### Abstract

- Top of page
- Abstract
- 1 Introduction
- 2 Data and Methodology
- 3 Results
- 4 Summary and Discussion
- Acknowledgments
- References
- Supporting Information

[1] This study develops a generalized extreme value (GEV) distribution analysis approach, namely, a GEV tree approach that allows for both stationary and nonstationary cases. This approach is applied to a century-long homogenized daily temperature data set for Australia to assess changes in temperature extremes from 1910 to 2010. Changes in 20 year return values are estimated from the most suitable GEV distribution chosen from a GEV tree. Twenty year return values of extreme low minimum temperature are found to have warmed strongly over the century in most parts of the continent. There is also a tendency toward warming of extreme high maximum temperatures, but it is weaker than that for minimum temperatures, with the majority of stations not showing significant trends. The observed changes in extreme temperatures are broadly consistent with observed changes in mean temperatures and in the frequency of temperatures above the ninetieth and below the tenth percentile (i.e., extreme indices). The GEV tree analysis provides insight into behavior of extremes with re-occurrence times of several years to decades that are of importance to engineering design/applications, while extreme indices represent moderately extreme events with re-occurrence times of a year or shorter.

### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 Data and Methodology
- 3 Results
- 4 Summary and Discussion
- Acknowledgments
- References
- Supporting Information

[2] Climate extremes and changes therein are of great scientific interest. Many major impacts of climate change arise from extreme events, which have been considered of sufficient importance for the Intergovernmental Panel on Climate Change (IPCC) to produce a Special Report on Extremes [*IPCC*, 2012].

[3] A large number of studies have analyzed changes in temperature extremes [e.g., *Alexander et al*., 2006; *Brown et al*., 2008], generally finding increased occurrence of warm extremes and decreased occurrence of cool extremes (with only limited localized exceptions). Such changes are associated with changes in the probability distributions [*Hansen et al*., 2012; *Donat and Alexander*, 2012]. Many of these studies have focused on analyses of extreme indices. The indices most commonly analyzed have been the occurrence of days above, or below, a percentile-based threshold (normally the ninetieth and tenth percentile, respectively). A few such studies have been carried out in Australia [e.g., *Plummer et al*., 1999; *Collins et al*., 2000; *Alexander et al*., 2007], finding an increase in warm extremes and a decrease in cold extremes over the post-1957 period.

[4] Extreme indices represent moderately extreme events with re-occurrence times (i.e., return periods) of a year or shorter. They do not reflect the most extreme extremes with return periods of several years or longer, such as one-in-20 year extremes (namely, extremes that are expected to occur once every 20 years, on average, i.e., are of a 20 year return period). Thus, an alternative approach to extreme analysis is to fit a statistical distribution to a data set to infer changes in the occurrences of extreme events with specific return periods. *Zwiers et al*. [2011] used such an approach to assess changes in the occurrence of a one-in-20 year temperature extreme over the period 1960–2009. *Brown et al*. [2008] used fitted distributions to assess changes in temperatures above the 98.5th or below the 1.5th percentile. In this study, we fit generalized extreme value (GEV) distributions. As detailed below, we develop a GEV tree approach that allows for both stationary and nonstationary distributions, which is similar to, but of higher complexity than, the nonstationary GEV approach of *Cannon* [2010].

[5] The underlying data are of great importance for any analysis of extremes. Many data sets used for extremes analysis are homogenized only at the annual and/or monthly mean level by adjusting the annual or monthly mean values. However, extremes do not necessarily respond to an inhomogeneity in the same way as means [e.g., *Trewin*, 2012], and hence, homogenization of an annual and/or monthly mean time series will not necessarily produce a homogeneous time series of extremes.

[6] Recently, a few methods [e.g., *Wang et al*., 2010; *Trewin*, 2012] have been developed for making quantile-dependent adjustments, which produce a homogeneous time series for extremes as well as for means. Such methods have been used to homogenize the Australian Climate Observations Reference Network - Surface Air Temperature (ACORN-SAT) data set [*Trewin*, 2012], which contains daily maximum (*T*_{max}) and daily minimum (*T*_{min}) temperature data from 112 Australian stations and forms the data basis for this study.

