Let us consider a system S that describes precipitation over a time period. The original data are expressed as a time series of precipitation intensity over fixed time steps. This time series alternates between a dry period (no precipitation) and a rainy one. A given event is characterised as dry, extreme or non-extreme with respect to the amount of precipitation during the event.

We denote by E the state space of the system S, with at least three states, i.e. E≥3. Also let D0, D1 and D2 be the non-empty sets of states of dry periods, non-extreme and extreme events, respectively, with D0=1, D1=d1≥1 and D2=d2≥1, i.e. there exists exactly one state for dry periods, D0=Ddry, but a different number of extreme and non-extreme states (d1 and d2, respectively) can be defined for both the non-extreme and extreme events. An extreme event can be further categorised according to the severity of the phenomenon, expressed in terms of the return period of the measured intensity. Non-extreme events can be categorised according to the season in which they appear. Hence, the state space E is partitioned into three disjoint subsets as follows: E=D0∪D1∪D2, where Di∩Dj=∅,i≠j,ij∈0,1,2. We link the non-extreme events to the seasonality of the phenomenon and thus D1=Dwinter,Dspring,Dsummer,Dautumn, that is d1=4. D2 can likewise be partitioned into one or several states appropriate for describing extreme precipitation which may have different return periods or different hydro-climatic origins. In this study, we use a partition based on return periods with D2=D2,D10,D100, referring to states that classify the extremes as either 2-, 10- or 100-year events based on return level.

By definition there is always a dry period between two events, and we assume that there is no dependence between consecutive events. We define the following processes that describe the evolution of a semi-Markov system (Barbu and Limnios, 2008):

J:=Jnn∈N is a Markov chain with state space E, where Jn is the state of the system at the nth event;

There is no unique way to assign a state to an extreme event. In the literature some studies apply hydro-climatic regimes for this classification (Gelati et al., 2010; Svoboda et al., 2016), while others apply event statistics (Madsen et al., 2009; Sørup et al., 2016a). For any given application, the most appropriate classification depends on the data available. In this paper, various methods based on the maximum mean intensities are used to define the event state. For all investigated methods the changes to extremes are evaluated by calculation of IDF curves based on return levels, zis, at event level for a selection of return periods, I (WMO, 2009). The return period (T) of individual events across all intensities is determined using the median plotting position (Rosbjerg, 1988): Tmedian=Ttotal+0.4rank-0.3, where Ttotal is the length of the time series and rank is the rank number of the individual event.

Using data with observations every minute and a minimum dry weather separation between events of 60 min, the mean maximum intensities over 5, 10, 30, 60, 180, 360 and 720 min are calculated for each event. At shorter time frames, e.g. 1 min, the variability of the observed extremes is expected to be very large due to the inherent sampling error (Fankhauser, 1998), and at very long time frames, e.g. 1 day (i.e. 1440 min), the extremes often consist of several events following one another and a different event definition would be necessary to ensure that the real extremes are identified (Madsen et al., 2009). A representative return period for the event is derived based on a mathematical comparison to regional IDF estimates (Fig. 3). This return period is then in turn used to define the state of the event. We test four different selection criteria which define the state of extreme events as either D2,D10 or D100. The selection criteria are listed in Sect. 3.3.

With each event of a time series classified according to a state, the time series can be perturbed using the following methodology linking the time series to the states of the individual events.

Let Rk, k∈N, be the precipitation intensity at time step k and R:=Rkk∈N the corresponding process describing these intensities. The process of perturbed precipitation in each time step k is denoted by R*:=Rk*k∈N.

Similarly to the state space E, we introduce the state space of the change factors, denoted by ECF, ECF=E. We can then write ECF=C0∪C1∪C2, with C0=1, C1=d1 and C2=d2.

