In recent decades, air pollution as a by-product of increased industrialization and urbanization, has been highly valued by the public and local government agencies in China. Air pollution regulation, air quality forecast and other related works have become critical issues to scientific researchers [1]. The sources of atmospheric pollutants are divided into natural and anthropogenic [2]. Anthropogenic sources include carbon and organic compounds emitted from heavy industries such as electric power, metal smelting and non-metallic mineral products, or emitted from the motor exhaust gas and coal combustion [3]. During 2013–2017, Beijing and other cities in north China suffered severe and persistent haze events caused by high fine particulate matter (PM_2.5) concentrations [4]. The Chinese State Council issued powerful policies to restrict pollution emissions [5]. So far, PM_2.5 concentrations have been reduced by 30% in China [5]. However, ozone (O₃) concentrations show an increasing trend in recent years [6]. PM_2.5 and O₃ have been the two major air pollutants in most Chinese cities. Direct or indirect exposure to air pollutions PM_2.5 and O₃ can seriously damage our physical and mental health, causing respiratory infections, various contact allergies and other diseases [7].

Time dependence and temporal predictability of time series are associated with memory behavior of time series. Nature time series such as earthquake and climate records have been found to widely exit the memory behavior [8–10]. A short-term memory process can be expressed by a first-order autoregressive process and quantified by fitting the parameter of AR (1) [11]. For nature time series, usually both short and long-term memory processes exit. Yuan et al. [12] showed that Antarctic sea ice extent is not a simple short-term persistence time series, but is actually a combination of short- and long-term memory processes. To quantify such long-term memory of time series, Peng et al. [13] proposed Detrended Fluctuation Analysis (DFA) under the fractal theory [14, 15] for the first time to study the memory behavior of internal molecular chains of DNA. This method can filter out the trend component of its own evolution, and the remaining deviation sequence is the component of its own fluctuation [16–19]. Thus, it can eliminate the unreal correlation caused by the non-stationary characteristics of time series [20]. The DFA method has been successfully applied to seismology, stock market, biology, climate and environment, etc. [21–24]. For example, Lennartz et al. [25] and Fan et al. [23] used the DFA method to find that the memory exists in inter-occurrence seismic records. Yang et al. [26] estimate the persistence of precipitation over a wider range of scales and show that precipitation persistence can be described as a varying by DFA. Yuan et al. [27] observed temperature records of 12 stations from Antarctica island, coastline, and continental areas are analyzed by means of DFA and they found different long-term climate memory (LTM) behaviors.

Also, previous studies have found significant long-term (short-term) memory in air pollutant concentrations in some regions [28–30]. Liu et al. [31] found that there are different self–organized criticality process for pollutions SO₂, NO₂, and PM₁₀ and the daily air pollution indices (API) associated with different power–law relations in Shanghai by using DFA and multifractal method. Also, Shi et al. [32] showed that the time series of three pollution indexes (SO₂, NO₂, and PM₁₀) and the daily air pollution indexes (APIs) in China have a strong long-term memory within a year by three different methods. For other countries, the similar long-term memory behaviors of air pollution were also found i.e., Nikolopoulos et al. [33] used DFA to analyze PM₁₀ time series in the Athens area (GAA) and found the long-memory patterns. Windsor et al. [34] examined the statistical characteristics of United Kingdom pollution time series and found evidences of high persistence and long-term memory of pollutant fluctuations up to 400 days. PM₁₀ and O₃ pollutants in the Caribbean region showed the multifractal nature with the significant Hurst parameter [35]. Wu et al. [36] studied the long-term persistence characteristics of several air pollutants (PM_2.5 and O₃) during the epidemic situation of COVID-19 by using multifractal detrended fluctuation analysis (MFDFA) and they found that the concentrations of three cities (Changsha, Zhuzhou, and Xiangtan) showed strong long-term persistence characteristics and multifractal structures. Shi et al. [37] comparatively analyzed the long-term persistence characteristics of PM_2.5 evolution for eight air monitoring stations of Chengdu and the results showed that the spatial and temporal evolution of PM_2.5 exhibit a long-term persistence. However, the above studies did not involve the spatial and temporal distribution characteristics of short- and long-term memory of air pollutants over China. In this paper, the DFA method is introduced to study the spatial evolution characteristics of memory of three pollutants (PM_2.5, PM₁₀, and O₃) in 366 cities of China. Moreover, we compare the memory behaviors between short- and long-term scales.

