As the Earth's mean climate changes under increased concentrations of human-emitted greenhouse gases, the intensity and frequency of extreme weather conditions may change as well (, chap. 11: , and references therein). Extreme events, while rare by definition, can have large impacts on both human society and environmental systems: present-day weather damages are dominated by rare events that happen only every several decades or less , and society's vulnerability to extreme events appears to be growing . In the United States, the frequency of climate and weather events with damages greater than USD 1 billion appears to be increasing at around 5 % per year . It remains unclear to what extent long-term climate trends contribute to that rise, but these factors have led to extensive efforts to understand the relationship between long-term climate change driven by greenhouse gas forcing and potential changes in climate extremes.

Over the past 2 decades, numerous studies have sought to identify changes in temperature extremes both in observations and in general circulation model (GCM) simulations of future climate . A number of studies find that extreme changes follow closely with changes in means and standard deviations e.g.,. , for example, argue that changes in temperature extremes in station data from Eurasia and the United States are largely explained by changes in the mean and standard deviation of daily temperatures within the relevant season (summer for warm extremes and winter for cold extremes). Other studies may suggest more complex changes. use quantile regression and clustering to show that the changes in the 5th percentiles and the 95th percentiles of daily air temperature over central Europe are not the same as changes in medians, although they do not explicitly examine to what extent these differences can be explained by changing standard deviations. find evidence that accounting for changes in skewness in marginal distributions allows for a more accurate representation of changes in cold extremes than can be obtained from just considering changes in mean and standard deviation.

Analysis of changes in extremes is complicated by the fact that there is no unique definition of “extremes”. One common definition is as exceedance of certain defined thresholds, with threshold values often defined based on past climate distributions e.g.,. Analyses then evaluate changes in the frequencies at which these thresholds are surpassed. , for example, used percentile-based thresholds for various climate metrics from 3 decades of global gridded observations (1961–1990) and then considered changes in a subsequent period (1991–2003). They found a significant increase in the occurrence of annual warm nights (defined as the 90th percentile of daily minimum temperature under the past climate) and a significant decrease in the occurrence of cold nights (defined as the 10th percentile of daily minimum temperature under the past climate). However, those studies do not characterize changes in the shape of a distribution by making use of information on the magnitude of exceedances above (or below) the defined threshold.

An alternative and potentially more useful definition of “extremes” is based on occurrences in the far tail of the distribution of the quantity of interest. Extreme value theory (EVT) provides a mathematical framework for studying these far tails. One common approach making use of EVT is based on “block extremes”, the maxima (minima) of some climate variable over given blocks of time. In climate studies, blocks of 1 year are common e.g.,, although use blocks of 45–50 days in order to get two blocks per season. Under some conditions, the magnitudes of extremes over sufficiently long blocks approximately follow a generalized extreme value (GEV) distribution (, , and see chap. 1 of , for details). In these cases, the tail behavior of the distribution can be described by a functional form involving only three parameters: the location, scale, and shape parameters of the GEV distribution. For example, the widely used measure of extreme events, the r-year return level, can be expressed as a function of the GEV parameters (see Sect.  for further review). Annual extremes are directly relevant to important societal impacts; for example, the coldest temperature in a winter can affect the abundance of the beetle population in the following spring . The behavior of annual extremes is difficult to discern from marginal distributions of daily temperatures. Figure  illustrates how changes in block extremes can differ from changes in the underlying overall distribution, using seasonal temperature minima from millennial-scale climate model runs. Figure  in Sect.  illustrates the effect on return levels of changing each individual GEV parameter.

The major limitation of the block extremes approach is that it discards all but the most extreme value within each block. In contrast, the peak over threshold (POT) method uses all data above a specified threshold. The POT method models exceedances over a (sufficiently high/low) threshold as a generalized Pareto distribution (GPD) . Point process theory as applied to exceedances over a threshold provides a unified way to derive both GEV and GPD distributions . While the POT method allows data on extremes to be used more efficiently, it requires choosing appropriate thresholds and declustering temporally dependent extremes. Threshold selection becomes complicated because temperature extremes show strong seasonality and temporal dependence. Furthermore, the temporal dependence in threshold exceedances means that it is not straightforward to convert inferences from the POT approach into inferences on annual extremes. We focus on the block extremes approach here because of its simplicity, its ease of interpretation, and its common usage in climate science.

