Extreme temperatures in a warming climate: modulation of the effect of changes in the mean and variance by the distribution skewness

Transcription

1 Extreme temperatures in a warming climate: modulation of the effect of changes in the mean and variance by the distribution skewness Working Paper 1* Lydia Stefanova and 1,2 Philip Sura 1* Center for Ocean-Atmospheric Prediction Studies, Florida State University, Tallahassee, FL 2 Department of Earth, Ocean, and Atmospheric Science, Florida State University, Tallahassee, FL *corresponding author: lstefanova@fsu.edu

2 Abstract Because of the non-linear dependence of the frequency of extremes on the distribution s moments and their changes, the relative importance of these factors remains poorly understood. In an effort to provide an estimate for their relative contributions, this study focuses on the change in frequency of temperature extremes between a neutral and an anomalously warm observed climate subsets for the contiguous United States. Theoretical extreme counts for the warm subset are calculated assuming a Gaussian distribution of daily temperatures, given the observed changes in the mean and variance relative to the neutral subset. These theoretical counts are shown to be qualitatively similar to the observed counts, yet substantial discrepancies are present in all seasons and most pronounced in summer (in terms of absolute difference) and winter (in terms of % difference). In the majority of instances, the observed counts exceed the Gaussian assumption estimates. It is shown that the discrepancy between the observed counts and the Gaussian estimates is closely tied to the skewness of the baseline distribution. In most seasons and locations daily temperatures are negatively skewed; as a result, observed counts exceed the Gaussian estimates. In the rarer cases of positively skewed distributions, such as near the coast of California, the Gaussian approach results in their overestimation. It is found that the relationship between observed counts and changes in the mean is approximately as strong as the one expected from Gaussian assumptions. The relationship between observed counts and changes in the standard deviation, on the other hand, is much weaker than Gaussian. This is attributed to a relatively strong relationship between baseline skewness and changes in the spread. 2

3 Introduction Research in recent years has solidified the realization that changes of the mean state are not the only and, depending on the application, not the dominant factor that needs to be considered when comparing climatic regimes. In a number of cases, changes in the statistics of temperature extremes may have much stronger impacts on species distribution (Zimmerman et al 2009), agricultural productivity (Tubiello et al 2007, Moriondo et al 2010) and human health outcomes (McMichael et al 2006), just to name a few examples, than changes in the average temperature. The existence of a trend for increasing frequency of hot extremes in step with an increasing mean has been documented for various regions of the world (Hartman et al 2013). A range of opinions exists on whether such increase in the frequency of hot extremes is attributable largely to shifts in the mean (e.g. Simolo et al 2011, Rhines and Huybers 2013), increased variability (e.g. Schär et al 2004, Hansen et al 2012), or to a combination of changes in mean, variance and skewness (e.g. Klein Tank and Konnen 2003, Donat and Alexander 2012, Volodin and Yurova 2013) It is well understood that if daily temperatures follow a normal distribution, and if climate change manifests through a simple shift in the mean, the number of hot extremes would increase. If at the same time the variance increases, the number of hot extremes would increase even more. The notion that, for a normally distributed variable, the number of extreme events is more sensitive to changes in the distribution s spread than to changes in its mean (Kats and Brown, 1992) has been widely accepted in the extreme event literature (e.g. Mearns et al 1997, Schär et al 2004, Zimmerman et al 2009). However, this only holds for small changes in the mean and spread. Furthermore, while the normal distribution is often considered an adequate approximation for daily temperatures, a closer look reveals that there are regions and seasons for which the distribution of daily temperatures is markedly non-normal (Smith and Sardeshmukh 2000, Shen et al 2011, Stefanova et al 2011), which has significant implications for the frequency of extreme events (Sardeshmukh et al 2015). For a negatively (positively) skewed distribution, for example, the same increase of the mean would lead to a relatively larger (smaller) increase in the number of hot extremes. Due to the non-linear influence of the mean, 3

