# Variation analysis of precipitation during past 286 years in

HYDROLOGICAL PROCESSES Hydrol. Process. 27, 2934–2943 (2013) Published online 22 June 2012 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/hyp.9388 Variation analysis of precipitation during past 286 years in Beijing area, China, using non-parametric test and wavelet analysis Miao Li,1,2 Jun Xia,1*,† Zhi Chen,3 Dejuan Meng1,2 and Chongyu Xu4 1 Key Laboratory of Water Cycle & Related Land Surface Processes, Institute of Geographic Sciences & Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China 2 Graduate University of Chinese Academy of Sciences, Beijing 100039, China 3 Department of Building, Civil & Environmental Engineering, Concordia University, Montreal, Canada 4 Department of Geosciences, University of Oslo, P.O. Box 1047 Blindern, N-0316 Oslo, Norway Abstract: Non-parametric methods including Mann–Kendall (M–K) test, continuous wavelet transform (CWT) and discrete wavelet transform analysis are applied in this paper to detect the trend and periodic trait of precipitation data series in Beijing area where the data set spans nearly 300 years from 1724 to 2009. First, the trend of precipitation variables is elaborated by the M–K test (Sequential M–K test). The results show that there is an increasing trend (the value of this trend is 1.98) at the 5%-signiﬁcance level and there are not turning points in the whole data series. Then, CWT and wavelet variance are used to check for signiﬁcant periodic characteristics of data series. In the plots of wavelet transform coefﬁcients and ﬁgure of wavelet variance, some periodic events affect the trend of the annual total precipitation series in Beijing area. 85-year, 35-year and 21-year periodic events are found to be the main periodic series of long-term precipitation data, and they are all statistically signiﬁcant. Moreover, the results of non-parametric M–K test are exhibited on seven different combinations of discrete wavelet components. D5 (32-year periodicity) periodic component is the effective and signiﬁcant component on data. It is coincident with the result (35-year periodic event as one part of main periodicity) by using CWT analysis. Moreover, approximation mode shows potential trend of the whole data set because it is the residuals as all periodicities are removed from data series. Thus, the mode A + D5 is responsible for producing a real basic structure of the trend founded on the data. Copyright © 2012 John Wiley & Sons, Ltd. KEY WORDS long-term data series; Mann–Kendall test; wavelet analysis; precipitation; Beijing area Received 19 August 2011; Accepted 27 April 2012 INTRODUCTION The need for better information about the variability exhibited by climatic and hydrological process has increased as a result of the changing climate (Kalra and Ahmad, 2011). The complexity of climatic and hydrological phenomena with nonstationary characteristics leading to diverse periodic events occurred in different time periods has been paid close attention by meteorologists and hydrologists (Buishand et al., 1988; Andreo et al., 2006; Partal and Kahya, 2006; Jiang et al., 2007; Kang and Lin, 2007; Huang et al., 2008; Kumar et al., 2009; Dibike et al., In press; Zhang et al., 2011). Detecting the period of observed data is used to be as a meaningful method to study climatic and hydrological changes. Both time series of adequate length and mathematical tools are necessary to optimize the results. Thus, the long-term time series are important because they permit to study climatic changes using the real data or to reconstruct longer series using different mathematical tools. In all factors of climatic and hydrological domain, the alteration of precipitation with increasing and decreasing trend *Correspondence to: Jun Xia, Key Laboratory of Water Cycle and Related Land Surface Processes, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China. E-mail: [email protected] † Present address: State Key Laboratory of Water Resources and Hydropower Engineering, Wuhan University, 430072, China Copyright © 2012 John Wiley & Sons, Ltd. has attracted more attention as the changing climate has caused several catastrophic ﬂood and drought events which already caused much disaster around the world to both nature and human beings (Karl and Knight, 1997; Zhang et al., 2009; Shen et al., 2011). Thus, it is indispensable to study the hydrologic regime especially the alternation of precipitation for a long-term data series. It would be remiss not to mention some notable works that have emphasized the changes in climatic and hydrological process especially precipitation (Lettenmaier et al., 1994; Hennessy et al., 1999; Luis et al., 2000). Lettenmaier et al. (1994) examined spatial patterns of mean streamﬂow and precipitation in the continental United States and found a signiﬁcant upward trend in both during September–December. Hennessy et al. (1999) used the Kendall tau test to detect trends in seasonal and total rainfall for 379 stations spread over the whole of Australia from 1910 to 1995 and compared the