A simple decomposition of European temperature variability capturing the variance from days to a decade
Abstract
We analyze European temperature variability from station data with the method of detrended fluctuation analysis. This method is known to give a scaling exponent indicating long range correlations in time for temperature anomalies. However, by a more careful look at the fluctuation function we are able to explain the emergent scaling behaviour by short time relaxation, the yearly cycle and one additional process. It turns out that for many stations this interannual variability is an oscillatory mode with a period length of approximately 7–8 years, which is consistent with results of other methods. We discuss the spatial patterns in all parameters and validate the finding of the 7–8 year period by comparing stations with and without this mode.
Keywords
Climate oscillations Interannual variability Timescale decomposition DFA Temperature pressure1 Introduction
Variability of temperature time series has been described in two ways. Some people concentrate on scaling behavior of correlations and its seeming power law behaviour (long range correlations). Popular methods are detrended fluctuation analysis (DFA) (Fraedrich and Blender 2003), and multifractal versions (Moon et al. 2018), power spectral densities (Fredriksen and Rypdal 2016) and wavelet analysis (Lovejoy 2018). Such results consider the complete power of climate fluctuations. They are useful for estimations of prediction errors (Massah and Kantz 2016; Ludescher et al. 2016) and the validation of more complicated general circulation models (Govindan et al. 2002).
Other projects have tried to identify characteristic timescales in the data (Ghil et al. 2002). Here people use for example singular spectrum analysis (SSA) (Plaut et al. 1995), versions of Monte Carlo SSA (Paluš and Novotná 2004), and again wavelet methods (Baliunas et al. 1997). Results of these methods also help to validate general circulation models and contribute to the physical understanding of the atmosphere as well as to make predictions if the component is significant like the El Nino–Southern Oscillation (Chekroun et al. 2011). They can explain observed slowdown (or fastening) of global warming (Steinman et al. 2015; Hu and Fedorov 2017).
The disadvantage of the first approach is that it neglects properties that are already known and reduces the information to one property which is difficult to interpret (Maraun et al. 2004). The disadvantage of the second approach is that it does not take the full power of the fluctuations into account (Lovejoy 2015).
While parts of the literature concentrating on characteristic timescales completely ignore the findings of power law scaling, others mention the so called continuum variability (Huybers and Curry 2006) and treat it as one of the features contained in temperature fluctuations. Recently interest in this continuum variability is growing again (Lovejoy 2018; Fredriksen and Rypdal 2017). Lovejoy (2015) suggest to filter the known frequency modes, just like it is normally done with the seasonal cycle, in order to understand the nature of the continuum better. Our approach in fact follows this spirit. We will apply a novel method based on DFA that in contrast to traditional usage of DFA concentrates on the characteristic timescales. Therefore we show connections between the two approaches mentioned at the beginning. However, we conclude that all the power of the fluctuations can be explained by a superposition of few simple processes for the time range under consideration.
We want to concentrate on climate data from Europe where a large number of stations with several decades of daily recordings exists. For this region there is already a long history of investigations on both individual stations or grid data. Most importantly, people have repeatedly reported a 7–8 year cycle in various stations and other records (Plaut et al. 1995; Paluš and Novotná 1998; Sen and Ogrin 2016), the origin of which is still unclear. Spacial patterns were described in Grieser et al. (2002), Pišoft et al. (2009) and Jajcay et al. (2016), however, to our knowledge there has not been an algorithmic investigation on a similar number of stations as we perform it.
We present the foundations of our method in Sect. 3, summarizing previous work in Meyer and Kantz (2019). In Sect. 4 we postulate our model for weather and macroweather variability in Europe and extend our method to processes with more than one characteristic timescale. In Sect. 5 we show results of a large scale algorithmic application of our method to station data of European temperatures. We are able to reproduce the 7–8 year cycle in part of this data. We show that our method is able to separate data with and without this frequency mode.
2 Data
We analyze station data of temperature and pressure in Europe (Tank et al. 2002). It can be downloaded from http://www.ecad.com. The resolution of the time series is one day. The datasets strongly vary in length. We restrict ourselves to long time series as described in Appendix 1.
3 The method
Meyer and Kantz (2019) have shown that this method has the ability to uncover characteristic timescales in time series. The idea is to fit the theoretical fluctuation function of autoregressive models of different orders: AR(1) models for relaxation times, and AR(2) for noisy oscillations. The method works well for data that is dominated by one characteristic timescale, like approximations to yearly global mean temperatures (Meyer et al. 2018) or an 11month averaged El Nino signal. We show the theoretical fluctuation functions in (16) and (23) in Appendix 3.
As a first example we want to show the application of the method to the time series of air pressure measured at Potsdam, Germany. We deseasonalize both the pressure and its variance in order to get a homogeneous time series. Therefore we calculate the long time climatological average pressure for each calender day and subtract it from each day. We also divide each value by the average variance for the calender day.
In the long time limit F(s) scales like 1/2, implying short range correlations. Figure 1 shows that the data can be described by an AR(1) model in a fairly good approximation. Obviously, for a complex system like the atmosphere, we would not claim that pressure is purely autoregressive and no other effects happen at characteristic timescales. However, we can see that those other timescales are not significant for a smoothed measure such as the DFAfluctuation function. The method shows us a data model that can be used for stochastic predictions and qualitative understanding of the dynamics.
4 Decomposition of one time series
In the examples above fitting fluctuation functions was exclusively applied to systems with one dominant characteristic timescale. What are we supposed to do if the fluctuation function implies a more complex dynamics? In this section, by analysing historic temperature data from Potsdam station, we show that our method still works for systems with more than one characteristic timescale if these timescales are sufficiently well separated.
Our model contains some simplifications. This is not only because it only consists of three processes described by few parameters, but also our assumption of independence of the three processes. This is not always true as shown in Paluš (2014) and Jajcay et al. (2016). However our focus in this text are not the crossscale interactions, but only the observation of the different timescales. We also do not discuss spatial correlations.
Due to the temporal separation of the three processes \(X_t\), \(Y_t\) and \(Z_t\), and because autoregressive models are low pass filters, the impact of the slower processes can be neglected for short times. For Potsdam the fit of Eq. (16) to \(F_T\) for small s (see Fig. 2) yields a relaxation time of 3.8 days. We subtract the fitted AR(1) fluctuation function \(F^2_X\) from \(F^2_T\).
For Potsdam and other stations we looked at, our fit seems to show good agreement with \(F^2_{TXY}\), although the fluctuation function is not as smooth as for shorter times. We obtain a period of 7.5 years. The complete fluctuation function \(F^2_{X+Y+Z}\) is shown in Fig. 1(right panel). It shows excellent agreement with the measured \(F^2_T\).
It has been suggested that climate can be described by a continuum variability with anomalous scaling (Lovejoy 2015) and quasiperiodic perturbations. For the timerange under consideration, which is admittedly shorter than in most studies that concentrate on this issue, we do not see any indicators that such a model would be necessary. The scale invariance of the anomalies visible in Fig. 1(right) is explained in our model by superposition of white noise driving at each identified timescale. This is because AR(2) is a low pass filter which ’hides’ the fluctuations of its driver for short times. Several AR(2) processes with different periods therefore lead to a successive increase of the background noise for longer time. This emergent scaling is an alternative interpretation to dynamical long range correlations, the origin of which is still not fully understood (Fraedrich et al. 2004; Fredriksen and Rypdal 2017; Meyer and Kantz 2017).
5 Analysis for European station data
The decomposition of the fluctuation function as described in the previous section was incorporated into an algorithm and applied to temperature data of stations around Europe. We obtain six parameters, r, \(\sigma _X\), \(\sigma _Y\), a, b, and \(\sigma _Z\), for each station, which model the fluctuations of timescales up to a bit more than a decade, describing weather and macroweather. The aim is to investigate the spatial dependence of these parameters and especially for which regions the AR(2) model (6) indicates an oscillatory mode on the timescale of several years. The period time \(\tau \) of this oscillation can be calculated from the parameters a and b.
The short time dynamics is described by the relaxation time r. We obtain it by fitting \(F_X\) to \(F_T\). The results are displayed in Fig. 3. r shows clear regional patterns, which are much stronger than the uncertainty of our method. The correlation time is short in the west and the north of Europe close to the Atlantic. It is longer in central and eastern Europe and around the Baltic sea. For most stations r is just a bit smaller than 4 days.
The dynamics of pressure data for short times turns out to be similar to that of temperature. Both describe the typical timescale of weather patterns. However, there seems to be no connection between \(r_T\) and \(r_p\) for one specific station. On the contrary the regional patterns of pressure strongly differ from those of temperature. Pressure values relax faster in the south compared to the north.

