1. Introduction

Even though there is unambiguous evidence for climate change (Bernstein et al. 2007), the identification of a nonstationary temperature trend attributable to an increase in greenhouse gas emissions is complicated by the fact that climate change is not happening in a monotonic and uniform way. Because of the interaction of the various climate system components, which have different intrinsic time scales and quasi-oscillatory climate modes, climate on a global as well as a regional scale will change in heterogenous and nonmonotonic ways (Ghil and Vautard 1991; Keenlyside et al. 2008; Knight et al. 2009). The ability to distinguish an anthropogenic-caused trend from genuine climate variability is hampered by the fact that even simple stochastic processes can produce time series that exhibit apparent trends over finite periods. Recent studies have shown that for some climate indices interannual variability and trends can also be attributed to climate noise; that is, they can arise from sampling variability of high-frequency atmospheric fluctuations (Madden 1976; Feldstein 2000, 2002; Franzke 2009).

This paper examines whether Antarctic temperature trends can be explained by climate noise. To do this, one needs to generate surrogate time series x_t (t = 0, … , N) with variance σ² mimicking the internal atmospheric fluctuations to assess the probability that an observed trend is within the expected range of apparent trends due to internal variability. In most climate-related studies, such surrogate time series are generated by an autoregressive process of first order [AR(1); von Storch and Zwiers 1999; Santer et al. 2000, 2008]. Such a short-range dependent process exhibits an exponential decay with time scale τ of the autocorrelation function

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thus, γ(k) ∼ exp(−k/π) with

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Here x denotes the climatological mean. On the other hand, there is mounting evidence that surface air temperature exhibits long-range dependence (Koscielny-Bunde et al. 1998; Huybers and Curry 2006; Gil-Alana 2005; Rybski et al. 2006; Vyushin and Kushner 2009; Thomas et al. 2009). Long-range dependence is usually characterized by the long-range dependence parameter d. A long-range dependent process has a hyperbolical decay of autocorrelation γ(k) ∼ k^2d−1, which indicates a much slower decay than the exponential decay of an AR(1) process. This property leads to the fact that the sum over the autocorrelation function——is equal to ∞ (Robinson 2003; Graves 2009). The sum (or integral) over the autocorrelation function is a measure of how quickly or slowly a perturbed system relaxes back to its equilibrium state. If the sum over the autocorrelation function is infinite, then this indicates that a perturbation to the system will never be forgotten and that a stationary system will need a very long time to relax back to its equilibrium state. This property has the effect that long-range dependent processes can display apparent trends over relatively long periods.

A stationary time series is also long-range dependent for a given β ∈ (0, 1) and a constant c > 0 when

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where f (ω) denotes the spectral density (Beran 1994). This property says that the spectral density of a long-range dependent process has a pole at the origin. It can be shown that, by setting ω = 0 in (1), both definitions are equivalent because the Fourier transform of the autocorrelation function is the power spectral density (Malamud and Turcotte 1999).

For the case in which climate variability is long-range dependent, it might be hard to distinguish between anthropogenically forced trends and intrinsic apparent trends because of the long-range dependence. In most studies on long-range dependence, it is implicitly assumed that climatic time series can be decomposed into a trend and fluctuations and that the trend does not influence the fluctuations. However, there are recent modeling studies that show an effect of global warming on storm activity (Leckebusch and Ulbrich 2004; Fischer-Bruns et al. 2005). This effect of increasing temperatures on the fluctuation properties is what makes it potentially hard to identify the causes of climate trends. Furthermore, as discussed by Santer et al. (2008), if the autocorrelation structure of an observed variable has a more complex structure and exhibits a slower decay than that of an AR(1) process or is even long-range dependent, then this would imply that a statistical significance test based on an AR(1) null model is more likely to indicate significant differences between modeled and observed trends for cases in which they do not exist.

One of the simplest “physical” models of long-range dependence is the aggregation of many short-range dependence models, for example, AR(1) models (Robinson 1978; Granger 1980). This is an attractive explanation for possible long-range dependency in the climate system where the atmosphere, ocean, land, cryosphere, and so on, can be thought to evolve according to short-range processes; however, the aggregate of all of these climate subcomponents would then be better described by a long-range dependent process.

