The ongoing increase in extremely warm temperature events across large areas of the globe is generally thought to be a signature of a more extreme climate. However, it is still unclear whether global warming is accompanied by changes in statistical properties beyond the mean, such as an increasing temperature variability. Here we shed light on this issue by uncovering the way probabilities of extremes are being influenced by temperature evolution. Focusing on Europe, we show how the behavior of warm and cold extremes can be determined to a high accuracy by statistically modeling daily temperatures and their changes. Detailed comparison with observations over the past decades puts forward the dominant role of the mean in explaining exceptionally hot events, and rules out contributions from potential changes in second and higher moments.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 Recent decades have witnessed an abrupt rise in global surface temperatures, accompanied by marked changes in the frequency of extremely warm and cold events [Alexander et al., 2006]. Current interest is typically focused on a broad class of extremes which are confined, roughly speaking, to the outermost ten down to a few percent tails of temperature probability density functions (PDFs). The rationale for this choice is to balance the inherent rareness of these phenomena with the existence of enough observations for a firm-based measurement. Empirical approaches are thereby suitable in this context, whilst extreme value theory is fit for analysis of multi-year return period events [see Simolo et al., 2010, and references therein]. Changes over time in the frequency of moderate (so-called soft) extremes have an obvious relevance for natural systems and human life, and provide useful hints for surveying basic mechanisms of climate warming, from natural modes of variability [Scaife et al., 2008; van den Besselaar et al., 2010] to anthropogenic influences [Christidis et al., 2005; Klein Tank et al., 2005].
 Both, observations and climate model projections indicate that Europe belongs to the most sensitive areas of climate changes, where the recent increase in extremely warm events is predicted to be even more severe in the near future [e.g., Giorgi, 2006]. A vexed issue is whether the observed warming over Europe is associated with long-term changes in the second and higher moments of temperature PDFs, as any of these changes does have a potentially strong impact on tail probabilities [Mearns et al., 1984; Katz and Brown, 1992]. Conjectures have been made regarding the possible role of an enhanced width of temperature PDFs in explaining the recent increase in extremely warm events over Europe [Schär et al., 2004; Della-Marta et al., 2007]. Likewise, tracks of changes in the width as well as the degree of asymmetry of PDFs have been inferred from discrepancies between the observed changes in the frequency of soft extremes and the behavior of tail probabilities entailed by a pure shift of the mean [Klein Tank and Können, 2003; Klein Tank et al., 2005]. Future evolution of European temperatures is further expected to be accompanied by an increase in daily or interannual variability, whereas a pure shift of the mean is generally thought to be unsuited to explain the projected increase in warm extremes [e.g., Kjellström et al., 2007; Fischer and Schär, 2009; Ballester et al., 2010; Hirschi et al., 2011]. Whether changes in the higher moments are currently under way or will be prominent in the future is still debated [e.g., Clark et al., 2006] and, strictly, there is no compelling evidence for such changes over the past decades, except for weak, seasonal effects only [e.g., Scherrer et al., 2005]. A paradigm change is thus required to unravel the interplay between the basic statistical properties of daily temperatures and the way probabilities of extremes evolve in a warming climate. Based on a flexible statistical model, we provide a theoretical description of time-evolving PDFs, that enables a faithful prediction of soft extremes under current climate conditions. We consider here the European case as a baseline, and show how the recent evolution of warm and cold extremes can be reconciled with the perspective of a mean temperatures shift, by exploiting the intrinsic non-linearity between changes in the mean and those induced on probabilities of extremes. A similar approach has been used in a previous work [Simolo et al., 2010] for discussing temperature evolution over Italy, and to which we refer for some technicalities.
 The paper is organized as follows. In section 2 data and their processing are presented. In section 3 distributional properties of observed PDFs and their changes are investigated year-by-year using a moment-based method. Results are then exploited in section 4 to model time-evolving PDFs and derive theoretical evolution of soft extremes. Comparison with observations and main findings are illustrated in section 5. Section 6 concludes and summarizes. Some technical details are given in the auxiliary material.
