Our case study site is the river Mangfall at gauge Rosenheim, shortly before it flows into the Inn river. Rosenheim is a city in Bavaria that has suffered severe flood losses from Mangfall flooding in the past (Wasserwirtschaftsamt Rosenheim, 2014). With an area of 1102 km2, the Mangfall is a medium-sized catchment exhibiting a highly heterogeneous topography. Elevations within the catchment range from 443 to 1988 m a.s.l. with a mean value of approximately 1000 m a.s.l., indicating the pre-alpine nature of the river basin. Southern sub-catchments in the Mangfall mountains are steep and rocky, resulting in a rapid runoff response. On the contrary, northern regions in the Alpine foothills show a more moderate discharge behavior due to gentle slopes. Thus, the discharge pattern of the Mangfall combines both characteristics of mountainous and lowland areas (Kunstmann and Stadler, 2005; RMD Consult, 2016; Magdali, 2015).

Precipitation in the catchment is strongly affected by the adjacent Alpine arch leading to annual mean amounts of 1800 mm in mountainous and 1000 mm in low-altitude zones. The watershed receives most precipitation in July, often in form of convective, high-intensity precipitation (Magdali, 2015; Deutscher Wetterdienst, 2018). Therefore, this study focusses on the uncertainty analysis for summer discharge, since it poses the greatest threat to the city of Rosenheim. Planning authorities give the 100-year design discharge at the Rosenheim gauge as 480 m3 s-1 (RMD Consult, 2016). Figure 2 shows the topography of the Mangfall catchment alongside its river network. The available historic record at the Mangfall gauge in Rosenheim is reproduced in Supplement S1.

Table 1 lists the projections available at the case study gauge. Several projections have identical modeling chains and differ only in the model run; 6 of the 10 regional climate models (RCMs) are nested in the same global climate model (GCM), ECHAM5; and all GCM–RCMs are based on the same IPCC SRES (Special Report on Emissions Scenarios) emission scenario, A1B. Furthermore, all climate models are coupled to the same hydrological model (WaSiM) and same downscaling technique (quantile mapping). The ensemble is limited in that it does not cover a wide range of modeling uncertainties, and it is imperfect in that the projections of the ensemble are not independent. Such a set of available projections is quite typical of what is encountered in flood protection planning. The projections are reproduced in Supplement S2.

The term “internal variability” describes the irreducible uncertainty component in extreme discharge: even with perfect knowledge, it cannot be predicted deterministically what the annual maximum discharge of a year will be, and thus how the design flood estimate will change. This is because discharge realizations occur spontaneously, due to interactions of components within the climate system (IPCC, 2013). In the available projections, the absolute amount of internal variability did not change in time significantly and is thus modeled as stationary. In projections of future discharge, however, the relative importance of internal variability decreases with time as climatic uncertainties increase with increasing projection horizon. In a small pre-alpine catchment, such as considered in our case studies, the internal variability is large and dominates the uncertainty spectrum, potentially masking existing trend signals in heavy precipitation (and thus extreme discharge) for the entire projection horizon up to the year 2100 (Maraun, 2013). Alternative terms for the internal variability are “inherent randomness” or “noise”.

Discharge projections are the result of a complex multistep climate modeling process. In the literature, this is often termed the climate modeling “chain”, which, as new uncertainties are introduced at each modeling step, leads to the “uncertainty cascade” (Mitchell and Hulme, 1999; Foley, 2010). It is worth pointing out that the uncertainty cascade does not necessarily lead to an increase in uncertainty at each step, as the modeling steps depend on each other in a nonlinear fashion. Just as uncertainties can add up, it is conceivable that they may not be relevant for future steps in the modeling chain (Refsgaard et al., 2013). In the following, we briefly introduce the individual modeling steps required to obtain projections of (extreme) discharge. The uncertainty from the interaction of consecutive steps in the modeling chain is called “interaction uncertainty” (Bosshard et al., 2013). The uncertainties in the climate modeling chain are in principle epistemic, yet it is debatable whether they can and will be reduced in the foreseeable future (Hawkins and Sutton, 2009).