The structure of this paper is as follows: in Section 2, the source of air pollution data and the data processing method are described. In Section 3, we give the specific steps of eliminating DFA method. Section 4 shows the results. Section 5 further summarizes our findings and draws the conclusion.

Hourly time series of three air pollutants concentrations (PM_2.5, PM₁₀, O₃) in 366 Chinese cities are downloaded from the website (https://quotsoft.net/air/). The time period is from 2015 to 2020. In this paper, we focus on air pollution concentrations in winter and summer over China. Here, we define November, December, January, and February as winter; and May, June, July, and August are defined as summer. The lengths of PM_2.5, PM₁₀, and O₃ time series in winter and summer are shown in Table 1.

First of all, we remove the seasonal and daily trend from the original data, which can be represented as X ( h , d , y ) as a function of hour h , day d   , and year y . The method to obtain detrend data X ' ( h , d , y ) is as follows: X ′ ( h , d , y ) = X ( h , d , y ) − X ¯ ( h , d ) (1) X ¯ ( h , d ) = 1 66 ∑ d 0 = − 5 5 ∑ y = 2015 2020 X ( h , d + d 0 , y ) (2) where X ¯ ( h , d ) represents the averaged seasonal and daily trend at day d   and hour h . Since there are some missing data in the original data, we use the linear interpolation method to fill the missing data. The arithmetic mean of the previous and next neighbors’ concentration to the vacancy position is used to supplement the missing data to ensure the integrity and continuity of the data.

Short- and long-term correlated time series can be well described as a generalized AR (1) [12]: X ′ H = a X ′ H − 1 + ε α   (3) where H is order of time step and a is the parameter of the first-order autoregressive process. The short-term memory of time series can be characterized by the parameter a. We can obtain the parameter a according to the least square method to estimate the AR (1) process of the detrended time series of pollution data [38]. ε α is the long-term correlated noise with Hurst index α , which characterizes the long-term memory of time series. The parameter is the DFA scale exponent can be obtained as follows.

In general, the DFA algorithm can be divided into five steps [26]:

1) For the detrended time series     X i ′ ( i = 1,2... ,   N ) , we calculate its cumulative deviation sequence y ( k ) :

First, we show samples of the detrended times series of PM_2.5, PM₁₀, and O₃ concentrations as Eq. 1 for Beijing, Shanghai, Chengdu, and Guangzhou in Figure 1. There are larger fluctuations for PM_2.5 and PM₁₀ than other cities, especially in winter as shown in Figure 1A. Due to the implementation of environmental protection policies of China in recent years, we can clearly see that the fluctuation of PM_2.5 is smaller in recent years in comparison to that of the earlier years. In Figure 1C, Guangzhou shows the smallest fluctuation for the PM pollution than other cities. This fluctuation can be strongly affected by meteorological factors [40–43]. The meteorological field is generally more intense in Northern China than Southern China. For the O₃ times series, it shows completely different features with PM. The largest O₃ fluctuation appears in summer related to the product of photochemical reactions. Furthermore, the O₃ fluctuation seems to be more intense in Guangzhou than other cities. Also, the detrended O₃ fluctuation does not show a significant increasing or decreasing trend with time for these four cities in Figure 1. Though the seasonal and daily cycles have been removed for the times series, we can still observe the large different characteristics between winter and summer such that we consider below the memory behaviors for winter and summer, respectively.

