Whether the CMIP5 Models Can Reproduce the Long-Range Correlation of Daily Precipitation?

In this study, we investigated the performance of nine CMIP5 models for global daily precipitation by comparing with NCEP data from 1960 to 2005 based on the detrended fluctuation analysis (DFA) method. We found that NCEP daily precipitation exhibits long-range correlation (LRC) characteristics in most regions of the world. The LRC of daily precipitation over the central of North American continent is the strongest in summer, while the LRC of precipitation is the weakest for the equatorial central Pacific Ocean. The zonal average scaling exponents of NCEP daily precipitation are smaller in middle and high latitudes than those in the tropics. The scaling exponents are above 0.9 over the tropical middle and east Pacific Ocean for the year and four seasons. Most CMIP5 models can capture the characteristic that zonal mean scaling exponents of daily precipitation reach the peak in the tropics, and then decrease rapidly with the latitude increasing. The zonal mean scaling exponents simulated by CMCC-CMS, GFDL-ESM2G and IPSL-CM5A-MR show consistencies with those of NCEP, while BCC_CSM1.1(m) and FGOALS-g2 cannot capture the seasonal variations of daily precipitation’s LRC. The biases of scaling exponents between CMIP5 models and NCEP are smaller in the high latitudes, and even less than the absolute value of 0.05 in some regions, including Arctic Ocean, Siberian, Southern Ocean and Antarctic. However, for Western Africa, Eastern Africa, Tropical Eastern Pacific and Northern South America, the simulated biases of scaling exponents are greater than the absolute value of 0.05 for the year and all four seasons. In general, the spatial biases of LRC simulated by GFDL-ESM2G, HadGEM2-AO and INM-CM4 are relatively small, which indicating that the LRC characteristics of daily precipitation are well simulated by these models.


INTRODUCTION
Precipitation changes not only affect the global hydrographic cycle (Trenberth, 2011;Ma and Zhou, 2015), but also play an essential role for human societal and economic development (Wang et al., 2012;IPCC, 2013;Zhang et al., 2018;Chen et al., 2020). Global climate models are widely used to reproduce the current climate and project future climate change (Zhou and Yu, 2006;Xu and Xu, 2012; Knutson et al., 2013;Kumar et al., 2013;Dong et al., 2018;He et al., 2019). It's crucial to evaluate and investigate the models' performance in simulating daily precipitation for developing adaptation strategies to reduce uncertainties of projecting precipitation in the future (Jiang et al., 2007;Jiang et al., 2009;Wang and Chen, 2013;Li et al., 2015;Li et al., 2018;Lin et al., 2019).
The Coupled Model Intercomparison Project Phase 5 (CMIP5) includes more comprehensive global climate models enabling researchers to address many scientific questions (Taylor et al., 2012). At present, assessment methods for different models' performance are transforming from traditional qualitative methods to quantitative methods (Sillmann et al., 2013;Jiang et al., 2016;Li et al., 2017). A lot of studies evaluate models based on some traditional statistical methods, such as linear trend analysis (Guo et al., 2013;Dong et al., 2018), the spatial correlation coefficients Tian et al., 2015), the standard deviation (STD) (Yang et al., 2014), signal-tonoise ratio (SNR) (Peng et al., 2019) and so on. However, these evaluation methods cannot reproduce the inner dynamical characteristics of climate system. Therefore, a nonlinear method, long-range correlation (LRC) is needed to understand the intrinsic dynamical characteristics of climate system (Koscielny- Bunde et al., 1998;Malamud and Turcotte 1999;Fu et al., 2016;He et al., 2016;Zhao et al., 2017).
The LRC method is characterized by a timescale and shows the scaling law of an autocorrelation function (Peng et al., 1994;Bunde et al., 2005). For a random system, it is uncorrelated in both temporal and spatial evolution, so the scaling exponent of its time series is 0.5. However, for the climate system, which is a nonlinear complex system with multi-scale interactions, the persistence of external forcing and transmission of energy and information between different subsystems make it no longer isolated. Moreover, the large scale system will have a continuous impact on the small scale system, which makes its time evolution nonrandom. Therefore, its previous state will have a strong or weak impact on its future evolution, which is the LRC revealed in this paper. The value of LRC reflects the strength of nonlinear interaction between different subsystems, and to some extent,   it reflects the internal dynamic characteristics of climate system Lennartz and Bunde 2011;Yuan et al., 2015). For example, the LRC in equatorial Pacific is larger than that in land, which shows relatively strong interaction between ocean and atmosphere (Yeo and Kim, 2014). Therefore, we try to address the question about whether the CMIP5 models can reproduce the LRC of daily precipitation. It is very important and urgent to consider the LRC besides the traditional statistical methods. The detrended fluctuation analysis (DFA) is a useful tool to estimate the LRC for assessing models' simulate performance (Kantelhardt et al., 2001;Kantelhardt et al., 2002;Blender and Fraedrich, 2003;Kumar et al., 2013;Zhao and He, 2014;He and Zhao, 2017). Govindan et al. (2002) found that seven models failed to reproduce the LRC of temperature. Kumar et al. (2013) assessed the performance of 19 CMIP5 models based on long-term persistence and concluded that these models show poor performance in the long-term persistence of precipitation, however, they show better performance in temperature. Different models show different performance in the LRC of climatic variables. Most continents exhibit the long-range correlation of temperature in global coupled general circulation models (Rybski et al., 2008). Daily precipitation also shows long-range correlation both for the Beijing Climate Center Climate System Model [BCC_CSM1.1(m)] and observational data in China . The LRC is present in many aspects of climate system, such as air temperature (Du et al., 2013;Yuan et al., 2015;Koscielny-Bunde, et al., 1998;Talkner and Weber, 2000), precipitation (Kantelhardt et al., 2006;He and Zhao, 2017), sea surface temperature (Zhang and Zhao, 2015), geopotential height (Tsonis et al., 1999), extreme climate events (Feng et al., 2009) and so on. Therefore, in terms of the LRC of climate system, it is an effective way to assess CMIP5 models' performance in global daily precipitation. Based on this, we will have a deeper understanding for intrinsic dynamical characteristics of the climate system and make contributions to improve models' development.
The remainder of this paper is organized as follows. The data sets and DFA method are introduced in Methods and Data. Results presents the features of the LRC for the year and four seasons based on NCEP and CMIP5 data. Moreover, the spatial differences of LRC from different regions are shown in Results. Finally, a summary and discussion are given in Discussion and Conclusion.

