1. Introduction
Global near-surface air temperature databases are standardly derived from long-term instrumental temperature measurements and are offered for public use to document and help understand historical and ongoing climate variations and change (Karl et al. 1993). The three most prominent groups that produce such databases are the NASA Goddard Institute for Space Studies (GISS), the NOAA/National Centers for Environmental Information [NCEI; formerly the National Climatic Data Center (NCDC)], and a joint effort of the Met Office Hadley Centre and the University of East Anglia Climatic Research Unit (with the corresponding dataset HadCRUT; Hansen et al. 2010; Morice et al. 2012; Smith et al. 2008). Input observations to these datasets are largely drawn from the same sources: the World Meteorological Organization (WMO) and Global Climate Observation System (GCOS) initiatives provide the bulk of the land station data (Morice et al. 2012; Jones et al. 2012), while the International Comprehensive Ocean–Atmosphere Dataset (ICOADS), a compilation of meteorological records collected by ships and drifting and tethered buoys, is the main source of the ground ocean data (Morice et al. 2012; Freeman et al. 2017). All the observational data are typically updated monthly (Morice et al. 2012). Despite their common sources of observational data, the datasets largely differ in how they handle issues such as incomplete spatial and temporal coverage or nonclimatic influences on a measurement station’s environment (Hansen et al. 2010; Morice et al. 2012). Further, methodological differences in the construction of datasets include usage or lack thereof of spatial infilling (Hansen et al. 2010), incorporation of satellite measurements (Reynolds et al. 2002), and estimation of near-surface air temperature above sea ice (Smith et al. 2008; Hansen et al. 2010). Finally, all prominent global temperature datasets are given in the form of temperature anomalies, calculated against different climatological reference periods, with different resolutions of spatial averaging and interpolation (i.e., with different sizes of corresponding grid elements; Hansen et al. 2010; Morice et al. 2012; Smith et al. 2008).
Comparative assessments of these datasets indicate their consistency regarding certain components of temperature variability, such as hemispheric or global trends (Hansen et al. 2010). However, a reliable quantification of consistency lacks the estimation of uncertainty in the long-range spatial and temporal temperature characteristics, originated by both intrinsic variability of data and the structural differences between the datasets. In this regard, scaling properties are known to characterize correlated randomness (Stanley 2005) that persists over a wide range of time scales. In this paper, we investigate scaling properties of the two main global temperature datasets: the current versions of the HadCRUT (HadCRUT4) and NASA GISS Land–Ocean Temperature Index (LOTI). By calculating power-law exponents of appropriately prepared statistical functions that describe the gridded monthly data time series, we determine the existence and forms of global patterns of the observed near-surface air temperature stochastic variability and assess the influence of structural uncertainties that arise from the choice of a particular dataset preparation methodology on the quantification of the long-range spatial and temporal order of the data.