The forcing of the climate through greenhouse gas emissions (GHGs) is the first element in the climate modeling chain. The future socioeconomic, political and technological development determines the amount of GHGs emitted. The SRES scenarios – first introduced by the IPCC in 2000 (IPCC, 2000) – were recently substituted by representative concentration pathways (RCPs), which directly refer to the amount of GHGs emitted rather than complex scenarios (Moss et al., 2010). For our case study, only projections based on SRES scenario A1B, a widely used scenario with moderate socioeconomic and technological changes, are available. Thus, we have to take into account the uncertainty of what the projection results might have been under other forcing scenarios. However, in Europe, forcing uncertainty only becomes relevant in the far future and is of particularly low significance for local extreme precipitation (Hawkins and Sutton, 2011; Maraun, 2013; Tebaldi et al., 2015).

For climate change impact studies, it is customary to use ensembles of multiple combinations of global and regional climate models (GCMs–RCMs) (Huang et al., 2014; Muerth et al., 2012; Rajczak et al., 2013). The differences in GCM–RCM output when driven by the same emission forcing are termed “model response uncertainty” or “model spread” (IPCC, 2013). Multi-model ensembles (MMEs) such as the one available for the case study reproduce part of this spread. They do not reproduce it completely because they consist of a finite number of possibly biased and dependent models that typically have to be chosen based on availability rather than on statistical considerations (Knutti et al., 2013; Tebaldi and Knutti, 2007). To mitigate this problem, some researchers assign weights to individual models, but there is an ongoing debate about this: some researchers are making a general case for the benefits of weighting (Ylhäisi et al., 2015) or its drawbacks (Aghakouchak et al., 2013), some are detailing when it may make sense on the basis of model performance (Refsgaard et al., 2014; Rodwell and Palmer, 2007) or genealogy (Masson and Knutti, 2011), but all approaches are disputed. The relative importance of model response increases with projection lead time and is particularly significant for extreme summer precipitation (Bosshard et al., 2013). Since flooding in the case study catchment is dominated by extreme summer precipitation, we expect model response to form the second most important uncertainty contribution (after internal variability).

The available projections underwent statistical downscaling using quantile mapping, which is often recommended for extreme events (Bosshard et al., 2011; Dobler et al., 2012; Hall et al., 2014; Themeßl et al., 2010). Statistical downscaling is frequently used to align GCM–RCM outputs with historic records, but its use is still controversial (Chen et al., 2015; Ehret et al., 2012; Huang et al., 2014; Maurer and Pierce, 2014). The uncertainty contribution of the downscaling is likely to be large (Hundecha et al., 2016; Sunyer et al., 2015b). It would be beneficial to use not one but several downscaling techniques, similarly to how one uses an ensemble of GCM–RCMs (Arnbjerg-Nielsen et al., 2013; Sunyer et al., 2015a), as well as several calibration datasets (Sunyer et al., 2013a).

Up to and including statistical downscaling, the climate modeling chain produces not discharge but various other climatic variables that are translated to discharge in a specific catchment through a hydrological model. Catchment parameters (such as surface roughness) are typically found in an elaborate calibration procedure (Labarthe et al., 2014; Li et al., 2012). The parameters are usually assumed to be stationary, but they might in fact be non-stationary (Merz et al., 2011). Furthermore, the calibration might mask model errors by tuning the catchment parameters to balance them. Thus, the parameter estimates strongly depend on the calibration period (Brigode et al., 2013). Several approaches exist to quantify the uncertainty stemming from the hydrological model (Götzinger and Bárdossy, 2008; Velázquez et al., 2013). Overall, however, the error from the choice of hydrological model is small, in particular for high-flow indicators (Velázquez et al., 2013). It is likely smaller than or comparable to forcing uncertainty (Wilby, 2005).

In this section, we estimate the relative contribution of climatic uncertainties, using internal variability as a reference. Note that this is done as a rough estimate, since uncertainty quantification is not the focus of this paper. As will become clear in Sects. 4.2 and 5; an exact quantification is also not necessary for the proposed decision-making process. To summarize the previous two sections, the following qualitative statements can be made about the contribution of relevant sources of uncertainty in the considered midsize pre-alpine catchments with floods driven by summer precipitation:

internal variability is dominant throughout most of the coming century;

