Event-based rainfall–runoff models used for design flood estimation and flood forecasting represent rainfall losses through parsimonious conceptual formulations whose parameters are calibrated on observed events, yet the engineering target is often floods far larger than any calibration event. O'Shea, Nathan, Wasko and Hill (2021) showed that four widely used two-parameter loss models — Initial Loss/Continuing Loss (ILCL), Initial Loss/Proportional Loss (ILPL), the SCS Curve Number, and the Probability Distributed Model — diverge sharply in extrapolation: ILCL, whose ongoing loss is a constant rate independent of rainfall magnitude, shows minimal dependence of performance on event magnitude, while ILPL, SCSCN and PDM, whose ongoing loss scales with rainfall, underpredict with a bias that grows with event size. The open question, stated in that study and still untested, is whether more complex infiltration-based loss formulations whose loss rates decrease during a storm independently of rainfall magnitude — such as the Green–Ampt and Horton equations — can improve both the simulation of historic events and, critically, extrapolation to floods larger than the calibration events, without their additional parameters introducing new equifinality problems. Joint-probability design flood estimation, ensemble operational forecasting and the non-stationarity of a warming climate are the applications and the motivation for which that extrapolation behaviour matters in the study.

Background

Event-based rainfall–runoff models are the workhorse of engineering design flood estimation and operational flood forecasting. Instead of simulating a catchment's water balance continuously, they simulate individual storm events: observed or design rainfall is converted into a flood hydrograph by first subtracting rainfall "losses" — the depth of rainfall intercepted, stored on the surface, or infiltrated that never appears as runoff — and then routing the remaining rainfall excess to the catchment outlet, usually with a lumped representation in which the catchment is treated as a single unit. The loss component is typically a parsimonious conceptual model with two or three calibrated parameters. Standard choices include: Initial Loss/Continuing Loss (ILCL), in which a fixed depth is absorbed before runoff begins and a constant continuing loss rate is deducted thereafter, so ongoing loss is independent of rainfall magnitude; Initial Loss/Proportional Loss (ILPL), in which a fixed fraction of rainfall is lost at each time step after the initial loss; the Soil Conservation Service Curve Number method (SCSCN), a US empirically derived relation between cumulative event rainfall, an abstract initial abstraction, and runoff depth; and the Probability Distributed Model (PDM), which represents catchment storage as a set of soil moisture stores whose capacities follow a probability distribution.

Because parameters are calibrated on the modest floods that happen to have been observed, while design applications target much rarer events (in Australia, dam safety assessments may require floods with annual exceedance probability — the chance per year of being exceeded — of 1 in 100 to beyond 1 in 10,000), extrapolation behaviour is decisive: how does a loss model behave, with parameters held fixed, when driven by rainfall far exceeding the calibration events? A second concern is equifinality — the situation, familiar from calibration of conceptual hydrological models, in which many different parameter sets reproduce observed events equally well, so that individual parameters are not identifiable and calibrated values cannot be trusted outside the calibration regime.

O'Shea et al. (2021) investigated these questions empirically. They embedded the four two-parameter loss models (ILCL, ILPL, SCSCN, PDM) in a common lumped event-based model, calibrated to 876 observed storm events across 35 small, rural Australian catchments, then fixed each model's parameters to "typical" catchment values (the median of observed parameter ranges, with each catchment's largest five events excluded) and re-simulated the record. ILCL displayed minimal dependence of performance (measured by the Nash–Sutcliffe efficiency, NSE, of simulated versus observed hydrographs) on event magnitude, whereas ILPL, SCSCN and PDM all underpredicted with a bias that increased with event size — behaviour traced to their representing ongoing loss as a proportion of rainfall. They concluded that loss models whose ongoing loss is independent of event magnitude are better suited to applications requiring extrapolation, a conclusion that becomes more consequential as climate change widens the gap between calibration magnitudes and the extremes that must be estimated. The study also reports severe parameter interaction in the two-parameter PDM, noting that multi-event calibration strategies that hold the PDM's maximum storage-capacity parameter (Cmax) fixed across events can reduce it, at the cost of higher-dimensional calibration.

The Green–Ampt and Horton equations are physically based descriptions of infiltration at a point: the Green–Ampt model represents ponded infiltration into soil behind a sharp wetting front, with infiltration rate f=K (1+Δθ S/F)f = K\,(1 + \Delta\theta\,S/F) depending on saturated hydraulic conductivity KK, wetting-front suction head SS (so that the product Δθ S\Delta\theta\,S is a length), soil moisture deficit Δθ\Delta\theta, and cumulative infiltration FF; the Horton model is an empirical exponential decay of infiltration capacity with time, f(t)=fc+(f0−fc) e−ktf(t) = f_c + (f_0 - f_c)\,e^{-kt}. In both, the loss rate decreases during a storm as the soil wets and is not proportional to rainfall intensity — structurally closer to the magnitude-independent behaviour of ILCL — but they carry more parameters than the tested two-parameter models and are point/plot-scale constructs that must be adapted for lumped catchment-scale use, raising calibration and identifiability questions of their own.