2. Data Processing
 Focusing on Central Europe, the observed evolution of temperatures and related extremes over about the past 50 years is derived from a bulk of station series of daily maximum (TX) and minimum temperatures (TN) of the European Climate Assessment (ECA) data-set [Klein Tank et al., 2002] (see Figure 1, red dots). We consider only raw (non-blended) data that meet necessary requirements of record length, completeness, quality and homogeneity (see section 1 of auxiliary material for details). The domain of interest is finally represented by 63 ECA series along with three more TX and TN series from a different source (Italian Air Force) used in a previous work [Simolo et al., 2010]. Station data, expressed as anomalies relative to the conventional base period (1961–1990), are gathered in three independent sub-domains, identified by a correlation-based Principal Component Analysis, namely a southern area extending from the Alpine to the Carpathian Chains (AC), the Atlantic (AT) and the Northern Sea (NS) areas (Figure 1). Data summarized into (unweighted) spatial averages for the above sub-domains allows the 1961–2007 time interval over the NS and AC sub-domains, and 1961–2004 over France (AT) to be covered safely.
 Temperature extremes are then measured in terms of exceedance probabilities (EPs), i.e., probabilities of exceeding fixed thresholds, using the well-known suite of indicators for climate extremes endorsed by the Working Group on Climate Change Detection [Alexander et al., 2006]. We consider here the percentile-based indicators TXNp with N = 90, 95, 99 and N = 10, 5, 1 for gradually smaller fractions respectively of the warm and cold tails in TX PDFs, and similar for TN. Observed EPs are evaluated on a yearly and seasonal basis, and long-term exceedance thresholds are derived for any day-of-the-year from empirical distributions across the full record period for removing possible inhomogeneities at the edges of the baseline period [see, e.g., Zhang et al., 2005].
3. Density Functions: Leading Features and Changes
 First, we summarize basic properties of observed temperature anomalies using the sample estimators (by the U-statistic) of the first four L-moments, lr, r = 1 … 4 [Hosking, 1990]: l1 and l2 are a measure of location (the usual mean) and scale (in place of the ordinary standard deviation) respectively, whereas the moment ratios tr = lr/l2, r = 3, 4 quantify the degree of asymmetry (skewness) and departures from the tailedness of a normal curve (kurtosis), respectively [see also Simolo et al., 2010]. Changes in TX and TN PDFs are then traced from changes over time in the moment estimators l1, l2, t3, and t4, derived from single-year sub-samples for all the years and all the sub-domains, with no theoretical assumptions about the underlying PDFs. The sample size here is a trade-off between statistical relevance and the need for removing spurious effects in the second and higher moments induced by trends in the mean.
 As frequently pointed out [e.g., Moberg et al., 2006] the most prominent changes in European temperatures concern increases in the mean, which are only roughly linear in time. In agreement with some recent findings [Klok and Klein Tank, 2009], average rates of increase are generally more pronounced for TX than TN anomalies (Figure 2a), indicating the western part (AT) as the European area affected by the strongest warming. As seen in Figure 2a, the accelerating phase (after the mid 1980s) which follows a nearly stationary period is more closely approximated by including a second order correction to the fits (slightly lower root-mean-square errors), especially for TX. Furthermore, residuals of annual mean anomalies from the fits fluctuate about zero with roughly the same variance across the whole period (see section 2 of auxiliary material), whereby interannual variability can be safely considered unchanged.