The role of stochasticity in climate state and variability has been extensively studied since the initial application of present-day scaling techniques in statistical hydrology (Hurst 1951; Mandelbrot 2001). Specifically, it was determined that observational and derived regional and global temperature data show strong natural long-term persistence. It can be described by the autocorrelation function 
In this paper, scaling properties of global temperature data are described through the scaling (or Hurst) exponent α of each temperature gridpoint time series. To determine α, we used second-order detrended fluctuation analysis (DFA2; see, e.g., Kantelhardt et al. 2001), where linear trends in the data are systematically removed. We used DFA2 in combination with the wavelet transform (WT) power spectral analysis to confirm the DFA2 results by determining the scaling exponent β of the wavelet power spectra (Blesić et al. 2003; Bashan et al. 2008). In addition, we used the wavelet transform power spectrum (WTS) to provide insight into the existence, positions, and amplitudes of significant periodic or nonperiodic cycles in the data (Sarvan et al. 2017; Stratimirović et al. 2018). To this end, we used Morlet wavelets of the sixth order as a wavelet basis for our analysis. The Morlet wavelets have been proven to possess the optimal joint time–frequency localization (Goupillaud et al. 1984; Torrence and Compo 1998) and can thus be effectively used to detect locations and spatial distribution of singularities in time series (Mallat and Hwang 1992; Zanchettin et al. 2008). We calculated the scaling exponents α for all the available gridpoint data of the two datasets, without restrictions regarding the amount of missing data. The purposes of this approach were to obtain global spatial pattern(s) of scaling, to examine its differences and similarities for two databases, to identify dissimilarities that stem from inhomogeneities due to data management (Karl et al. 1993; Peterson et al. 1998; von Storch et al. 2012), and to test the robustness of our methods against data nonuniformity (Hu et al. 2001; Chen et al. 2002; Rust et al. 2008). Our results may be compared with other methods of data analysis, such as the Fourier transform power spectral analysis or the calculation of the autocorrelation function, through direct dependence (Talkner and Weber 2000; Höll and Kantz 2015) and scaling relations given below. Finally, our approach did not hypothesize any particular underlying physical process as a source of scaling. It can nevertheless be compared to the outputs of approaches based on other functional forms and/or specific model assumptions, such as with the structure functions analysis based on the concept of scale invariance in turbulence (Schertzer and Lovejoy 1987, 1990; Talkner and Weber 2000; Lovejoy and Schertzer 2013); for the comprehensive assessment of links of structure function analysis to DFA, please see Talkner and Weber (2000) and Kantelhardt et al. (2006).
Our paper is structured as follows. In section 2, we give a brief overview of the sources of data and of the general methodological framework of the DFA and the WTS analysis. In section 3, we present the results of the usage of DFA2 and WTS to study scaling properties of the HadCRUT4 and NASA GISS LOTI datasets. This includes our findings that concern possible sources of the observed anti-autocorrelated (with scaling exponents 
2. Data and methods
a. Data
We used the NASA GISS LOTI gridded monthly temperature anomalies data available on the GISS Surface Temperature Analysis (GISTEMP) website (GISTEMP Team 2017). We used the LOTI data derived from the analysis that combines the Extended Reconstructed Sea Surface Temperature (ERSST) version 4 (Huang et al. 2015; Liu 2012; Huang et al. 2016) dataset with optimum interpolation (OI) of the satellite data and with 1200-km spatial smoothing for insufficient coverage (Hansen et al. 2010). In GISTEMP, the grid boxes are 2° latitude × 2° longitude. We also used the Met Office Hadley Centre observational gridded dataset HadCRUT4, which provides median temperature anomalies from the 100 ensemble members in each grid box (Morice et al. 2012), available on the Met Office website (Met Office Hadley Centre 2010). In HadCRUT4, the grid boxes are 5° latitude × 5° longitude. For parts of our analysis that compare results obtained within particular grid elements with those obtained using the source observational data of the same grid cell, we used land station data provided by Google Earth for the HadCRUT4 land temperature dataset CRUTEM4 (Jones et al. 2012; Osborn and Jones 2014; CRUTEM4 Team 2017) and the NCDC Global Historical Climatology Network (GHCN; version 3) land station monthly data (Lawrimore et al. 2011; GHCN Team 2017). To compare our results for the gridded data with marine observations, we considered the ICOADS version 2.5 time series provided by the Royal Netherlands Meteorological Institute (KNMI) Climate Explorer web application (KNMI Team 2017). Whenever possible, we used both unadjusted and adjusted land station or marine measurements to account for the effects of data homogenization (Rust et al. 2008). Finally, as a source of satellite temperature measurements, we used the University of Alabama in Huntsville (UAH) satellite temperature analysis (Christy et al. 2003; Christy and Spencer 2017) in combination with the NCEI OI sea surface temperature (OISST; Banzon et al. 2016; Reynolds et al. 2007; OISST Team 2017); in UAH, the grid boxes are 2.5° latitude × 2.5° longitude, while the OISST dataset has a resolution of 1° latitude × 1° longitude. An overview of our data sources is given in Table 1.