results using linear regression tests. They concluded that changes are signiﬁcant for total rainfall but show insigniﬁcant changes during different seasons. Luis et al. (2000) used non-parametric Mann–Kendall (M–K) and Spearman’s rho tests to evaluate the trends in rainfall for 97 rain gauge stations in the region of Valencia (eastern Spain). They observed a decrease in annual rainfall and showed a signiﬁcant increase in the interannual variability. Moreover, discrete wavelet transform (DWT) analysis recently was also used to detect the period of hydro-meteorological factors. Partal VARIATION ANALYSIS OF PRECIPITATION IN BEIJING AREA and Küçük (2006) studied the precipitation of 64 years in Marmara region, Turkey through using DWT analysis. The data set was decomposed into series of approximation and ﬁve details following Mallat’s algorithm, and they found 16 years periodic component (DW4) was the effective component and was responsible for producing a real trend founded on the data. Xing et al. (2010) found that it is a fast and effective way to detect and compare the similarity of different hydrological series by using the DWT analysis. There are lots of studies to focus upon the time series trends analysis of precipitation (Widmann and Schar, 1997; Partal and Kahya, 2006; Mekis and Vincent, 2011) Widmann and Schar (1997) detected 113 series of the long-term record during 1901–1990 spread all over Switzerland. They found a wintertime increase in precipitation by up to 30 percent per 100 years in the western and northern parts of Switzerland, and this trend is statistically signiﬁcant at the 90 percent level. In most parts of southeastern Switzerland, winter precipitation increased as well but by a smaller rate and at a slightly lower statistical signiﬁcance level. Partal and Küçük (2006) determined trends in the long-term annual mean and monthly total precipitation series (1929–1993) with 96 stations across Turkey using non-parametric methods. They found that a noticeable decrease in the annual mean precipitation was observed mostly in western and southern Turkey. Beijing is the political, economic and cultural centre of China, with a history of over 2500 years and the residents of more than 20 million. Several researches have investigated the trend of precipitation in Beijing area. Li et al. (2010) studied the precipitation variation in Beijing area, and they considered that the trend of precipitation in Beijing area is linearly declined in the latest 51 years, and some other studies also came to the same conclusions (Xie et al., 2000; Kuang et al., 2000). For studying the periodic ﬂuctuation of precipitation in Beijing area, Li et al. (2010) found three periods of 7-year, 14-year and 26-year as main periods of 51-year data series. Xie et al. (2000) studied wavelet characteristics of climate change in Beijing area since the 19th century, and they concluded that in summer, 35-year period of precipitation was found to be a main period in around 150 years data series, and in spring and autumn, the periodicity of around 20 years controlled the whole series. Kuang et al. (2000) determined three dominating periods of precipitation, 6–7 years, 21–22 years and 35–36 years, using data of 117 years in North China Plain. However, all these researches mentioned above did not consider the precipitation variations of much longer time series. It still remains unknown about the alternation of increasing and decreasing of precipitation of nearly 300 years in Beijing area. Therefore, it will have positive and far-reaching signiﬁcance to take a research on trend and ﬂuctuation of precipitation in large time scales of 300 years for well-known Beijing city with such a long history. This study aimed to detect the possible trends of annual precipitation data series by using the non-parametric methods such as the M–K test and the wavelet analysis. First, the M–K test was applied to detect the trend and turning points of longterm precipitation data series in Beijing area, and then the continuous Morlet wavelet analysis is applied to ﬁnd the periodicity oscillation strength of series and the methods of wavelet variance and wavelet power spectrum are used to Copyright © 2012 John Wiley & Sons, Ltd. 2935 Figure 1. The annual total precipitation plot of the Beijing meteorological station in China determine the main periods and its statistical signiﬁcance. Finally, we made a discrete wavelet analysis based on the scales 2m (m = 1, . . . , 7) to ﬁnd which periodicities are mainly responsible for the trend of measurement series. DATA AND METHODOLOGY Data Annual total precipitation data which cover a time series of 286 years (from 1724 to 2009) in Beijing, China were examined for trend and wavelet detection (Figure 1). The Beijing meteorological station is located at 39º 480 latitude and 116º 280 longitude which mostly controlled by semiarid climate. Data consistency check. The total precipitation data for 180 years (1724–1904) in Beijing area has been compiled by Chinese meteorologists1 (1975) according to the ‘Clear and Rain