# On the Difference of Scaling Properties for Temperature and Precipitation over China

Authors: Lei Jiang; Liqing Zhao; Zihao Zhao
Journal: Advances in Meteorology
 (2017)

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## Abstract

The daily air temperature and precipitation records of four meteorological observation stations over China are used to investigate the differences of scaling property employing the detrended fluctuation analysis (DFA) method. The results show that the values in DFA-exponent for temperature are higher than those for precipitation compared by different orders DFA1–3. A 95% significance test is also applied to verify LRCs by resampling the temperature and precipitation records 10000 times in Beijing. The values of scaling exponent from original temperature and precipitation records are larger than the upper range value of the interval threshold after shuffling the data records, which implies there are positive LRCs. For temperature records, the value of scaling exponent calculated by FA is greater than those by DFA1–3 at all four stations. This indicates that the FA curve overestimates the scaling behavior due to the effect of trends. By contrast, the values of scaling exponent in precipitation are almost the same by using FA and DFA1–3 for all time scales, respectively. Furthermore, there are crossovers on short time scales in different orders DFA1–3 for the temperature records, while the slopes keep almost consistent on all time scales for the precipitation records.


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## Body

## 1. Introduction

Variations in air temperature and precipitation records are important indicators of climate change. Some studies have exhibited the variation trends of temperature and precipitation at different spatial-temporal scales [1]. Variable persistence represents memory characteristics within data sets at various time scales. The temperature and precipitation time series display different self-similar structures and long range correlations (LRCs). The persistence at various time scales is governed by different physical processes and sometimes even affected by the trends. There is little research on the difference of scaling properties for temperature and precipitation time series over China.Temperature and precipitation time series show obvious nonstationary characteristics in climate system. In fact, it is not feasible to detect intrinsic dynamic properties of variables due to nonstationarity and nonlinearity by using some traditional approaches, such as the power spectrum and correlation analysis. Therefore, the detrended fluctuation analysis (DFA) method with different orders is applied to investigate scaling behaviors, which may eliminate the data trends often masked by nonstationarities.Peng et al. successfully developed the DFA method [2]. Since then, many studies have used the DFA method in atmospheric temperature [3–8], cloud breaking [9], wind speed [10], relative humidity [11, 12], air pollution [13], global tropopause [14], ozone variations [15], sea surface temperature [16–18], and so on. In China, different fractal characteristics of air temperature and ground surface temperature have been also studied [19]. Jiang et al. found that most of the values of ground surface temperature in DFA-exponent are higher values than those of air temperature over China [20]. Meanwhile, Jiang et al. also studied LRCs of precipitation over China [21]. The precipitation time series, as an important fluctuation indicator in climate system, exhibits weak persistence characteristics. A case study of temperature and precipitation records can provide a basis and deepen our understanding for examining seasonal, interannual, and interdecadal variability and trends over China.The first aim of this paper is to study the differences of LRCs for temperature and precipitation time series over China. The paper is organized as follows. In Section2, the acquisition of the temperature and precipitation records and the FA and DFA methods are described. In Section 3, the difference of LRCs for temperature and precipitation is investigated and compared using FA and DFA over China. The conclusion and discussions are summarized and drawn in Section 4.


## 2. Data and Methods

### 2.1. Data Records

The daily air temperature and precipitation data records used in this paper come from the Chinese National Meteorological Information Center. Most of data length is from 1951 to 2009. The long data records have been investigated in various aspects. Yuan et al. used the DFA method to analyze the LRCs of five kinds of daily temperature records and found that there are power law correlated characteristics [22]. The high-quality air temperature and precipitation records are also applied to detect the persistence characteristics and trends at various time scales over China [20, 21, 23]. We removed the seasonal cycles from the raw time series to obtain the air temperature and precipitation anomaly records. The anomaly records can be expressed by Δxi=xi-xid, i=1⋯N, where xi denotes the raw data, N is the data length, and xid represents the mean value for a given calendar day.