An important advantage of using any EVT-based approach for the study of climate extremes is that it allows estimation of the probability of events that are more rare than the “moderate extremes” analyzed in threshold exceedance studies. The threshold exceedance study of , for example, used a threshold equivalent to the 90th percentile of temperature calculated over a 5-day sliding window, which corresponds to events whose present mean recurrence interval is only 50 days. EVT allows examination of rare events of much longer recurrence intervals, and in principle even allows estimation of return levels longer than the observed range of the available time series together with a measure of statistical uncertainty. EVT has been widely applied in hydrology, for example, to estimate 100-year floods using several decades of data .

In the past 2 decades, EVT has been applied in numerous studies to climate variables, generally temperature and precipitation including. Studies have evaluated GEV distributions both in observational data and in output from general circulation models (GCMs) and regional climate models (RCMs). evaluated GCM ensembles from the CMIP3 and CMIP5 archives under several projected emission scenarios. In both studies, they found asymmetry in extreme changes, with high temperature extremes in most regions following changes in the mean summer temperature, while low temperature extremes warmed substantially more than mean winter temperatures.

Prior GEV studies are limited to some extent by two factors: the length and non-stationarity of the analyzed time series. For observational studies, existing data records are relatively short (often only several decades). Model runs may be longer, but the model runs used in these studies typically extend only around 100 years from the present. For some models and scenarios, “initial condition ensembles” are available, i.e., multiple runs of the same model and forcing scenario that differ only in their initial conditions. Such ensembles obviously provide further information about extreme events, but they generally include only a few model realizations. The length of the time series has a large influence on the ability to detect changes in extremes, and a record that is too short can lead to large uncertainty in estimated return levels at long periods (see Sect. ). Furthermore, at present and for the foreseeable future, the Earth's climate is not in equilibrium, but is evolving (transient), so the GEV model must be extended to account for non-stationarity in climate extremes. The means of extension are not trivial. The most commonly used nonstationary GEV model e.g., assumes that location and scale parameters change linearly in time and the shape parameter is time-invariant. A more flexible approach to modeling nonstationarity can be achieved by using generalized additive temporal structure of the parameters for extreme value distributions . Given the limited information about extreme events, it is a challenging and important problem to decide on an appropriate degree of flexibility in a nonstationary model.

In this study we avoid the limitations of short transient runs by using three long (millennial) climate model runs in which climate is fully equilibrated. Although numerical simulations of future climates provide only suggestions for possible changes in climate variables, not direct evidence of changes, they are important complements to observational studies. We use temperature output from the widely used Community Climate System 3 (CCSM3) model at three different CO2 levels. For each scenario, the climate model is run for a sufficiently long warmup period to ensure the climate has fully responded to forcing changes, so that any time series of block extremes effectively forms a stationary sequence. Millennial stationary runs allow a more accurate determination of any changes in GEV distributions than do the shorter runs of the CMIP archives. The long model runs also allow us to assess, based on the max-stable property of GEV distributions (see Sect.  for details), the appropriateness of the choice of annual blocks that have been traditionally used with shorter (century-scale) climate model runs. Finally, millennial runs allow us to evaluate empirically the sampling errors when estimating extreme return levels in shorter model runs, because we can use estimates based on entire long runs as “ground truth”.

The paper is structured as follows: in Sect. , we describe the climate model output used in this work; in Sect. , we provide a background for the univariate extreme value theory we employ; in Sect. , we describe the changes in extreme value distributions and corresponding return levels, and compare them with changes in climate means. In Sect. , we assess the assumption that annual blocks are sufficiently long for the GEV approximation to be valid, and in Sect.  we assess the sensitivity of estimates of return levels to the series length. We conclude with a discussion of the implications of these results.