Monthly data sources with major parameters and number of data points N used for scaling analysis.
Whenever the time series were given in absolute temperatures, and in order to correctly compare their DFA2 and WTS results with the corresponding outputs of HadCRUT4 and NASA GISS LOTI time series, we used conventional deseasoning to define their anomaly time series. In these instances, the seasonal means for the entire record, instead of for the particular reference period, have been removed (Livina et al. 2011; Torrence and Compo 1998). In this paper, we refer to such deseasoned records as the “raw data.” This method of seasonal detrending has been proven appropriate for the purpose and design of our study [i.e., the assessment of (monofractal) scaling and consistency of cycles in data; Livina et al. 2011; Ludescher et al. 2011; Bunde et al. 2013a]; it dampens the amplitude of the annual cycle in the amount sufficient to enable the assessment of underlying long-term correlation properties. If the original absolute temperature date were used, this would lead to a remarkable change in DFA2 results; in the range of scales of interest to this paper, the seasonal trend will dominate DFA2 (and WTS) behavior in such a profound way that the estimation of scaling will be impossible, and DFA2 functions will be almost indistinguishable for different observational records (Hu et al. 2001).
b. Methodology
We used the DFA and the WTS approaches for data analysis. DFA was introduced as an appropriate scaling analysis to deal with nonstationary records that contain some trends of unknown form (Peng et al. 1994). In DFA, the procedure of detrending was devised so as to eliminate such trends. The resulting remarkable performance of this method in data analysis critically stems from this highly effective detrending solution, as shown by numerous systematic studies that investigate the effects of trends, nonstationarities, and nonlinearities (Hu et al. 2001; Chen et al. 2002, 2005), as well as the effects of extreme data loss (Ma et al. 2010) on the DFA function form, and compare DFA with other detrending methods (Xu et al. 2005; Bashan et al. 2008) or other independent methods of data analysis (Alvarez-Ramirez et al. 2008b; Rodriguez et al. 2014). Recently, a new mathematical insight further illuminated how DFA operates on nonstationary data series with nonstationarity due to their intrinsic dynamics (Höll et al. 2016).
The advantages of using DFA over the more conventional statistical approaches (such as the calculation of the autocorrelation functions or the Fourier power spectra) for the analysis of records from complex systems are twofold, and both stem from the method design. First, DFA takes any typical time-dependent discrete data series—which is, in general, likely nonstationary and with unknown trends—and produces a series that fluctuates much less than the original by subtracting local trends at different time window lengths. The remaining time series has the same statistical properties as the original (Stanley 2000) but is now prepared in a way that greatly helps clarify its dynamic behavior. Second, direct calculations of the autocorrelation function, or of Fourier power spectra, are hindered by the level of noise present in a typical natural record by the possible nonstationarities in the data. DFA, however, calculates the fluctuation function, which is, by definition, a sum over autocorrelations (Höll and Kantz 2015), and thus fluctuates less. As a result, one uses a function that is entirely defined by the autocorrelation function but is more stable (Bunde et al. 2013b), allowing for clearer (or less noisy) presentation and interpretation of the results on (log–log) graphs. In the online supplemental material, we provide a graphical illustration of these claims for the statistical functions of the time series of HadCRUT4 global temperature anomalies.
Pure, long-range autocorrelated behavior rarely occurs in natural records. The corresponding DFA2 functions, depicted on the log–log graphs, are thus rarely ideal linear functions. Instead, they tend to display transient crossovers in scaling that stem from occurrences of irregular phenomena of different types (Mallat and Hwang 1992; Hu et al. 2001). Of those, climate records are likely to embed effects of mixtures of cyclic components that can cover a whole band of frequencies and locally perturb scaling (including DFA2) analysis (Mandelbrot and Wallis 1969). It has been shown (Hu et al. 2001) that these perturbations present in a form of peak-like structures, superposed on the DFA functions of a pure long-range correlated signal, with widths wider than those that would be expected from single sharp periodic waves. The spread of any such perturbation, and the length of scales that it covers until asymptotically resuming to the DFA behavior dominated by the long-range correlated noise, depends on the scaling exponent α and the period and/or amplitude of the hypothetical periodic trend and is generally much less visible for the greater values of α [see detailed explanations and theoretical relations by Mandelbrot and Wallis (1969) and Hu et al. (2001)]. When the effects of such irregularities are visible on DFA2 curves but are not comparatively strong to change the global behavior of DFA2 functions, we use WT analysis to investigate them.
