Multifractal Detrended Cross ‐ Correlation Analysis of Global Methane and Temperature

: Multifractal Detrended Cross ‐ Correlation Analysis (MF ‐ DCCA) was applied to time series of global methane concentrations and remotely ‐ sensed temperature anomalies of the global lower and mid ‐ troposphere, with the purpose of investigating the multifractal characteristics of their cross ‐ correlated time series and examining their interaction in terms of nonlinear analysis. The findings revealed the multifractal nature of the cross ‐ correlated time series and the existence of positive persistence. It was also found that the cross ‐ correlation in the lower troposphere displayed more abundant multifractal characteristics when compared to the mid ‐ troposphere. The source of multifractality in both cases was found to be mainly the dependence of long ‐ range correlations on different fluctuation magnitudes. Multifractal Detrended Fluctuation Analysis (MF ‐ DFA) was also applied to the time series of global methane and global lower and mid ‐ tropospheric temperature anomalies to separately study their multifractal properties. From the results, it was found that the cross ‐ correlated time series exhibit similar multifractal characteristics to the component time series. This could be another sign of the dynamic interaction between the two climate variables.


Introduction
Climate change studies depend, to a great extent, on the examination of the relationship between different key components of the climate system. Atmospheric temperature and methane (CH4) are two of the most important parameters examined in climate research. Air temperature variability and trends at various atmospheric levels represent a crucial indicator of the warming or cooling of the Earth's atmosphere both at a global and regional scale [1][2][3][4]. In addition, adequate and in-depth knowledge of temperature fluctuations is essential for validating the accuracy of climate models' simulations [5][6][7]. The development of a climate data record of suitable length and reliability is required for the reliable detection of changes in the Earth's atmospheric temperature [8]. In this respect, microwave soundings from space have proven successful in providing long-term temperature observations of the Earth's atmosphere. Indeed, since the beginning of the satellite era in the late 1970s, atmospheric temperatures have been routinely monitored from space, until 2006 by the Microwave Sounding Unit (MSU), and later by its successor, the Advanced Microwave Sounding Unit (AMSU), which is fully operational since 1998. The MSU/AMSU microwave sounders are crossscanning instruments onboard polar-orbiting weather satellites measuring the profile of temperature throughout the Earth's atmosphere. Satellite observations of temperature at different levels of the the thermal microwave radiance that is emitted by oxygen (O2). Measurements of molecular oxygen's microwave emission are used to estimate the weighted averages of temperature at three extensive atmospheric layers, namely the lower troposphere (LT), the mid-troposphere (MT) and the lower stratosphere. In general, oxygen concentration remains relatively constant in the atmosphere and, consequently, O2 is considered a secure temperature tracer. Details concerning the instruments' calibration, data adjustments, and new processing methodologies, which are associated with the version 6.0 of the dataset, are further discussed in [51].
Globally-averaged mean monthly data of atmospheric CH4 dry air mole fractions, expressed as parts per billion (ppb), were also directory acquired from NOAA's Earth's System Research Laboratory (ESRL) for the same period . The global average values of CH4 are estimated through weekly measurements from a subset of ESRL's global network of air sampling sites [52]. This subset includes surface stations at remote sites with a well-mixed marine boundary layer (MBL), whereas stations, where altitude and anthropogenic or natural sources and sinks could affect the observations, are excluded. The averaging methodology [53] involves fitting a smooth curve to the data at each site to diminish the impact of synoptic atmospheric phenomena and data gaps, plotting the latitudinal distribution of the smoothed values for weekly time steps per year and finally calculating the global means from the latitude plot. Detailed information regarding the data uncertainties and processing strategy can be found in [52,54].
The CH4 values and LT and MT temperature anomalies data were initially detrended, using polynomial regression analysis (of third order) prior to the implementation of the MF-DFA and MF-DCCA methodologies. Furthermore, the annual and semi-annual seasonal components that were identified in the CH4 time series were eliminated, using the well-established Wiener filter [55]. Figure  1 depicts the initial and processed time series of global CH4 and global LT and MT temperature anomalies.