 In turn, the time series of the L-scale l2 (insets of Figure 2a) and the moment ratios t3 and t4 of daily PDFs do not exhibit significant changes over the period of concern (p-values well above 0.10 in most cases), but random fluctuations only. Few exceptions emerge only at the seasonal level, with significant though weak increasing tendencies in the TN L-scale of the AC sub-domain during autumn (∼0.07 K/decade) and the TX L-scale of AT during summer (∼0.05 K/decade). In the latter case, similar effects were observed in previous studies [Della-Marta et al., 2007; Scherrer et al., 2005]. As shown in Figure 2b, finally, the moment ratios t3 and t4 outline departures of TX and TN PDFs from a normal behavior, which are mainly related to the degree of asymmetry. The largest effects concern TN anomalies with generally left-skewed distributions (t3 < 0), whereas TX asymmetries are less pronounced. Discrepancies from normal tailedness (t4) are generally of minor importance, though less (more) than Gaussian kurtosis in isolated single-year sub-samples of TX (TN) data are observed.
4. Modeling Evolution of Extremes
 Relationships between changes in the distribution moments and those observed in the frequency of extremes are inherently non-linear, since the flow of probabilities across fixed thresholds is controlled by the functional form of the underlying density. For normal distributed data, for instance, rates of changes in EPs scale exponentially in time for a uniform shift of the mean. In practice, as departures of real-world observations from a normal behavior can be relevant, we incorporate non-zero skewness in a theoretical description of evolving EPs by modeling TX and TN anomalies with a simple three-parameter extension of the normal density ϕ, the skew-normal density [Azzalini, 2005; Simolo et al., 2010]
The factor Φ(αz) determines the degree of asymmetry through a shape parameter α, by slanting the density to the left (right) for positive (negative) values (see inset of Figure S1 in the auxiliary material), whereas the standard normal is retrieved for α = 0. A location ξ and a scale parameter ω are restored by the usual transformation x = ωz + ξ (see Azzalini  for details). The benefit of including a shape parameter, the slant α, in the representation of temperature anomalies is seen by model fits to observed data, performed over single-year sub-samples (for all the years and all the sub-domains). Although observed asymmetries are quite moderate (∣α∣ ≤ 2), the skewed density yields a substantial improvement over the normal density, as witnessed by the χ2 per degree of freedom associated with the fits (see Figure S1 in the auxiliary material). Since, as discussed in section 3, only a change in the mean is of concern here, we can determine the theoretical evolution of EPs induced by a forward, rigid shift of the skewed density (1), as
for the left and right tail respectively. Fixed thresholds p and 1 − p are uniquely determined by the initial condition F−(t0) = F+(t0) = p for given probability p.
 By way of illustration, a linear dependence on time can be imposed replacing z in equation (1) by a shifting variable z − vt with constant velocity v, while keeping the slant α unchanged. The inherent non-linearity in the evolution of EPs as given by equation (1) and (2) is clearly seen in Figures 3a and 3b, where the marked dependence on the shift velocity v is shown for the standard normal (α = 0). The ratio of the absolute rate in the right to the left EP, in particular, is far from being constant, even over short time intervals (Figure 3b). The evolution is slightly modified by the degree of asymmetry, with left (right) skewness having a major impact on the right (left) EP. Such corrections are of growing importance over time, ranging from a few up to several tens of percents.