### 2 Data and Methodology

- Top of page
- Abstract
- 1 Introduction
- 2 Data and Methodology
- 3 Results
- 4 Summary and Discussion
- Acknowledgments
- References
- Supporting Information

[7] This study uses the homogenized *T*_{min} and *T*_{max} data from 52 long-term stations in the ACORN-SAT data set [*Trewin*, 2012]. These stations have daily temperature observations for at least 90 years out of the period 1910–2011. The annual minimum of *T*_{min}, denoted as *T*_{Nn}, is obtained for each year that has 310 or more valid daily *T*_{min} (i.e., at least 85% of non-missing daily observations in the year), and so is the annual maximum of *T*_{max} obtained, denoted as *T*_{Xx}. Two coastal stations which meet the requirements, Albany and Port Macquarie, are excluded because their extreme high temperatures are found unhomogenizable [*Trewin*, 2012].

[8] In this study, we focus on one-in-20 year extremes (namely 20 year return values). This is chosen as a compromise between the rareness of the event of interest and uncertainty in the estimated return values (see the Supporting Information and auxiliary Figure S1 for such uncertainty expressed as confidence intervals).

[9] Extremes can be studied by fitting a GEV distribution to block maxima (e.g., annual maxima) or fitting a generalized Pareto distribution (GPD) to excesses of peaks-over-threshold [*Coles*, 2001]. The GEV approach is more straightforward, although the GPD approach makes better use of the available data and should be preferred when the available sample is small. In this study, we have over 90 years of data for estimating 20 year return values, in which case the GEV and GPD estimates are comparable in terms of stability (see Figure S1a). Therefore, we take the GEV approach.

[10] A nonstationary distribution is a distribution with time-dependent parameter(s), while a stationary distribution has time-invariant parameters. In this study, we use two approaches: one is based on fitting a stationary GEV distribution to data in a series of 51 year running windows, and the other is based on applying a GEV tree analysis to data for the entire period (90+ years). Both approaches are detailed below.

[11] In light of the changing climate, stationarity cannot always be assumed, especially for a centennial period such as in the present study. Thus, for an extreme statistic (*T*_{Nn} or *T*_{Xx}) at each station, we fit a stationary GEV to data in each running 51 year window that has at least 44 years of non-missing values, obtaining time series of the GEV parameters and of the 20 year return value. We chose a window width of an odd number (51) of years so that we have an exact mid-window year to refer the results to.

[12] There exist different methods for fitting a GEV distribution (i.e., estimating the parameters), such as the method of probability-weighted moments (PWM) [*Hosking et al*., 1985], the maximum likelihood (ML) estimator, method of moments or L-moments, etc. The resulting estimates are different in terms of bias and efficiency [e.g., *Caires*, 2007]. Results of our supplementary analysis (see the auxiliary material) suggest that the PWM estimator provides much more stable estimates (with much narrower confidence intervals) than does the ML estimator when the sample size is less than twice the return period of the extreme to be estimated (e.g., less than 40 years of data for estimating a 20 year return value). However, for estimating a 20 year return value from a sample that is larger than 40 years, the GEV fits with the PWM or ML estimator have a stability that is comparable to that of fitting a stationary GPD to excesses of peaks-over-threshold (see Figure S1a), although the GEV fits with the ML estimator are least stable for estimating rarer extremes such as 50 or 100 year return values (see Figures S1b and S1c). Thus, we use the PWM method to estimate a stationary GEV from 44–51 years of data and use the ML estimator to fit a GEV tree to data over the whole period (90–102 years), as described below. We chose to use the ML estimator because it is more flexible than the other estimators, especially for fitting a nonstationary GEV, such as in a GEV tree analysis.

