1. Introduction

The word “extreme” is unusually associated with physically severe conditions or events. However, extreme can also be defined in statistical terms as “the largest and the smallest element of a set” (http://en.wikipedia.org, s.v. “extreme”). Moreover, extremes are represented by the tails of a variable's distribution and are usually very rare by nature (IPCC 2002). One difficulty encountered when studying extremes is that statistical analysis is very limited by a low number of events for finite and relatively short observational data series. Weather and climate extremes can involve various time scales, from tornadoes and hail storms lasting only minutes or hours to droughts of years' duration. Extreme precipitation over longer periods can have adverse effects, both in terms of flooding and droughts, and the focus here will be on precipitation on the monthly scale. It is usually hard to discern long-term temporal changes, or trends, in the behavior of extremes when the number of events is small.

2. Methods and data

Trends have often been analyzed in terms of a linear regression or the difference in the mean of two intervals. Such analyses give an indication of how the mean values change but may not give reliable figures for extremes (López-Díaz 2003; Frei and Schär 2001). One approach to examining trends extremes can be to test whether the upper tails of a given variable's probability distribution function (PDF) is changing. Based on similar ideas as discussed in Vogel et al. (2001), Benestad (2003) proposed a simple test for the upper tails of the PDF by examining the recurrence of record events. This test, which adopts the null hypothesis that the variable is independent and identically distributed (IID) and for which the PDF is a constant, is discussed further in Benestad (2004).

The definition for a record event here is when a given value in a chronological series is greater than all the preceding values (x_i > max[x_j] and j < i). We can express the occurrence of a record-breaking event by following the mathematical expression ϒ(x_i) = H(x_i − max[x_j]), j < i, H being the Heaviside function, and ϒ(x_i) ∈ [0, 1]. The empirical estimate of the number of records in a series is obtained simply by counting the number of record-breaking events in chronological order, expressed mathematically as ε̂(n) = Σⁿ_i=1ϒ(x_i). The theoretical recurrence of new record events in an IID series follows a simple mathematical law: The probability that the last value x_i is greater than the preceding i − 1 values is p_i (record event) = 1/i (assuming no ties for the record value). The expected number of records for one given IID time series of length n is ε(n) = Σⁿ_i=11/i (Benestad 2003), which is the sum of the products between the value for an event occurring [ϒ(x_i) = 1] and its probability p_i (record event). Here the notation is the same as in Benestad (2003, 2004) with ε̂(n) denoting the empirical value and ε(n) denoting the theoretical value. It is easy to derive the null distributions for a given sequence of real numbers and length n and compare this to the actual number of record events in the sequence.

Single IID series do not give statistically significant results unless they are very long, but it is nevertheless possible to improve the signal-to-noise ratio by analyzing many parallel but independent series expected to exhibit the same behavior. An aggregated statistic can be estimated for N parallel series each of length n according to 〈ε̂(n)〉 = Σ^N_k=1ε̂_k(n)/N. The number of records was calculated for each of the calendar months; then the average was taken in order to obtain the aggregated statistic for each grid box. The mean was calculated for N_s [= 3 or 12] calendar months representing either a given season [December–February (DJF), March–May (MAM), June–August (JJA), or September–November (SON)] or the whole year (January–December). The ensemble mean was subsequently estimated from the N_GCM (=[31, 9, or 14] depending on the batch of model runs) different models, and the total number of parallel series was N = N_s × N_GCM.

Gridded monthly precipitation from 17 different general circulation models (GCMs) was used here (details listed in Table 1), and the model results were taken from 54 different experiments comprising three different types of transient runs following the (a) A1b (N_GCM = 31), (b) B1 (N_GCM = 9), and (c) A2 (N_GCM = 14) Special Report Emission Scenarios (SRES; Houghton et al. 2001, p. 532). The analysis was repeated for all A1b, A2, and B1 separately. The comparison between the different emission scenarios showed that main features were robust, and only the results for the SRES A1b scenario are therefore shown here. The precipitation from the simulations were interpolated to a common 5° × 5° grid (170°W, 165°W, . . . , 180°E; 60°S, 55°S, . . . , 70°N) for the 99-yr period 2001–2099, and for each of these interpolated model results, time series from each grid box were tested for the number of records, ε̂(n).

Despite the simplicity of the IID test, a Monte Carlo approach was adopted to obtain a reliable estimate of the confidence interval as this interval is sensitive to the number of parallel sequences (Benestad 2004). The analysis was carried out in R (R Development Core Team 2004), and the stochastic series were obtained through R's random number generator¹ for N_GCM × 1000 × N_s sequences of the same length as the original data. The Monte Carlo simulations consisted of replacing the gridbox series with the stochastic series and repeating the same procedure as for the GCM results: the null distribution was generated by (a) ε̂(n)_MC for a 99-points-long sequence (representing a single month's series), then (b) repeating (a) N_s times (accounting for the different calendar months), and (c) taking the mean of the N_GCM estimates (representing different GCMs). The empirical distribution function (EDF), which is the empirical version of a cumulative distribution function (CDF; von Storch and Zwiers 1999), for the null distribution was then synthesized by repeating (a)–(c) 1000 times. This EDF was then used for determining the 95% confidence region. The 2.5% and 97.5% percentiles (two-tailed distribution) of the null distribution obtained from the Monte Carlo simulations were 4.78 and 5.60, respectively, for the 3-month-long seasons, whereas for the whole year (12 months), the 95% confidence interval was 4.98–5.38.