5. Comparing Modeled and Observed Changes
 As seen in section 3, observed changes in European temperatures are essentially driven by a non-uniform shift of the mean which preserves higher moments of daily PDFs, that is, long-term changes in variability and skewness are ruled out by the data. Thus, we assume here equation (1) as the fixed-shape PDF underlying temperature anomalies, with scale and slant parameters kept constant across the full period and equal to their long-term averages, and respectively, depending on the local variable only. The time dependence is imposed through the location parameter ξ which, for constant scale and slant, accounts for changes in the mean only [see, e.g., Simolo et al., 2010, Appendix B]. Hence
with coefficients ξi derived from the fits of the mean (Figure 2a) and the fixed values and , for each variable and each sub-domain. A second order correction is introduced to stay as close as possible to the observed evolution of the mean over the period of concern, given the high sensitivity of EPs to the shift velocity (Figures 3a and 3b). The time-dependent theoretical EPs are obtained from the unstandardized form of equations (1) and (2) with p = 0.10, 0.05 and 0.01, i.e., for gradually less moderate extremes, and the same choice of PDF parameters in the three cases. Theoretical EPs nicely fit the observed evolution of both warm and cold extremes, for all the TX and TN variables, outlining the actual trends in the probability of extremes down to p = 0.01. This is seen in Figure 4a for TX of the AT sub-domain (and Figures S2–S4 in the auxiliary material for all the other cases), where theoretical expectations are shown together with their overall uncertainty related to the ω and α parameters, which in turn reflects stochastic fluctuations in variability and skewness. Mean values of the slant are ≃ −0.9, 0.5, 1.1 (−2.0, −1.5, −1.7) for TX (TN) and the AC, AT and NS sub-domains respectively. In addition, for a detailed comparison between modeled and observed changes in the frequency of extremes, year-by-year theoretical EPs are obtained by adding back to equation (3) residuals from fits of the mean, thereby using year-by-year annual mean values (t) for the location parameter. The remarkable agreement between theoretical and observed EPs (Figure 4a) puts forward the dominant role of the mean in explaining recent evolution of warm and cold extremes, as a large fraction of observed variance is explained by model predictions, that is, on average over all sub-domains R2 ∼ 0.79, 0.69, 0.50 (0.79, 0.65, 0.40) for TX (TN) and p = 0.10, 0.05, 0.01 respectively. Thus, observations can be reasonably reproduced by exploiting only the inherent non-linearity in the time evolution of EPs caused by variations in the mean (long-term changes and stochastic fluctuations). As seen in Figure 4b, the model tailored to the AT summer season (i.e., based on JJA daily PDFs) is able to capture well-known record-breaking events such as the 2003 summer heat wave that dramatically hit Western Europe. Even though there is a weak signal of increasing summer variability in the AT sub-domain (section 2), extremely warm events are largely controlled by the shift of the mean in both TX (Figure 4b) and TN (Figure S2b in the auxiliary material). The variance explained by the model in the JJA season is slightly larger than in the full-year case, expressing the higher temporal coherence of distributional moments during summer (on average R2 ∼ 0.86, 0.77, 0.49 (0.88, 0.79, 0.68) for TX (TN) and p = 0.10, 0.05, 0.01 respectively).
 Remaining discrepancies between observations and model predictions, both at the yearly and seasonal level, can be ascribed to higher moments fluctuations, not accounted for by the choice of mean values for the scale and slant parameter. In all cases, these discrepancies remain generally moderate throughout the observed period (they show no pattern), signifying that predictions are not plagued by potentially underestimated trends in higher moments. Finally, though resolving PDFs skewness properties moderately affects evolution of EPs over the observed period, as can be argued by Figure 3a, it slightly reduces underestimation of actual increasing rates in warm extremes, entailed by the assumption of normality.
6. Concluding Remarks
 To summarize, we have first illustrated a simple parametric approach to bridge the gap between the observed behavior of soft temperature extremes and average distributional properties of daily anomalies, providing a correlated understanding of the long-term changes in PDFs and the evolution of EPs. Secondly, against commonly-held beliefs, we have shown that the changes in the frequency of warm and cold extremes observed in the past decades over Europe can be well explained by a simple, non-uniform shift of mean temperatures, whereas changes in the higher moments contribute through random fluctuations only. Indeed, under current climate conditions, only the evolution of annual mean temperatures is needed in this framework for a faithful prediction of soft extremes.
 Furthermore, the model envisaged can be seamlessly adjusted to account for trends in second and higher moments, if any, paving the way for a consistent treatment of present and future behavior of temperature extremes in a changing climate.
 ECA data are archived at the database: http://eca.knmi.nl. The Italian Air Force is gratefully acknowledged for Italian data. This study has been carried out in the framework of the EU project ECLISE (265240).
 The Editor thanks the two anonymous reviewers for their assistance in evaluating this paper.