METHODS AND DATA Data
The global daily precipitation datasets used in this study are composed of reanalysis data from the National Centers for Environmental Prediction and National Center for Atmospheric Research (NCEP) (Kalnay et al., 1996). The performance of NCEP reanalysis dataset has been assessed based on LRC characteristics (He and Zhao, 2017;Zhao et al., 2017), which are similar to the results of the observation. So we can use NCEP dataset as the benchmark to evaluate CMIP5 models' LRC characteristic of daily precipitation.
The simulated daily precipitation data is retrieved from the Earth System Grid (ESG) data portal for nine CMIP5 models (https://esgf-node.llnl.gov/search/cmip5/) (Taylor et al., 2012), which are from historical experiments. Only one realization of each model is analyzed. The more detailed information of each model is listed in Table 1. The horizontal resolution is different in different models. In order to facilitate model intercomparison and validation against observation, the inverse distance weighting method is used to regrid the model outputs to 2.5°× 2.5°grid. Considering the length of time series both for reanalyzed and simulated data, we chose 1960-2005 as the study period.
To reveal the geographical heterogeneity of DFA for the daily precipitation in the world, we divided the global world into 34 regions, including 12 ocean basins and 22 sub-continental land regions ( Table 2 and Figure 1). The 22 sub-continental regions are defined based on Giorgi (2002), and the 12 ocean basins are modified based on Chan and Wu (2015). We calculated the area-averaged LRC in each region for NCEP and model data, then the differences between NCEP and CMIP5 models are compared.

