## 1. Introduction

The fingerprint of human activity has been detected in the recently observed warming on global (Tett et al. 2002; Stott et al. 2006) and continental scales (Karoly et al. 2003; Stott 2003; Zwiers and Zhang 2003). Much of the work done in this area employs optimal detection (Hasselmann 1979; Allen and Tett 1999), a formal statistical technique adapted for use in the study of climatic changes (IDAG 2005; Hegerl et al. 2007). Research on the changes in extreme temperatures, however, in the context of optimal detection, is still in its early stages. Extreme events are of great importance because of their impacts on human health, ecosystems, and economies. Links between climate change and extremes are often highlighted, especially in the aftermath of devastating events, such as the European heat wave of 2003. In the first study that applied formal detection and attribution methodology to an extreme event, Stott et al. (2004) quantified the change in the probability of a heat wave in Europe similar to the one in 2003 under the influence of human activity.

Warming in extreme temperatures during the last few decades is evident in analyses of observed records (Frich et al. 2002; Alexander et al. 2006; Caesar et al. 2006; Brown et al. 2008), as well as climate model simulations (Kharin and Zwiers 2000; Tebaldi et al. 2006; Kharin et al. 2007). Although these studies are indicative of the importance of human influence, attribution to anthropogenic forcings would require a combination of information from both observations and models. Kiktev et al. (2003) were the first to do this, and they explained the warming in extreme temperatures by increasing greenhouse gas emissions. Taking a more formal approach, Hegerl et al. (2004) attempted to apply optimal detection in a feasibility study in which they examined whether it was possible to detect changes in indices that describe extreme temperature and precipitation in a perfect model configuration. They showed that for their indices, detection of changes in extremes is no more difficult than detection of changes in the mean. Christidis et al. (2005, hereafter CSBHC05) provided the first optimal detection analysis with real observations using the same indices for temperature extremes and detected significant warming in all events apart from the warmest days of the year. Shiogama et al. (2006) confirmed these findings using a different climate model.

The first aim of this paper is to examine whether the anthropogenic influence on the warming in warm-day extremes is also detectable when using indices that are potentially better for detection and attribution studies than those in CSBHC05. Second, a global attribution analysis is performed that partitions the response between external forcing components. Finally, regional estimates of changes in the nonstationary parameter of the extreme distribution employed here are derived using constraints from the global attribution analysis. The data used in the analysis and the methodology are discussed in the next section. Results are presented in section 3, and section 4 summarizes the main points and provides some discussion.

## 2. Data and methodology

The data used in this work are the same as in CSBHC05, where more details can be found. Daily observations of the maximum daytime temperature *T*_{max} with a quasi-global coverage from the gridded dataset of Caesar et al. (2006) are used here for the period 1950–99. Model data for the same period come from three experiments: the third climate configuration of the Met Office Unified Model (HadCM3) (Johns et al. 2002), forced externally with anthropogenic (ANTHRO) and natural (NAT) forcings; and a combination of the two (ALL). Each experiment is a four-member ensemble, with each member starting from well-separated points of a long control simulation. Although there are observations available up to 2007, we did not include more recent years in this study, as most of the model simulations required for the attribution analysis were not extended beyond 1999. This also allows for a direct comparison of the results with CSBHC05. Internal climate variability is estimated using 3500 yr of the control experiment and also intraensemble variability from runs with external forcings. Spectral analysis tests on time series of daily values of *T*_{max} averaged over the observational area show that HadCM3 gives a realistic representation of the internal variability (CSBHC05).

The observations and the model data employ the same 2.5° × 3.75° grid. There are, however, differences between the observations and HadCM3 in the spatial and temporal sampling of the data that provide the gridded values. The *T*_{max} value at each grid point in the Caesar et al. (2006) dataset is influenced by the 10 nearest stations to a degree that depends on their distance and is estimated using the angular-distance weighting algorithm described in New et al. (2000). In HadCM3, the gridpoint values are representative of the actual gridbox *T*_{max}, as there was no provision for subgrid land surface parameterizations. It may also be that the temporal aspects of the different observations could prevent an exact equivalent gridbox-average maximum daytime temperature from being calculated. However, it is beyond the scope of this study to investigate the effect of such sampling issues in detail. We assume that the effect is minor, given that one of the incentives for the development of the observational dataset was to validate model data. Moreover, the spatial smoothing applied to the data used in our detection analysis (discussed later) is expected to mitigate, to a certain extent, the effect of such sampling inconsistencies.