Records’ in the Forbidden City in the mid-1970s, and it extended sequence of precipitation in Beijing for 117 years. Linking up all the data with observed modern instrumental precipitation data (1841–2009) acquired from the Beijing Meteorological Station, China Meteorological Administration made precipitation data series in Beijing reaching up to 286 years. In the precipitation data series for 286 years, the compiling method of the former 117-year precipitation data is as follows: First, both series of number of hours for precipitation and quantity of precipitation of the overlapped sections are found according to the two precipitation series between the number of hours for precipitation in the ‘Clear and Rain Records’ and the quantity of precipitation from instrumental observation in Beijing station. Then, a relationship between them is found; Second, according to the established relationship, the hours for precipitation of the former 117 years based on ‘Clear and Rain Records’ are calculated into precipitation data. Hence, it can be considered that two data series of precipitation are consistent. 1 Institute of China Academy of Meteorology. 250 years Precipitation in Beijing (1724–1973). 1975. Hydrol. Process. 27, 2934–2943 (2013) 2936 M. LI ET AL. The details of speciﬁc conversion method for reconstructing annual precipitation and the relevant statistical parameters are presented in Appendix. Data standardized. Before the DWT analysis, a process of data standardized should be carried out by the following equation. x’t ¼ ðxt xÞ s (1) where x’t and xt are the standardized precipitation data and are the the observed data for t years, respectively; x and s mean value and the mean square deviation of precipitation data series, respectively. Mann–Kendall test The rank-based non-parametric M–K statistical test (Mann, 1945; Kendall, 1975) has been commonly used for trend detection (Yue and Wang, 2002; Zheng et al., 2007) because of its robustness for non-normally distributed and censored data, which are frequently encountered in hydroclimatic time series. This method deﬁnes the test statistic Z as 9 8 S1 > > > > p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ i f S > 0 > > > > VarðSÞ = < (2) Z¼ 0 if S ¼ 0 > > > > Sþ1 > > > ; : pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ i f S < 0 > VarðSÞ with S¼ n1 X n X sgn xj xi and Wavelet analysis Wavelet analysis is recently paid more attention as a useful method and is applied in wide ranges of ﬁelds in science and engineering. Compared to the Fourier transform, the wavelet transform analysis is considered to be more effective in studying non-stationary time series. Moreover, the wavelet transformation variance which expresses the energy of ﬂuctuations with distribution of time scales could distinguish the main time scales of time series. Continuous wavelet analysis. Assuming a continuous time series x(t), t 2 (1, 1), a wavelet function c() that depends on a non-dimensional time parameter can be written as t t (5) cðÞ ¼ cðt; sÞ ¼ s1=2 c s where t is time, tis the time step in which the window function is iterated and s 2 (0, 1) is the wavelet scale. c() must have zero mean and be localized in both time and Fourier space (Meyer, 1993). The continuous wavelet transform (CWT) is given by the convolution of x(t) with a scaled and translated c(), nðn 1Þð2n þ 5Þ Var ðSÞ ¼ i¼1 jiþ1 where n is the data record length and xi and xj are the sequential data values. The function sgn(x) is deﬁned as 8 if x > 0 <1 sgnðxÞ ¼ 0 if x ¼ 0 (4) : 1 i f x < 0 Equation (3) gives the standard deviation of S with correction for ties in the data, with ei denoting the number of ties of extent i. The upward or downward trend in the data is statistically signiﬁcant if |Z| > u1 a/2, where u1 a/2 is the (1 a/2) quantile of the standard normal distribution (Kendall, 1975) and when a = 0.05 and u1 a/2 = 1.96. A positive Z indicates an increasing trend in the time series, and a negative Z indicates a decreasing trend. Sequential Mann–Kendall test Also, the sequential values u(t) from the progressive analysis of the M–K test (Sneyers, 1990) were determined in order to see the change of trend with time. u(t) is similar Copyright © 2012 John Wiley & Sons, Ltd. to the Z value. Herein, u(t) is a standardized variable that has zero mean and unit standard deviation. Therefore, its sequential behavior ﬂuctuates around the zero level. u(t) is the same as the Z values that are found from the ﬁrst to the last data point (Partal and Kahya, 2006). n X iði 1Þð2i þ 5Þ i¼1 (3) 18 W ðt; sÞ ¼ s1=2 Z þ1 1 xðt Þc t t dt s (6) where * indicates the conjugate complex function. By smoothly varying both s and t, one can construct a twodimensional picture of wavelet power,|W(t, s)|2, showing the frequency (or scale) of peaks in the spectrum of x(t), and how these peaks change with time (Drago and Boxall, 2002). The lower scales refer to a compressed wavelet, and the abrupt changes or high frequency component of a signal can be identiﬁed by the lower scales. On the contrary, slowly progressing occurrences or low-frequency component of the signal should be presented by the corresponding coefﬁcients of the higher scales composed by the stretched version of a wavelet. Wavelet variance. Taking an integral to square of wavelet coefﬁcients in time domain could get the wavelet variance. Z VarðsÞ ¼ þ1 1 Wf ðs; tÞ2 dt (7) Hydrol. Process. 