We consider the process CF: =CFnn∈N with state space ECF, where CFn is the change factor at the nth event. Let W:=Wkk∈N be the chain, with state space ECF, of change factors in time steps k∈N, that is Wk=CFN(k) with N(k) to be the counting process defined in (Eq. 3). Under the above notation, the original and perturbed sequences of precipitation, Rk and Rk*, are written as Rk*=WkRk. This means that, for a sequence of events, some events will be perturbed more than others, and for extreme cases some might be reduced while others are increased, depending on the local expectations to climate change. Figure 2a shows an example where a non-extreme summer event is perturbed to a lesser volume than the original while a winter non-extreme is increased marginally and both 2- and 100-year extremes are increased considerably more (both in absolute numbers as well as in relative percentages). Figure 2b shows how the state space changes if these four events were to happen chronologically in time with the state jump times marked on the x axis.

The extreme part of precipitation is only expected to constitute a smaller fraction of the total precipitation volume on an annual basis (Sørup et al., 2016b), but as extreme precipitation is often associated with a particular season (see e.g. Sørup et al., 2012), the volumetric part of the extremes might be higher for sub-annual considerations. This implies that situations where the expectations for changes to the extremes are very different from the expectations to changes to seasonal precipitation have to be handled through volumetric corrections in order to accommodate the fact that both expectations to changes in extremes and overall seasonal changes are correct. How to do this best will be very much dependent on the local conditions. In our case this is described in Sect. 3.4.

The evaluation of the perturbed time series is done against the original time series and against the expected changes.

The average percent-wise difference between the perturbed return levels, zi,j,m*, of the modelled time series, Rk*, perturbed with the time-dependent change factors, Wk, against the same return levels, zi,j,m, of the original time series, Rk, multiplied by the target change factor, CFej, can be defined as Φi,j,m=1-zi,j,m*zi,j,mCFej100%, across all IDF points, i, all extremity levels and seasonality, j, and all perturbed time series, m. A combined skill score, Φ, across all considered metrics that describe the average deviance from the expectations, can then be defined as Φ=∑i∈I∑j∈J∑m∈M1-Φi,j,mIJM, with IJM being the product of the total number of IDF points, I, the total number of extreme levels considered plus seasonality, J, and the total number of time series perturbed, M, as a normalization factor.

The robustness of the methodology is tested by evaluating its sensitivity to the actual magnitude of the target parameters for both extreme and seasonal changes. Low (L), mean (M) and high (H) scenarios are constructed and paired in all possible combinations to assess both the individual and combined influence of these (Table 1). As this increases the number of scenarios with which the precipitation time series substantially are perturbed, this is not done until after an initial evaluation of the state selection criteria.

For Denmark the state space (Eq. 1) is defined with a total of eight states based on the expectations to climate change listed in Tables 3 and 4 with four seasonal states defined for the non-extreme events and three states for the different extreme event levels: E={Ddry,Dwinter,Dspring,Dsummer,Dautumn,D2,D10,D100}. Correspondingly the change factors used to perturb the time series are, as a starting point, determined based on the mean expectations listed in Tables 3 and 4.

For the determination of the state of the individual extreme events four different selection criteria are investigated, with the purpose of defining a representative return period for each event. All points mentioned refer to the return periods of the events intensity points, T={T5,T10,T30,T60,T180,T360,T720}, shown in a situation as depicted in Fig. 3:

Criterion A

The maximum return period is used to define the return period of the whole event (based on one point);

Criterion C

The mean of all the return periods is used to define the events (based on all seven points):

Following these selection criteria, four different systems, Si,i∈A,B,C,D, are constructed and analysed.

Options SA to SC are straightforward based on Eqs. (11)–(13), but option SD is determined specifically for the case study. Table 5 summarises the methodology for option SD used in this study; specifically it is reflected that for very extreme events, fewer durations have to be extreme for the event as a whole to be considered extreme compared to the more moderate 2-year return level.

