Dan Rosbjerg

1 Water resources managers leverage the complexity of natural and human water systems for equitable sharing of water among competing users and beneficiaries. Natural water systems, such as rivers, ...

The lumped conceptual rainfall-runoff model NAM was published in Nordic Hydrology (now Hydrology Research) in 1973 in the same issue of the journal as the Swedish HBV model. It was developed at the Technical University of Denmark (DTU), but its worldwide applications can be ascribed primarily to DHI. DHI adopted the model and implemented it in their river modelling software, and it has been used in numerous consulting projects and further developed to become an element in distributed hydrological models.Like HBV, NAM is a soil-moisture accounting model. The overall structure will be briefly presented, and the main difference to HBV highlighted. Also, its role as an element in numerous operational flood forecasting systems worldwide and, more recently, as part of DHI’s global hydrological model will be touched upon. Finally, a discussion of recent developments of machine-learning based rainfall-runoff models will be discussed, which perhaps will bring the era of the lumped conceptual rainfall-runoff models to an end.

For Danish conditions, discrete rain intensity climate factors for a 100-year time horizon are generalized to continuous curves as functions of the return period. Assuming the tails of the distribution functions for T-year events to be approximately exponential, a general formula for projecting current return periods into future return periods is developed. Moreover, the uncertainty of the future T-values due to uncertainty in the climate factor is approximately assessed using first order analysis.

The sensitivity of expected annual damage, EAD, is analytically analysed by applying a log-linear relation between return periods and corresponding damages. It is found that the smallest return period for damage should be estimated as precisely as possible, that the percentage uncertainty in the damage estimate is transformed into the same percentage uncertainty in the EAD estimate, and that it is possible to extrapolate beyond the largest return period with corresponding damage assessment. The precision of the estimate of EAD is investigated in detail in the case of only few available data, and it is found that two different methods for numerical integration may result in strongly diverging results. By applying a piecewise log-linear damage function, it is shown that the log-linear model provides a trustworthy estimate of EAD, also in the case of few available data. Finally, the modifications needed in the special case of threshold exceedance data instead of annual maxima data are presented.

As an alternative to straightforward numerical integration of annual damage costs as function of either the return period or the exceedance probability, the expected annual damage, EAD, can be estimated analytically by using a log-linear model for the damage curve. With this model, it is found that it is important to assess the return period where damage begins as precisely as possible, and that the relative error in damage data gives rise to an error in EAD of the same order of magnitude. Moreover, the model allows for extrapolation above the largest return period for known damage costs. The precision of the EAD estimation is analysed in detail, and it is found that usual numerical integration over either the return period or the exceedance probability can give contradictory results, when only few data is available for the damage function. Using a piecewise log-linear damage function is show to be less sensitive to limited data availability. Finally, the necessary corrections when damage occurs below the 1-year return period are determined.

The distribution functions for large rain events Xc in current climate is denoted F(x) = P{Xc ≤ x} and for large rain events Xf in a future climate G(x) = P{Xf ≤ x}. A climate factor k is introduced, and it is assumed that P{Xc ≤ x} = P{Xf ≤ k x} corresponding to G(k x) = F(x). If we further assume that the distribution functions F and G have exponential tails, the following simple transformation of the return period in current climate Tc to the corresponding return period in future climate Tf can be deducedTf = Tc1/kApplying a first order analysis on this equation with k as independent variable leads to a relation between the uncertainties of k and Tf. In terms of the coefficient of variations we getCV{Tf} ≈ 1/ lnTc CV{k}This equation reveals that even with moderate uncertainty in k, the uncertainty in Tf is notably increased.