Stochastic investigation of daily air temperature extremes from a global ground station network

Abstract

Near-surface air temperature is one of the most widely studied hydroclimatic variables, as both its regular and extremal behaviors are of paramount importance to human life. Following the global warming observed in the past decades and the advent of the anthropogenic climate change debate, interest in temperature’s variability and extremes has been rising. It has since become clear that it is imperative not only to identify the exact shape of the temperature’s distribution tails, but also to understand their temporal evolution. Here, we investigate the stochastic behavior of near-surface air temperature using the newly developed estimation tool of Knowable (K-)moments. K-moments, because of their property to substitute higher-order deviations from the mean with the distribution function, enable reliable estimation and an effective alternative to order statistics and, particularly for the outliers-prone distribution tails. We compile a large set of daily timeseries (30–200 years) of average, maximum and minimum air temperature, which we standardize with respect to the monthly variability of each record. Our focus is placed on the maximum and minimum temperatures, because they are more reliably measured than the average, yet very rarely analyzed in the literature. We examine segments of each timeseries using consecutive rolling 30-year periods, from which we extract extreme values corresponding to specific return period levels. Results suggest that the average and minimum temperature tend to increase, while overall the maximum temperature is slightly decreasing. Furthermore, we model the temperature timeseries as a filtered Hurst-Kolmogorov process and use Monte Carlo simulation to produce synthetic records with similar stochastic properties through the explicit Symmetric Moving Average scheme. We subsequently evaluate how the patterns observed in the longest records can be reproduced by the synthetic series.

Appendices

Appendix 1

To facilitate the understanding of the theory behind K-moments, we explain some basic notions of statistics in this appendix.

Let $\underline {x}$ be a stochastic variable and $\underline {x}_{1}$, $\underline {x}_{2}$, …, $\underline {x}_{p}$ be copies of it, independent and identically distributed, forming a sample. The maximum of all, which is identical to the pth order stochastic, is by definition:

$$\underline {x}_{\left( p \right)} := {\text{max}}\left( {\underline {x}_{1} , \underline {x}_{2} , \ldots , \underline {x}_{p} } \right)$$

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It is readily obtained that if $F\left( x \right)$ is the distribution function of $\underline {x}$ and $f\left( x \right)$ its probability density function, then those of $\underline {x}_{\left( p \right)}$ are distributed by:

$$F^{\left( p \right)} \left( x \right) = \left( {F\left( x \right)} \right)^{p} ,\,\,f^{\left( p \right)} \left( x \right) = pf\left( x \right)\left( {F\left( {\underline {x} } \right)} \right)^{p - 1}$$

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where the former is the product of $p$ instances of $F\left( x \right)$ (justified by the independent and identically distributed assumption), while the latter is the derivative of $F^{\left( p \right)} \left( x \right)$ with respect to $x$. The expected maximum order of $p$ of $\underline {x}$, i.e. the expected value of $\underline {x}_{\left( p \right)}$, is therefore:

$${\text{E}}\left[ {\underline {x}_{\left( p \right)} } \right] = {\text{E}}\left[ {{\text{max}}\left( {\underline {x}_{1} , \underline {x}_{2} , \ldots ,\underline {x}_{p} } \right)} \right] = p{\text{E}}\left[ {\left( {F\left( {\underline {x} } \right)} \right)^{p - 1} \underline {x} } \right]$$

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It is worth to stress that the variables $\underline {x}_{1}$., $\underline {x}_{2}$, …, $\underline {x}_{p}$ considered here, are not meant in temporal succession and, in this respect, do not form a stochastic process, but are rather regarded to be an ensemble of copies of $\underline {x}$. In other words, the possible dependence in time of a stochastic process is not considered to be prerequisite for the application.