A methodology to quantify the size of the internal variability, model response and forcing uncertainty in mean precipitation and corresponding results for different regions and seasons has been presented in Hawkins and Sutton (2009, 2011). We base our estimate of these components on equivalent results for summer precipitation in Europe obtained from Ed Hawkins (email communication, 17 February 2017). We consider precipitation results to be transferable to discharge in the given catchment since extreme summer precipitation has in the past been the dominant trigger of high discharge in the Mangfall. A comparison of uncertainty shares for mean versus extreme discharge is available in Bosshard et al. (2013) and is used to adapt the results. Quantitative estimates of the shares of model response, downscaling, hydrological model and interactions for a different pre-alpine catchment are also provided in Bosshard et al. (2013). We combine the quantitative results with the catchment-specific qualitative knowledge to produce the estimate. The uncertainty spectrum is shifted towards the later projection horizon to account for the longer dominance of internal variability in a pre-alpine catchment with small-scale, extreme summer precipitation as the flood-triggering process. This results in a near-term contribution of the internal variability of at least 80 % of total uncertainty, as expected. The shift also reduces the uncertainty share attributed to model response and emission forcing, which, following Ed Hawkins (email communication, 17 February 2017), explained over 90 % of total uncertainty by the end of the century. The shares are adjusted to better represent the particular modeling and topography: the share of model response is set to peak at around 40 %. For downscaling, shares of up to 25 % are expected. Uncertainties stemming from interactions are anticipated to lie on the order of 10 %. Contributions attributed to hydrological modeling are set to remain below 5 % over the whole time horizon. The results of the estimation are shown in Fig. 3. Figure 3a shows the resulting relative uncertainty shares and Fig. 3b the resulting absolute uncertainties for the projection CCLM. Forcing, downscaling, hydrological model and interaction components are hidden uncertainties in the case study. As will be shown in Sect. 3, the sum of hidden uncertainties rather than individual components is used in the Bayesian estimation. Thus, it does not matter if the share of any one of these uncertainties has been slightly over- or underestimated. The question of sensitivity will be discussed further in Sect. 4. The estimated variance shares of the hidden uncertainty components and internal variability with respect to total uncertainty for Rosenheim are given in Supplement S3.

Statistical modeling of extreme discharge Q is commonly based on fitting a suitable extreme value distribution to the available data, e.g., a Gumbel or a generalized extreme value (GEV) distribution. These are described by their probability density function (PDF), fQ|θ(q|θ), in which θ is the set of parameters of the distribution function that are estimated from the data. Estimating θ from finite data will result in a probability distribution over θ, which describes parameter uncertainty (Kennedy and O'Hagan, 2001).

The discharge q(T) of a design flood associated with a return period T is defined as a function of θ as 11-FQ(t)|θq(T)|θ=1T↔q(T):=FQ(t)|θ-11-1T|θ, where FQ(t)|θ is the cumulative distribution function (CDF) and FQ(t)|θ-1 is the inverse CDF of the annual maximum discharge Q(t). In a Bayesian setting, the posterior joint PDF of the parameters θ can be estimated from N years of annual maximum discharges q = [q1, …, qN] (from historic record or projections) as follows: 2fθ|Q(t)(θ|q)∝L(θ|q)fθ(θ), where fθ(θ) is the prior distribution of the parameters and L(θ|q) is the likelihood describing the discharge data q. The discharge maxima can be assumed to be independent between individual years (Coles, 2004). Neglecting measurement error, the likelihood function in Eq. (2) can hence be formulated as 3L(θ|q)=∏t=1NfQ(t)|θqt|θ. With increasing number of records of annual maximum discharges qt, the uncertainty in the parameters θ is reduced.

The Bayesian method requires the selection of a prior distribution fθ(θ) in Eq. (2). For the application to flood protection planning, one may wish to select a prior distribution that is only weakly informative in q(T). We propose to use the following distribution for this purpose (dropping the time dependence t for readability): 4fθ(θ)∝1fQ(T)q(T)=1fQ(T)FQ(T)|θ-11-1T|θ, where fQ(T)(q(T)) is the PDF of q(T) based on a prior distribution that is uniform in θ and Eq. (1) has been applied in the equality.

Different categorizations of uncertainty have been proposed in the literature. In Sect. 2.3–2.6 for example, we have presented the uncertainties in extreme discharge by source. Another common way to categorize uncertainties is the distinction between aleatory (irreducible) and epistemic (reducible) uncertainties (Der Kiureghian and Ditlevsen, 2009; Refsgaard et al., 2013). This categorization is useful in that it underlines in which areas future research could lead to uncertainty reduction. Other authors focus their categorization on the different effects of uncertainties (Merz et al., 2015).