The rarity of extreme events is expected to hamper the detection of significant changes as it decreases the signal-to-noise ratio. To get round this problem, investigators usually examine “moderate” extremes, such as the most extreme event in a year. Numerous indices have been introduced to describe these events (e.g., Frich et al. 2002). Here we follow a different approach and employ extreme value theory to derive a more sophisticated measure that describes warm-day extremes. We adopt a time-varying location parameter as introduced by Brown et al. (2008), who computed its value on each grid point by fitting point-process extreme value distributions to anomalies of daily *T*_{max} data (max Δ*T*_{max}). The anomalies were calculated by removing an estimate of the mean annual cycle. Hence, the resulting distribution describes anomalously warm daytime temperatures throughout the year, which could appear in any season. Here, we also study the actual warmest daytime temperatures of the year (expected in the local summer), by applying the extreme value distributions to the actual daily *T*_{max} data (max *T*_{max}). The extreme value distributions employ a threshold exceedances model and provide the expected number of exceedances per year above any level greater than the threshold, as functions of the location, scale, and shape parameters (Coles 2001). The scale parameter does not depend on the chosen threshold, and the three parameters are equivalent to those in the generalized extreme value (GEV) distribution for maximum annual values. A time-varying threshold is employed for each grid point by fitting a smooth spline through the daily gridpoint values and shifting the fitted line until no more than 2% of the data lie above it. Brown et al. (2008) also considered cold-day, warm-night, and cold-night extremes; however, here we focus solely on changes in warm-day extremes, as these have not until recently been established in detection and attribution studies. Changes in the location parameter with time provide a measure of changes in the extreme temperatures. Assuming that the temporal variations of this parameter are small within a decade, we allow it to have different values for each 10-yr segment while keeping the other distribution parameters constant for the entire 50-yr period; that is, for the reference period 1950–90, we get five location values, while for the model control experiment we get 350 values. Brown et al. (2008) found that both globally and in continental regions, the observed trends in the location parameter for the max Δ*T*_{max} extremes lie outside the 5%–95% range of the variability from the HadCM3 control simulation. This indicates that the new measure of extremes may have a high enough signal-to-noise ratio to enable detection of changes in this kind of daily extremes in a formal detection analysis.

Figure 1 illustrates the patterns of the observed and modeled trends in the location parameter during 1950–99 for the two types of warm-day extremes (max Δ*T*_{max} and max *T*_{max}). The trends are given by the slope of the line that provides the best least squares fit to the time series of the decadal mean location parameter values. The modeled patterns are given by the ensemble mean for each of the three HadCM3 experiments and bear less of the effect of internal climate variability as a result. The observations show more areas of cooling for the warmest days of the year than the anomalous warm days. In both cases, the signal is more uniform in the ALL and ANTHRO experiments, presumably because of the suppressed internal climatic variations, while the NAT patterns show an overall cooling. The time series of the decadal values of the location parameter averaged over the entire observational area are plotted in Fig. 2. An estimate of the 5%–95% range of the internal variability is superimposed on the time series. This was calculated by applying a normal distribution to the decadal global mean values of the location parameter from the control experiment. The use of the normal distribution is justified because in our study, we employ maximum likelihood estimates of the location parameter, which are asymptotically normal (Cox and Hinkley 1974). The plotted time series of the observations and the experiments with anthropogenic forcings show a warming that emerges above internal variability in the last decade of the reference period, while the NAT time series show a small decrease in the location parameter with time.

Time series of the location parameter plotted as anomalies relative to the 1950–99 mean and averaged over the entire observational area for extremes of *T*_{max} with the annual cycle (a) removed and (b) retained. Time series were calculated from the observations (black line), and the ensemble mean of the ALL (red line), ANTHRO (green line), and NAT (blue line) experiments. Gray band illustrates the 5%–95% range of internal climate variability estimated from the CONTROL experiment. Overall trend during the analysis period is marked on each panel.