4 variance and higher moments on the frequency of extremes (and their changes), the relative importance of these factors remains poorly quantified. The analysis presented here is intended to provide insight into the reasons for the disagreement across different studies regarding the relative roles of changes in the mean and changes in the variance in determining change in extreme frequency. We begin by quantifying the contribution of non-gaussianity to frequency of extremes under observed warmer-mean conditions for the contiguous United States using data and methodology described in section 2. The results are presented in section 3, where we demonstrate that within the warm subset hot extremes are generally more frequent than expected based on Gaussian statistics and that their counts are more closely related to changes in the mean than to changes in the variability. The few exceptions to the latter arise in regions/seasons for which daily temperatures have positive skewness and whose warm-state variance is increased relative to climatology. These results are discussed and summarized in section Data and methods We use historical observations over the United States to define a subset of abovenormal monthly temperature means, and analyze the properties of the daily temperature distribution and the extreme counts within this subset in comparison to Gaussian-based expectations. Daily gridded temperatures are obtained from the Maurer data set (Maurer et al 2002) for the entire period of the dataset s availability, This dataset is formed from National Oceanic and Atmospheric Administration (NOAA) Cooperative Observer (COOP) station measurements gridded to a degree resolution accounting for elevation. For each calendar month and for each grid point, the twenty years with the highest monthly mean temperatures are used to define a subset of the 20 warmest years and 20 coolest years, and the remaining 23 years are designated as neutral. The neutral subset is used to calculate the baseline mean μ 0, standard deviation σ 0 and skewness s 0 of daily temperatures for each calendar month and each grid point. It is also used to define the 95 th percentile of daily temperature values for each 4

5 calendar month and each grid point. The number of days exceeding the 95 th percentile for each month is the number of hot extremes for that month. Given this definition, the average count (H 0 ) of hot extremes within a month in the neutral subset is 0.05 times the number of days in that month, i.e., 1.55 days for a 31-day month. Assuming daily temperatures follow a normal distribution with a mean of μ 0 and standard deviation of σ 0, the probability p 0 of exceedance of a threshold c is given by Rearranging, p 0 = p(c, μ 0, σ 0 ) = [1 + erf (c μ 0 2σ 0 )] c = μ 0 + 2σ 0 erfinv(1 2p 0 ) Suppose the distribution s mean is changed by μ and its variance by σ. Now the probability of exceeding c is given by p Gauss = p(c, μ 0 + μ, σ 0 + σ) = [1 + erf (c (μ 0 + μ) 2(σ 0 + σ) )] Using the expression for c as a function of p, p Gauss = [1 + erf (erfinv(1 2p 0) Δμ 1 + Δσ 2(1 + Δσ ) )] where Δμ = μ σ 0 and Δσ = σ σ 0. With hot extremes defined as those above the 95 th percentile, p 0 = The average count of hot extremes per month is H Gauss = Np Gauss, where N is the number of days in that month. The quantities μ and σ are calculated, respectively, as the difference between the warm subset s mean and the neutral subset s mean, and the difference between the warm subset s standard deviation and that of the neutral subset. To evaluate how important observed departures from Gaussianity are for the counts of extremes, we compare the observed counts H Obs its Gaussian-assumption based 5

