Discrete Dynamics in Nature and Society

Volume 2016 (2016), Article ID 4269431, 7 pages

http://dx.doi.org/10.1155/2016/4269431

## Nonlinearity and Fractal Properties of Climate Change during the Past 500 Years in Northwestern China

^{1}Yangzhou Meteorological Office, Yangzhou 225009, China^{2}Yangzhou University, Yangzhou 225009, China^{3}College of Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China^{4}National Climate Center, CMA, Beijing 100086, China

Received 25 November 2015; Accepted 2 March 2016

Academic Editor: Amit Chakraborty

Copyright © 2016 Shiquan Wan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By using detrended fluctuation analysis (DFA), the present paper analyzed the nonlinearity and fractal properties of tree-ring records from two types of trees in northwestern China, and then we disclosed climate change characteristics during the past 500 years in this area. The results indicate that climate change in northwestern China displayed a long-range correlation (LRC), which can exist over time span of 100 years or longer. This conclusion provides a theoretical basis for long-term climate predictions. Combining the DFA results obtained from daily temperatures records at the Xi’an meteorological observation station, which is near the southern peak of the Huashan Mountains, self-similarities widely existed in climate change on monthly, seasonal, annual, and decadal timescales during the past 500 years in northwestern China, and this change was a typical nonlinear process.

#### 1. Introduction

The climate system exhibits a long-term memory. However, no one knows how long this memory is. The memory plays an important role in climate prediction, and thus the study of this memory is crucial. The short-term memory of a dynamic system is usually present on a timescale of certain duration and it is accompanied by the exponential decay of an autocorrelation function. The diverging timescale existed in a time series with long-term memory which exhibits the scaling properties of the autocorrelation function. However, this long timescale does not mean that the system has a memory of infinitely long duration [1–3]. Many natural and social-economical phenomena display a scale distribution, such as price changes in the economic market, fluctuations in human’s heart rate, electrical signals, water levels of the Nile River, and DNA sequences. By studying fluctuations in financial market indices with different timescales, the scaling of their distributions shows long persistence in the volatility [4], and the cumulative distribution of the volatility is consistent with a power-law asymptotic behavior [5]. The heartbeat time series also have fractal scaling properties, which fluctuate in an irregular and complex manner, even under resting conditions. The fractal analysis may provide a new approach to recognize deceased states by studying changes in the scaling properties [6]. Long-range correlations are present only in the REM phase [7]. An application of detrended fluctuation analyses on the electric signals shows that its power-law exponents are related with the non-Markovian character, which are consistent with the existence of long-range correlations [8]. The return intervals of extremes above some threshold also exhibit long-range correlation [9]. A similar phenomenon was also founded in the temperature and heavy rainfall records [10–14]. Based on the values of an autocorrelation function of a time series, these phenomena can be divided into three categories: an autocorrelated sequence, an anticorrelated sequence, and a noncorrelated sequence. The essential difference between price fluctuation and a temperature record is that price fluctuation contains a white-noise spectrum lacking intrinsic correlation, whereas a temperature record exhibits a long-range correlation.

The study of global climate change is a crucial scientific field of the International Geosphere-Biosphere Programme, and a great deal of information on past climate changes is contained in various types of proxy data. For example, ice cores, stalagmites, lake sediments, and tree rings all recorded some information of temperatures and rainfall amounts to some extent. Because the existing observational data span a relatively short time period and are scarce, proxy climate data are particularly important for the research of long-term climate change. The growth of tree rings is significantly affected by local climatic condition, and tree-ring data have several advantages, including being accurately dated and continuous and having high resolution. The measurements of tree-ring growth possess relatively higher precision than other proxy data and the identical tree-ring chronologies are easy to acquire in different time-instants. Therefore, tree rings can be used to extract a great deal of information regarding climate and environmental changes, which are more than that in other types of proxy data, and it is true for the reliability of the tree-ring data. In a technical report of the International Climate Dendrochronology Academic Conference in 1982, it was noted that, among the proxy data related to the change of various climate factors in most areas on the Earth, the tree-ring data are far more reliable than any other proxy data measured in terms of years [15].