The WT was introduced in order to circumvent the uncertainty principle problem in classical signal analysis (Stratimirović et al. 2018) and achieve better signal localization in both time and frequency than classical Fourier transform approaches (Morlet 1983; Grossmann and Morlet 1984). In WT, the size of an examination window (equivalent to the size of a sliding segment in DFA) is adjusted to the frequency analyzed. In this way, an adequate time resolution for high frequencies and a good frequency resolution for low frequencies is achieved in a single transform (Bračič and Stefanovska 1998).
In this paper, we found it convenient to use the standard set of Morlet wavelet functions as a wavelet basis for our analysis (Morlet 1983; Grossmann and Morlet 1984). The Morlet wavelet, a plane wave modulated by a Gaussian, is a complex nonorthogonal wavelet function (Torrence and Compo 1998) that is recommended for use in time series analysis in instances where smooth, continuous variations in wavelet amplitude are expected (Torrence and Compo 1998). We choose to use the Morlet wavelet of order six, so as to also be able to utilize its shape for localization of singular time events (Bračič and Stefanovska 1998). It has been shown that that this wavelet transform is particularly well adapted to estimate the local regularity of functions (Mallat and Hwang 1992); namely, in the local wavelet power spectra, the Morlet wavelet is narrow in spectral (scale) space and broad in the time space, which produces very well-localized, relatively sharp peaks in the global WT spectra, the averages of local spectra over time (Torrence and Compo 1998). This choice provides us with a possibility to investigate effects of influence of both periodic and nonperiodic cycles on the dynamics of our data, together with the effects of occurrences of significant singular events (e.g., volcanic eruptions). Finally, by construction, the Morlet wavelet scale is almost equal to the Fourier scale (Torrence and Compo 1998), which makes the two power spectra comparable.
We calculated DFA2 fluctuation functions (DFA2ff) and WT power spectra (WTS) for the temperature anomalies data series and plotted them on double logarithmic time/scale axes so that the exponents α or β are estimated by linear fit. We took into consideration only the values of DFA2ff between the minimum time scale of 
3. Results
We calculated the DFA2ff and the WTS and their corresponding scaling exponents α and β for the HadCRUT4 and NASA GISS LOTI global average time series of temperature anomalies, and the DFA2 exponents α for each gridpoint time series, for the two temperature anomaly products. In Fig. 1, we present the combined DFA2–WTS results for the global average HadCRUT4 and NASA GISS LOTI series, together with their raw data; here, and hereafter, DFA2–WTS represents the abbreviated notation for the results independently derived from DFA2 and WTS. In Fig. 1, the DFA2ff and the WTS are depicted in the form 
Results of the DFA2–WTS analysis of the time series of global average temperature anomalies of the HadCRUT4 and NASA GISS LOTI datasets. (top) Data and (bottom) DFA2ff (solid lines) and WTS (filled circles) functions, together with linear fits to the DFA2ff and WTS curves (pink solid lines). The DFA2ff and the WTS are depicted on a log–log graph, in the form 
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-17-0823.1
a. Global pattern of scaling: HadCRUT4
Figure 2 shows the global scaling pattern of the time series of HadCRUT4 gridded temperature anomalies. Depicted are values of the scaling exponent α for the time series of each grid point on the 5° × 5° grid, together with the latitudinal averages of the exponent α. Long-range correlated behavior is found in all of the grid points belonging to the ocean regions and in nearly all grid points on land (in 27 land grid cells, we found 
(left) DFA2 exponents α calculated for all available gridpoint time series of temperature anomalies in the HadCRUT4 dataset. Values of 
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-17-0823.1
Figure 2 also shows latitudinal averages of the DFA2 exponent α, calculated along the 36 latitude rows of the HadCRUT4 dataset. It presents a weak latitude dependence of α, with lower LTP in high latitudes; this pattern seems to follow the global distribution of land and is probably somewhat affected by the strong influence of El Niño–Southern Oscillation (ENSO) teleconnections in the midlatitudes (Graf and Zanchettin 2012). The global averages of the scaling exponent α—the normal average, calculated as the average of all the DFA2 exponents of all the grid cells (when it is 
Results of the DFA2–WTS analysis of the time series of global average temperature anomalies of the HadCRUT4, together with the average temperature anomalies for the Northern and Southern Hemispheres and the tropics, presented as in Fig. 1. Dotted vertical lines at t = 12, 40, 72, and 110 months are given as visual guides.