Citation: Journal of Climate 24, 7; 10.1175/2011JCLI4150.1

Time series of the location parameter plotted as anomalies relative to the 1950–99 mean and averaged over the entire observational area for extremes of *T*_{max} with the annual cycle (a) removed and (b) retained. Time series were calculated from the observations (black line), and the ensemble mean of the ALL (red line), ANTHRO (green line), and NAT (blue line) experiments. Gray band illustrates the 5%–95% range of internal climate variability estimated from the CONTROL experiment. Overall trend during the analysis period is marked on each panel.

Citation: Journal of Climate 24, 7; 10.1175/2011JCLI4150.1

Time series of the location parameter plotted as anomalies relative to the 1950–99 mean and averaged over the entire observational area for extremes of *T*_{max} with the annual cycle (a) removed and (b) retained. Time series were calculated from the observations (black line), and the ensemble mean of the ALL (red line), ANTHRO (green line), and NAT (blue line) experiments. Gray band illustrates the 5%–95% range of internal climate variability estimated from the CONTROL experiment. Overall trend during the analysis period is marked on each panel.

Citation: Journal of Climate 24, 7; 10.1175/2011JCLI4150.1

The same optimal detection methodology as in CSBHC05 is employed here. The algorithm regresses model signals (given by the ensemble mean) against the observations, taking into account the noise associated with both. It returns scaling factors together with their 5%–95% uncertainty range that need to be applied to the model fingerprints to bring them in best agreement with the observations. Scaling factors consistent with zero imply no detection. Consistency with one and a small uncertainty range imply good agreement between the model and the observations. To enable the optimization of the regression, the analysis is carried out in the space defined by the leading eigenvectors of the control variability. The exact number of the eigenvectors retained in the analysis is selected on the basis of a standard testing procedure that compares the modeled unforced variability with that of the regression residuals (Allen and Tett, 1999). Here we employ 5–15 leading modes of variability (which explain 94.7%–99.8% of the variability) and find that in all cases, the results vary little in the vicinity of the chosen truncation.

The signals used in the regression are three-dimensional (spatiotemporal) patterns that comprise five time slices of gridpoint location parameter values, one for each of the five decades in the reference period. The parameter values are computed as anomalies relative to the 1950–99 mean. The model patterns are masked with the observations, and therefore areas with no observational data are excluded from the analysis. The patterns are also spatially smoothed using spherical harmonics at T8 truncation, which filters out scales smaller than about 2500 km. The model signals are first regressed individually against the observations to examine whether the effect of the forcings included in each signal (ALL, ANTHRO, or NAT) can in itself be detected in the observations (single fingerprint analysis). A two-fingerprint analysis is also carried out. In this case, the regression employs a linear combination of the ANT and NAT signals and aims to partition the climate response between its anthropogenic and natural components.

Regional estimates of the 1950–99 trends in the parameter with and without the influence of anthropogenic forcings are then estimated in continental-scale regions using constraints from the two-fingerprint optimal detection analysis over the quasi-global observational area. The methodology was introduced in Christidis et al. (2010), where more details can be found. In summary, we construct the probability density functions (PDFs) of the regional location parameter trends by taking the following three steps:

The global anthropogenic and natural patterns of the location parameters are scaled by the factors from the optimal detection analysis. Given the uncertainty associated with the scaling factors, we use a set of values to represent their distribution rather than a single value. We therefore have one distribution of scaled patterns for the anthropogenic response and one for the natural. The convolution of the two gives the distribution of the total response.

We extract regions from the total and the natural response patterns and compute the regional mean trend in the location parameter during 1950–99. As we have two distributions of patterns (i.e., several possible realizations due to the uncertainty in the scaling factors from optimal detection), one with and one without anthropogenic influences, we end up with two corresponding distributions of the regional trends.

We convolve each of the two distributions from step 2 with a distribution that represents the effect of internal climate variability. This is estimated from patterns extracted from the control experiment, also smoothed with spherical harmonics. The result is two distributions of the regional trend in the location parameter, one with and one without the influence of anthropogenic forcings.