The GCM output used is part of an ensemble of climate simulations completed by the Center for Robust Decision Making on Climate and Energy Policy (RDCEP) e.g.,, using the Community Climate System Model version 3 (CCSM3) , a fully coupled model with full representation of the atmosphere (CAM3), land (CLM3), sea ice (CSIM5) and ocean (POP 1.4.3) components. The model was run at the relatively coarse T31 spatial resolution (3.75∘ × 3.75∘ grid) , which made the lengthy runs used here possible. We use here the last 1000 years of output from each of three multimillennial runs with different atmospheric CO2 concentrations: 289 ppm (pre-industrial), 700 ppm (3.4 ∘C increase in global mean temperature (GMT)), and 1400 ppm (6.1 ∘C increase in GMT). In each case, solar forcing, aerosol concentrations, and concentrations of greenhouse gases other than CO2 are kept at their pre-industrial values. The final 1000 years of each model run are very nearly in equilibrium: net global radiative imbalances at the surface and at the top-of-atmosphere (TOA) are smaller than 0.1 Wm-2. The annual extreme temperatures therefore effectively form stationary time series. In this work, we consider annual maxima (from 1 January to 31 December) of daily maximum temperature (Tmax) and annual minima (from 1 July to 31 June) of daily minimum temperature (Tmin) over the contiguous United States and adjacent ocean regions (see Fig. ).

The GEV distribution is widely applicable in the sciences because it arises, at least approximately, in many cases of natural data. The simplest situation where a GEV distribution can arise is in distributions of maxima taken from sequences of n independent and identically distributed (i.i.d.) random variables (Y1,⋯,Yn), with “block length” n sufficiently large. The extremal types theorem says that if the maxima Mn=max⁡(Y1,⋯,Yn), after normalization (i.e., Mn-bban, an>0), converge in distribution as n→∞ (see Appendix ), then they converge to a GEV distribution .

In this work, when studying high temperature extremes, the random variables Y1,i,⋯,Yn,i are the daily maximum temperatures in year i, and the block length n is 365 days, so that Mn,i is the maximum for year i. Of course time series of daily temperatures are neither independent (they are somewhat autocorrelated) nor identically distributed (their distributions vary within a year). There is however theoretical justification for the use of GEV distributions in our case: it has been shown that the independence assumption can be relaxed for weakly dependent stationary time series , and extended the theory to non-identically distributed observations with conditions on the tail distributions (e.g., distributions share a common absolute maximum). In this work, we explicitly assess whether inferred GEV distributions actually provide an adequate description of the data set in question. The CCSM3 temperature time series studied here do seem to meet this condition: quantile–quantile plots show that the GEV approximation fits reasonably well for most locations in our study area (see Figs.  and  in Appendix ). We therefore assume that annual maxima and minima of daily temperatures in our model output can be approximated by GEV distributions. We will revisit the assessment of the GEV approximation in Sect.  when we test whether our block length of 1 year is sufficient.

By fitting the annual maxima and minima to a GEV distribution (see Appendix  for details), we can identify changes in the distribution of extremes in possible future warmer climates, and evaluate how the characteristics of those changes alter return levels of extreme events. We show results here for all three model runs (pre-industrial, 700 and 1400 ppm CO2).

In this analysis, as in many climate applications of EVT, we have assumed that an annual block is long enough that the GEV distribution is approximately valid. Our long time series allow us to explicitly evaluate this assumption. The evaluation relies on the fact that the GEV distribution has the property of “max-stability”. In the context of distributions of block maxima (Mn), max-stability means that the distribution after normalization is identical for any block size larger than n.

In our case, if an annual block is long enough that the GEV distribution is approximately valid for a given time series, then the shape parameter estimate ξ^n obtained with annual blocks should be approximately the same as that obtained with a larger block size n′ (ξ^n′). The validity of the GEV approximation using annual extremes can therefore be assessed by checking the consistency of estimated shape parameters using longer block sizes (i.e., multiple years).