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-17-0823.1
To understand how the obtained results are affected by the data loss in regions where large amounts of source data are missing, were removed due to artifacts in the observational records (Fraedrich and Blender 2003), or underwent a considerable adjustment due to inhomogeneous observations (Menne and Williams 2009; Alexandersson 1986), we compared the DFA2ff and WTS behavior of the raw (unadjusted) and the adjusted data for several such land stations. Because of the large amount of data, we could not investigate these effects for all records; we made a choice to focus our analysis on land stations that are the sole source or one of the few sources of observations available in the considered grid cell. Our results, shown in Fig. 4 for two illustrative examples, demonstrate that in such cases, the DFA2 exponents for the adjusted data in the gridded dataset can be slightly or even substantially smaller than for the raw data. The corresponding WTS reveals that this is probably due to the modulation of the annual cycle, as well as to the strong dampening of interannual and decadal fluctuations in the adjusted data. Artificial reduction of LTP by data adjustment seems to be a general feature. The opposite behavior (i.e., an increase of LTP by data adjustment, as shown in Fig. 5) only occurs in several polar or subpolar stations. There, systematic lack of data for entire seasons yields DFA2 exponents of the adjusted series that are slightly higher than those of the corresponding raw series, probably as a result of superposition of seasonality to the data. These findings indicate that the true DFA2 exponents for a largely predominant part of the HadCRUT4 grid, where there is a large percentage of missing values (Fraedrich and Blender 2003), are likely higher than those estimated from the actual gridded data and illustrated in Fig. 2. Our conclusion about a likely underestimation of the DFA2 exponent is in line with previous findings on effects of homogenization (Rust et al. 2008) on artificial data. The results also suggest that, excluding polar and parts of subpolar regions for substantial data inhomogeneity, the HadCRUT4 global temperature is long-range correlated (i.e., all the gridded DFA2 exponents are likely equal to or higher than 0.5).
