# Multifractal Detrended Cross-Correlation Analysis of Global Methane and Temperature

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{4}) 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 cross-scanning 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 Earth’s atmosphere are considered to be a valuable tool in climate change studies, since they provide high-resolution measurements with global coverage over a multidecadal time period [9].

_{2}) [13,14], it remains in the Earth’s atmosphere for a relatively short period of time [15,16,17]. Methane is produced by both anthropogenic activities and natural processes. The greatest natural formation of CH

_{4}occurs in wetlands due to the metabolic activity of anaerobic microorganisms under hypoxic conditions. CH

_{4}production in natural wetlands is responsible for approximately 30% of its total emissions [18]. The anthropogenic sources of CH

_{4}include fugitive releases from solid materials used in the energy production industry, discharges from gas mining and its supply chain, as well as emissions that are related to agricultural activity, livestock farming, and waste treatment [19]. Anthropogenic activities are responsible for more than half of the CH

_{4}production from land and oceans [18]. On the other hand, the reaction with the hydroxyl radical (OH) is considered to be the basic sink of atmospheric CH

_{4}, a process that is subject to interannual fluctuations [20]. Apart from destroying CH

_{4}, this reaction is one of the main sources of water vapor in the stratosphere [21]. Atmospheric CH

_{4}concentrations have been systematically monitored through a global network of surface air-sampling stations [22]. An overall increase by a factor of 2.5 has been observed during the industrial era [23]. However, for reasons that still remain under scientific debate [24,25,26], global CH

_{4}concentrations remained stagnant for almost a decade between the late 1990s and 2006 and experienced continuous growth ever since [27,28,29,30].

_{4}and remotely-sensed global temperature anomalies, both in the lower and mid-troposphere. Thus, MF-DCCA was adopted for these two significant components of the climate system to examine whether positive correlation exists between them throughout the Earth’s atmosphere.

## 2. Materials and Methods

#### 2.1. Data

_{2}). 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, O

_{2}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].

_{4}dry air mole fractions, expressed as parts per billion (ppb), were also acquired from NOAA’s Earth’s System Research Laboratory (ESRL) for the same period (1984–2018). The global average values of CH

_{4}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].

_{4}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 CH

_{4}time series were eliminated, using the well-established Wiener filter [55]. Figure 1 depicts the initial and processed time series of global CH

_{4}and global LT and MT temperature anomalies.

#### 2.2. Methodology

#### 2.2.1. Multifractal Detrended Fluctuation Analysis (MF-DFA)

_{k}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.

_{S}= 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 2N

_{S}segments in total.

_{S}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.

^{2}(s,v) is then calculated:

^{th}order fluctuation function:

_{q}(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].

_{q}(s) is computed for all values of s. The scaling behavior of F

_{q}(s) is examined through the plot of log(F

_{q}(s)) against log(s) for each moment q. For time series that are long-range correlated, F

_{q}(s) follows a power law:

_{max}equals 1. When the above condition is satisfied, α is called the dominant Hurst exponent α

_{0}and corresponds to the prevailing scaling behavior. Along with α

_{0}, the spectral width is also a key feature. It can be estimated by fitting a second-order polynomial around α

_{0}, as proposed by [61] and measuring the distance between α

_{max}and α

_{min}, the two points where the fitting curve intersects the horizontal axis:

#### 2.2.2. Multifractal Detrended Cross-Correlation Analysis (MF-DCCA)

_{k}and y

_{k}, 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:

^{th}order fluctuation function of the detrended covariance is calculated:

_{q}(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.,

_{xy}represents the cross-correlation exponent. For x = y, that is, the two time series are identical, the procedure is simplified to MF-DFA. For h

_{xy}> 0.5, increasing values of one time series are expected to be followed by increasing values of the other (persistent behavior). For 0 < h

_{xy}< 0.5, increasing values of one time series are expected to be followed by decreasing values of the other (antipersistent behavior). Finally, for h

_{xy}= 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.

## 3. Results and Discussion

#### 3.1. MF-DFA Results

_{4}. The three types of plots are the plot of log(F

_{q}(s)) against log(s) (Figure 2a), the plot of h(q) against q (Figure 2b), and the multifractal spectrum f(a) against α (Figure 2c). The time scales used both in the MF-DFA and MF-DCCA range between 30 months (s ≈ 10

^{1.5}) and N/5, i.e., 84 months (s ≈ 10

^{1.9}), where, by N, the length of the time series is represented. The values of q also range from −5 to +5 by steps of 0.1. From the examination of Figure 2a, it is observed that log(F

_{q}(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 CH

_{4}display multifractal characteristics.

_{q}(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 CH

_{4}. In addition, h(q) > 0.5 for all moments q, and, therefore, long-range positive correlations are identified in the CH

_{4}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].

_{4}, LT, and MT temperature anomalies (Table 1). The values of α

_{0}for the three climate variables are 1.714, 1.387, and 1.441, respectively. Thus, α

_{0}> 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 CH

_{4}. 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].

#### 3.2. MF-DCCA Results

#### 3.2.1. Lower Troposphere

_{4}and temperature anomalies of the global LT. Figure 3a portrays the way that log(F

_{q}(s)) changes with log(s) for a multitude of moments q. It can be observed from the plot that log(F

_{q}(s)) increases linearly with log(s). Therefore, F

_{q}(s) and s are related with a power-law mechanism. The values of h

_{xy}(q) were obtained, in a similar way to the MF-DFA procedure using a linear fit. Figure 3b depicts the graphical representation of h

_{xy}(q) against q. It is clear that h

_{xy}(q) decreases with q. This is a sign of the presence of multifractality in the cross-correlations. All of the h

_{xy}(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 CH

_{4}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 CH

_{4}global cycle. Increased natural CH

_{4}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 h

_{xy}(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 depicts the main multifractal characteristics and their values. It can be observed that both values of α

_{ο}and w of the cross-correlated time series lie between the corresponding values of the initial time series.