In the context of estimating flood extremes under climate change with limited information, we distinguish between the following:

Visible uncertainty, which is known and can be quantified. For an ensemble of discharge projections, this would, for example, be the internal variability, the model response uncertainty and parameter uncertainty. Parameter uncertainty is also visible in that it is straightforward to quantify, but it is not a climatic uncertainty.

When using discharge projections, it is important to account for uncertainty and bias within them. As discussed in Sect. 2, climatic uncertainties increase with the projection horizon and thus the information value of a projection made late on the horizon is smaller than that of an earlier one. For example, a projection for the year 2100 is associated with higher uncertainty than one that is made for the coming year and should have less weight when estimating the parameters θ of the distribution of annual maximum discharge from climate projections. In the following, we develop a methodology that accounts for this.

We introduce the standard deviation σi,t(u), in which the superscript (u) describes which type of uncertainty is considered (internal or hidden), the subscript i denotes the projection and the subscript t the time dependence. The internal variability in a projection, [σi(internal)]2, can be quantified following Hawkins and Sutton (2009). Note that the subscript t is excluded here since internal variability is assumed to be independent of time. Relative variance shares of the individual uncertainties, including hidden ones, can be estimated using literature (Bosshard et al., 2013; Hawkins and Sutton, 2011) and expert judgement, as was done in Sect. 2.5. The share of an individual uncertainty component in the total variance is here labeled ηt(u), with the indexing as for σ. The uncertainty shares are assumed to be general for a given location, independent of the projection. Thus, the absolute value of the hidden uncertainty can be found from the absolute internal variability and the uncertainty variance shares of Sect. 2.5 (reproduced numerically in Supplement S3) as 5σi,t(hidden)=σi(internal)ηt(hidden)ηt(internal). For estimating the joint PDF of the parameters θ of the annual maximum discharge distribution, we treat the i = 1, …, M discharge projections pi = [pi,t=1, …, pi,t=N′] as samples of the true future discharge τt with a bias Δi,t: τt = pi,t - Δi,t. We express the likelihood Li,t(θ|pi,t, Δi,t) describing the annual maximum discharge of projection i in year t as 6Li,tθ|pi,t,Δi,t=fQ(t)|Θpi,t-Δi,t|θ, where fQ(t)|Θ is the PDF of the extreme value distribution describing Q(t). The likelihood Li,t(θ|pi,t, Δi,t) determines the estimation of the PDF of parameters θ from projections, similar to Eq. (2).

The bias Δi,t is modeled as a normal random variable with zero mean and standard deviation σi,t(hidden): 7Δi,t=zσi,t(hidden)=zσi(internal)ηt(hidden)ηt(internal), with z being a standard normal random variable. By modeling all Δi,t as a function of the same z, it is assumed that the Δi,t are fully dependent within one projection i. This treatment is conservative, since it minimizes the amount of learning from projected discharges. Due to the large impact of the projection on the bias, it is a better depiction of reality than the assumption of independent Δi,t within one projection i. From this follows the likelihood for a complete projection time series pi as 8Liθ|pi=∫-∞∞∏t=1N′fQ(t)|Θpi,t-zσi,t(hidden)|θν(z)dz, where ν is the standard normal PDF. Internal variability is included in Eq. (8) naturally via pi,t, as is parameter uncertainty, which is a function of the length of projections. The estimate of hidden uncertainty, as from Sect. 2.5, is included via σi,t(hidden). While we are focussing on climate uncertainty here, in principle, any kind of additional uncertainty can be incorporated via the hidden uncertainty parameter σi,t(hidden) in Eq. (8). Model response uncertainty is included in the combination of the likelihoods Li(θ|pi) from different projections i, as described in the following section.

Individual projections are not independent. Hence, one cannot combine Li(θ|pi) into a joint likelihood L(θ|p) via a simple product over projections pi. Dependence among multiple projections is due to common model biases, be it because they, for example, share code from the same institution or because our understanding of climate processes is not perfect (Knutti et al., 2013; Tebaldi and Knutti, 2007). Consequently, confidence in the prediction variance should not increase linearly with the number of projections in an ensemble. Instead, the ensemble should be seen as consisting of an effective number I of quasi-independent projections (adding independent pieces of knowledge) that is smaller than the ensemble size M (Pennell and Reichler, 2011; Sunyer et al., 2013b). We thus partition the ensemble into J sets of I projections, where J is the integer quotient of MI. For each of these sets, the likelihood function can then be formulated as the product of the likelihoods Li(j)(θ|pi) of the set members, since they are assumed to contain independent information: 9L(j)(θ|p)=∏i=1ILi(j)θ|pi. Climatological rationale is applied to determine the division of the ensemble into sets: in line with the concept of effective projections, the projections in each set should be as distinct as possible, adding a maximum of additional information.