Two examples of the (top) DFA2ff and (bottom) WTS calculated for the raw and adjusted temperature records of stations from the HadCRUT4 gridded dataset with (left) considerable amount of missing data and (right) observations that were preprocessed for data homogenization. In the DFA2ff graphs, the values of scaling exponents are given for both raw (unadjusted) data 
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-17-0823.1
Two examples of the (top) DFA2ff and (bottom) WTS calculated for the raw (unadjusted) and adjusted temperature HadCRUT4 records of (left) polar or (right) subpolar stations that systematically miss data for entire seasons. In the DFA2ff graphs, the values of scaling exponents are given for both raw data 
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-17-0823.1
Finally, we inspected the DFA2ff that have scaling exponent values larger than 1 to determine whether they display crossovers and thus the existence of intrinsic nonstationarities (Höll et al. 2016) that should then be explored and understood further. None of the HadCRUT4 grid points with 
Examples of the calculated DFA2ff (solid lines) and WTS (filled circles) functions for the grid points in the HadCRUT4 dataset that have scaling exponents 
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-17-0823.1
b. Global pattern of scaling: NASA GISS LOTI
Figure 7 shows the DFA2 global pattern for the NASA GISS LOTI 2° × 2° gridded time series, together with latitudinal averages over 90 GISS latitudes, which produce the normal average over all grid cells of 
(left) DFA2 exponents α calculated for all available gridpoint time series of temperature anomalies in the NASA GISS LOTI dataset. Values of 
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-17-0823.1
c. Understanding differences in scaling between HadCRUT4 and NASA GISS LOTI
1) Land: Effects of the 1200-km rule
Here, HadCRUT4–NASA GISS LOTI differences in values of scaling over the land are assessed, accounting for the different approaches employed to solve the problem of incomplete spatial coverage in their construction. HadCRUT4 does not employ any form of spatial infilling, and as a result, gridbox anomalies can readily be traced back to observational records (Morice et al. 2012). NASA GISS LOTI instead interpolates among station measurements and extrapolates anomalies as far as 1200 km into regions without measurement stations (Hansen et al. 2010). To probe whether the spatial infilling that is employed in the construction of the NASA GISS dataset determines the observed difference in scaling over land between NASA GISS and HadCRUT4, we compared the average DFA2ff and WTS of the raw adjusted station records that contribute to a HadCRUT4 grid point, with the DFA2–WTS HadCRUT4 (adjusted) results and the NASA GISS LOTI DFA2–WTS results within the corresponding grid point. We repeated this procedure for several sparsely filled (in terms of number of recording stations) and several densely populated grid points. Examples of our findings are given in Fig. 8. Our results show that in sparsely filled grid cells, the procedure of spatial interpolation of station data, which is the only data processing performed in the HadCRUT4 dataset, lowers the scaling exponent α due to the modulation of the interannual and multidecadal variability and the flattening of noise at scales higher than annual. This finding is not universal for all spatially averaged HadCRUT4 data; it depends on the relative influence of the high LTP records present within the considered grid box. In NASA GISS LOTI, the additional procedure of spatial infilling within a 1200-km radius from the selected grid point increases this effect (i.e., it further decreases the value of α; see left panels in Fig. 8). Moreover, in the case presented in Fig. 8, the surrounding land grid points have significantly different scaling exponents, so that the process of extrapolation as far as 1200 km integrates spurious correlations that are entirely location related (i.e., dependent on the scaling of the nearest-neighbor grid cells). For this reason, changes in the values of α over land introduced by the 1200-km rule in sparsely filled grid boxes cannot be viewed or corrected as for the systematic bias. Finally, the observed discrepancy between HadCRUT4 and NASA GISS LOTI scaling does not appear at grid points sufficiently populated with recording stations (see example in the right panels of Fig. 8).
An example of the effect of a spatial infilling procedure on DFA2–WTS calculations for (left) sparsely infilled and (right) sufficiently populated HadCRUT4 grid points. Depicted are average DFA2ff of the station records that compose grid points (pink solid line), the average DFA2ff of the four NASA GISS LOTI grid points encompassed by the grid point analyzed (green filled circles), and the DFA2 HadCRUT4 results (gray filled circles), with the corresponding WTS given in the figure insets.
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-17-0823.1
2) Ocean: Effects of inclusion of satellite data
NASA GISS LOTI constructs ocean data (Hansen et al. 2010) as an integration of the Met Office Hadley Centre analysis of SSTs (HadISST1; the sole basis of HadCRUT4 ocean data; Rayner et al. 2003) for the 1880–1981 period, where measurements are ship based, and satellite SST measurements (OISSTv2; Reynolds et al. 2007) from 1982 to the present. Satellite measurements in the NASA GISS dataset are additionally calibrated with the help of ship and buoy data (Hansen et al. 2010). To understand how this methodological difference affects scaling over ocean regions in both datasets, we calculated and compared DFA2ff and WTS of several HadCRUT4 marine grid points with the matching average (within the same HadCRUT4 grid cell) NASA GISS LOTI, average OISSTv2, and average UAH satellite temperature for the lower troposphere (TLT) scaling. An example of the obtained findings is given in Fig. 9, showing the TLT UAH data scale as white noise (with 
Comparison of DFA2–WTS HadCRUT4 gridpoint scaling results (violet-filled circles) with the matching average of four (within the same HadCRUT4 grid point) NASA GISS results (green-filled circles), the average of 25 OISSTv2 results (pink solid line), and the average of four UAH satellite temperature results for TLT scaling (gray solid line). The results are given for the HadCRUT4 grid box centered at 7.5°N, 167.5°W.