Multifractal Detrended Fluctuation Analysis (MF-DFA)
The multifractal properties for each parameter were investigated with MF-DFA [31]. The basic steps of the method are the following: 1. The profile X(i) is firstly constructed: where by xk and < x > the time series and its mean value are designated, respectively. The upper bound of summation i takes values from 1 to N, which corresponds to the length of the time series.
2. X(i) is partitioned into an integer number of NS = int(N/s) non-intersecting segments, all of which have the same length s, i.e., time scale. However, for time series with length N not divisible exactly by s, not all profile points will be considered. For this reason, the segmentation procedure is also repeated for the retrograde time series of the profile. Thus, we get 2NS segments in total.
3. Within each segment, a second-order (m = 2) polynomial is fitted to the profile, representing the local trend, where v = 1,…,2NS is the number of each segment. The local trend is then subtracted from the profile and, thus, second-order trends are eliminated from the profile.
4. The detrended variance F 2 (s,v) is then calculated: 5. Considering the average of all segments, we get the q th order fluctuation function: For q = 0, we have, Fq(s) is determined only for s ≥ m+2. For q = 2, the MF-DFA results are identical to the DFA procedure [56][57][58][59][60]. 6. Fq(s) is computed for all values of s. The scaling behavior of Fq(s) is examined through the plot of log(Fq(s)) against log(s) for each moment q. For time series that are long-range correlated, Fq(s) follows a power law: For monofractal time series, the scaling exponent h(q) remains constant and it is equal to the Hurst exponent H. For multifractal time series, h(q) depends strongly on q, i.e., the scaling behavior is different for fluctuations of different magnitude. In these cases, h(q) is the generalized form of the Hurst exponent. For q > 0, h(q) reflects the scaling characteristics of partitions with large fluctuations. On the other hand, for q < 0, h(q) reflects the scaling properties of partitions with small fluctuations.
Ignoring the dependency of h(q) on q and supposing that h(q) = H, for values of H between 0 and 0.5, the time series is characterized by long-range negative correlation, denoting an anti-persistent character; for H > 0.5, it is characterized by long-range positive correlation (persistent behavior); for H = 0.5 it is considered to be uncorrelated, i.e., white noise. Using the relationship τ(q) = qh(q)−1 and applying a Legendre transformation, we get (6) Additionally, The entity α describes the singularity strength, while f(α) represents the subset of the time series that is characterized by α. The plot of f(α) against α is the multifractal spectrum and it provides insights regarding the multifractal traits of the time series. The value of f(α) reaches its peak when the derivative of f(α) with respect to α equals zero, i.e., d f(α)/dα = 0. From equation 7, this happens for q = 0 and, thus, f(α)max equals 1. When the above condition is satisfied, α is called the dominant Hurst exponent α0 and corresponds to the prevailing scaling behavior. Along with α0, the spectral width is also a key feature. It can be estimated by fitting a second-order polynomial around α0, as proposed by [61] and measuring the distance between αmax and αmin, the two points where the fitting curve intersects the horizontal axis: (8) A multifractal spectrum with a broad width indicates rich multifractality in the time series. On the other hand, smaller widths are associated with a more monofractal character of the time series. Furthermore, another aspect of the multifractal spectrum's shape is its truncation type. The left side of the curve corresponds to positive values of q, whereas the right side of the curve is related to the negative values of q. Thus, a left-truncated spectrum indicates a multifractal structure with insensitivity to the large local fluctuations. A right-truncated spectrum, on the other hand, indicates a multifractal structure that is insensitive to the small local fluctuations. Finally, a symmetrical multifractal spectrum suggests that the time series shows the same sensitivity to both small and large local fluctuations.