So far only two studies (Stephenson et al. 2000; Percival et al. 2001) have attempted to address whether a short- or long-range dependent model fits observed climate variability better. These studies conclude that even century-long data are not sufficient to demonstrate the superiority of one model over the other. Thus, here only how the choice of the null model influences the statistical significance of climate trends will be examined—in particular, whether the trends can be explained to arise from climate noise.

2. Long-range dependence of Antarctic temperatures

In this study daily temperature data from eight Antarctic stations (Bellingshausen (14 824 days), Faraday–Vernadsky (20 513 days), Halley (18 901 days), McMurdo (18 596 days), Molodezhnaya (13 271 days), Neumayer (9524 days), Rothera (11 871 days), and South Pole (16 009 days) are used: see Fig. 1), taken from the Reference Antarctic Data for Environmental Research (READER) dataset (Turner et al. 2004), and Table 1. The mean annual cycle is subtracted. It is computed by taking the calendar average for each day (all of the results listed below are insensitive to the subtraction of the annual cycle).

To examine whether Antarctic temperature time series are long-range dependent, the Geweke and Porter-Hudak (GPH) semiparametric estimator (Geweke and Porter-Hudak 1983; Hurvich and Deo 1999; Vyushin and Kushner 2009) is used to estimate the long-range dependence parameter d. This estimator performs a least squares regression of the logarithm of the periodogram of the time series against the logarithm of the frequency in a specified neighborhood of the origin (Geweke and Porter-Hudak 1983; Hurvich and Deo 1999; Graves 2009; Vyushin and Kushner 2009). The plug-in selector (Hurvich and Deo 1999) is used for setting the largest frequency used in the regression. Similar periodogram estimators have been developed by Künsch (1986), Robinson (1995), Velasco (1999), and Kim and Phillips (2006). If the value of the long-range dependence parameter d is not significantly different from 0, then the temperature variations are short-range dependent; if d is significantly different from 0 and positive, then there is evidence for persistent long-range dependence in Antarctic temperature variations. The inferred parameters d are statistically significantly different from zero at the 2.5% level (Table 2) for all considered daily temperature time series. The three stations on the Antarctic Peninsula are the ones with the largest d values of about 0.26–0.28, and all other stations have lower d values.

3. Nonlinear trends

Most studies addressing possible trends in climate datasets assume a priori that the trends are linear and consequently use linear least squares fits to extract the trend. Also, the period over which trends are assessed is in many cases subjective and amounts to “eyeballing” (Percival and Rothrock 2005). There is no reason why trends in temperature have to be linear. Thus, in this study the ensemble empirical mode decomposition (EEMD) method (Wu and Huang 2009; Huang et al. 1998; Huang and Wu 2008) is used to analyze temperature data from Antarctic stations and to extract their possible nonlinear trends (Wu et al. 2007; Franzke 2009). The EEMD decomposes a time series into a finite number of intrinsic mode functions (IMF) and an instantaneous mean:

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where the jth IMF ψ_j can be written in polar coordinates as ψ_j(t) = γ_j(t) sin[θ_j(t)] and γ_j is the jth amplitude, θ_j is the jth instantaneous frequency, and R is the residual. Both the amplitude and the frequency are time dependent. An IMF is different from Fourier modes in which both γ_j and θ_j are time independent. An IMF is defined by the following two properties: 1) each IMF ψ_j has exactly one zero crossing between two consecutive local extrema and 2) the local mean of each IMF ψ_j is zero. To avoid mode mixing, EEMD adds white noise to the observed time series before the sifting process of the standard empirical mode decomposition (EMD; Huang et al. 1998; Huang and Wu 2008; Wu et al. 2007; Franzke 2009) and treats the mean of the ensemble as the final IMF. Thus, EEMD is a noise-assisted data analysis method. Here an ensemble of 1000 realizations with an amplitude of 0.1 standard deviation of the original time series are used; see Wu and Huang (2009) for more details. While there is no unambiguous definition of a trend, in this study the instantaneous mean is interpreted as a trend that is possibly nonlinear (Wu et al. 2007; Franzke 2009). Note that all definitions of a trend depend on the procedure that is used to extract it and that all trends identified in this study do not necessarily correspond to the “true” trend. The EMD method has been shown to be a powerful method for extracting trends from noisy and nonlinear time series (Wu et al. 2007; Huang and Wu 2008; Franzke 2009).