### 2.2. Methodology

Compared with the traditional approaches such as the power spectrum technique and autocorrelation functions analysis, detrended fluctuation analysis systematically eliminates the trends of recorded data and reveals intrinsic dynamic characteristics often masked by nonstationarities. The DFA methodology has been successfully applied in detecting long range correlations of data records such as temperature time series.Such technique was established by Peng et al. [2] and extended by Bunde et al. [24] and Kantelhardt et al. [25]. Three key steps are described in the DFA method.(1) Obtain the profile by integrating the air temperature and precipitation anomaly records withN samples:(1)yi=∑k=1iΔxk,i=1,…,N,where y(i) is the profile, Δxk is deseasoned records, and N is data length.(2) Divide the profile into nonoverlapping segments of equal lengths, indexed by k=1,…,Ns with Ns≡[N/s]. A least-square fit yν(k) is used to determine the local polynomial trend in each segment.(2)y′i=yi-yνi.In each of these segments, the local trendyν(k) is calculated by least-square fits in the interval. In the first-order detrended fluctuation analysis (DFA1), the detrended variability F(s) can be determined from the linear fit of the profile for each window s. The possible linear trends are eliminated by this step. In DFA2, possible linear and parabolic trends are removed by subtracting quadratic fits of the profile. Likewise, in DFAn, possible n order trends are eliminated in the profile. However, in FA, any kind of trend is retained in the profile.(3) The fluctuation functionF(n)(s) is calculated by the profile subtracting the local polynomial fit in each segment length s: (3)Fns=12Ns∑k=12Nsyk-ysk2.There is double logarithmic linear relationshipF(s)~sα between the fluctuation function F(s) and time window s. This represents the correlation characteristic can be depicted by the scaling exponent α. The data signal exhibits random behaviors for α=0.5, positive LRCs for α>0.5, and anticorrelated behaviors for α<0.5, respectively.In different ordern of DFA, the nth-order trend of time series is eliminated by subtracting polynomial fittings such as linear, quadratic, cubic, and higher order corresponding to DFA1, DFA2, DFA3, and higher order DFA. DFAn removes n-order trend of the profile and n-1-order trend of the original data records. The trend strength of temperature and precipitation time series can be estimated by using different orders of DFA.


## 2.1. Data Records

The daily air temperature and precipitation data records used in this paper come from the Chinese National Meteorological Information Center. Most of data length is from 1951 to 2009. The long data records have been investigated in various aspects. Yuan et al. used the DFA method to analyze the LRCs of five kinds of daily temperature records and found that there are power law correlated characteristics [22]. The high-quality air temperature and precipitation records are also applied to detect the persistence characteristics and trends at various time scales over China [20, 21, 23]. We removed the seasonal cycles from the raw time series to obtain the air temperature and precipitation anomaly records. The anomaly records can be expressed by Δxi=xi-xid, i=1⋯N, where xi denotes the raw data, N is the data length, and xid represents the mean value for a given calendar day.


## 2.2. Methodology

Compared with the traditional approaches such as the power spectrum technique and autocorrelation functions analysis, detrended fluctuation analysis systematically eliminates the trends of recorded data and reveals intrinsic dynamic characteristics often masked by nonstationarities. The DFA methodology has been successfully applied in detecting long range correlations of data records such as temperature time series.Such technique was established by Peng et al. [2] and extended by Bunde et al. [24] and Kantelhardt et al. [25]. Three key steps are described in the DFA method.(1) Obtain the profile by integrating the air temperature and precipitation anomaly records withN samples:(1)yi=∑k=1iΔxk,i=1,…,N,where y(i) is the profile, Δxk is deseasoned records, and N is data length.(2) Divide the profile into nonoverlapping segments of equal lengths, indexed by k=1,…,Ns with Ns≡[N/s]. A least-square fit yν(k) is used to determine the local polynomial trend in each segment.(2)y′i=yi-yνi.In each of these segments, the local trendyν(k) is calculated by least-square fits in the interval. In the first-order detrended fluctuation analysis (DFA1), the detrended variability F(s) can be determined from the linear fit of the profile for each window s. The possible linear trends are eliminated by this step. In DFA2, possible linear and parabolic trends are removed by subtracting quadratic fits of the profile. Likewise, in DFAn, possible n order trends are eliminated in the profile. However, in FA, any kind of trend is retained in the profile.(3) The fluctuation functionF(n)(s) is calculated by the profile subtracting the local polynomial fit in each segment length s: (3)Fns=12Ns∑k=12Nsyk-ysk2.There is double logarithmic linear relationshipF(s)~sα between the fluctuation function F(s) and time window s. This represents the correlation characteristic can be depicted by the scaling exponent α. The data signal exhibits random behaviors for α=0.5, positive LRCs for α>0.5, and anticorrelated behaviors for α<0.5, respectively.In different ordern of DFA, the nth-order trend of time series is eliminated by subtracting polynomial fittings such as linear, quadratic, cubic, and higher order corresponding to DFA1, DFA2, DFA3, and higher order DFA. DFAn removes n-order trend of the profile and n-1-order trend of the original data records. The trend strength of temperature and precipitation time series can be estimated by using different orders of DFA.