In geophysical processes, it is justifiable to assume that the variance $\mu_{2} \equiv \sigma^{2}$ is finite, because an infinite variance would translate to an infinite amount of eney to materialize, which is absurd. However, high-order classical moments $\mu_{p}$ diverge to infinity beyond a certain $p$ (i.e., in heavy-tailed distributions). That is not the case for the K-moments, here a significant part of the moment is calculated using the always finite distributi function (Koutsoyiannis 2019a), which is the reason from which their knowability stems.

To derive knowable moments for high orders $p$, in the expectation defining the pth moment, we raise $\left( {\underline {x} - \mu } \right)$. to a low power $q < p$ and for the remaining $\left( {p - q} \right)$ multiplicative terms, we replace $\left( {\underline {x} - \mu } \right)$ with $\left( {2F\left( {\underline {x} } \right) - 1} \right),$ where $F\left( x \right)$ is the distribution function. This leads to the following definition of central K-moment of order $\left( {p,q} \right)$ (Koutsoyiannis 2019a):

$$K_{pq} : = \left( {p - q + 1} \right){\text{E}}\left[ {\left( {2F\left( {\underline {x} } \right) - 1} \right)^{p - q} \left( {\underline {x} - \mu } \right)^{q} } \right],\quad p \ge q$$

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(Likewise, the non-central K-moment of order $\left( {p,q} \right)$ is defined (Koutsoyiannis 2019a):

$$K^{\prime}_{pq} : = \left( {p - q + 1} \right){\text{E}}\left[ {\left( {F\left( {\underline {x} } \right)} \right)^{p - q} \underline {x}^{q} } \right], \quad p \ge q$$

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The quantities $\left( {F\left( {\underline {x} } \right)} \right)^{p - q}$ and $\left( {2F\left( {\underline {x} } \right) - 1} \right)^{p - q}$ are estimated from a sample, without the use of powers of $\underline {x}$, thus making the estimation more reliable. Specifically, for the ith element of a sample $x_{\left( i \right)}$ of size $n$, sorted in ascending order, $F\left( {x_{\left( i \right)} } \right)$ and $\left( {2F\left( {x_{\left( i \right)} } \right) - 1} \right)$ are estimated as:

$$\hat{F}\left( {x_{\left( i \right)} } \right) = \frac{i - 1}{{n - 1}}, \ldots 2\hat{F}\left( {x_{\left( i \right)} } \right) - 1 = \frac{2i - n - 1}{{n - 1}}$$

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taking values in [0,1] and [− 1,1], respectively, irrespective of the values $x_{\left( i \right)}$.. Hence, the estimators of K-moments are:

$$\hat{K}^{\prime}_{pq} = \frac{p - q + 1}{n}\mathop \sum \limits_{i = 1}^{n} \left( {\frac{i - 1}{{n - 1}}} \right)^{p - q} \underline {x}_{\left( i \right)}^{q}$$

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$$\hat{K}_{pq} = \frac{p - q + 1}{n}\mathop \sum \limits_{i = 1}^{n} \left( {\frac{2i - n - 1}{{n - 1}}} \right)^{p - q} (\underline {x}_{\left( i \right)} - \hat{\mu })^{q}$$

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The rationale of the definition is very relatively easy to grasp. Assuming that the distribution mean is close to the median, so that $F\left( \mu \right) \approx {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}$ (this is precisely true for a symmetric distribution), the quantity whose expectation is taken from the finition of the central K-moment of order $\left( {p,q} \right)$ is: $A\left( {\underline {x} } \right){ \sim } := \left( {2F\left( {\underline {x} } \right) - 1} \right)^{p - q} \left( {\underline {x} - \mu } \right)^{q}$ and its Taylor expansion is:

$$\begin{aligned} A\left( {\underline {x} } \right) &= \left( {2f\left( \mu \right)} \right)^{p - q} \left( {\underline {x} - \mu } \right)^{p} + \left( {p - q} \right)\left( {2f\left( \mu \right)} \right)^{p - q - 1} f^{\prime}\left( \mu \right)\left( {\underline {x} - \mu } \right)^{p + 1} \\&\quad+ O\left( {\left( {\underline {x} - \mu } \right)^{p + 2} } \right) \end{aligned}$$