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-17-0823.1
4. Discussion and conclusions
We used detrended fluctuation analysis of second-order (DFA2) and wavelet-based spectral (WTS) analysis to investigate and quantify the global pattern of scaling in major datasets of observed near-surface air temperature anomalies and to better understand cyclic behavior as a possible underlying cause of the observed long-term scaling behavior. Both methods allow us to overcome problems related to nonlinearity and partially nonstationarity of natural data series. We focused our analysis on two prominent sources of global temperature data, namely, the Met Office HadCRUT4 and the NASA GISS LOTI gridded historical records. Our approach allowed us to characterize the global pattern of temperature scaling and to investigate the relevance and the extent of possible influences of real or artificial (i.e., originated by data processing) cycles upon global scaling. In particular, we investigated how DFA2ff and WTS can be affected by data processing compensating for the issue of inhomogeneity of data linked to the scarcity of records or to the changes of data-recording practices. Finally, we studied the possible structural sources of dissimilarities in global pattern of scaling that we found to exist between the HadCRUT4 and NASA GISS LOTI datasets.
We found that the global temperature pattern is likely long-range autocorrelated, except for polar and parts of subpolar regions, where data inhomogeneity is substantial. We confirmed the existence of a land–ocean contrast in persistence (Bunde and Havlin 2002; Fraedrich and Blender 2003), with marine data showing an appreciably more pronounced long-range persistence than land data. Four prominent cyclic influences, or characteristic times of underlying processes, emerged in the time range of analysis of our data. They appear at periods of 12, ~40, 72, and 110 months. The first two cycles that we found in our data can be attributed to the seasonal cycle and probably to the influence of the leading ENSO eigenmode (Penland and Matrosova 2006; Compo and Sardeshmukh 2010) on sea surface and land temperatures. The other two characteristic times are difficult to attribute to any individual or canonical source of climate variability. We refer to research showing that the period of approximately 6 years can be related to the variance of ENSO indices such as Niño-3.4 SSTs (Penland and Matrosova 2006) or to the first harmonics of decadal variabilities (Zanchettin et al. 2013), while the near-decadal period of 110 months can emerge as a response to nonperiodic strong events of volcanic eruptions (Rypdal 2012; Lovejoy and Varotsos 2016) or as a reflection of decadal climate variability originated either by internal processes (Liu 2012) or forced by external natural factors (Zanchettin 2017). A systematic assessment of the observational records that will explore the universality of appearance of the range of cycles obtained here remains a task for future research. Understanding of universality and of the nature of these irregular structures, be they periodic or nonperiodic phenomena or even significant singular events (Mallat and Hwang 1992; Zanchettin 2017), may be used as a tool to objectively differentiate between climate scaling regimes (Stratimirović et al. 2018) or as an additional source of information in climate modeling efforts (Lima and Lall 2010).