We refit GEV distributions for each grid cell in our study region with block maxima/minima sizes of 2, 5 and 10 years, and compare shape parameters. We find that for annual maxima, the differences in ξ with block size are distributed around zero, suggesting that an annual block size may be sufficient (Fig. , left panel). For annual minima, we see effects of block size depending on latitude: longer blocks produce larger ξ at high latitudes, where changes in extremes are largest (Fig. , right panel). These results suggest that 1 year may not be sufficiently long for the GEV approximation to be accurate. These results also suggest that quantile–quantile plots (see Fig.  in Appendix ) may not be sufficient to check for the appropriateness of the GEV distribution for the purpose of predicting events with a longer return period than the data length. We observe however that annual blocks may be long enough for the study of the changes in extremes, since changes in return levels are fairly consistent with block length (see Figs.  and ).

The 1000-year model runs used in this work provide fairly accurate estimates of changes in return levels even for long return periods. Estimated uncertainties in return level changes – the bootstrapped envelopes in Fig.  – are generally within 20 % of the estimated changes even for 100-year return periods. This data set therefore allows us to study empirically how well GEV methods work when applied to model output at the shorter lengths (decades to centuries) more commonly used in climate studies.

To assess how uncertainties increase with shorter model runs, we divide the pre-industrial and 700 ppm time series into segments of 20 or 50 years and refit the GEV parameters for each pair of segments (so, for example, pairing the first 20 years of the pre-industrial run with the first 20 years of the 700 ppm run). The resulting distributions of estimates of changes in return levels for warm and cold extremes are shown in Figs.  and .

The results show that sampling error can be large, as expected, when using climate simulations of length comparable to the return periods of interest. In both warm and cold extremes, the distribution of estimates of return level changes derived from short model segments are centered around their “true” values but with large spread (Figs.  and ). In these examples, the sampling errors are comparable to their “true” changes in return level when 20-year segments are used. Unsurprisingly, spreads are largest when using short model segments to predict long return periods. For example, for the Idaho test location, when changes in annual minimum 100-year return levels are estimated using 20-year segments (Fig. , lower left panel), those estimates can differ by ±10 ∘C, while the “true” change is only 8.4 ∘C. Uncertainties can be affected by the choice of estimation method. suggest that estimation based on the method of probability weighted moments (PWMs) has better small sample properties than maximum likelihood, as we use here. Figure  shows that results are generally very similar for the two methods, although a PWM does avoid the few cases of highly unrealistic estimates seen in Fig.  in the TX example location.

The example locations shown in Figs.  and  are representative of land locations in the entire study area. In Fig. , we show the spread in estimates for all model grid cells. We show two cases: estimating changes in 20-year returns (both warm and cold extremes) with 20-year segments and changes in 50-year returns with 50-year segments. The length of the model segment is especially important for temperature extremes, because their distribution has a bounded tail. For inland locations, the spread in estimates is comparable to the true change for the 20-year case but somewhat lower for the 50-year case. Ocean locations show much smaller sampling error, presumably because changes in extremes are produced predominantly by shifts in the location parameter and not the scale or shape parameters, which are more difficult to estimate. The locations with the largest uncertainties are indeed those with the largest changes in shape parameter: a few Midwest locations for warm extremes and some locations in Mexico for cold extremes.

It is important to note that spatial modeling approaches may be able to reduce estimation variability. (See further discussion in the following section.) When not using such approaches, this assessment suggests that when using single model runs or observational data, sampling error may be quite large when the length of the series is comparable to the return period of interest.

In this study, we have used extreme value theory to study changing temperature extremes in model projections of future climate states. Following prior work, we assume the GEV model provides a reasonable approximation for the distribution of annual temperature extremes (see Figs.  and ) and that changing temperature extremes can therefore be studied by estimating changes in GEV parameters and resulting implied changes in return levels. We use millennial-scale equilibrated simulations of pre-industrial and 700 and 1400 ppm CO2 concentrations (producing changes in global mean temperature of 3.4 and 6.1 ∘C) to explore both the physics of climate model projections and the utility of statistical approaches based on GEV distributions in different contexts.

