return periods, and risk under nonstationarity. Water Resources Research, 51(8), 6381–6398. doi:10.1002/2015WR017089, https://doi.org/10.1002/2015WR017089
3. Serinaldi, F. (2015). Dismissing return periods! Stochastic Environmental Research and Risk Assessment, 29(4), 1179–1189. doi:10.1007/s00477-014-0916-1, https://doi.org/10.1007/s00477-014-0916-1
4. Serinaldi, F., & Kilsby, C. G. (2015). Stationarity is undead: Uncertainty dominates the distribution of extremes. Advances in Water Resources, 77, 17–36. doi:10.1016/j.advwatres.2014.12.013, https://doi.org/10.1016/j.advwatres.2014.12.013
5. Cheng, L., & AghaKouchak, A. (2014). Nonstationary precipitation intensity-duration-frequency curves for infrastructure design in a changing climate. Scientific Reports, 4, 7093. doi:10.1038/srep07093, https://doi.org/10.1038/srep07093
6. Kim, H., Kim, S., Shin, H., & Heo, J.-H. (2017). Appropriate model selection methods for nonstationary generalized extreme value models. Journal of Hydrology, 547, 557–574. doi:10.1016/j.jhydrol.2017.02.005, https://doi.org/10.1016/j.jhydrol.2017.02.005
7. Qi, W. (2017). A non-stationary cost-benefit analysis approach for extreme flood estimation to explore the nexus of 'Risk, Cost and Non-stationarity'. Journal of Hydrology, 554, 128–136. doi:10.1016/j.jhydrol.2017.09.009, https://doi.org/10.1016/j.jhydrol.2017.09.009
8. Schlef, K. E., François, B., Robertson, A. W., & Brown, C. (2018). A general methodology for climate-informed approaches to long-term flood projection — illustrated with the Ohio River basin. Water Resources Research, 54(11), 9321–9341. doi:10.1029/2018WR023209, https://doi.org/10.1029/2018WR023209
9. Salas, J. D., Obeysekera, J., & Vogel, R. M. (2018). Techniques for assessing water infrastructure for nonstationary extreme events: a review. Hydrological Sciences Journal, 63(3), 325–352. doi:10.1080/02626667.2018.1426858, https://doi.org/10.1080/02626667.2018.1426858
10. Serinaldi, F., Kilsby, C. G., & Lombardo, F. (2018). Untenable nonstationarity: An assessment of the fitness for purpose of trend tests in hydrology. Advances in Water Resources, 111, 132–155. doi:10.1016/j.advwatres.2017.10.015, https://doi.org/10.1016/j.advwatres.2017.10.015
11. Buchanan, M. K., Kopp, R. E., Oppenheimer, M., & Tebaldi, C. (2016). Allowances for evolving coastal flood risk under uncertain local sea-level rise. Climatic Change, 137(3-4), 347–362. doi:10.1007/s10584-016-1664-7, https://doi.org/10.1007/s10584-016-1664-7
12. Read, L. K., & Vogel, R. M. (2016). Hazard function analysis for flood planning under nonstationarity. Water Resources Research, 52(5), 4116–4131. doi:10.1002/2015WR018370, https://doi.org/10.1002/2015WR018370
13. François, B., Schlef, K. E., Wi, S., & Brown, C. M. (2019). Design considerations for riverine floods in a changing climate — a review. Journal of Hydrology, 574, 557–573. doi:10.1016/j.jhydrol.2019.04.068, https://doi.org/10.1016/j.jhydrol.2019.04.068
14. Volpi, E. (2019). On return period and probability of failure in hydrology. WIREs Water, 6(3), e1340. doi:10.1002/wat2.1340, https://doi.org/10.1002/wat2.1340
15. Byun, K., & Hamlet, A. F. (2020). A risk-based analytical framework for quantifying non-stationary flood risks and establishing infrastructure design standards in a changing environment. Journal of Hydrology, 584, 124575. doi:10.1016/j.jhydrol.2020.124575, https://doi.org/10.1016/j.jhydrol.2020.124575
16. Yan, L., Xiong, L., Jiang, C., Zhang, M., Wang, D., & Xu, C.-Y. (2021). Updating intensity–duration–frequency curves for urban infrastructure design under a changing environment. WIREs Water, 8(3), e1519. doi:10.1002/wat2.1519, https://doi.org/10.1002/wat2.1519
17. Slater, L. J., Anderson, B., Buechel, M., Dadson, S., Han, S., Harrigan, S., Kelder, T., Kowal, K., Lees, T., Matthews, T., Murphy, C., & Wilby, R. L. (2021). Nonstationary weather and water extremes: a review of methods for their detection, attribution, and management. Hydrology and Earth System Sciences, 25(7), 3897–3935. doi:10.5194/hess-25-3897-2021, https://doi.org/10.5194/hess-25-3897-2021
18. Awasthi, C., Archfield, S. A., Ryberg, K. R., Kiang, J. E., & Sankarasubramanian, A. (2022). Projecting flood frequency curves under near-term climate change. Water Resources Research, 58(8), e2021WR031246. doi:10.1029/2021WR031246, https://doi.org/10.1029/2021WR031246
19. Coelho, G. de A., Ferreira, C. M., Johnston, J., Kinter, J. L., Dollan, I. J., & Maggioni, V. (2022). Potential impacts of future extreme precipitation changes on flood engineering design across the contiguous United States. Water Resources Research, 58(4), e2021WR031432. doi:10.1029/2021WR031432, https://doi.org/10.1029/2021WR031432
20. Hu, Y., Liang, Z., Peng, A., Wang, K., Wang, J., & Li, B. (2023). Nonstationary hydrological frequency analysis in light of model parameters and climate projections uncertainty. Journal of Hydrology, 617, 129120. doi:10.1016/j.jhydrol.2023.129120, https://doi.org/10.1016/j.jhydrol.2023.129120
21. Laux, P., Weber, E., Feldmann, D., & Kunstmann, H. (2023). The robustness of the derived design life levels of heavy precipitation events in the pre-alpine Oberland region of southern Germany. Atmosphere, 14(9), 1384. doi:10.3390/atmos14091384, https://doi.org/10.3390/atmos14091384
22. Xiong, B., Zheng, S., Ma, Q., Fu, C., Wen, T., He, Z., Li, L., & Xu, C.-Y. (2024). Robustness of design flood estimates under nonstationary conditions: parameter sensitivity perspective. Stochastic Environmental Research and Risk Assessment, 38(6), 2297–2314. doi:10.1007/s00477-024-02680-9, https://doi.org/10.1007/s00477-024-02680-9
23. Barbaux, O., Naveau, P., Bertrand, N., & Ribes, A. (2025). Integrating non-stationarity and uncertainty in design life levels based on climatological time series. Weather and Climate Extremes, 50, 100807. doi:10.1016/j.wace.2025.100807, https://doi.org/10.1016/j.wace.2025.100807
24. Bhushan, M. (2026). A review on nonstationary flood frequency analysis under climate and catchment change. NG Civil Engineering, 2(1), 45–54. doi:10.66132/ngce20260105, https://doi.org/10.66132/ngce20260105