## 3. Analysis

Optimal detection is applied first by regressing a single fingerprint from each of the experiments with external forcings (ALL, ANTHRO, and NAT) against the observations, to examine whether a significant signal emerges above internal climate variability. The scaling factors with their 5%–95% uncertainty range are plotted in the left section of the panels in Fig. 3. Experiments that include the effect of anthropogenic forcings (ALL and ANTHRO) give a detectable change in the location parameter. The ALL scaling factor for max Δ*T*_{max} is greater than unity, indicating that the model underestimates the response, which therefore needs to be scaled up to match the observations, whereas the opposite is the case for max *T*_{max}. The max *T*_{max} index is determined by extreme high values of the actual *T*_{max}, which are expected to occur during the local summer, whereas max Δ*T*_{max} comes from extremely high *T*_{max} anomalies that can occur in any season. Some possible reasons for the different model response to ALL forcings between the two indices could be the model representation of climatic forcings with a seasonal signature, such as aerosols (Zhang et al. 2005); the simulated ground wetness in the summer (Conil et al. 2009); or the model representation of modes of variability that may affect seasonal extremes (Gillett et al. 2003). The ANTHRO experiment produces more warming than the ALL for max Δ*T*_{max} (Fig. 1), and its scaling factor is consistent with unity. Natural forcings cannot explain the overall observed change on their own for both types of warm-day extremes, as indicated by the NAT scaling factors, which are consistent with zero.

Scaling factors from optimal detection analyses for (a) max Δ*T*_{max} and (b) max *T*_{max} extremes. (left) Best estimate and 5%–95% uncertainty range from single fingerprint analyses with the fingerprint taken from the ALL (red), ANTHRO (green), and NAT (blue) experiments. (right) Results from a two-fingerprint analysis that separates the ANTHRO (green) and NAT (blue) contributions to the response. Ellipse marks the two-dimensional 90th percentiles, and horizontal and vertical bars mark the one-dimensional 5%–95% confidence interval for each of the two signals.

Citation: Journal of Climate 24, 7; 10.1175/2011JCLI4150.1

Scaling factors from optimal detection analyses for (a) max Δ*T*_{max} and (b) max *T*_{max} extremes. (left) Best estimate and 5%–95% uncertainty range from single fingerprint analyses with the fingerprint taken from the ALL (red), ANTHRO (green), and NAT (blue) experiments. (right) Results from a two-fingerprint analysis that separates the ANTHRO (green) and NAT (blue) contributions to the response. Ellipse marks the two-dimensional 90th percentiles, and horizontal and vertical bars mark the one-dimensional 5%–95% confidence interval for each of the two signals.

Citation: Journal of Climate 24, 7; 10.1175/2011JCLI4150.1

Scaling factors from optimal detection analyses for (a) max Δ*T*_{max} and (b) max *T*_{max} extremes. (left) Best estimate and 5%–95% uncertainty range from single fingerprint analyses with the fingerprint taken from the ALL (red), ANTHRO (green), and NAT (blue) experiments. (right) Results from a two-fingerprint analysis that separates the ANTHRO (green) and NAT (blue) contributions to the response. Ellipse marks the two-dimensional 90th percentiles, and horizontal and vertical bars mark the one-dimensional 5%–95% confidence interval for each of the two signals.

Citation: Journal of Climate 24, 7; 10.1175/2011JCLI4150.1

A two-fingerprint analysis is carried out next, whereby the regression combines the fingerprints from the ANTHRO and NAT experiments and attempts to partition the observed response between these two components. The scaling factors are shown in the right section of the panels in Fig. 3. The anthropogenic signal is detected in the observations for both types of extremes. The natural signal is detected for max Δ*T*_{max} and only marginally for max *T*_{max}, though the uncertainty range of the NAT scaling factor is in both cases much wider than the ANTHRO. The successful detection of human influences on warm-day extremes demonstrates the advantage of the new measure of extremes over the simpler ones employed in CSBHC05.