27, 2934–2943 (2013) VARIATION ANALYSIS OF PRECIPITATION IN BEIJING AREA 2937 Process of wavelet variance with scaletcan get a map of wavelet variance. It reﬂects the distribution of ﬂuctuation energy with time scales, thus it could determine the main time scale in a time series which is called the main period (Heng et al., 2002; Lu et al., 2010). Discrete wavelet transform. It takes amount of work when calculating the wavelet coefﬁcients at every possible scale, and it also generates lots of data. Obviously, it is more efﬁcient and will not lose accuracy when using powers of two (dyadic scales and positions) to replace calculating every scale of wavelet coefﬁcients which is called DWT that has the form of t t t nt0 sm m=2 0 ¼ s0 c (8) cm;n sm s 0 Where m and n are integers that control, respectively, the wavelet dilation (scale) and translation (time), s0 is a speciﬁed ﬁxed dilation step greater than 1 and t0 is the location parameter and must be greater than zero. From this equation, it can be seen that the translation step nt0 sm 0 depends on the dilation sm 0 . The most common (and simplest) choices for the parameters s0 and t0 are 2 and 1 (time steps), respectively. This power of two logarithmic scaling of the translations and dilations is known as dyadic grid arrangement and is the simplest and most efﬁcient case for practical purpose (Mallat, 1989). For a discrete time series xi, where xi occurs at discrete time i (i.e. here integer time steps are used), the DWT becomes Wm;n ¼ 2m=2 N 1 X xi cð2m i nÞ (9) i¼0 where Wm, n is wavelet coefﬁcient for the discrete wavelet of scale s = 2m and location t = 2mn. Therefore, the DWT informs variation in a time series at different scales and location as deﬁned above. RESULTS AND DISCUSSION Mann–Kendall results The M–K test is applied to detect the trend and turning points of precipitation data series during past 286 years in Beijing area. The values calculated by M–K test algorithm are shown by the respective solid line, dotted line in Figure 2, and the horizontal dashed lines correspond to the conﬁdence limits at the 5%-signiﬁcance level (1.96). If the solid line passes over the dashed line, it means there is a statistically signiﬁcant trend of increasing or decreasing. The value of M–K test of the whole data series is 1.98, which presents a signiﬁcant increasing trend for long-term precipitation data series in Beijing. The solid line calculated by the sequential M–K test in Figure 2 demonstrates temporal changes of the annual precipitation in Beijing area. It is clearly seen that the long-term data series is divided into two parts on 1777. Before this year, a decreasing trend of precipitation happened in Beijing area. Copyright © 2012 John Wiley & Sons, Ltd. Figure 2. Statistics values using Mann–Kendall test for Beijing precipitation data series It is beyond the threshold of signiﬁcant level U0.05 = 1.96 over the period of 1745–1767, and even beyond the signiﬁcant level U0.001 = 2.56 from 1747 to 1762. Therefore, it shows the signiﬁcant trend of precipitation in Beijing area. However, meaningful long-term trends are identiﬁed in increasing trend from in end of 1760s to 2009, and the trend always keeps increasing. For a very long time period, the trend of precipitation exhibited signiﬁcance because the values of M–K test are more than 1.96 and even more than 2.56. From the ﬁrst decade of the 21st century, it became insigniﬁcant (the values of M–K test are less than 1.96), but it is still an increasing trend according to the M–K values of precipitation in Beijing area. There are two points of intersection in 1767 and 1999 in Figure 2 which are zoomed in and shown in Figure 3 for a clear seeing. Because two intersect points are not at the thresholds of signiﬁcant level (U0.05 = 1.96) which represents as dashed lines in Figure 2 and Figure 3, therefore, both two intersect points are not turning points. Continuous-time Morlet wavelet analysis Before the wavelet transform, the preprocessing of data should be carried out by anomaly for ﬁltering one-year natural cycle of precipitation series. Multiple time scale analysis of precipitation. The continuous Morlet wavelet analysis is applied, which allows the completion of time-scale representation of localized and transient phenomena occurring at different time scales. The time-scale discrimination is achieved in a more satisfactory way than time-frequency decompositions such as the windowed Fourier method. In Figure 4, the real part time-frequency distribution of Morlet wavelet transform coefﬁcients of the whole precipitation data series in Beijing area is shown. It is a time-scale plot of the signal where the x-axis represents position along the signal (time), the y-axis represents a periodicity scale and the solid contour represents the positive phase which means the coefﬁcients of real part is greater than or equal to 0.0; and Hydrol. Process. 