For our model runs, the results suggest that for the contiguous United States, much of the behavior of temperature extremes in higher CO2 climate states appears to be a straightforward consequence of changes in the mean and standard deviation of the underlying temperature distributions. Annual warm extremes generally shift simply in accordance with mean shifts in summer temperatures. Annual cold extremes warm more strongly than do winter mean temperatures, but largely do so as expected given decreased variability in wintertime temperatures. In both cases, the changes in the location parameter of annual extremes appear well explained by shifts in overall seasonal temperature means and standard deviations (Fig. ). In most inland locations in the winter, changes in the scale and shape parameters can provide an additional complication, producing substantial differences in return level changes at longer return periods (Fig. ). Note that these results are from a single climate model, and there is no guarantee that any model will capture all aspects of changes in extremes well e.g.,. Model studies are, however, important given the difficulty in evaluating changes in extremes from the short observational record.

The millennial-scale model runs used here allow us to assess the validity of the use of annual block sizes in studies of GEV distributions of temperature extremes. Our results suggest that annual blocks are sufficiently long for representing warm extremes, but may be insufficient for cold extremes, especially for inland locations at high latitudes. In these locations, altering the block length alters the shape parameter of the estimated GEV distribution (Fig. ). In general, increasing block size reduces biases, but can increase sampling error in estimates if the time series is limited. The choice of block size therefore requires consideration of inherent trade-offs.

Millennial-scale model runs also enable detailed investigation of the relationship between data length and GEV sampling error. Previous studies have suggested that extrapolation should be carried out with caution e.g.,. Our long runs allow us to better quantify estimation uncertainties resulting from short model runs. We find that over inland locations, estimation uncertainty is indeed large when using series length is comparable to the return period of interest (Fig. ). Ocean locations show much smaller sampling errors, in large part because temperature variability is much lower over oceans than over land. When using very short series (e.g., 20 years), lack of data mandates care in choosing the appropriate estimation procedure . Our 1000-year model runs allow us to concretely demonstrate the large uncertainties in estimates of changes in extremes that result when time series length is comparable to the return period of interest.

The computational demands of millennial-scale climate simulations restrict us here to examining a single climate model run at fairly coarse resolution. A single model should not be taken as robust guidance on how temperature extremes may change in future climate conditions. However, few modeling groups have performed runs of comparable length, and the multi-model data in public archives are not ideal for estimating changes in extremes: run lengths are much shorter (≈ 100 years), the number of realizations of any scenario is small, and climate conditions in the simulations are evolving rather than stationary. Estimating GEV distributions that are changing over time is also more challenging than in the equilibrium setting, making the limitations of short model runs even more pertinent. If understanding changes in extremes in a changing climate is a research priority, larger ensembles of model runs would be helpful.

In the absence of large ensembles, one could potentially overcome some of the challenges of limited data by exploiting the spatial structure of temperature extremes (see Fig. ) to “borrow strength over space”. Numerous prior studies do explicitly model the spatial structure of a climate variable of interest, e.g., , , , and . However, spatial modeling is not ideal: it may introduce bias, it greatly complicates the computation, and spatial structure in GEV parameters can be difficult to distinguish from spatial dependence in observed extremes. We do not use these approaches in this work since the millennial-scale model runs allow us to estimate the GEV parameters and their changes accurately by fitting them to each model grid cell separately. However, where only shorter time series (decades to centuries) are available, such as in analyses of observations, appropriate spatial modeling may be essential to reduce estimation uncertainty.

Extreme value theory (EVT) has become popular for studying climate extremes, but our results here suggest that one should be aware of the underlying assumptions, the corresponding implications, and the potential limitations when applying it. EVT using block extremes can potentially allow characterization of tail behavior that differs from that of the underlying distribution, but it involves a corresponding penalty, as fitting a distribution of block extremes necessarily involves throwing out most of one's data. For short time series, care must be taken to ensure that the drawbacks do not outweigh the benefits. In the climate model output studied here, for example, shifts in the location parameter for both warm and cold extremes are well explained by changes in the mean and standard deviation of the underlying temperature distribution. The millennial time series used here allow us to also identify significant changes in the scale parameter, especially for cold extremes, but with shorter time series, sampling errors can be too large for estimated return levels to be of much practical value. Long model runs such as those used here therefore provide an important tool for the study of climate extremes, helping both in clarifying the contexts in which EVT should best be used and in devising approaches for working with shorter time series.