## 3. Results

The temporal evolution of the daily temperature and precipitation anomaly records is shown in Figure1 at four selected weather stations over China. The four stations include Beijing, Langfang, Tangshan, and Baoding. The temperature and precipitation anomaly records exhibit irregular high frequency variations for all weather stations. The fluctuations of the precipitation anomaly records are more intense than that for the temperature anomaly records. The probability of extreme events for precipitation is higher than that for temperature due to the large fluctuations.Figure 1
Temporal evolution of the temperature and precipitation anomaly records of four meteorological observation stations over China. ((a) and (e)) Station of Beijing. ((b) and (f)) Station of Baoding. ((c) and (g)) Station of Tangshan. ((d) and (h)) Station of Langfang.See Figure2; profiles of temperature and precipitation anomaly time series vary with the totally different tendency. There are obvious persistence characteristics of temperature anomaly time series at four stations in Figure 2(a). The temperature anomaly records firstly decrease below the value 0 and then gradually increase to 0 and show a similar letter shape “v” and positive linear trends. These indicate that the persistence characteristics of temperature profile at such four stations are stronger than those of precipitation records shown in Figure 2(b).Figure 2
(a) Profiles of the temperature anomaly records at four weather stations. (b) Profiles of the precipitation anomaly records.
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(b)To characterize the LRCs, the FA and DFA methods are applied to analyze the daily temperature and precipitation anomaly records of four weather stations over China. Figure3 exhibits double log plots of the FA and DFA1–3 results for temperature anomaly time series. The fluctuations of temperature display the characteristics of long range power law correlations across all four stations. For the temperature anomaly records, the scaling exponent obtained from the FA method is greater than the values from the DFA1–3 at whole time scales. Large value in scaling exponent indicates the results using the FA method may overestimate the LRCs due to the existence of trends in temperature time series. Urbanization as one of the factors might affect the significant difference of scaling property in temperature at short time scales over China.Figure 3
Log-log plots of power law relationship in temperature anomaly records using FA and DFA1–3 (black, red, green, and blue solid circles, resp.) at four stations. (a) Beijing. (b) Langfang. (c) Tangshan. (d) Baoding. Solid lines are linear fits.
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(d)Similarly, long range power law relationships of precipitation anomaly time series are shown in Figure4. For the precipitation anomaly records, the curves are straight lines with approximately the same scaling exponent across all time scales using FA and DFA1–3, implying that there are no obvious nonlinear trends. It seems that the scaling behavior of precipitation over China is almost not affected across time scales. There are weak LRCs in precipitation anomaly time series.Figure 4
Log-log plots of power law relationship in precipitation anomaly records using FA and DFA1–3 (black, red, green, and blue solid circles, resp.) at four stations. (a) Beijing. (b) Langfang. (c) Tangshan. (d) Baoding. Solid lines are linear fits.
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(d)The values of scaling exponent in temperature anomaly records are significantly higher than those in precipitation. There are diversity characteristics for the curves in Figure3 compared with Figure 4. An interesting finding is that the FA curves in temperature time series are approximately a straight line across all time scales at all weather stations. Additionally, the slope of FA curve seems to be consistent with the slopes of DFA1–3 curves at short time scales. However, the slopes of temperature anomaly records are significantly different at short time scales using the DFA1–3 method.Frequently, the data records do not show the same scaling behaviors for all time scales and their temporal correlation characteristics are different. So one or one more crossovers are detected by the DFA method [26]. For instance, the long-term persistence of data records might become stronger on small time scales than that on large time scales. Moreover, the crossovers might exist in the fluctuation functions with different orders of detrending due to the effect of trends. In addition, there are crossovers for all DFA1–3 curves of temperature at short time scales. Crossovers in the DFA1–3 curves imply that there are different correlation characteristics at various time windows. The existence of the linear trend may be a reason why the value of scaling exponent in the FA curve is higher than in the DFA1–3 curves.For all four meteorological stations, there are similar persistence characteristics of temperature and precipitation anomaly records, respectively. Next, Beijing, as a representative observation station, is used to analyze LRCs of data records. To further confirm our findings, the slopes of DFA1–3 for temperature records in Beijing are separated by different time windows in Figure5. The values of scaling exponent gradually decrease with the increase of the order n of DFA at short time scales. The smaller the time scales are, the greater the influence is. However, the values of scaling exponent have small changes in long time scales to some extent. This indicates that local trend effects on LRCs of temperature in Beijing hold within certain time ranges. There are nearly similar slopes for three different orders DFA1–3 in temperature at long time scales.Figure 5