Based on their genealogy, we partition the available projections (Table 1) as follows.

When using two sets of five effective projections:

Set 1: CLM1, CCLM, REMO2, HadGM, RACMO.

Protection requirements (“criterions”) are based on the T-year discharge q(T) (see Eq. 1), most commonly the 100 year discharge. Since the estimate of q(T) – the peak of the PDF – changes as new data become available, the capacity of the flood protection system will be re-evaluated in the future and possibly be adjusted. The probability that adjustment becomes necessary is determined by the level of uncertainty: the higher the uncertainty in the future extreme discharges, the more likely it is that an adjustment of the protection system will become necessary in the future. To understand why this is, consider Fig. 4: after initial planning, new discharges are observed (lilac dots). If, as pictured here, the observed discharges are higher than expected, the design flood estimate q(T) will increase. (We show the estimate of q(1) to be able to display observations on the same scale. Note that the 99th percentile of the shown PDF does not correspond to q(100).) If the uncertainty is large at the time of initial planning – as is the case here, visualized with the blue, original PDF – then the additional information from the new observation has a larger weight in predicting future extreme discharges. The change in q(T) is larger than if the distribution of extreme discharges had been more informative (i.e., more “certain”, less “spread out”). In practice, the protection will only be adjusted when a significant change in q(T) has occurred that cannot be compensated by the freeboard and planning margin present (represented by the protection level boundary from which onward adjustment is needed). To avoid the need for frequent adjustments and increase robustness, the optimization framework of Dittes et al. (2018) thus recommends a higher planning margin when the system is constructed under higher uncertainty initially, as will become apparent in the results.

Because the – as yet uncertain – future discharge realizations determine future decisions, they have an impact on the optimality of the initial decision. Therefore, it is sensible to model protection planning as sequential, with probabilistic future discharge observations, updating of the discharge PDF and corresponding decisions on adjustment in regular time intervals. A Bayesian network approach doing so for decisions on adapting infrastructure to a changing climate has been presented by Nishijima (2015) and a POMDP (partially observable Markov decision process) approach applied to flood protection, using climate scenarios, has been described by Špačková and Straub (2016). An alternative sampling-based approach, which takes the full joint parameter PDF into account, has been proposed by Dittes et al. (2018). The planning horizon is divided into a number of time periods. After each period, the current protection level is re-evaluated and possibly adjusted based on the annual maximum discharges that have been observed during that period or – more precisely – based on the q(100) as resulting from the updated distribution of annual maximum discharges. To probabilistically model this future updating (before these data are actually available), future realizations of annual maximum discharge q are sampled from the discharge distribution fQ(t)|Θ(q|θ) estimated initially. Optimal decisions are then identified via backwards induction optimization (Raiffa and Schlaifer, 1961), which works by first determining the system that should be installed at the last adjustment, conditional on the existing protection and discharges observed by then. The obtained recommendation is then used to find the system that should be installed at the second to last adjustment and so forth until arriving at a recommendation for the system that should be installed initially. We employ this optimization framework in the following case study.

Note that, since there is often a discrepancy between the level of observed past discharge at a specific gauge and the corresponding regional climate projections, we take the commonly used approach (Fatichi et al., 2013; Pöhler et al., 2012) of computing relative rather than absolute values from the climate projections. Here, this means that we find a planning margin γ based on the projection ensemble and uncertainty estimates from the literature, which may then be applied to the absolute protection (100-year flood) as estimated from historic records.