Next, we use the least square method to estimate the AR (1) process of the detrended time series of PM_2.5, PM₁₀, and O₃ to obtain the autoregressive parameter a in Eq. 3. We show the spatial distribution of the autoregressive parameter a of 366 Chinese cities for summer and winter respectively in Figure 2. By comparing the autoregression parameters a of PM_2.5, PM₁₀, and O_3, we find that the short-term memory effects of pollutants have seasonal and regional differences. And the short-term scale is hour-to-hour. For summer, particulate matters in central and eastern China show stronger short-term memory behavior, especially around the Sichuan Basin and Hubei Province. The values of individual cities are as high as 0.98. For most winter cites, the short-term memory effects of PM_2.5 and PM₁₀ were stronger than those shown in the summer. Compared with PM_2.5 and PM_10, the short-term memory of O₃ varies less, and the a value is stable at around 0.9 or lower in different regions and seasons. The results suggest that compared with O₃, PM_2.5, and PM₁₀ are more dependent on the local meteorological conditions, which strongly influenced by the short-term memory behavior. In winter, the meteorological conditions are stronger and the boundary layer is lower than in summer leading to the stronger short-term memory behaviors of PM_2.5 and PM₁₀. The higher value of a with stronger short memory in PM_2.5 and PM₁₀ over southeast China corresponds to higher intrinsic predictability [44]. Studies also show that other meteorological variables over southeast China exhibit higher intrinsic predictability [45]. The results also show the contrast performance of a-values in South and North China, which will be discussed further in later chapters.

Then, we study the long-term memory of pollutions and implement the DFA analysis to the detrended time series of PM_2.5, PM₁₀, and O₃ for the four cities. And the long-term scale is month-to-month. According to Eq. 7, we can obtain the fluctuation F(t) as a function of time scale t for summer (see Figure 3). Previous studies [46, 47] pointed out that due to the effect of limited length of time series, the beginning part of F(t) may be affected by short-term memory effect and lead to a overestimated DFA index. For the AR (1) process, the reliable time scale t to fit scale index α should be above the crossover s x = 15 a ( 1 − a ) ( 1 + a ) (a is the parameter of AR (1) process) theoretically [46]. We calculate the crossover s _x and plot in Figure 2 (Red dashed lines). Indeed, the short time scale range below s _x shows some different scale behaviors with that above s _x in Figure 2 which is affected by the short-term memory effect. Thus, we only fit scale exponent α for the time scale t above the crossover s _x . Here we show the results of four typical cities, and further summarize the DFA scaling exponent α in Table 2. For the four cities, they all show 1 / 2 < α < 1 with the strong long-term memory. In summer, all DFA index α are higher than 0.7, except for Beijing α = 0.66 for PM_2.5. In summer, the boundary lay is high so that it benefits to the diffusion of air pollution, especially in Beijing.

We show the results of winter in Figure 4. There are clear differences between the time scales below and above s _x . The index α values of four cities show stronger long-term memory effect in winter. We find that except for PM_2.5 and PM₁₀ in Shanghai, the α values of the other three cities are higher than 0.7 in winter (see Table 2). The memory of pollutants in winter is stronger associated with more persistent weather systems in general. But Shanghai is different, which is located on the west coast of the Pacific Ocean and belongs to the subtropical marine monsoon climate. Compared with summer, the persistent low temperature and northerly wind in winter in Shanghai [48]. They couple with the increase of local emission sources and frequent cold air activities and other meteorological factors, leading to the intensification of pollutant concentration changes. Therefore, these factors may lead to lower persistent characteristics of PM concentrations in winter for Shanghai.

To comprehensively understand spatial patterns of the memory of pollutants in China, we further study the spatial distribution of the DFA index α for PM_2.5, PM₁₀, and O₃ in 366 Chinese cities. We repeat the above DFA analysis to obtain the index α for each city. The spatial distributions of α are presented in Figure 5 for summer and winter respectively. We find that the scale exponent α of most cities is between 0.7 and 1 in summer and winter, indicating that the three pollutants in Chinese cities have strong long-term memory. There are largest α values located in Central and Eastern China for PM_2.5 and PM₁₀ in Figures 5A–D. Moreover, Northern China has a smaller α value than Southern China for PM_2.5 and PM₁₀ and summer has a smaller α value than winter for PM_2.5, PM₁₀, and O₃. The cause of long-term memory difference between north and south might be the influence of long-term meteorological factors. For winter, the largest α value with strongest memory locates in the Sichuan basin for Figures 5B,D of PM_2.5 and PM₁₀. In the basin, the stable temperature inversion and low boundary layer are easy to be formed in winter leading to the accumulation of PM_2.5 and PM₁₀ on surface and causing the long-persistent haze Meteorological conditions related to PM_2.5 and PM₁₀ have been found to be different between Northern China and Southern China, i.e., Zhang et al. found that the Rossby waves can influence PM_2.5 in Northern China but not Southern China [43]; relative humidity in Northern and Southern China is positive and negative correlated with PM_2.5 respectively [49]. Moreover, strong northwest winds and cold fronts can rapidly disperse the PM pollutants leading to the lower memory in Northern China. In general, the memory of PM pollutants in winter is stronger than in summer. The temperature decreases significantly due to nighttime radiation. If the surface atmospheric temperature is lower than the upper atmospheric temperature, the inversion layer will form [50]. This prevents convection up and down the air, making it difficult for pollutants to accumulate in winter. And the air pollution caused by coal consumption is more serious in winter.