Log-log plots of power law relationship between the fluctuationF(s) and piecewise time window s for temperature time series in Beijing using DFA1–3 (square, hollow circle, and solid circle, resp.). Solid lines are linear fits.A numerical test is applied to eliminate the trends and correlations by resampling the temperature and precipitation records 10000 times in order to verify LRCs at different time ranges in Beijing. Figure6 shows the probability density distribution of scaling exponent by resampling the temperature and precipitation records inside a loop for 10000 times applying the DFA2 method. Figure 6(a) exhibits the frequency distributions of the scaling exponent for temperature. The range of the interval threshold is from 0.47 to 0.53 for temperature records at 95% significant level. Similarly, the interval threshold is the same range for precipitation records shown in Figure 6(b). The maximum frequencies of scaling exponent in shuffled temperature and precipitation records are centered on 0.50 and take on the normal distributions. The value α=0.70 for temperature and the value α=0.57 for precipitation in Beijing are significantly larger than the upper threshold of scaling exponent α=0.53 at 95% confidence level. Therefore, there are the LRCs for temperature and precipitation records in Beijing. Moreover, the Gauss fits of scaling exponent in temperature are almost consistent with that in precipitation shown in Figure 6(c). The Gauss fit curve in precipitation is slightly higher toward the maximum value than that in temperature. This indicates increasing extreme events occurrence for precipitation anomaly time series.Figure 6
The frequency distributions of the scaling exponent using DFA2 after shuffling 10000 times in Beijing. (a) Temperature. (b) Precipitation. (c) Gauss fits of temperature and precipitation.
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(c)The changes of the slopes for temperature and precipitation time series in Beijing are almost consistent when the annual cycles are eliminated by employing DFA1–3. So we further investigate the LRCs of temperature and precipitation using the DFA2 method. The long range power law relationships of log-log plots between the fluctuationF(s) and the window scale s are given in Figure 7. For temperature records, the value of scaling exponent α=0.66 above one month is consistent with that of the earlier finding [27]. However, the slope below one month α=0.91 is in the extremely high level, which indicates there is the trend effect of local climate change. For precipitation records, the value of scaling exponent α=0.57 shows weak LRCs for all time scales. Moreover, LRCs of precipitation records are not affected by local influencing factors.Figure 7
Log-log plots of power law relationship for temperature and precipitation time series in Beijing using DFA2 (black, green, and blue solid circles, resp.). Solid lines are linear fits.


## 4. Conclusion and Discussions

Many studies are on the scaling behaviors of temperature or precipitation, but few studies are on the difference of scaling properties between temperature and precipitation. Moreover, traditional approaches cannot detect natural variability due to nonstationarity of climate variables. In this letter, long-term daily air temperature and precipitation records of four weather stations over China are investigated by applying the FA and different orders DFA methods. The results show that there are LRCs of temperature and precipitation records for all weather stations.Different characteristics of LRCs are found after resampling the data records many times through a 95% significance test. LRCs of temperature decrease with the increase of time window by employing different orders DFA. However, LRCs of precipitation records are not obvious across all time scales. This may be caused by the factors of complexity and discontinuity of precipitation. The value of scaling exponent for temperature records is significantly higher than that for precipitation records, which implies that there are stronger LRCs for temperature over China.In addition, there are similar slopes for temperature and precipitation records by comparing different orders DFA1–3. The value in DFA-exponent for temperature time series is pretty higher at short time scales than that at long time scales, which implies the LRCs can be affected by some local trends, while precipitation records show similar slopes across all time scales. The scaling behaviors of the temperature and precipitation anomaly records might be affected by many different factors at different time scales. It is necessary to perform more studies on the difference of scaling behaviors between temperature and precipitation in the future.



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