In previous studies using the SVK data set, it has been shown that

the extreme events account for at most 25 % of the total rainwater volume on an annual basis (Sørup et al., 2016b), and

Furthermore, in the summer season the expected seasonal change (-10 %) differs mostly from the expected change in extremes (+20–40 %); see Tables 4 and 3, respectively. Based on this information the seasonal change factor for non-extreme summer events has to be adjusted to reach the overall change factors reported in Table 4. We estimate a partition between non-extreme and extreme events of fnon-extreme,fextreme=0.8,0.2, and the change factor for 2-year events, CF2, is used to represent the extremes as the largest seasonal volume by far is for the more frequent extremes (Sørup et al., 2016b). In this way the change factor for summer, CFsummer, can be adjusted from its value listed in Table 4 (0.9) as CFadjustedsummer=CFsummer-CF2fextremefnon-extreme=0.9-1.2⋅0.20.8=0825. In other words the change factors for non-extreme summer events are modified from -10 to -17.5 % in order to compensate for the positive change of +20–40 % to the extremes occurring in the summer period. For the other seasons such an adjustment is not needed.

The 10 time series are perturbed using the four different state selection criteria (SA-SD) and the evaluation metric is calculated using Eq. (9) with the extreme events having return periods closest to 2, 10 and 100 years (Table 6). Overall, state selection criterion SD outperforms the other alternatives even though all selection criteria seem reasonable, as all estimated deviances are below 13 % of the expected changes.

In order to study the performance for each state, we construct the skill score variable of Eq. (8) and plot it against the duration for the individual extremes and against months for seasonal precipitation (Fig. 4). Plotted this way 100 % represents a perfect fit, 0 % represents no change and everything positive represents a change in the right direction. For the 2-year return levels both state selection criteria SB and SD perform similarly and with a relative change close to 100 %. State selection criterion SA overestimates the 2-year return level by approximately 10 % on average and state selection criterion SC underestimates it with a similar magnitude, which still corresponds to a positive change for the events (Fig. 4a). For the 10-year return level, all state selection criteria perform similarly very well (Fig. 4b). When the 100-year return level is evaluated, the reason for criterion SD's better overall performance becomes clear: it is the only criterion that does not systematically underestimate this return level (Fig. 4c). Even so, all criteria produce results where the direction of change is correct. Given the inherent uncertainty in estimating the actual levels of such events, obtaining close to 85 % of the expected change is considered good. With respect to the seasonal behaviour, all state selection criteria have approximately the same performance at a level close to 100 % (Fig. 4d).

The performance of all the state selection criteria drops when considering durations that are both shorter and longer than the durations used in the state selection methodology (5–720 min). At the minute scale, this is of minor importance, but at 2 days (2880 min) the tendency is very robust across different state selection criteria and extremity levels. This is most likely because these average extreme events are caused by several events with dry periods in between. Hence the individual events are each assessed to be non-extreme and they are adjusted towards lower volumes, even though combined they are rather extreme.

The sensitivity analysis is carried out for the best state selection criterion only, i.e. criterion SD. The resulting skill scores for the nine individual sensitivity scenarios are listed in Table 7. The highest sensitivity is found when changing between the different extreme precipitation scenarios, with a large increase in the metric when moving from low to mean scenarios and also a notable increase when moving from mean to high scenarios. As such the performance of the methodology drops with the magnitude of the expected changes to extremes, but even for the high extremes the performance is similar to the performance of state selection criteria SA to SD in Table 6. The methodology, on the other hand, shows very little sensitivity to the variation in expectations to seasonal changes, not even for the combination where the difference between expectations to seasonal summer precipitation (-20 %) and the extremes become very high.

For all extreme (+45–100 %) indices (Fig. 5a–c), the sensitivity of the expected change in extremes is notable and, especially for the 100-year return level, it is clear that performance drops with increased magnitude of the expected changes to extremes (Fig. 5c), but only to levels comparable to that of the state selection criteria SA-SC as shown in Fig. 4. Again, the performance for 2-day events (2880 min) is worse than average, as also seen in Fig. 4. For seasonality (Fig. 5d), the general picture is that the sensitivities of both expectations to seasonality and extremes are of less importance and at a similar level, which in general is a lower level than the one observed for the three extreme indices.