[13] In light of the changing climate, it might also be sub-optimal to assume stationarity over a 51 year period. A better alternative is to fit each time series of annual extremes for the whole period with a most suitable GEV model that is chosen from a GEV tree. The tree has the stationary model at its root and various nonstationary models at different levels up. Here, the parameters can have linear or nonlinear temporal trends. Let 0, 1, and 2 denote no temporal trend, linear trends, and nonlinear trends, respectively. Let G000 denote a stationary GEV (all parameters are time-invariant), G100, a GEV with a linear trend in the location *μ*_{t} but constant scale and shape parameters (*σ*, *ξ*), …, and G222, a GEV with a nonlinear trend (*a* + *bt* + *ct*^{k} where *k* ∈ {1.1,1.2, …,2.0}) in each of the three parameters. This set of *k* values is chosen from the results of our fitting experiments with the data. Thus, G000 is the simplest model, and G222, the most complicated model (the model at the tree top). Different nonstationary GEV models are at different levels between the root and the top of the tree. For the same parameter, a linear trend is always introduced before introducing a nonlinear trend with *k* = 1.5 (we use 1.5 as the initial value of *k*; this will be refined later if the most suitable model involves a nonlinear trend).

[14] Considering that return value estimates are much less sensitive to estimation errors in the location than in the scale and shape parameters (they are most sensitive to shape errors) and that estimates of the location are usually much more stable than those of the scale and shape, a trend in the location parameter is first introduced if it is significant at the 5% level, and the model selection procedure proceeds to the next level up without testing whether or not there is a significant trend in the scale and/or shape. Similarly, a significant trend in the scale parameter is introduced without testing whether or not there is a significant trend in the shape parameter (namely, a trend would first be introduced to the scale). Also, a trend in the scale or shape parameter is introduced only if it is significant at 0.1% level. Consequently, a trend is most likely to be included in the location parameter and least likely in the shape parameter. This results in much better fits than allowing a trend to be equally likely in any of the three parameters, as shown in Figures 2a and 2b.

[15] In the GEV tree analysis, a likelihood ratio test is used to compare the fit of a simpler model with each of the models in the next level up to check whether or not the fit of the best model in the next level up is significantly better than that of the simpler model. If none of the models at the next level up provides a significantly better fit, the simpler model is chosen as the most suitable model. Otherwise, the best model in the next level up is chosen as the next simpler model, and its fit is compared with the fits of the models in its next level up. This process continues until either the current simpler model is chosen as the most suitable model or the tree-top level is reached (there is no next level up).

[16] If the resulting most suitable model involves a nonlinear trend, a likelihood ratio test is similarly used to choose the most suitable *k* from the trial values {1.1,1.2, …,2.0}. As a result, we find that a nonlinear trend with *k* ranging from 1.1 to 2.0 is chosen for *T*_{Nn} at 13 stations and for *T*_{Xx} at 2 stations (Table S1 in the Supporting Information).

[17] Changes in the annual temperature extremes are inferred from the chosen most suitable GEV model fitted to the data in the whole period. These changes are compared with those obtained from fitting stationary GEVs to data in the first and last 51 year windows: 1910–1960 and 1961–2011. In the stationary GEV approach, the two-sample two-sided Kolmogorov-Smirnov test is applied to check whether or not the distribution of *T*_{Nn} (or *T*_{Xx}) in the first 51 year period is significantly (at 5% level) different from that in the last 51 year period. The results are shown in Figures 1 and 2 and discussed below.

### 3 Results

- Top of page
- Abstract
- 1 Introduction
- 2 Data and Methodology
- 3 Results
- 4 Summary and Discussion
- Acknowledgments
- References
- Supporting Information

[18] The 20 year return values (RV20yr) of the annual minima *T*_{Nn} and the annual maxima *T*_{Xx} as estimated for the climate condition of year 2010 (namely, the time-dependent GEV parameters take their values at the time that corresponds to year 2010) using the GEV tree approach and their changes over the last century (from 1910 to 2010) and last half century (from 1960 to 2010) are shown in Figure 1 (also in Table S1). As would be expected, inland areas have colder cold extremes and warmer warm extremes than do the coastal areas (Figures 1a and 1b). For example, the annual minimum temperature that is expected to occur once every 20 years is about 3.7°C at Sydney and about −8.9°C at Bathurst, an inland station west of Sydney (Figure 1a). In terms of RV20yr of *T*_{Nn} (see Table S1), Darwin is the least cold site (RV20yr = 12.2°C), while Inverell is the coldest site (RV20yr = −9.0°C). Similarly, the least hot site is Low Head in Tasmania (RV20yr = 27.4°C), while the hottest site is Marree in South Australia (RV20yr = 48.0°C).