The PDFs of the trends in the location parameters are computed next over the entire observational area and also in four continental-scale regions, namely, North America (10°–80°N, 50°–170°W), Europe (30°–80°N, 25°W–50°E), Asia (10°S–80°N, 50E°–180°), and Australia (10°–50°S, 110E°–180°). The scaling factors from the two-fingerprint max Δ*T*_{max} analysis (right section of Fig. 3a) are plotted again as PDFs in Fig. 4a. We subsample these factors over sets of values that represent the two PDFs, apply them to the ANTHRO and NAT fingerprints, and compute the mean trend attributed to the respective forcings over a selected region. In the case of the entire observational area, the resulting ANTHRO and NAT PDFs of the trends are represented by the green and the blue lines in Fig. 4b, respectively. Their convolution gives the total trend represented by the red line. The black dashed line, labeled as CONTROL on the same plot, represents the PDF of internal climate variability computed from the control, as mentioned in the previous section. Finally, we convolve the CONTROL with the ALL and also the CONTROL with the NAT distributions to get the PDFs of the trend with and without the effect of the anthropogenic forcings. The convolution gives the distributions of the trends, including the effect of both external forcings and internal variability. These are illustrated in the regional plots on the map shown in Fig. 5a, and the observed trends are also marked on the plots. Results for the max *T*_{max} extremes are shown in Fig. 5b. In all regions, anthropogenic forcings shift the distribution of the trends from mostly negative to mostly positive values; however, the effect is more pronounced for the max Δ*T*_{max} extremes, for which the PDFs are more separated. The observed trends lie in the tails of the distributions with natural forcings only; however, in most cases, they are within the vicinity of the peak of the distributions with the anthropogenic forcings included.

Illustration of the methodology used to calculate the distributions of the 1950–99 trend in the location parameter with and without the effect of human influences. (a) PDFs of the anthropogenic (green line) and the natural (blue line) scaling factors from the two-fingerprint global optimal detection analysis. (b) PDFs of the ANTHRO (green) and NAT (blue) trends over the entire observational area calculated from the model fingerprints scaled by the corresponding factors from optimal detection. Convolution of the ANTHRO and NAT distributions gives the PDF attributed to ALL (red line). Dashed black line gives the PDF that corresponds to internal climate variability estimated from 50-yr-long segments of the CONTROL experiment using *t* statistics. Distribution of the trend with and without the effect of human influence is estimated by convolving CONTROL with ALL and CONTROL with NAT, respectively (convolved distributions not plotted here). The *y* axis gives the normalized likelihood.

Citation: Journal of Climate 24, 7; 10.1175/2011JCLI4150.1

Illustration of the methodology used to calculate the distributions of the 1950–99 trend in the location parameter with and without the effect of human influences. (a) PDFs of the anthropogenic (green line) and the natural (blue line) scaling factors from the two-fingerprint global optimal detection analysis. (b) PDFs of the ANTHRO (green) and NAT (blue) trends over the entire observational area calculated from the model fingerprints scaled by the corresponding factors from optimal detection. Convolution of the ANTHRO and NAT distributions gives the PDF attributed to ALL (red line). Dashed black line gives the PDF that corresponds to internal climate variability estimated from 50-yr-long segments of the CONTROL experiment using *t* statistics. Distribution of the trend with and without the effect of human influence is estimated by convolving CONTROL with ALL and CONTROL with NAT, respectively (convolved distributions not plotted here). The *y* axis gives the normalized likelihood.

Citation: Journal of Climate 24, 7; 10.1175/2011JCLI4150.1

Illustration of the methodology used to calculate the distributions of the 1950–99 trend in the location parameter with and without the effect of human influences. (a) PDFs of the anthropogenic (green line) and the natural (blue line) scaling factors from the two-fingerprint global optimal detection analysis. (b) PDFs of the ANTHRO (green) and NAT (blue) trends over the entire observational area calculated from the model fingerprints scaled by the corresponding factors from optimal detection. Convolution of the ANTHRO and NAT distributions gives the PDF attributed to ALL (red line). Dashed black line gives the PDF that corresponds to internal climate variability estimated from 50-yr-long segments of the CONTROL experiment using *t* statistics. Distribution of the trend with and without the effect of human influence is estimated by convolving CONTROL with ALL and CONTROL with NAT, respectively (convolved distributions not plotted here). The *y* axis gives the normalized likelihood.