3. Results

Figure 1 shows how the multimodel ensemble mean number of monthly precipitation record events 〈ε̂(n)〉 varies geographically. The regions where 〈ε̂(n)〉 is outside the 95% confidence region of the null distribution are marked as light and dark gray. The light gray regions with high 〈ε̂(n)〉 indicate a shift toward greater values in the upper tail of the PDF for monthly precipitation (wetter). Conversely, the dark gray regions indicate a shift in the upper tail of the PDF toward lower values (drier). For an IID sequence of 99 random data points, the expected 〈ε̂(n)_MC〉 derived from the Monte Carlo simulations is 5.2 (rounded off to one digit), which agrees well with the theoretical expectation value: ε(n)= 5.2 (rounded off). There is a clear indication of a high number of monthly record events in terms of monthly precipitation over most of the region poleward of 50°S and 50°N (light gray). There are also indications of a high number of records near the intertropical convergence zone (ITCZ) in the Tropics and over the south Asian monsoon region. In contrast, the model results indicate a low number of new monthly record event (dark regions) regions where there typically is persistent subsidence (high pressure), such as over the Azores, the Mediterranean, and the maritime regions off the Chilean coast, between Africa and Australia and over the south Atlantic. The different panels in Fig. 1 show the four different seasons, all of which show the same general picture regarding the high number of records (light gray). There may be some indications, however, of a slightly greater fractional area with significantly low record numbers (dark gray) in the summer hemisphere (Figs. 1a,c), however, subregions of low record counts are also found in the winter hemisphere.

A comparison between the results on an annual basis from the A1b (representing a more moderate emission scenario) and the A2 (larger emissions) scenarios indicated that the A2 results did give a slightly smaller area with the number of monthly record events outside the 95% confidence region (not shown). The results from the B1 scenario (lowest emissions; not shown) suggested a similar type of pattern but with a much smaller area with a significantly high or low number of monthly record events.

4. Discussion and conclusions

A robust pattern of an anomalously high number of record-breaking values for monthly precipitation was found in the mid- and high latitudes. Although only marginally different, the moderate A1b scenario indicated a slightly larger area with significantly high ε(n) than the A2 scenario, contrary to expectations. Part of this can be attributed to the fact that the ensemble size for the A2 results was less than half that of A1b, hence the A2 results being more prone to sampling uncertainties. The results for the B1 scenario were also obtained using a smaller ensemble, but these were associated with a substantially smaller region with high ε(n). These results therefore could suggest that there is a smaller difference between moderate and high emission scenarios than between low and moderate, and that the response of ε(n) to a climate change is nonlinear.

The reason for why there could be a nonlinear response is impossible to unveil without an in-depth and more detailed analysis of each individual model and thus is outside the scope of this short note. However, one could give reasons for why the precipitation becomes less sensitive to the warming as the global temperature increases through “hand waving” arguments. It is expected from the Clausius–Clapeyron equation that an increase of temperature will result in higher-saturation vapor pressure (Fleagle and Businger 1980), hence providing a potential for the enhancement of the moisture source required for increased precipitation. However, an increased saturation vapor pressure just by itself may not be sufficient for a sustained increase in precipitation (P) because an increase in evaporation must follow an acceleration in the entire chain of processes associated with the hydrological cycle if the atmospheric moisture content is to be sustained. If the global water mass budget for the atmosphere is ∫_AE dA = ∫_V(dM_w/dt) dV + ∫_AP dA, where ∫_AE dA is the evaporation rate for the earth's surface area, ∫_V dM_w/ dt dV is the mass of water in both liquid and gas form integrated over the volume occupied by the global atmosphere, and P is the rate of precipitation, then an increase in ∫_A P dA must be balanced by an increase in evaporation rate or a change in atmospheric water. A response in the hydrological cycle is expected to be complex involving a number of feedback processes that cause a transition from ∫_AE dA to ∫_V(dM_w/dt) dV and then from ∫_V(dM_w/dt) dV to ∫_AP dA, many of which may involve small spatial scales that are represented by parameterizations in the climate models. If the rate at which this transition occurs is constrained, either numerically or physically, then the acceleration of the hydrological cycle may diminish with increased warming. On the other hand, atmospheric water may play a role as both enhanced vapor pressure as well as increased cloud cover affect the radiative balance: water vapor is a potent greenhouse gas, and clouds affect the albedo as well as the longwave radiation. Thus, a substantial change in ∫_VM_w dV cannot take place without affecting the radiative balance.