[19] Changes over the past century are much more extensively significant in the cold extremes *T*_{Nn} than in the warm extremes *T*_{Xx} (Figures 1c and 1d). Significant increases in *T*_{Nn} are identified at 31 out of the 52 sites, with 20 sites being found to have no significant change (Figure 1c) and Hobart showing a nonlinear trend with the 1960s being the coldest decade (Figure 2c). Significant increases in *T*_{Xx} are identified at only 13 sites, with 36 sites being found to have no significant change (Figure 1d). In particular, none of the stations in New South Wales are found to have a significant change in *T*_{Xx}, and Queensland is found to have become less susceptible to extreme heat in general, with a significant decrease in *T*_{Xx} being identified at three sites and no significant increase anywhere in the state (Figure 1d). The results for the 1960–2010 period are very similar to those for 1910–2010; for *T*_{Nn}, 31 of the 52 sites show a significant increase, and only 1 a significant decrease, while for *T*_{Xx}, the numbers are 12 and 4 respectively, with 36 sites showing no significant change (Table S1).

[20] Most of the increases over the 1910–2010 period in RV20yr of *T*_{Nn} are 1.1°C–4.1°C per century (at 26 out of the 31 sites; Figure 1c). The greatest warming in *T*_{Nn} is seen at Cairns (4.1°C), Mackay (4.0°C), and Richmond (3.6°C) in Queensland (Table S1). Overall, the results suggest that Queensland has become much less cold in winter and also slightly less hot in summer. As a result of the warming, an annual minimum temperature that was expected to occur once every 20 years in the 1910 climate can hardly be expected to occur in the 2010 climate at 19 out of the 31 sites of warming *T*_{Nn} (the return period has become more than 1,000 years; Table S1). At the other 12 sites of warming *T*_{Nn}, the return period has become about 52–575 years (Table S1).

[21] The significant *T*_{Xx} increases are mainly in Tasmania and the southwest corner of Western Australia. Most of the increases in RV20yr of *T*_{Xx} are 1.3°C–2.7°C per century (Table S1). The greatest warming in *T*_{Xx} is seen at Cape Leeuwin (3.9°C) and Wilsons Promontory (3.5°C; Table S1), both of which are very exposed coastal locations with highly positively skewed distributions of summer maximum temperatures. Overall, Tasmania and southwestern Australia seem to have become less cold in winter and hotter in summer. As a result of the warming, an annual maximum temperature that was expected to occur once every 20 years in the 1910 climate can be expected to occur about once every 2–4 years in the 2010 climate.

[22] Changes inferred from the GEV tree approach are generally consistent with the corresponding changes inferred from the stationary GEV fits to data in the first and the last 51 year windows, namely, from the stationary GEV approach (see Figures 1e, 1f, and 2). However, fewer stations are found to have a significant change in the distribution of *T*_{Nn} between the two non-overlap 51 year windows (Figures 1e and 1f). Note that the stationary GEV approach assumes that *T*_{Nn} in each of the 51 year window is stationary, which is more likely to be invalid than valid in light of the changing climate. Thus, the GEV tree approach, which allows for stationary or nonstationary model (whichever is more suitable), is arguably preferable for estimating changes in extremes and their statistical significance. This is why we focus on the GEV tree analysis results in this study.

[23] Time series of the *T*_{Xx} and *T*_{Nn} and their 20 year return values as estimated using the GEV tree and the stationary GEV approaches are shown in Figure 2 for six selected stations, along with the 95% confidence intervals (CIs) of the 20 year return value. The CIs are estimated using the adjusted bootstrap method [*Coles and Simiu*, 2003] for the stationary GEV approach (gray shadings) and using the asymptotic variance-covariance of the parameter estimates [*Coles*, 2001] for the GEV tree approach (dark gray hatchings in Figure 2). Generally, the two sets of RV20yr estimates are in reasonable agreement, although the stationary GEV estimates are more variable due to higher sampling variability (data in a sequence of 51 year running windows).