Citation: Journal of Climate 24, 7; 10.1175/2011JCLI4150.1

PDFs of the 1950–99 trend in the location parameter in different regions constrained by the global attribution analysis in a climate forced with all external forcings (red lines) and with natural forcings only (blue lines) for (a) max Δ*T*_{max} and (b) max *T*_{max} extremes. Observed trends are represented by the vertical black line, and their value is also marked on each panel. The *y* axis gives the normalized likelihood.

Citation: Journal of Climate 24, 7; 10.1175/2011JCLI4150.1

PDFs of the 1950–99 trend in the location parameter in different regions constrained by the global attribution analysis in a climate forced with all external forcings (red lines) and with natural forcings only (blue lines) for (a) max Δ*T*_{max} and (b) max *T*_{max} extremes. Observed trends are represented by the vertical black line, and their value is also marked on each panel. The *y* axis gives the normalized likelihood.

Citation: Journal of Climate 24, 7; 10.1175/2011JCLI4150.1

PDFs of the 1950–99 trend in the location parameter in different regions constrained by the global attribution analysis in a climate forced with all external forcings (red lines) and with natural forcings only (blue lines) for (a) max Δ*T*_{max} and (b) max *T*_{max} extremes. Observed trends are represented by the vertical black line, and their value is also marked on each panel. The *y* axis gives the normalized likelihood.

Citation: Journal of Climate 24, 7; 10.1175/2011JCLI4150.1

## 4. Discussion

This work provides a formal detection and attribution analysis of changes in warm-day extremes in the second half of the twentieth century. The first attempt for such an analysis on different types of daily extremes (CSBHC05) succeeded in detecting human influences in recent changes in daily cold extremes and warm-night extremes but not in extremely warm days. In this follow-up work, we make use of a more sophisticated measure of extremes with a higher signal-to-noise ratio than the simple indices in CSBHC05, computed as a time-varying location parameter within an extreme value distribution. We find that detection of changes in both anomalously warm days as well as in the warmest days of the year becomes possible with the new measure. The fingerprint of the anthropogenic component of the global response is detected in the observations. The natural fingerprint is also detected, though only marginally for max *T*_{max}. We also examine regional changes in warm-day extremes, which are more useful to inform adaptation planning than the global-scale change. We find that in continental-scale regions, anthropogenic forcings tend to shift the 1950–99 trend in the location parameter from negative to positive values. With a projected intensifying warming of several degrees in the global mean temperature during the twenty-first century (Meehl et al. 2007), temperature extremes are expected to become increasingly more severe, which would result in adverse economic impacts and stress on human health in absence of adaptation (McMichael et al. 2006).

The use of the location parameter in detection and attribution studies of daily extremes has also recently been employed in the study of Zwiers et al. (2011). In their work, the authors fit generalized extreme value distributions to annual maximum and minimum values of daily maximum and daily minimum temperatures for each grid point. The time evolution of the location parameter is represented in a different way from our work, namely, the parameter is constrained to be proportional to its simulated change in each decade relative to 1961–70, with the change computed from an experiment with anthropogenic forcings only. The scaling factor that describes this dependence is calculated by minimizing the sum of a negative log-likelihood function across all the grid boxes, and a bootstrap technique is used to estimate its uncertainty. The analysis by Zwiers et al. (2011) confirms the detection of significant anthropogenic influence on all types of daily extremes previously shown in CSBHC05 and also illustrates the first successful detection for extremes related to warm days. They find that the scaling factors related to max *T*_{max} extremes are smaller than one, indicating that the models used in their analysis overestimate the signal, as is also the case with HadCM3 shown here. Our work not only provides independent evidence from a different model of the detection results for max *T*_{max} extremes shown in Zwiers et al. (2011) but also takes a step further by separating the anthropogenic and natural components of the observed change by means of a two-fingerprint analysis. The results from these studies mark an important advance in the attribution of extreme changes. It is, however, important that in future work, more models are employed to assess more rigorously these first indications that the observed warming trends in warm-day extremes are significant and linked to human activity rather than internal climate variations.

## Acknowledgments

We are grateful to the three reviewers for their constructive comments. We thank G. S. Jones for useful discussions. This work was supported by the Joint DECC and Defra Integrated Climate Programme (DECC/Defra) (GA01101).

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