Another reason for why A2 and A1b are similar but B2 gives a weaker signal is the change in sensitivity of the IID test as ε(n) increases, which is related to the fact that the probability of new record-breaking values occurring diminished with 1/n. To see why this is the case, one should consider whether a shift in ε(n) from say ∼5 to 6, as seen in Fig. 1, really is of any significance. We can set this question in perspective by noting that ε(n) = 5.2 here corresponds to a 99-yr-long IID series, but ε(n) = 6 would require ∼230 yr on average assuming IID conditions. To really appreciate the implications of changes in ε(n), consider that ε(n) = Σ^M_i=11/i = 10 requires M > 12 000. Thus, an increase in ε(n) can be viewed as a change on a logarithmic scale (Benestad 2004). This fact is due to the diminishing recurrence of record events in an IID process and the record-level shifts each time an old record is broken. A high value for ε(n) is an indication of a change in the extreme monthly rainfall amount due to a shift in the upper tails toward higher values. The probability for monthly rainfall exceeding thresholds that are considered extreme at present will also increase faster than the rate at which new records are set, and this rate depends on the particular shape of the PDF (such an analysis is beyond the scope of the present paper). Since the upper tail of the PDF is not constant, it would be inappropriate to use a traditional return value analysis, such as a general extreme distribution (GEV), to make extrapolations for the future in this case.

There are also some physical interpretations of the results presented in Fig. 1, as there is a tendency of the high ε(n) associated with the ITCZ to be more prominent in the Southern Hemisphere in winter (Fig. 1a) and Northern Hemisphere in summer (Fig. 1c). This observation is consistent with the notion of an intensification of deep tropical convection, thus bringing more rain, and an enhancement of the Hadley cell. Regions with anomalously low ε(n) in the subtropics are also indicators of a shift in the monthly rainfall statistics toward drier conditions as the upper range of monthly rainfall is lowered. Hence, these observations are also consistent with the interpretation of an enhanced Hadley cell in terms of more pronounced subsidence. Another interesting region is south Asia, characterized with high ε(n) in summer and autumn (Figs. 1c,d) that could suggest more extreme monsoon rainfall on a monthly scale, although this should be more of a summer response rather than autumn.

In the northern midlatitudes, there is a tendency of greater area with high ε(n) during northern winter and spring, suggesting that the precipitation increase is primarily associated with stratiform clouds more typical for the winter season. It is important to keep in mind that these results are from model simulations and that we assume here that the models are capable of describing the various cloud types and associated atmospheric conditions. Stratiform clouds have typically lower intensity but longer durations than convective processes. Moreover, a global warming seems to result in an enhancement of tropical convection, but there is also a clear signal of increased precipitation associated with stratiform clouds in the midlatitudes.

Frequent recurrence of record monthly rainfall amounts can have implications for the risk of flooding, and these scenarios point to greater flooding risks for the future. The analysis also found indications of low-number monthly record events in some regions that were slightly more pronounced during the summer season. Since the present analysis focuses on the upper tails of the statistical distribution, it is premature to draw any conclusion about droughts from these results, other than noting that a low number of monthly record events is consistent with conditions that are more drought-like. Since this analysis was applied to the monthly time scales, it is not possible to make any conclusions about precipitation on a shorter time scale (hours and days) that is typically associated with the term “extreme precipitation.”

One great advantage of the IID test proposed by Benestad (2003) is that it is insensitive to the exact distribution of the variable as long as there are no cases where the record value is tied (Benestad 2004). This fact makes it ideal for comparing different series with different constant values and different amplitudes, and hence the precipitation at different locations and different times of the year. Here the test serves to identify regions where global climate models (GCMs) indicate that the upper tails of the PDF for precipitation are being shifted toward higher values (i.e., “being stretched”).

The present analysis has not addressed the issue of spatial correlation (Benestad 2003 2004; Vogel et al. 2001; Bairamov and Eryilmaz 2000). The reason is that each grid box is treated entirely separately, and it is not assumed that its sequence is independent to the neighboring ones. Of greater concern is the question of whether or not there is any autocorrelation in the different series, as this would affect both the IID test and the implications of the Monte Carlo simulations. The autocorrelation was estimated for the whole study region and was found to be low except for a few regions in the Tropics (Fig. 2). An autocorrelation with a magnitude of ≈0.3 in some regions may have some implications for the confidence intervals for record precipitation associated with the ITCZ, but the low values in the extratropics should have a negligible effect.

The main objective of this study was to assess a range of the most advanced and up-to-date climate models and document the response of precipitation patterns to a global warming scenario. This study has also been partly motivated by the need to create greater awareness about a simple and elegant analytical method, the IID test, that has not been used much in the climate research community.