[24] The six selected stations represent six different types of temperature changes (Figure 2). Hobart experienced a nonlinear trend in *T*_{Nn}, with the 1960s being the coldest decade in the record but the record low temperatures are seen in the 1970s, while it shows no significant change in *T*_{Xx}. The results of the stationary GEV fits suggest an insignificant decrease in *T*_{Nn} at Hobart between the first and last 51 year periods (Figure 1f). Cairns shows a linear increase in *T*_{Nn} but no significant change in *T*_{Xx}. Contrarily, Darwin shows no significant change in *T*_{Nn} but a linear increase in *T*_{Xx}. Alice Springs experienced a linear increase in both *T*_{Nn} and *T*_{Xx}. Carnarvon shows a stabilizing increasing trend in *T*_{Nn} but no significant change in *T*_{Xx}. Esperance experienced a nonlinear trend in *T*_{Nn}, with the 1930s being the coldest decade in the record, followed by a steady significant warming; it also experienced a linear increase in *T*_{Xx} (Figure 2h).

### 4 Summary and Discussion

- Top of page
- Abstract
- 1 Introduction
- 2 Data and Methodology
- 3 Results
- 4 Summary and Discussion
- Acknowledgments
- References
- Supporting Information

[25] In this study, a GEV tree approach for analyzing changes in extremes has been developed and applied to Australian temperature extremes. This new approach is data adaptive. It provides insight into the behavior of extremes with return periods of several years to decades that are of importance to engineering design/applications. Thus, it is a useful alternative to analyzing extreme indices. The latter are based on events that occur relatively frequently (with return periods of 1 year or shorter) and thus has the advantage to allow for more robust identification of related changes.

[26] The results show strong warming in extreme low minimum temperatures through most of Australia over the last 50 to 100 years, with significant increase for both time periods for the majority of stations across the country. The strongest increase is generally seen in northeastern Australia, and the weakest (particularly in the last 50 years) is in southwestern Australia, a region which has seen a substantial decrease in rainfall over the last 40 years [*CSIRO and Bureau of Meteorology*, 2010].

[27] Extreme high maximum temperatures also show a tendency toward warming, with significant warming trends outnumbering significant cooling trends, but the results are much weaker than those for extreme low minimum temperatures, with the majority of stations showing no significant trends in either direction. Trends are consistently weak or negative through most of inland eastern Australia, with the strongest warming evident in the west and south.

[28] At a national scale, the results are broadly consistent with those for less extreme events (summer maxima above the ninetieth percentile and winter minima below the tenth percentile) reported in *Trewin and Smalley* [2013]. They found that for warm summer maxima, 19 of 40 stations showed no significant trends over 1910–2011, and 55 of 80 for 1960–2011, although warming trends are predominant amongst those stations which did show significant trends. (The proportion of stations with significant trends was lower for warm summer maxima than for any other index or season analyzed, except for warm autumn maxima).

[29] The cooling of extreme high maxima in inland eastern Australia is mirrored to some extent for summer maxima above the ninetieth percentile over the 1910–2010 period, but not over the post-1960 period, where *Trewin and Smalley* [2013] did not find significant cooling at any Queensland station.

[30] The strong warming in extreme low minimum temperatures generally corresponds with decreases in the frequency of winter minima below the tenth percentile, which were found for 30 of 46 stations for 1910–2011 and 42 of 83 stations for 1960–2011 [*Trewin and Smalley*, 2013]. The geographic patterns are also similar, with the strongest warming found in northeastern Australia for both elements.

[31] The differences between the results for extreme high maximum and low minimum temperature are also broadly consistent with changes in mean temperature. Mean summer maximum temperatures show a warming trend averaged over Australia of 0.51°C over the century (somewhat less than the annual mean maximum trend of 0.75°C), while mean winter minimum temperatures show a stronger warming trend, 0.79°C over the century. As for extremes, Queensland generally shows the strongest trends for mean winter minima [*Fawcett et al*., 2012]. Along with inland New South Wales, it also shows the weakest trends for mean summer maxima, although decreasing trends in mean summer maxima are not as common as those for extreme high maxima.