Problem Statement

As stated in the Discussion of O'Shea et al. (2021), it remains to be tested whether more complex infiltration-based loss formulations that also represent ongoing loss independently of rainfall — such as the Green–Ampt or Horton equations (the study also names the Modified Holton–Overton equation), whose infiltration rates decrease during a storm — can improve the simulation of observed flood events and, critically, whether their additional parameters translate into better extrapolation performance for events larger than those used for calibration, without introducing new equifinality problems. The single overarching objective is to determine which loss-model structures can be trusted when an event-based rainfall–runoff model is applied to floods beyond the calibration record; the requested results — (i) reproduction of observed events, (ii) magnitude-stratified extrapolation performance with parameters fixed at typical catchment values, and (iii) parameter identifiability and interaction diagnostics for the added infiltration parameters — are the jointly necessary components of that objective, since a structure that simulates well but extrapolates poorly, or extrapolates well but with non-identifiable parameters, fails it.

Two further points of context: joint-probability design flood estimation and ensemble operational forecasting are the applications for which the study's magnitude-independence finding has implications, and the non-stationarity of a warming climate is the motivation that makes extrapolation behaviour more pertinent.

References

  1. O'Shea D, Nathan R, Wasko C, Hill P (2021). Implications of event-based loss model structure on simulating large floods. Journal of Hydrology 595:126008. doi:10.1016/j.jhydrol.2021.126008, https://doi.org/10.1016/j.jhydrol.2021.126008
  2. Nathan R (2025). Improving event-based methods for modelling flood risk in a variable and non-stationary climate. Philosophical Transactions of the Royal Society A 383(2302):20240292. doi:10.1098/rsta.2024.0292, https://doi.org/10.1098/rsta.2024.0292
  3. Ho M, Wasko C, O'Shea D, Nathan R, Vogel E, Sharma A (2023). Changes in flood-associated rainfall losses under climate change. Journal of Hydrology 625:129950. doi:10.1016/j.jhydrol.2023.129950, https://doi.org/10.1016/j.jhydrol.2023.129950
  4. Ho M, Nathan R, Wasko C, Vogel E, Sharma A (2022). Projecting changes in flood event runoff coefficients under climate change. Journal of Hydrology 615:128689. doi:10.1016/j.jhydrol.2022.128689, https://doi.org/10.1016/j.jhydrol.2022.128689
  5. O'Shea D, Nathan R, Wasko C, Ho M, Sharma A (2024). Evaluation of key flood risk drivers under climate change using a bottom-up approach. Journal of Hydrology 640:131694. doi:10.1016/j.jhydrol.2024.131694, https://doi.org/10.1016/j.jhydrol.2024.131694
  6. Ho M, O'Shea D, Wasko C, Nathan R, Sharma A (2025). The impact of climate change on dam overtopping floods in Australia. Hydrology and Earth System Sciences 29(21):5851–5870. doi:10.5194/hess-29-5851-2025, https://doi.org/10.5194/hess-29-5851-2025
  7. Dykman C, Wasko C, Nathan R, Sharma A (2026). Robust Calibration of Hydrological Models for Simulating Unseen Flood Extremes. Water Resources Research 62(6):e2025WR040134. doi:10.1029/2025wr040134, https://doi.org/10.1029/2025wr040134
  8. Bahramian K, Nathan R, Western AW, Ryu D (2023). Probabilistic Conditioning and Recalibration of an Event-Based Flood Forecasting Model Using Real-Time Streamflow Observations. Journal of Hydrologic Engineering 28(4):04023003. doi:10.1061/(ASCE)HE.1943-5584.0002236, https://doi.org/10.1061/(ASCE)HE.1943-5584.0002236
  9. Kemp D, Alankarage Guna Hewa (2023). Benchmarking Three Event-Based Rainfall-Runoff Routing Models on Australian Catchments. Hydrology 10(6):131. doi:10.3390/hydrology10060131, https://doi.org/10.3390/hydrology10060131
  10. Gazda F, Nagy ED, Négyesi K, Farkas D (2025). Comparing infiltration loss methods for event-based rainfall-runoff modelling in a Hungarian catchment. Hydrological Sciences Journal. doi:10.1080/02626667.2025.2571452, https://doi.org/10.1080/02626667.2025.2571452