Introduction

The state of climate prevails through various physical processes, each of these processes have very large fluctuations in different time-space scales, in this case, climate shows a hierarchy of structures (Lin & Fu, 2008). Air temperature, which is considered as a physical variable for measuring the warmth and coldness of the air (Jiang et al., 2015), is also considered as the most important indicator of fluctuations or changes in the climate system (da Silva et al., 2020; Folland et al., 2001; Hasselmann, 1993; Hegerl et al., 2006) which appears as a fractal phenomenon by adopting a self-similar structure in multiple time scales (Lin & Fu, 2008).
 

Methodology

In this study, the daily temperature data of Khorramabad synoptic station with time coverage of 1401-1329 were used. In order to identify the nature of dynamic behavior and the presence of long-term correlations in this time series, Hurst's range-scale approach (R/S) and detrended fluctuation analysis (DFA) were used.

Results

By considering the median base of 13140 days and incremental steps of 10 days, 1317 daily time series (n) of air temperature were produced. Then by calculating the mean and standard deviation (S) of these sub-series and subtracting each of the daily data of these sub-series from the average of that sub-series, their cumulative signals were created. Finally, the ratio of the range of changes (R) of each of the 1317 cumulative signals to the standard deviation (S) of the initial data of these 1317-time series was calculated by fitting the natural logarithm (R/S) against the natural logarithm of the statistical period of these subseries. Hurst and the dynamic nature of temperature time series was identified. Two intersection points that indicate the change in the behavior of the fluctuation function with respect to the time scale were identified. The slope and the dynamic pattern of the distribution curve are changed before and after these intersection points. In another step, under the DFA2 approach, it was performed on the temperature signal. The distribution of the fluctuation function obtained from DFA2 against different time scales shows that the amount of fluctuation increases as the scale of the time series increases. After all, the change rate of the fluctuation function in different time scales is not the same and shows the multi-scale mode. Also, the result of DFA2 on this seasonally detrended signal determined that the fluctuation function increases with the increase of the time scale. In this regard, the Hurst exponent was obtained as 0.77.
 

Discussion

Intersection points in the Hurst diagram occurs due to changes in signal correlation characteristics in different time scales (Movahed et al., 2006). As these points of intersection indicate the scale behavior of a very complex time series whose different parts have different scale profiles? On the other hand, these intersection points can be caused by the effects of instabilities with parabolic, sinusoidal and power law trends on the scaling behavior of long-term signal correlations (Chen et al., 2007; Chen et al., 2002; Hu et al., 2001). Similar to the studies of (Eichner et al., 2008; Lin & Fu, 2008; Rybski et al., 2008; Yuan et al., 2010), seasonal trends were removed from the temperature signal. Discussion. Fluctuation function increases with the increase of the time scale implies the fractal property of the temperature time series, because the existence of an ascending power relationship between the fluctuations and the time scale is the most important identifier for determining the fractal nature. It is considered a time series (Kantelhardt et al., 2001). The Hurst exponent was obtained in result indicates the existence of correlation and long-term memory in the daily temperature time series of Khorramabad synoptic station.
 

Conclusion

The time series of air temperature of Khorramabad synoptic station was investigated and analyzed using R/S and DFA methods. The results of the application of the R/S method showed that this signal has severe instability due to the presence of annual and seasonal parabolic and sinusoidal trends, which have caused the complexity of the temperature signal; While identifying important angles of the temperature signal, this approach is unable to meaningfully calculate the long-term correlation and memory of this signal due to the complex instabilities of the temperature time series, and perhaps the correlation obtained from this method is false. On the other hand, the results of DFA2 also showed two points of intersection in the temperature signal that the correlation profile has changed around these points and in other words, the scale behavior of the temperature signal has changed in different scales. In this regard, by removing the seasonal trends, it was found that the change in the correlation and the change in the scaling behavior of the fluctuations was not due to the inherent and natural fluctuations of the temperature signal. It should be noted that the results of this study contain the important point that despite the difference in the accuracy of the results of the mentioned methods, each of these methods is able to clarify angles from the temperature signal, which contain important information about the structure are the dynamics of the time series of temperature.
Acknowledgments
The present research is the result of the scientific activity of the author.