We conduct our case study for the Mangfall river in Rosenheim, which has been introduced in Sect. 2.1. We consider the designed flood protection systems to have a lifetime of 90 years and to be designed in such a way as to protect from the 100-year flood, with design discharge q(100). The decision on the protection capacity will be revised every 30 years, taking into account the discharge records that will be available at these points in time. When estimating climate parameters – especially trends – from a time step, 30 years is an often used compromise between the desire to minimize statistical uncertainty and that to capture recent climate developments (IPCC, 2013; Kerkhoff et al., 2015; Laprise, 2014; Pöhler et al., 2012). The protection requirement corresponds to the maximal required protection during the time step in question. As in Dittes et al. (2018), a square root function describes the cost of the construction/extension of the protection system and a discounting rate of 2 % annually is employed. In Dittes et al. (2018), we considered a measure of flexibility which describes how costly it is to adapt measures later in their lifetime. In this contribution, we give results for the non-flexible case only, which implies that future adjustments to the system are expensive. Introducing some flexibility into the protection system would lead to lower planning margin results than those obtained here.

Following model plausibility testing on the projections (MacKay, 1992), a GEV distribution is chosen to model the annual maximum discharges. It is described by shape parameter k, scale parameter β>0 and location parameter μ. We employ a linear trend in the scale and location parameters, which is common practice in the literature (Coles, 2004; Delgado et al., 2010; Hanel and Buishand, 2011; Maraun, 2013). The scale is expressed as β = β0 + β1t and the location as μ = μ0 + μ1t (Coles, 2004; Hanel and Buishand, 2011). Thus, θ = (kβ0μ0, β1μ1).

The joint PDF of parameters of annual maximum discharge estimated from the climate projections is used as the basis for future updating with discharge realizations. To obtain this PDF, the climate projections are estimated on a prior distribution that is weakly informative in the 100-year design discharge of the first time step (years 1–30) as by Eq. (4). Computationally, the prior distribution is constructed by uniform sampling of parameters over a large space, computing the respective 100-year flood estimate for the first time step for each sampled parameter vector, and performing rejection sampling to obtain samples on the order of 6 × 105 following Eq. (4).

To find the optimal flood protection considering the full sequential decision process, it is necessary to simulate future discharge data, from which new flood estimates will be estimated (see Sect. 3.4). For this purpose, we used 300 samples of annual maximum discharge in the period 1–30 years and 70 samples of annual maximum discharge in the period 31–60 years. Using fewer discharge samples in later periods is computationally preferable and still comes with a high accuracy, as the absolute number of samples in the second period overall is 300 × 70 = 21 000. This choice of number of samples leads to a relative error of less than 4 % in the protection recommendation.

Figure 5 shows the 100-year discharge PDF (weighted mean) from the initial parameter distribution for the first 30 years of planning when estimated from the 39-year-long historic record versus 10, 5, 3 and 1 effective projections of 90-year length. Ten effective projections correspond to multiplying all posteriors and one effective projection corresponds to averaging all posteriors. For five and three effective projections, we split the projections into sets as given in Sect. 3.3.

The PDFs shown in Fig. 5 are used as input to the optimization framework of Dittes et al. (2018) to obtain recommendations for the planning margin. Section 3.4 gave an intuitive understanding of how these relate to the 100-year PDF. The planning margin that is recommended when estimating based on the historic record only is 111.8 %, versus 81.9, 16.5, 12.5 and 2.6 % for 1, 3, 5 and 10 effective projections, respectively. These results are summarized in Table 2.

Using a similar ensemble of climate projections over Denmark, Sunyer et al. (2013b) established that an ensemble of 10 projections corresponds to 5 effective projections for 20-year heavy summer precipitation. Despite some issues with transferability – as will be discussed in Sect. 5 – we thus use five effective projections and hence a planning reserve of 12.5 % as the recommended protection margin from the extreme summer precipitation floods observed at the Mangfall in Rosenheim.

To investigate the effect of hidden uncertainty on the protection recommendation, we repeated the optimization, once using no hidden uncertainty and once doubling the hidden uncertainty variance shares estimated in Sect. 2.5 (reproduced numerically in Supplement S3), with an effective model number of five. The recommended planning margins lay in the expected order, with the “no uncertainty” recommendation the smallest at 8.1 % and the “double uncertainty” recommendation the largest at 13.8 %. Finally, we studied the effect of changing the trend in the projections of annual maximum discharge. Detrending the projected annual maxima lead to a recommendation of 12.2 %. We then used the projected annual maxima with doubled trend: from the observed average of 0.25 m3 s-1 per year (corresponding to an 11 % rise in mean annual maximum discharge during the 90-year lifetime) to 0.5 m3 s-1 per year. The recommended planning margin increased only very slightly, from 12.5 to 12.7 %. The results are summarized in Table 3.