However, the seasonal differences could be affected by the monsoon system. In winter, China is controlled by Siberian high, and the air mass is dry, which is not conducive to precipitation. The atmospheric stratification is relatively stable, and it is easy to accumulate aerosols in the lower atmosphere, which is not conducive to pollutant diffusion. We find that the α values significantly increase in Eastern and Northern China for O₃ in Figure 5F. It indicates that there is a strong long-term memory for winter there in contrast to PM_2.5 and PM₁₀. It is well known that O₃ is a photochemical oxidant formed by a series of photochemical reactions of nitrogen oxides and volatile organic compounds in the atmosphere [51]. There are more sunny days in winter of Eastern and Northern China. We find that some places, e.g., Sichuan Basin with the short-term memory in Figure 2 also are associated with the long-term memory in Figure 5 in winter. We calculate the spatial correlation coefficient between short (Figure 2) and -term memory (Figure 5) for China, and find that the coefficients are 0.273, 0.226, and 0.164 in summer and 0.303, 0.319, and 0.041 in winter for PM_2.5, PM₁₀, and O₃ respectively. There are more similarities in space between the short- and long-term memory in winter than that of summer for PM_2.5 and PM₁₀. For O₃, the short- and long-term memory are very different and the spatial correlation coefficients are smallest.

In this study, we use the AR (1) process and the DFA method to quantify the memory characteristics of the time series of three air pollutants (PM_2.5, PM₁₀, O₃) in China. According to the AR (1) process, we obtain the spatial distribution of the autoregressive parameter a of 366 Chinese cities for summer and winter respectively which is associated with the short-term memory behavior. The short-term memory of particulate matter in most parts of southern China is stronger, especially in the Yangtze River Basin, while the short-term memory of O₃ is slightly different between the north and the south. Moreover, the short-term memory of PM_2.5 and PM₁₀ in winter are stronger than in summer related to the stronger short-term and local meteorological field. In the previous studies [31, 32], they found two different scaling exponents of DFA in short- and long-term scales for PM₁₀ in Shanghai in consistent with us. However, our results suggest that using the scaling exponent of DFA in short-term scale to quantify memory is problematic, since the DFA index can by strongly affected by the limited length of time series [46, 47]. Thus, it is better to use the autoregressive parameter of the AR (1) process.

To quantify the long-term memory behavior, we study the DFA index of different pollutants in Beijing, Shanghai, Chengdu and Guangzhou in winter and summer. We find that the three pollutants in four representative cities have strong long-term memory (the DFA index α greater than 0.5). In comparison to previous studies on the memory analysis of some regions or cities in China [31, 32, 36, 37], our results show the similar long-term memory behaviors. Furthermore, our results suggest that the long-term memory behaviors strongly depend on seasons and locations in China, which are lost in previous studies. To show the spatial pattern of long-term memory in China, we further analyzed the DFA index of pollutant time series in 366 cities across China in summer and winter. It is found that the long-term memory of air pollutants in southern cities is generally higher than those of northern cities for PM pollutions. The long-term memory of O₃ does not show significant differences between southern and northern. The spatial differences of long-term memory of PM pollutions are mainly related to different long-term meteorological conditions in Southern and Northern China. Also, the long-term memory measures of air pollutants in Chinese cities are higher in winter than in summer associated with the longer persistent weather system in winter. There are some spatial similarities between the short- and long-term memory measures for PM_2.5 and PM₁₀. But for O₃, the spatial similarity is very low and irregular. In our further work, we will consider relationships of the memory between air pollution and meteorological factors or human activities from the perspective of dynamic characteristics.