Method
The DFA method is often used to estimate the LRC of time series and an index of power law exponent, namely scaling exponent, can be used to quantitatively quantify the strength of LRC, which could be obtained by DFA (Peng et al., 1994;Bunde and Havlin, 2002;Bunde et al., 2005). DFA has been extensively applied to investigate LRC in climate variability (Talkner and Weber, 2000;Kantelhardt et al., 2006;Gan et al., 2007; . For a giving time series, {Xi, i 1, 2, . . ., N}, the departures of X i is calculated to eliminate the periodic seasonal trends in the climate system. In this study, X i is the daily mean value for each calendar date i. For example, X i in 1st January can be obtained by averaging Frontiers in Environmental Science | www.frontiersin.org June 2021 | Volume 9 | Article 656639 5 the daily temperature on 1st January of all years in the records. Then, cumulative sum y (k) of the time series x (i) is calculated (Eq. 2), which is called profile.
Next, the profile y(k) is divided into n int(N/τ) non-overlapping segments of equal length τ. In each segment, we apply a polynomial function, y τ (k), to fit the local trend. For order l of DFA (DFA1 if l 1, DFA2 if l 2, etc.), the l-order polynomial function should be used for the fitting. Thus, profile y(k) is detrended by subtracting the local trend y τ (k) in each segment, and the fluctuation function (F(τ)) of each segment is calculated by Typically, F(τ) will increase with the segment length τ. A linear relationship on a log-log plot indicates the presence of the power law. In this case, fluctuations functions can be characterized by a scaling exponent a.
F(τ) ∼ τ α (4) If 0.5<a<1, the time series {Xi, i 1, 2, . . ., N} is long range correlation. If a 0.5, the time series is uncorrelated. If 0<a<0.5, the series {X i } has anti-persistent correlation. In this study, the DFA2 method is used to estimate the scaling exponent in a time series.  Figure 3A). In spring, the scaling exponent of NCEP precipitation is 0.66 at this point. Except for CMCC-CMS, FGOALS-g2 and MPI-ESM-MR, the other models show greater scaling exponents than 0.59, and even the value for GFDL-ESM2G is 0.7 ( Figures 2B, 3A). In summer, the LRC of NCEP daily precipitation at this point is the strongest, and the scaling exponent reaches 0.84. Except for CMCC-CMS, FGOALS-g2 and MPI-ESM-MR, the other models' scaling exponents are greater than 0.6. In autumn, the scaling exponent of NCEP precipitation is 0.6, which is the smallest among four seasons ( Figure 3A). The scaling exponents of CMCC-CMS and MPI-ESM-MR are smaller (0.55), while that of INM-CM is the largest (0.7) among nine models. In winter, the scaling exponent for daily precipitation of NCEP is 0.63. Except for BCC_CSM1.1(m), the LRC value of most CMIP5 models are underestimated.

The LRC Characteristics of Daily Precipitation Based on NCEP and CMIP5 Models
In general, the scaling exponent of NCEP daily precipitation at the central of North American continent (110°W, 35°N) is the biggest in summer and the smallest in autumn (Figures 2, 3A). The seasonal variations of scaling exponents simulated by CNRM-CM5, GFDL-ESM2G, HadGEM2-AO and IPSL-CM5A-MR are similar to those of NCEP. These four models can capture the main characteristics that the scaling exponents are the largest in summer and the smallest in autumn, while the seasonal differences of scaling exponents simulated by the other models are various.
At the grid point (175°W, 0°) of the equatorial central Pacific Ocean, the scaling exponent of NCEP precipitation is 0.96 for the whole year, and the values simulated by nine CMIP5 models range from 0.71 to 1.0 ( Figure 4A). In spring, the scaling exponent of NCEP precipitation is 0.81. The scaling exponents of daily precipitation simulated by CMCC-CMS and MPI-ESM-  Figure 4B). The scaling exponent of NCEP precipitation in summer is 0.76, which is slightly lower than that in spring. Except for BCC_CSM1.1(m) and FGOALS-g2, the scaling exponents of the other models are greater than 0.7, among which CMCC-CMS, CNRM-CM5 and IPSL-CM5A-MR are greater than 0.9 ( Figures 3B, 4C). The scaling exponent of NCEP in autumn is 0.76, which is the same as that in spring. For the results of models, the scaling exponents of BCC_CSM1.1(m) and MPI-ESM-MR are less than 0.7, while those of CMCC-CMS and CNRM-CM5 are greater than 0.9 ( Figures 3B, 4D). In winter, the scaling exponent of NCEP precipitation is 1.0. The scaling exponents of CMCC-CMS, CNRM-CM5, INM-CM4 and IPSL-CM5A-MR are close to 1.0 value, meanwhile, CMCC-CMS shows the biggest scaling exponent among nine models, which is 1.06 ( Figures 3B, 4E). We also calculated the median values of the scaling exponents for year and four seasons (Figure is not shown). The median values of the scaling exponents' biases throughout the year range from −0.04 to −0.02, and most of them are closer to zero. From the 5% and 95% ranking values, GFDL-ESM2G and HadGEM2-AO show smaller biases band, FGOALS-g2 shows bigger biases band. The differences of scaling exponents of global daily precipitation simulated by models are smaller in spring than those of other seasons.
Generally, the scaling exponents of daily precipitation in the equatorial central Pacific Ocean are the smallest in summer, followed by spring and autumn. While for winter and the whole year, the scaling exponents fluctuate around the value of 1.0 ( Figure 3B). The seasonal differences of scaling exponents simulated by BCC_CSM1.1(m), CMCC-CMS and FGOALS-g2 are smaller than the other models in summer.