These proxy data usually display complex evolutionary trends. We often know little about their underlying dynamic processes, and it is very difficult to distinguish the dynamic processes of a system from any superimposed external disturbances in the proxy data. Therefore, linear or nonlinear features of potential dynamic processes represented by the proxy data are important in developing time-sequence models, which can simulate the characteristics of climate change in these data. Consequently, there is an urgent need to qualitatively classify the properties of proxy data. It remains an opening question on the definition of nonlinearity. For example, some researchers define nonlinearity based on the response of a system to an external disturbance: if the response of a system to a disturbance is linear (nonlinear), the system is thus linear (nonlinear). Others distinguish linearity from nonlinearity based on the equation representing the dynamics of the system: if this equation contains linear (nonlinear) terms, the system is classified as linear (nonlinear). In order to deal with this problem, Ashkenazy et al. [1] proposed the following definition of a nonlinear time series: if the incremental sequence of a time series displays a long-range correlation, the time series is nonlinear; otherwise, it is linear. Because there is often no correlation between the incremental sequences of a linear time series, this definition is widely applied. In this study, we first conducted a scaling analysis of an incremental sequence in a tree-ring width chronology of Huashan pines on the southern peak of the Huashan Mountains based on the definition of nonlinearity presented by Ashkenazy et al. A clear scaling region is found and the scale exponent is greater than 0.5, which indicates that the original time series of the incremental sequence is nonlinear. Because the radial growth of trees is primarily affected by climatic and environmental factors, their growth rings display a significant response to temperatures and precipitation and may be expected to exhibit a long-range correlation with variations in temperature and precipitation. By analyzing tree-ring data from northwestern China, we studied climate change during the past 500 years in this region. The results indicate that there was long-range persistence in climate change over the past 500 years in northwestern China, and it was a typical nonlinear process with fractal behaviors. Further analysis indicates that the persistence of climate change in northwestern China can span a period of 100 years and even possibly longer.

#### 2. Data and Methods

The tree-ring width chronologies used in this study were obtained from the Tree-Ring Data Center of China. We analyzed tree-ring data from two common types of trees at three locations in northwestern China: Huashan pines on the southern peak (elevation 2020 m) and western peak (elevation 2030 m) of the Huashan Mountains (110°05′E, 34°29′N), Shaanxi Province, with the tree rings during the periods of 1515–1992 A.D. and 1359–1992 A.D., respectively, and Qilian junipers in the De Halin area of Qinghai (97°56′E, 37°27′N, elevation 3500–3900 m) with the tree-ring chronology during the period of 980–2001 A.D. The records of daily average maximum temperatures were obtained from the National Meteorological Information Center, Chinese Meteorological Administration.

The incremental sequence of a tree-ring sequence , where is the sample size, is defined as the absolute value of the difference between the neighboring tree-ring index values in the sequence:To a nonlinear time series, if the incremental sequence of a time series (namely, the volatility of the sequence) displays a scaling feature, the original time series is nonlinear [1]. In this study, we analyzed the incremental records of the tree-ring sequences.

The persistence of a time series can usually be determined by calculating the autocorrelation function of the sequence: where and is the average of the sequence . If there is no persistence in the sequence , there is no correlation in , and the autocorrelation function is . If there is persistence in the scale of , the autocorrelation function is greater than 0 when it is smaller than a critical scale , and when the scale exceeds the critical scale , this persistence disappears. However, there is nonstationary variation in most of observational data, such as noises and various trends, which often affect the reliability of calculations of the correlation function using (2).

Detrended fluctuation analysis (DFA) is a scaling analysis method which was designed to investigate the long-range fluctuation correlation in a given time interval, where it is typically assumed that the type of correlation is unknown. Here, we briefly introduced the DFA algorithm [16, 17]. Considering a time series, (), firstly, we integrate the time series ,wherewhich is the average of . Next, the integrated time series is divided into nonoverlapping boxes of equal length . In each box of length , we fit the integrated time series by using a polynomial function, , which is called the local trend. For order of DFA (DFA1 if , DFA2 if , etc.), the -order polynomial function should be applied for the fitting. In the third step, we detrend the integrated time series, , by subtracting the local trend in each box, and the root-mean-square fluctuation of this integrated and detrended time series is calculated byThis computation is repeated over all timescales (box sizes ) to characterize the relationship between , the average fluctuation, and the box size . Typically, will increase with box size. A linear relationship on a log-log plot indicates the presence of power law. Under such conditions, the fluctuations can be characterized by a scaling exponent , the slope of the line relating to . If , there is no correlation and time series behaves as a random series (Brownian noise), indicates anticorrelations, and indicates long-range correlations.

#### 3. Results

Figure 1(a) shows the tree-ring width chronology for the Huashan pines on the southern peak of the Huashan Mountains, which spans the 478 years from 1515 A.D. to 1992 A.D. The corresponding incremental sequence is shown in Figure 1(b).