#### 3.2.2. Mid-Troposphere

_{4}and temperature anomalies of the global MT to explore the cross-correlations of global CH

_{4}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(F

_{q}(s)) plot against log(s) (Figure 3d). Furthermore, h

_{xy}(q) decreases with increasing q (Figure 3e). Thus, h

_{xy}(q) depends on q, which is a strong manifestation of multifractality. Moreover, h

_{xy}(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 Figure 3c,f is that the former is right truncated, i.e., the cross-correlations between global CH

_{4}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 cross-correlations 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 CH

_{4}and temperature in the LT and MT as well as the above-mentioned different multifractal characteristics of LT and MT temperature.

#### 3.2.3. Origins of Multifractality

_{4}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 CH

_{4}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 Figure 4a and Figure 5a,d, it can be seen that the plot of log(F

_{q}(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 h

_{xy}(q) on q is very small. Their values in all shuffled time series are close to 0.5 (Figure 4 and Figure 5b,e). This also highlights the absence of rich multifractal properties. Furthermore, the width of the multifractal spectra is greatly reduced in the shuffled time series (Figure 4c and Figure 5c,f). In Table 2, the values of w and α

_{0}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.

## 4. Conclusions

_{4}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 CH

_{4}. Furthermore, the values of the exponents h(q) of CH

_{4}and tropospheric temperature along with the exponent h

_{xy}(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 α

_{0}value of the cross-correlated time series lies between the α

_{0}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 CH

_{4}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.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Time series of initial (

**a**–

**c**) and processed (

**d**–

**f**) global CH

_{4}and temperature anomalies of the lower troposphere (LT) and mid-troposphere (MT).

**Figure 2.**Multifractal Detrended Fluctuation Analysis (MF-DFA) results for global CH

_{4}; (

**a**) Plot of log(F

_{q}(s)) against log(s); (

**b**) Plot of h(q) against q; and, (

**c**) Multifractal spectrum f(a) against a.

**Figure 3.**Multifractal Detrended Cross-Correlation Analysis (MF-DCCA) results between global CH

_{4}and global LT (left) and MT (right) temperature anomalies; (

**a**,

**d**) Plots of log(F

_{q}(s)) against log(s); (

**b**,

**e**) Plot of h

_{xy}(q) against q; and, (

**c**,

**f**) Multifractal spectrum f(a) against a.

**Figure 4.**Multifractal Detrended Fluctuation Analysis (MF-DFA) results for the shuffled series of global CH

_{4}; (

**a**) Plot of log(F

_{q}(s)) against log(s); (

**b**) Plot of h

_{xy}(q) against q; (

**c**) Multifractal spectrum f(a) against a.

**Figure 5.**Multifractal Detrended Cross-Correlation Analysis (MF-DCCA) plots for the shuffled series between global CH

_{4}and global LT (left) and MT (right) temperature anomalies; (

**a**,

**d**) Plot of log(F

_{q}(s)) against log(s); (

**b**,

**e**) Plot of h

_{xy}(q) against q; and, (

**c**,

**f**) Multifractal spectrum f(a) against a.

**Table 1.**Multifractal characteristics for CH

_{4}and temperature anomalies of the global LT and MT along with their cross-correlations (CC).

Parameter | CH_{4} | Temperature Anomalies (LT) | Temperature Anomalies (MT) | CC (LT) | CC (MT) |
---|---|---|---|---|---|

α_{0} | 1.714 | 1.387 | 1.441 | 1.558 | 1.582 |

w | 0.790 | 1.108 | 1.128 | 0.887 | 0.757 |

**Table 2.**Multifractal characteristics for the shuffled time series of global CH

_{4}and the temperature anomalies of the global LT and MT and their cross-correlations (CC).

Parameter | CH_{4} | Temperature Anomalies (LT) | Temperature Anomalies (MT) | CC (LT) | CC (MT) |
---|---|---|---|---|---|

α_{0} | 0.508 | 0.527 | 0.508 | 0.524 | 0.500 |

w | 0.288 | 0.265 | 0.280 | 0.229 | 0.317 |

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## Share and Cite

**MDPI and ACS Style**

Tzanis, C.G.; Koutsogiannis, I.; Philippopoulos, K.; Kalamaras, N.
Multifractal Detrended Cross-Correlation Analysis of Global Methane and Temperature. *Remote Sens.* **2020**, *12*, 557.
https://doi.org/10.3390/rs12030557

**AMA Style**

Tzanis CG, Koutsogiannis I, Philippopoulos K, Kalamaras N.
Multifractal Detrended Cross-Correlation Analysis of Global Methane and Temperature. *Remote Sensing*. 2020; 12(3):557.
https://doi.org/10.3390/rs12030557

**Chicago/Turabian Style**

Tzanis, Chris G., Ioannis Koutsogiannis, Kostas Philippopoulos, and Nikolaos Kalamaras.
2020. "Multifractal Detrended Cross-Correlation Analysis of Global Methane and Temperature" *Remote Sensing* 12, no. 3: 557.
https://doi.org/10.3390/rs12030557