The Spatial Distribution of LRC for nine CMIP5 Models' Daily Precipitation
The zonal average scaling exponents of NCEP daily precipitation are smaller in middle and high latitudes ( Figure 5A). The zonal mean scaling exponents decrease rapidly from the equator to middle latitudes and decrease to about 0.6 near 30°S and 30°N. Subsequently, the reduction rate slows down and the zonal average scaling exponents range from 0.5 to 0.6 in the high latitude regions. The scaling exponents of daily precipitation simulated by CMIP5 models also show similar characteristics, the zonal average scaling exponents are smaller in middle and high latitudes. However, the scaling exponents of CMIP5 models' daily precipitation are underestimated, especially in the tropics. The zonal mean scaling exponents simulated by CMCC-CMS, GFDL-ESM2G and IPSL-CM5A-MR are closer to those of NCEP, while BCC_CSM1.1(m) and FGOALS-g2 show relatively poor performance.
In spring, the zonal mean scaling exponents of NCEP are larger in the northern hemisphere than those in the southern hemisphere, and reach a peak value more than 0.7 in the equatorial region ( Figure 5B). The zonal average scaling exponents in the northern hemisphere vary slightly from extratropical areas to high latitudes. In the southern hemisphere, the zonal average scaling exponents reach the minimum near 40°S, and then increase to 0.6 with the increase of latitudes. The zonal mean scaling exponents simulated by most models in the mid-latitude region are closer to those of NCEP, while the differences simulated by BCC_CSM1.1(m), CNRM-CM5, FGOALS-g2 are greater in the tropical region. The zonal mean scaling exponents of daily precipitation simulated by CMIP5 models are generally smaller than those of NCEP in low and middle latitudes. Seasonal characteristics in summer and autumn are similar to those in spring ( Figures 5C,D). INM-CM4 performs worse at the middle latitudes in summer. In winter, the scaling exponents of NCEP daily precipitation reach the minimum near 60°S in the southern hemisphere, and then increase rapidly ( Figure 5E).
In conclusion, most CMIP5 models can capture the characteristic that zonal mean scaling exponents of daily precipitation reach the peak in the tropics, and then decrease rapidly with the latitude increasing. Among nine CMIP5 models, the zonal mean scaling exponents simulated by CMCC-CMS, GFDL-ESM2G and IPSL-CM5A-MR are similar to those of NCEP, while BCC_CSM1.1(m) and FGOALS-g2 cannot capture the feature of seasonal variations.
According to the annual average scaling exponents of daily precipitation in each region, the differences between CMIP5 models and NCEP are generally no more than the absolute value of 0.25. In addition, the differences are larger in the middle and low latitudes ( Figure 6A). In AO, SIB, ALA, GRL, MED, CAS, NPO, ENA, NAO, SIO, SPO, SAO, SO and ANT regions, the differences of scaling exponents between CMIP5 models and NCEP are less than the absolute value of 0.05. While the scaling exponent biases are greater than the absolute value of 0.05 in TIB, EAS, EAF, TEP and NSA.
In spring, the differences of scaling exponents between NCEP and CMIP5 models are less than the absolute value of 0.05 in most of the world, while the differences are greater in tropical areas ( Figure 6B). In MEX, WAF, EAF, TEP, NSA, TAO, more than half of the models show the absolute value of biases more show the absolute value of differences greater than 0.05 ( Figure 6C). In autumn, the absolute value of the model's biases are less than 0.05 in the most extratropical areas, while the absolute value of more than half of simulated models' biases are greater than 0.05 in WAF, EAF, SEA and TEP areas ( Figure 6D). In winter, the absolute value of the CMIP5 Frontiers in Environmental Science | www.frontiersin.org June 2021 | Volume 9 | Article 656639 9 models' simulated biases are less than 0.05 in most parts of the northern hemisphere and middle and high latitudes in the southern hemisphere. In WAF, EAF, TIO, TCP, TEP, NSA and SAF areas, there are more than half of the models' absolute value of biases greater than 0.05 ( Figure 6E).
The global daily precipitation of NCEP shows LRC characteristic in most parts of the world. The scaling exponents are generally range from 0.65 to 0.9 in tropical areas and even above 0.9 in the tropical middle and east Pacific Ocean, which are significant at a significance level of 0.05 ( Figure 7A). Compared with NCEP data, the scaling exponents simulated by most models are smaller in the tropics. Seven models overestimate the LRC in the equatorial western Pacific except for BCC_CSM1.1(m) and HadGEM2-AO. There are larger biases in Northwest Africa, while smaller biases in the extratropical areas for most models. Overall, the biases of

