Return periods in current and future climate

. 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.


Introduction
In Denmark, recommended climate factors for design rain intensities looking 100 years ahead are given as discrete values for return periods T equal to, respectively, 2, 10 and 100 years (SVK, 2014).Multiplication of the current T -year event with the climate factor results in an estimate of the future T -year event.The recommended Danish climate factor values are based on simulations with climate models (Christensen et al., 1998;Gregersen et al., 2013) and therefore not exact values, but nevertheless useful for design of urban drainage structures.To account for the uncertainty, they are provided both as standard climate factors and as high climate factors.To facilitate design efforts, continuous curves are presented below.Climate factors are telling how much a given T c -year event in current climate will increase in the future climate.Another, but related question is which future return period, T f , corresponds to a T c -year event in the current climate.Below a convenient, general formula for the future return period as function of the climate factor and the current return period is developed.In addition, by means of first analysis the uncertainty of T f is assessed talking into account the uncertainty in the climate factor.x T ,f denotes the T -year event in the future climate, the climate factor, k, is defined as the ratio between the events.Thus The climate factor depends on several different other factors, where the return period T by far is the most important one.
The dependence of T is shown in Table 1 both for standard climate factors and for high climate factors.
A generalisation of the discrete values into a continuous curve might be done using a simple log-linear approximation.Here, however, it is chosen to use a second order polynomium approximation, which exactly corresponds to the three given climate factor values in Table 1.The relation for the Danish standard climate factor is found to be  which is shown in Fig. 2. Both the relations are developed for 2 ≤ T c ≤ 100, but can be assumed valid also for values of T c greater than 100.

Projection of the current return period into a future return period
A wide class of distribution functions possesses exponential tails, and it is here assumed plausible that this is also the case for the distributions of x T ,c and x T ,f .Thus for the extreme value regions we get for x T ,c From Eq. (1) we have  Combining with Eqs. ( 4) and ( 5) we get By insertion into Eq.( 5) we then obtain leading to the below general relation between T f and T c The relation is exemplified in Fig. 3 using Danish standard climate factors.It is, e.g., seen that a 100-year value in the current Danish climate becomes a 27-year event in the future climate.
In Table 2, the values obtained using Eq. ( 9) are compared to values published in MST (2021) obtained by using results from climate models.The correspondence is found reasonable.Moreover, it can be seen from the table that the use of high climate factors gives rise to a dramatic change in the assessment of future return periods.

Uncertainty of projected future return periods
Due to uncertainty in the climate factor, the projection of a return period into the future will become uncertain as well.For fixed values of T c , the uncertainty of T f is assessed in terms of coefficient of variations.Interpreting Eq. ( 9) as T f = f (k) and applying first order analysis lead to the general equation The relation is shown in Fig. 4. It can, e.g., be seen that a 10 % uncertainty in the climate factor implies a 33 % uncertainty in the future projection of a 100-year return period.

Conclusions
Discrete climate factors are for Danish conditions generalized to continuous functions of the return period for both standard and high climate factors.Using exponential tail approximations for the distribution function of current and future T -year events, a general analytical expression for projecting the current return period into the future return period as function of the climate factor is developed and evaluated.Finally, the uncertainty of the projected future return period due to uncertainty in the climate factor is assessed revealing a notable uncertainty even in case of a moderate uncertainty in the climate factor.

Figure 1 .
Figure 1.Standard climate factor as function of return period.

Figure 2 .
Figure 2. High climate factor as function of return period.

Figure 3 .
Figure 3. Projected future return period as function of current return period using standard climate factors.

Figure 4 .
Figure 4. Coefficient of variation of the future return period as function of the current return period assuming the coefficient of variation of the climate factor equal to 10 %.

Table 1 .
Climate factors as function of return periods.

2 Continuous climate factor curves If
x T ,c denotes the T -year event in the current climate, and

Table 2 .
Current return periods, corresponding climate factors and projected future return periods.