The EEMD algorithm is applied to monthly-mean temperatures, which are derived from the daily data, because I want to focus on the trends here. In Fig. 2 is displayed the IMF modes for the Faraday–Vernadsky station data. As can be seen, the modes display oscillatory behavior with varying amplitudes and frequencies. The subtraction of the successive IMFs leads to a smoothing of the time series and reveals the low-frequency behavior of the time series (Fig. 2). The IMFs of the other stations show qualitatively similar characteristics (not shown). In Fig. 3 is presented the instantaneous means of the monthly-mean temperature time series. The instantaneous means show temperature increases at Bellingshausen, Faraday–Vernadsky, McMurdo, and Rothera and temperature decreases at South Pole and Neumayer, while Halley and Molodezhnaya are almost constant. The signs of these trends are consistent with previous studies (Turner et al. 2005; Steig et al. 2009) in that they show an approximate separation in the sense of temperature change between western and eastern Antarctica (Table 1). Although the trends at Faraday–Vernadsky, Rothera, and South Pole can be reasonably well approximated by a linear least squares fit [as used in Turner et al. (2005) and Steig et al. (2009); Fig. 3b], the EEMD trends at Bellingshausen, Halley, McMurdo, Molodezhnaya, and Neumayer show differences from straight lines. For example, EEMD reveals a possible warming trend at Bellingshausen (Fig. 3a) starting in 1970 to about 1998 and more constant temperatures afterward.

4. Climate noise

Now I want to examine whether the trends can be explained by climate noise, that is, internal atmospheric variability. I also want to compare the effect of the long-range dependence with the traditionally assumed short-range dependence for climate variability. While the above analysis provides evidence for long-range dependence in Antarctic temperatures, I want to illustrate here that the choice of null model really matters in practical applications. In most previous climate studies, only an AR(1) model has been considered (Santer et al. 2000, 2008; Percival and Rothrock 2005), and, thus, I want to explicitly compare the short-range dependent AR(1) with a long-range dependent model. For this purpose the two paradigmatic models that are also parsimonious are used: 1) an AR(1) process as the paradigmatic short-range dependent model (von Storch and Zwiers 1999) and 2) a fractionally integrated process FARIMA(0, d, 0) as the paradigmatic long-range dependent model (Hosking 1981; Stoev and Taqqu 2004). An AR(1) is defined as x_t+1 = αx_t + σζ_t, where σ is the lag-1 autocorrelation coefficient and σ denotes the variance of the Gaussian white-noise process ζ with zero mean and unit variance. The FARIMA(0, d, 0) model is given by Δ^dx_t = ζ_t, where the long-range parameter |d| ≤ ½ is real valued and, thus, Δ^d denotes a fractional difference operator (Beran 1994). Both processes are nested in the more general FARIMA(p, d, q) model. By setting p = 1, d = 0, and q = 0, one recovers the AR(1) model; by setting p = 0 and q = 0, one recovers the FARIMA(0, d, 0) model. The long-range dependency parameter d is inferred by the Geweke and Porter-Hudak semiparametric estimator (Geweke and Porter-Hudak 1983; Hurvich and Deo 1999). Both the AR(1) and FARIMA(0, d, 0) can produce apparent trends over finite periods.

The question now is, Are the observed temperature trends larger than the spurious trends one can expect from the corresponding AR(1) and FARIMA(0, d, 0) processes as proxies for the internal atmospheric variability? To answer this question, a Monte Carlo approach is used by creating an ensemble of 1000 members of AR(1) and FARIMA(0, d, 0) for each corresponding temperature time series. In this Monte Carlo approach, proper account is also taken of the estimation errors that are displayed in Table 2 by sampling the parameters from a Gaussian distribution with the same mean as the estimated mean and a standard deviation corresponding to the 5% confidence bounds (±1.96σ). These ensembles correspond to daily data that are then averaged to produce surrogate monthly-mean time series to compare directly with the observed monthly-mean temperature data. Because the algorithm used to simulate the FARIMA(0, d, 0) (Stoev and Taqqu 2004) does not allow the specification of the noise variance, the simulated surrogate time series is rescaled to have the same variance as the respective observed temperature time series. Fitting the data from daily data and then averaging the surrogate data amounts to testing the trends against a climate-noise null hypothesis; that is, trends would be caused by sampling fluctuations of intraannual variability (Feldstein 2002; Franzke 2009). To assign statistical significance to the instantaneous mean (trend), the Monte Carlo approach of Franzke (2009) is followed and how often the null models create trends that exhibit a range that is larger than the observed trends is estimated. If 97.5% of the trends of the surrogate data produced by AR(1) and FARIMA(0, d, 0) are smaller than the observed trends, then one can claim that the corresponding IMF or instantaneous mean is significantly different at the 2.5% level from the corresponding null model and cannot be explained by climate noise or internal atmospheric variability.