6 counterpart H Gauss given observed pairs of Δμ and Δσ. Analyses are presented for representative winter (January), spring (April), summer (July) and fall (October) months. 3. Results Qualitatively, H Gauss provides a reasonable first-order estimate for H Obs (Fig 1, first two columns). Hot extremes in the warm subset are most frequent in July, particularly in the southern states, and least frequent on the east side of the Rockies in winter and fall. Despite this qualitative similarity, the difference between the two (Fig 1, right column) is quite non-negligible. With a few exceptions, H Gauss tends to underestimate H Obs. In absolute terms, the discrepancy between the two ranges from about -5 (California s coast in summer) to +5days/month (Texas in summer). The difference in temperature means between warm and neutral subsets (Fig 2, x- axis) ranges from about +0.25σ 0 to +1.5σ 0, with the largest values found in July. The averaged difference between the means is [3.0/1.9/1.4/1.7] C in [January/April/July/October], respectively. The difference in variance (Fig 2, y- axis) ranges from about -0.5σ 0 to 0.5σ 0, with the largest negative values (i.e., reduction of variance) found in January. Warmer means are associated with a reduction of spread at [80/59/41/58]% of grid points in [January/April/July/October] respectively. The values of H Gauss as a function of the observed shift and spread changes from the neutral to the warm subset are shown by the coloring in Fig 2. The climate change-extreme events literature has widely accepted the idea that for normal distributions, the change in extreme events is more sensitive to changes in the variance than to changes in the mean. Katz and Brown (1992) show that for an extreme event defined as the exceedance of a threshold c, p σ c μ p = σ μ They pointed out that, since for c to be a considered an appropriate extreme threshold it is necessary that c > μ + σ, it follows the ratio (c μ)/σ is greater than 1, or, in other words, that p changes faster with σ than with μ. However, it is 6

7 important to realize that this argument holds only in the vicinity of the original μ and σ values, i.e., μ 0 and σ 0. Once the distribution shifts and/or spreads enough to provide Δμ + Δσ > (c μ 0 ) σ 0 1, the inequality no longer holds. With the definition of extreme events used here (exceedance of 95 th percentile), (c μ 0 ) σ , indicated by the line in Fig 2. Below this line, changes in the spread have stronger effect on the number of extremes; above it, changes in the mean are more influential. The observed shift and spread changes are such that, assuming normal distribution, in most cases the warmer subset would have more frequent hot extremes than the neutral subset. An exception to this is seen at a few grid points in January, where a relative small increase in the mean paired with a relatively large decrease in variance would, under assumptions of normality, result in a reduction of hot extremes. The observed values of H Obs as a function of the observed shift and spread changes from the neutral to the warm subset (Fig 3) share the general characteristics of H Gauss (Fig 2). In both cases, hot extreme counts generally increase with increasing Δμ and Δσ. The strengths of the relationship between H Gauss or H Obs and Δμ are similar (Fig 4). However the relationship between H Obs and Δσ is quite a lot weaker in reality than under assumptions of Gaussianity. The percent difference between the two (Fig 5) clearly reflects this: in all seasons, a decrease in spread is associated with H Obs > H Gauss and vice versa. The largest discrepancies between H Obs and H Gauss tend to be associated with the largest reduction of spread. An explanation for this is suggested by the distributions baseline skewness s 0 and its relationship to Δσ (Fig 6). As expected, a larger negative (positive) skewness is associated with a greater increase (decrease) in hot extreme frequencies relative to that of a normal distribution (Fig 7). The inverse correlation between s 0 and D is rather strong (Fig 8, black bars). A more interesting finding is the moderately strong relationship between s 0 and Δσ (Fig 8, grey bars), which implies that for regions strongly negatively skewed distributions, warmer regimes are associated with a reduction in variance which in turn implies that the warming for such regions is, at least partially, achieved through a shortening of the left tail and a decrease of the 7