Multifractal Detrended Cross-Correlation Analysis (MF-DCCA)
The MF-DCCA method [47] is used to measure the long-range cross-correlations between two time series, xk and yk, of equal length N. Steps 1-3 are similar to MF-DFA and they are applied for each time series, separately. Subsequently, their detrended covariance is determined: While considering the average of all segments, the q th order fluctuation function of the detrended covariance is calculated: For q = 0, we get Similarly to the MF-DFA analysis, the scaling behavior of the covariance fluctuations is examined by analyzing the plot of log(Fq(s)) versus log(s). A long-range correlation is inferred if the fluctuation function of the covariance is related to time scale via a power law, i.e., ~ In this case, hxy represents the cross-correlation exponent. For x = y, that is, the two time series are identical, the procedure is simplified to MF-DFA. For hxy > 0.5, increasing values of one time series are expected to be followed by increasing values of the other (persistent behavior). For 0 < hxy < 0.5, increasing values of one time series are expected to be followed by decreasing values of the other (antipersistent behavior). Finally, for hxy = 0.5, the two time series are considered to be long-range uncorrelated. The discussion concerning the multifractal spectrum characteristics, mentioned at the MF-DFA method description, is also valid for the MF-DCCA spectrum.
Finally, the source of multifractality is an important issue that should be examined when coping with multifractal systems. The multifractal characteristics of a time series may derive either from a broad probability density function or from different long-range correlations. The origins of multifractality can be investigated by randomly reordering (shuffling) the values of the two time series prior to applying MF-DFA and MF-DCCA. If the multifractal properties are strongly preserved, then multifractality in the time series originates from a broad probability density function. On the other hand, if the multifractality is prominently diminished, then multifractal properties mainly originate from long-range correlations of different fluctuation magnitudes.

MF-DFA Results
Herein, MF-DFA is applied on the time series of global methane and global LT and MT temperature anomalies and the findings are discussed. Indicatively, Figure 2 Figure 2a, it is observed that log(Fq(s)) linearly increases with log(s). It is important to mention that the slopes are different for each q; this is a sign that the time series of CH4 display multifractal characteristics.
Furthermore, by computing the slopes of Fq(s) for each q, the values of h(q) are estimated. From Figure 2b, it can be observed that h(q) depends on q. This also reveals the multifractal character of CH4. In addition, h(q) > 0.5 for all moments q, and, therefore, long-range positive correlations are identified in the CH4 time series. Moreover, it can be noted that the values of h(q) for q < 0 are greater than the corresponding values for q > 0. This result permits to assume that the small fluctuations (q < 0) display more abundant multifractal features and, therefore, a greater level of complexity than the large fluctuations (q > 0). The application of MF-DFA to the time series of both LT and MT temperature anomalies produced similar results. The presence of scaling features and multifractality in temperature time series is well documented [38,39,62,63].
Significant information concerning the multifractal properties can also be acquired from the singularity spectra of CH4, LT, and MT temperature anomalies ( Table 1). The values of α0 for the three climate variables are 1.714, 1.387, and 1.441, respectively. Thus, α0 > 0.5 for all parameters, which signifies that they have positive long-range correlations. The corresponding w values are 0.790, 1.108, and 1.128, respectively, which reflects that the temperature anomalies time series show higher levels of multifractality when compared to CH4. This finding further highlights the importance of examining their cross-correlation scaling properties for the study of the climate system dynamics. The global temperature is regulated by global energy balance and its fluctuations are influenced by multi-scale complex interactions of atmospheric processes and a number of climate factors (e.g., solar radiation, greenhouse gasses, aerosols, planetary albedo, and clouds) [64][65][66][67]. On the other hand, methane affects the climate directly and indirectly via chemical reactions [68].

Lower Troposphere
In this section, the results of the application of the MF-DCCA procedure are presented. MF-DCCA was initially applied to reveal the multifractal structure of the cross-correlations between global CH4 and temperature anomalies of the global LT. Figure 3a portrays the way that log(Fq(s)) changes with log(s) for a multitude of moments q. It can be observed from the plot that log(Fq(s)) increases linearly with log(s). Therefore, Fq(s) and s are related with a power-law mechanism. The values of hxy(q) were obtained, in a similar way to the MF-DFA procedure using a linear fit. Figure 3b depicts the graphical representation of hxy(q) against q. It is clear that hxy(q) decreases with q. This is a sign of the presence of multifractality in the cross-correlations. All of the hxy(q) values were found to be greater than 0.5, which designates the positive persistence of the cross-correlated time series. At this point, it should be noticed that CH4 global concentration increases evoke a temperature increase. However, according to [69,70], the increase of temperature due to global warming is also likely to increase methanogenesis in wetlands, which is, among others, a significant source in the CH4 global cycle. Increased natural CH4 emissions from wetlands may trigger further increases in global temperature within this feedback loop. It can also be observed from Figure 3b that the values of hxy(q) are greater for negative moments q. Therefore, it is inferred that the cross-correlated time series show a stronger multifractal character for small fluctuations in contrast to large fluctuations. Figure 3c provides further information regarding the multifractal properties of the cross-correlated time series. More precisely, the parabolic shape and the multifractal spectral width confirm the presence of rich multifractality. Additionally, no prominent signs of truncation are evident from a visual inspection of the plot. The symmetric shape of the parabola denotes that the cross-correlated time series exhibits equal sensitivity to the small and large local fluctuations. Table 1