This Monte Carlo approach shows that only the trends on the Antarctica Peninsula are statistically significant at the 2.5% level against the short-range dependent null model and that only the Faraday–Vernadsky station is also statistically significant against the long-range dependent null hypothesis (Table 1). All other considered stations show no statistically significant trends, and therefore the null hypothesis that the trend is caused by climate noise cannot be rejected. The results for the short-range dependence null model are similar to Turner et al. (2005), although here it is found that the nonlinear trends at Rothera and Bellingshausen are also significant against the short-range dependent null model. Turner et al. (2005) also find a huge variation in overlapping 30-yr trends. This implies the existence of either slowly varying climate modes or nonlinear trends.

5. Robustness of methods

To estimate the reliability and robustness of the methods, various sensitivity studies were done. As can be seen in Fig. 2, the Faraday–Vernadsky station temperature is skewed toward cold temperatures. This skewness could potentially affect the estimation of the trends and long-range parameters. To check the robustness of the trend estimation, 100 sensitivity experiments were run of estimating the EEMD trend by using surrogate time series of a linear trend corresponding to a 1° increase over the length of the time series superimposed on randomized time series of the Faraday–Vernadsky station temperature, thus destroying all correlation structure and trends but retaining the same skewness as Faraday–Vernadsky. Then the root-mean-square error (RMSE) between the linear trend and the estimated EEMD trend was calculated and an ensemble mean RMSE = 0.24 was obtained. The ensemble mean standard deviation of the time series is about an order of magnitude larger than the RMSE. Thus, the EEMD trend is a very good approximation of the linear trend, which is confirmed by a visual inspection (not shown). Qualitatively similar results were obtained for a quadratic trend and the other stations as well as when a noise from a σ-stable distribution (Stoev and Taqqu 2004; Gardiner 2009) with α ∈ (1.5, 2) was used in the tests. The α-stable distributions are leptokurtic and heavy tailed. Utilizing such a distribution demonstrates that detecting trends with EEMD is robust against heavy tails and/or outliers. Thus, the EEMD method is able to reliably extract trends from skewed and non-Gaussian time series.

As shown in Taqqu and Teverovsky (1998), the estimation of the long-range dependence parameter with GPH is insensitive to non-Gaussianity. The time series were also randomized, thus destroying the correlation structure while keeping the exact distribution. GPH estimates of these randomized time series are all very close to zero (Table 2), thus showing that deviations from Gaussianity do not lead to false detection of long-range dependence. This result provides further evidence of the robustness of the results.

A periodic AR(1) was also fitted to the anomalous daily temperature time series, which captures the non-Gaussian behavior of the observed time series better than the standard AR(1) process with constant parameters. However, using this model gives results for the trends that are very similar to those of the standard AR(1) null model (not shown).

6. Conclusions

This study presents evidence for long-range dependence in Antarctic station temperature data. The finding of statistically significant long-range dependence is at odds with the short-range dependent model AR(1). Because the AR(1) is the overwhelmingly used model in climate science for significance testing of trends, the study also explicitly examines how the choice of null model influences the statistical significance. It is shown that most observed trends in Antarctic temperatures can be explained as arising from climate noise. The only exception is the Antarctic Peninsula, where there were three stations showing significant trends that cannot be explained as arising from internal atmospheric fluctuations when an AR(1) model is used. On the other hand, if a long-range dependent model is used for the background atmospheric variability, then two out of these three trends turn out to be not significant and can be explained as arising from climate noise.

Thus, the results emphasize the fact that the significance of trends depends crucially on the null hypothesis one uses for the internal atmospheric variability, as was also noted in Santer et al. (2008). Because of the mounting evidence for long-range dependence in surface temperatures all over the world (Koscielny-Bunde et al. 1998; Huybers and Curry 2006; Vyushin and Kushner 2009), any statistical significance test should include a test for long-range dependence and should not a priori assume short-range dependence. This implies that greater attention needs to be given to assessing the correlation properties of climate and, in particular, whether it is long-range dependent.