if the slope is \(\propto s^{0.5}\) or ever more flat the algorithm indicates an oscillation with a very short period

if the slope is very steep (\(\propto s^{1.5}\)) the algorithm indicates an oscillation with a very long period

for stations in between the AR(2) model has real roots indicating no oscillation.
For 51% of the analyzed stations we observe an oscillation, see Fig. 5. We denote the set of these stations by \(\varOmega _+\). The categorization into the two subgroups uncovers regional patterns with few exceptions which might be interpreted as false classifications. The frequency mode is significant in an area including England, Belgium, the south of Scandinavia, central Europe north of the Alps and parts of eastern Europe. The values for \(\tau \) that we obtain do not show clear regional patterns, which indicates that we indeed only see the known 7–8 year cycle and the deviations are the uncertainty of our method. On average the period length is 7.6 years with large standard deviation of 1.8 years.
6 Conclusions
We introduce a method for the detection of characteristic timescales in time series based on detrended fluctuation analysis that works for data where the dominant timescales are sufficiently well separated. It is an intuitive tool for the derivation of approximate dynamics that accounts for the full power of the fluctuations in the system. In this way we bridge the gap between spectral methods that are applied for the detection of oscillations and scaling methods that detect colored noise. The method does not detect nonlinear effects like crossscale interactions but only covers the characteristic timescales individually.
While pressure data usually only shows exponential decay of correlations, temperature data shows a much slower decay, often interpreted as long range correlations. We model these time series with a simple model, a superposition of short range decay, a seasonal cycle and a potentially oscillating noisy component for longer times. This model is able to describe the observed fluctuation functions accurately. DFA is a smoothing method that is not designed to find every tiny structure in the time series. For timescales up to a decade it is not necessary to claim a colored background noise, as many authors do, who look at longer timescales. The increase of the background noise can be explained by the input noise of the interseasonal variability. By fitting the parameters we obtain the power of each component for each station. more importantly we are able to reproduce a previously found 7–8 period. Our method is with only few false classifications able to distinguish between stations where this component is significant, and stations, where it is not.
Notes
Acknowledgements
Open access funding provided by Max Planck Society. We acknowledge the data providers in the ECA&D project. Tank et al. (2002). Data and metadata available at http://www.ecad.eu. Map background made with Natural Earth. Free vector and raster map data @ http://www.naturalearthdata.com.
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