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where $f\left( x \right)$ is the probability density function of $\underline {x}$. Clearly then, $K_{pq}$ depends on $\mu_{p}$ as well as on classical moments of $\underline {x}$ of order higher than $p$. The independence of $K_{pq}$ from classical moments of order smaller than $p$ is the reason why it is a competent surrogate of the unknowable $\mu_{p}$. In addition, as $p$ becomes large, by virtue of the multiplicative term $\left( {p - q + 1} \right)$ in the definition of K-moments, $K_{pq}$ shares similar asymptotic properties with $\hat{\mu }_{p}^{{{\raise0.7ex\hbox{$q$} \!\mathord{\left/ {\vphantom {q p}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$p$}}}}$ (the estimate, not the true $\mu_{p}^{{{\raise0.7ex\hbox{$q$} \!\mathord{\left/ {\vphantom {q p}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$p$}}}}$). To illustrate this for $q = 1$ and for independent variables $\underline {x}_{i}$, we consider the variable $\underline {z}_{p} { \sim } = {\text{max}}_{1 \le i \le p} \underline {x}_{i}$ and denote $f\left( \right)$ and $h\left( \right)$. the probability densities of $\underline {x}_{i}$. and $\underline {z}_{i}$ respectively. Then (Papoulis 1990):

$$h\left( z \right) = pf\left( z \right)\left( {(F\left( z \right)} \right)^{p - 1}$$

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and thus, by virtue of the definition of non-central K-moment of order $\left( {p,q} \right)$:

$${\text{{\rm E}}}\left[ {\underline{z} _{p} } \right] = p{\text{E}}\left[ {\left( {F\left( {\underline{x} } \right)} \right)^{{p - 1}} \underline{x} } \right] = K'_{{p1}}$$

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On the other hand, for positive $\underline {x}$ and large $p \to n$,

$$\begin{aligned} \left( {{\text{E}}\left[ {\underline{{\widehat{{\mu '}}}} _{P} } \right]} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$p$}}}} & = \left( {{\text{E}}\left[ {\left( {\frac{1}{n}\mathop \sum \limits_{{i = 1}}^{n} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} _{i} ^{p} } \right)} \right]} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$p$}}}} \approx \left( {{\text{E}}\left[ {\left( {\frac{1}{n}{\text{max}}_{{1 \le i \le n}} \left( {x_{i}^{p} } \right)} \right)} \right]} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$p$}}}} \\ & \approx n^{{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$p$}}}} {\text{E}}\left[ {{\text{max}}_{{1 \le i \le n}} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} _{i} } \right)} \right] \approx {\text{E}}\left[ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} _{n} } \right] \\ \end{aligned}$$

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It is also worth noting that the multiplicative term $\left( {p - q + 1} \right)$ in the definitions of central and non-central $K_{pq}$ and $K^{\prime}_{pq}$ makes K-moments generally increasing functions of $p$.

Appendix 2

The Climacograms of the three parameters of the near-surface air temperature (average, maximum and minimum) are presented in the following figures. Note that the climacogram derived from the empirical data is depicted in blue color, while the climacogram of the synthetic data is in green color respectively. Solid lines represent the mean of each dataset (empirical and synthetic), while dashed lines represent the 5th and 95th percentile (90% confidence levels) of the respective distributions. The climacogram derived from the optimally fitted theoretical model is depicted in red colored solid line.

It is worth noting that the range between the 5th and 95th percentiles of the synthetic data in each of the three climacograms is narrower than the expected one from the respective empirical data. This is probably caused by the use of the same model (imposed by the same Hurst and Mandelbrot parameters) in the production of the synthetic timeseries for each of the three parameters of near-surface air temperature (Figs. 14, 15, 16)

Fig. 14

Climacogram of the average air temperature

Fig. 15

Climacogram of the maximum air temperature

Fig. 16

Climacogram of the minimum air temperature