Mid-Troposphere
MF-DCCA was also applied between global CH4 and temperature anomalies of the global MT to explore the cross-correlations of global CH4 and temperature anomalies at higher levels in the Earth's atmosphere and compare their multifractal characteristics with those of the lower troposphere. A power-law mechanism can be distinguished from the inspection of the log(Fq(s)) plot against log(s) (Figure 3d). Furthermore, hxy(q) decreases with increasing q (Figure 3e). Thus, hxy(q) depends on q, which is a strong manifestation of multifractality. Moreover, hxy(q) > 0.5 for all values of q, which implies the existence of positive persistence in accordance to the previous findings. Finally, the shape of the multifractal spectrum (Figure 3f) verifies the existence of multifractality. However, a notable distinction between Figures 3c and 3f is that the former is right truncated, i.e., the cross-correlations between global CH4 and global mid-tropospheric temperature anomalies have a structure that displays a greater sensitivity to the large local fluctuations. In addition, the spectral width in Figure 3f is smaller (w = 0.757) when compared to the width estimated in Figure 3c (w = 0.887) ( Table 1). From this finding, it can be deduced that the temporal fluctuations of the crosscorrelations between global methane and global temperature anomalies possess richer multifractal properties and, therefore, greater complexity in the LT than in the MT. This could be associated with the different degree of interaction between CH4 and temperature in the LT and MT as well as the above-mentioned different multifractal characteristics of LT and MT temperature. In the preceding section, we verified the multifractal structure of the global CH4 concentrations and the global LT and MT temperature anomalies along with their cross-correlations. Herein, the possible origins of the exhibited multifractality are investigated. Generally, multifractality stems either from long-range correlations or a broad probability density function. The time series of CH4 and temperature anomalies were randomly reordered prior to the implementation of MF-DFA and MF-DCCA to detect the origins of the multifractal characteristics of the examined time series. The shuffling procedure was repeated 100 times and the average values of the produced parameters were calculated. From Figures 4a, 5a, and 5d, it can be seen that the plot of log(Fq(s)) against log(s) consists of straight parallel lines, which implies that the multifractal structure in the shuffled time series is destroyed. In addition, the dependence of h(q) and hxy(q) on q is very small. Their values in all shuffled time series are close to 0.5 (Figures 4, 5b and 5e). This also highlights the absence of rich multifractal properties. Furthermore, the width of the multifractal spectra is greatly reduced in the shuffled time series (Figures 4c, 5c and 5f). In Table 2, the values of w and α0 for the shuffled time series are represented. Thus, while taking the above findings into account, it is deduced that a random shuffling of the values of the two climate parameters and their cross-correlations significantly weakens their multifractality. Consequently, it can be assumed that multifractality mainly stems from different long-range correlations for small and large fluctuations for the examined time scales.

Conclusions
In this work, global CH4 and remotely-sensed tropospheric temperature anomalies are analyzed using the MF-DCCA to study the multifractality of their cross-correlations. Initially, the LT and MT temperature anomalies were found to exhibit stronger multifractality when compared to global CH4. Furthermore, the values of the exponents h(q) of CH4 and tropospheric temperature along with the exponent hxy(q) of their cross-correlated time series, are all larger than 0.5 for all moments q, which is a clear indication of their positive persistent behavior. This indicates that the impact of past events has an influence on the succeeding values of the parameters. The spectral width is also indicative of the multifractality of the examined time series. Another important finding is that the α0 value of the cross-correlated time series lies between the α0 values of the two variables. Similarly, the value of the spectral width of the cross-correlated time series is between the corresponding values of the two parameters. This could be another sign of the interaction between the two climate variables from a nonlinear perspective. The scaling behavior of the cross-correlation between CH4 and the temperature anomalies of the global MT is weaker when compared to the LT. Finally, the outcome of the shuffling procedure in the cross-correlated time series suggest that multifractality originates mainly from different long-range correlations for small and large fluctuations.
Funding: This research received no external funding.