We found that the spatial average of scaling of the global gridded temperatures is significantly lower than the scaling of the spatially averaged global temperature time series and argued that this is an effect of the disproportionate influence of the high LTP series, particularly those in the midlatitudes, on regional, hemispheric, and global averages. We showed that the global temperature scaling is in this way dominated by the scaling of the Southern Hemisphere, which in turn is possibly significantly determined by the scaling in the tropics. This effect may explain why our values of DFA2 exponents averaged along parallels, particularly along the midlatitudes, differ from the corresponding averages calculated for the global coupled general circulation models by Rybski et al. (2008). Finally, these observations may indicate that the spatial resolution of global temperature products can affect their local (individual grid cells) and global scaling behaviors and that the spatial scaling may be important for understanding the dynamics underlying the observed climate variability. There is probably a need in climatically diverse regions for a more detailed sampling of the different areas (in both datasets) in order to account for their different scaling regimes in the regional estimate and to accurately determine regional dynamics. Sea ice dynamics seem to have a strong effect on scaling, as demonstrated by the sharp edges in the DFA2 exponents consistently detected in NASA GISS LOTI and HadCRUT4 between areas affected and not affected by sea ice. Whether the low persistence observed in the sea ice regions originates from the strong seasonal cycle of sea ice, rather than from other processes of the coupled ocean–atmosphere–sea ice system, remains to be determined.
Our results unraveled the nonuniformity of scaling within ocean or land data and the pronounced differences of such nonuniformity in the two datasets. Our findings suggest that the observed nonuniformity of scaling can reflect a number of different natural (Fraedrich and Blender 2003), as well as methodological, causes, whose individual contribution is difficult to disentangle. We found that for the still-predominant part of the analyzed datasets affected by a large percentage of missing values, the real values of the scaling exponents are likely higher than those calculated. This result is in accordance with assessments of artificial data with similar properties (Rust et al. 2008). We found instances of amplification of cyclic influence or even introduction of new cycles, sometimes coupled with the reduction of noise, in both datasets and due to the homogenization and optimization of the raw (unadjusted) temperature time series; these effects are probably more pronounced in cases of corrections due to the actual data loss (Chen et al. 2002; Ma et al. 2010). Since there is no apparent universal solution to this problem, we avoid conclusively asserting the exact nature of the dynamics underlying the temperature time series for such locations.
We also assessed structural uncertainties that arise from methodological choices made in the two temperature analysis products. We showed instances where spurious scaling is introduced in the NASA GISS dataset through spatial infilling procedure, or where reinforcement of the annual cycle is introduced due to the optimization of integrated satellite records. This highlights once more the need to consult in detail how data are prepared before assessing climate dynamics based on data analysis (von Storch et al. 2012). Nevertheless, keeping in mind the stochastic nature of climate (Hasselmann 1976; Franzke et al. 2012; Watkins 2017) and the current lack of an effective model capable of capturing long-range interactions between large numbers of interacting parts that would mimic LTP as an output from various climate systems (Ludescher et al. 2017), the observed global temperature pattern of scaling can serve as a nontrivial test (Monetti et al. 2003) for dynamic properties of current climate models.
Our results do not settle the debate about nature and origins of scaling properties of temperature or of the observed natural nonuniformity of scaling (Levine and McPhaden 2016; Markonis and Koutsoyiannis 2013; Bunde and Lennartz 2012; Rypdal 2012; Fraedrich et al. 2004; Stanley 1999; Press 1978). Instead, they point to the heterogeneity of scaling as an important area for further investigation in this context. This seems to be crucial for progress in our understanding of the critical problem of detection and attribution of trends and other climate change evidence (Crok et al. 2014; Zanchettin 2017). Specifically, if we assume that the observed temperature evolution, similar for both datasets (IPCC 2013, their Fig. 2.20), is a realization of a long-term autocorrelated process, then the appropriate statistical approaches and underlying theories must be applied to the detection problem. Current analytical approaches and numerical estimations (Lennartz and Bunde 2009, 2011) indicate the DFA2 scaling exponent α, along with the observed linear trend and the standard deviation around the data regression line, to be an important quantity to estimate anthropogenic trends. The heterogeneous scaling of global temperature reported in our study, and especially the presented evidence of weakly correlated or even random (with 
Acknowledgments
Suzana Blesić would like to thank Prof. Armin Bunde for the invaluable introductory advice on and insights into the analysis of time series of climate data. Her work received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Grant Agreement 701785. We thank Djordje Stratimirović and Darko Sarvan for very useful discussions. Finally, we thank the anonymous reviewers for stimulating comments and particularly for pointing out the critical issue of scaling of spatial averages.
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