DISCUSSION AND CONCLUSION
Based on the DFA method, this paper evaluates the performance of nine CMIP5 models for global daily precipitation from 1960 to 2005. The DFA results of NCEP daily precipitation present long-term correlation characteristics in most regions of the world. The scaling exponents of precipitation in the central part of North America are the largest in summer. The seasonal variations of daily precipitation's scaling exponents simulated by CNRM-CM5, GFDL-ESM2G, HadGEM2-AO and IPSL-CM5A-MR are similar to those of NCEP, which can capture the characteristics that scaling exponents are the biggest in summer and the smallest in autumn. The scaling exponents of precipitation in the equatorial central Pacific are the smallest in summer, indicating the LRC in this region is the weakest in summer. Moreover, the scaling exponents in winter are around 1.0 value. The zonal average scaling exponents of NCEP daily precipitation are smaller in middle and high latitudes. In spring, the zonal mean scaling exponents of NCEP are larger in the northern hemisphere than those in the southern hemisphere. The zonal average scaling exponents in the northern hemisphere vary slightly with the latitudes increasing and the scaling exponents are around 0.6. In the southern hemisphere, the zonal average scaling exponents reach the minimum near 40°S, and then increase to 0.6 with the increase of latitudes. Seasonal characteristics in summer and autumn are similar to those in spring. In winter, the scaling exponents of NCEP reach the minimum near 60°S in the southern hemisphere and then increase rapidly. Most CMIP5 models can capture the characteristics that zonal mean scaling exponents of daily precipitation reach the peak in the tropics and then decrease rapidly with the latitudes increasing. The zonal mean scaling exponents simulated by CMCC-CMS, GFDL-ESM2G and IPSL-CM5A-MR are similar to those of NCEP, while BCC_CSM1.1(m) and FGOALS-g2 cannot capture this feature of seasonal variations.
The global daily precipitation of NCEP shows LRC in most parts of the world, in which the scaling exponents are generally bigger and above 0.9 over the tropical middle and east Pacific Ocean for the year and four seasons. The differences between models and NCEP are larger in the middle and low latitudes. In AO, SIB, SO and ANT regions, the differences of scaling exponents' absolute value between CMIP5 models and NCEP are less than 0.05. While in WAF, EAF, TEP and NSA, the absolute value of scaling exponents' biases are greater than 0.05 for the year and all four seasons. The biases of GFDL-ESM2G, HadGEM2-AO and INM-CM4 are relatively small, which means that the dynamical characteristics of climate systems are well simulated by these models.
The present study provides a reference for different CMIP5 models' performance in simulating the LRC of global daily precipitation. Comparing the individual models for certain regions reveals that most CMIP5 models can capture the dynamical characteristics of climate system, while there are inter-model differences in various regions. Therefore, appropriate models should be selected according to different research regions.

DATA AVAILABILITY STATEMENT
The datasets presented in this study can be found in online repositories. Further inquiries can be directed to the corresponding author.

AUTHOR CONTRIBUTIONS
TD: Draft paper. Approve the final edition of the paper to be published WH: Make important revisions to the paper. Provide overall thinking. Approve the final edition of the paper to be published SZ, YM, XX, and SW offered the suggestion. All authors contributed to the article and approved the submitted version.