8 asymmetry of the distribution. The presence of an inverse relationship between s 0 and s (Fig 8, white bars) is consistent with this interpretation. 4. Discussion and Summary From simple statistical arguments, it is undisputable that: a) given an increased mean, a greater increase in variance will result in more numerous extremes, b) given the same change in variance, a greater increase in the mean will result in more numerous extremes, and c) for a non-gaussian distribution, if warming is the result of simple translation, a negatively (positively) skewed distribution will see a greater (smaller) increase in hot extremes than a neutrally skewed one. It has been previously (Katz and Brown 1992) analytically demonstrated that for a normally distributed variable changes in variance and mean, extremes are more sensitive to changes in the spread. Here, we have analytically defined the extreme-threshold-dependent range of changes for which this holds, and beyond which the reverse is true. Once the changes exceed this range, for a Gaussian distribution, changes in the mean become more influential than changes in spread. Comparing the observed warm subset to the observed neutral subset we find that for the continental US, for most grid points and in most seasons temperatures are negatively skewed, with the most notable exceptions seen along the west coast in all seasons but winter, and around the Great Lakes in the transitional seasons. As anticipated, relative to the neutral subset, the warm subset has more numerous hot extremes. When we compare the observed change in the number of hot extremes to the change computed under assumption of Gaussianity, we find that, while the latter is a reasonably good first-order estimate for the former, there remain significant discrepancies. The number of hot extremes in the warm subset usually exceeds the number predicted by Gaussian estimation. This exceedance is by up to five days (in parts of Texas in summer) in absolute terms, or by up to 300% (in parts of the Northwest in winter) in relative terms. There is a very strong relationship between the discrepancy between observed and Gaussian-predicted extremes and the baseline skewness of the distribution (i.e., a large negative skewness results in more numerous extremes). We find a strong relationship between skewness and changes in the spread from the 8

9 neutral to the warm subset. The relationship between the distribution s baseline skewness and the changes in spread observed in warmer regimes (strongly negatively skewed distributions tend to see reduction in spread in a warmer regime) suggests that that the left tail is shortened without a matching lengthening of the right tail. This is consistent with the finding of Peterson et al (2008) that since the mid-60s, averaged over North America, the highest minimum and maximum temperatures have warmed by ~1 C while the coldest minimum and maximum temperatures have warmed by ~3.5 C, as well as with our finding of an inverse relationship between baseline skewness and changes in skewness under warming. As a consequence of the relationship between skewness and changes in the spread, since the largest reductions (increases) in spread are associated with the most negatively (positively) skewed distributions, the larger (smaller) increases in extremes are associated with larger reductions (increases) in spread. This counteracts the effect seen under the Gaussian assumption, where the larger reductions (increases) in spread are associated with smaller (larger) increases in the number of extremes. The net outcome is that skewness of the temperature distribution reduces the relative impact of spread changes on the count of extremes. The observed shifts in the mean between the warm and neutral subsets are on par with those projected from climate change simulations for the 21 st century. While the mechanisms for this warming may be different in nature from those arising under increased radiative forcing, the subset comparison provides an estimate of the relative impact of the mean, variance and skewness within temperature changes comparable to those projected for the foreseeable future. The observations-based analysis above provides a reconciliation for the wide range of findings regarding the sensitivity of extreme counts to changes in the mean and spread associated with warmer regimes of the past, and underscores the importance of adequate representation of the shape of temperature distributions in historical climate simulations that are used as baseline for future projections. 9

10 Figures Figure 1 H Obs H 0 and (left column), H Gauss H 0 (middle column), and H Obs H Gauss (right column) [days/month] for a) January, b) April, c) July, and d) October 10

11 Figure 2 Warm subset H Gauss as a function of observed unitless Δμ and Δσ for a) January, b) April, c) July, and d) October [days/month]. For points below the blue line, H Gauss is influenced more strongly by Δσ than by Δμ ; the opposite is true for points above this line. 11

12 Figure 3 As in Fig 2 but for H Obs 12

13 Figure 4 Correlation between: H Gauss and H Obs (blue); H Gauss and Δμ (black); H Gauss and Δσ (grey); H Obs and Δμ (black hatched); H Obs and Δσ (grey hatched. 13

14 Figure 5 As in Fig 2 but for D = (H Obs H Gauss 1) 100 [%] 14

15 Figure 6 As in Fig 2 but as a function of observed s 0 and Δσ [%]. 15

16 Figure 7 Geographical distribution of s 0 [unitless] (left) and of D = (H Obs H Gauss 1) 100 [%] (right) for a) January, b) April, c) July, and d) October 16

17 Figure 8 Correlation between s 0 and D (black); s 0 and Δσ (grey); s 0 and Δs (white) 17

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