27, 2934–2943 (2013) 2938 M. LI ET AL. Figure 3. Mann–Kendall values of intersect points zoomed in of Figure 2 Figure 4. Real part time-frequency distribution from Morlet wavelet transform coefﬁcients Figure 5. Modulus square time-frequency distribution from Morlet wavelet transform coefﬁcients the dotted contour indicates a negative phase which means coefﬁcients of real part is less than 0.0. Figure 4 clearly shows the ﬂuctuation characteristics of the real part of the wavelet transform coefﬁcients, which reﬂects alternative variation of precipitation (more than normal or less than normal) in Beijing area. The ﬂuctuations of time scale for 21-year, 35-year and 85-year are distinctly shown in Figure 4 in which positive and negative phase happened with an alternation pattern. The modulus square time-frequency distribution through the Morlet wavelet transform is shown in Figure 5, and it represents the signal strength of characteristic time scales. There are three primary time scales of ﬂuctuation variation giving impact on precipitation in the whole time domain. The time scale of 85–95 years exhibits the strongest signal whose oscillation centre is around 1790, and it happens throughout the entire time domain. It is also the most prevailing time scale affecting the future precipitation in Beijing area. It has a stronger signal in time scale of 30–35 years whose oscillation centres are located at 1870 and 1960, respectively, and the time scale of 30–35 years happened primarily after 1830. The time scale of 20–25 years which happened mainly after 1850 is relatively weak, and the oscillation centres are in 1885 and 1980. In order to make a further illustration of the ﬂuctuation characteristics of precipitation in Beijing area, we made three cutting lines parallel to x-axis on the values mentioned above (21 years, 35 years and 85 years) respectively in Figure 4; and then the intersection points on each cutting line with Figure 4 formed a time series of real parts in Morlet wavelet coefﬁcients in the period of 1724–2009; and at last, we added the three time series in the annual precipitation anomaly series, as shown in Figure 6. It is found that, for a speciﬁc year, different time scales were studied, different conclusions whether the precipitation is over or short would be got. That is because in different time scales, the corresponding phase of the same year is diverse. In essence, small-scale alternating variations are nested into large-scale ones with macro-structure. It is also proved the necessity to study the dynamic trend of precipitation by combining with the time scales. As shown in Figure 6, the dynamic variations and mutation points of precipitation can be distinguished through the wavelet coefﬁcients line of real part of all the Copyright © 2012 John Wiley & Sons, Ltd. Hydrol. Process. 27, 2934–2943 (2013) VARIATION ANALYSIS OF PRECIPITATION IN BEIJING AREA Figure 6. The annual precipitation anomaly in Beijing area and real part variability of wavelet transformation main periods. The amplitude of 21-year time scale becomes gradually larger in the period of 1724–1860 and stabilizes in the period of 1860–2009. For 35-year time scale, the amplitude in the period of 1724–1820 is not strong and takes a signiﬁcant increase after 1820. In addition, the ﬂuctuation of 85-year time scale is stable in the entire calculation domain. However, it shows a slow trend after 1970. For the alternation of increasing and decreasing precipitation, taking 85-year time scale as an example, there exists a positive real part wavelet phase at periods of 1773–1816, 1861–1902 and 1944–1984 with sufﬁcient precipitation, while negative phase happened in the periods of 1729–1772, 1817–1860, 1903–1943 and 1985–2009 with scarce rainfall. Thus, it shows alternative characteristics with decreasing and increasing in the whole time series of precipitation in Beijing area. According to the periodical features, it can be concluded that the period of 2009–2030 is at a less precipitation one. The primary cycle analysis of precipitation time series. The wavelet variance is applied to conﬁrm the main periods of precipitation data series. The wavelet coefﬁcients calculated are substituted into Equation (6) to calculate the wavelet variance of annual precipitation anomaly series, and the plot (Figure 7) of wavelet transformation Figure 7. Wavelet transformation variance Copyright © 2012 John Wiley & Sons, Ltd. 2939 variance would be drawn with the wavelet transformation variance Var as the vertical axis and the time scales a as the horizontal axis. The wavelet transformation variance reﬂects the energy of ﬂuctuations with distribution of time scales, those with stronger energy (peaks in Figure 7) are called the main periods. Three peaks of the wavelet variance are marked in Figure 7 whose values are located at the scale of 21 years, 35 years and 85 years. The apex value of wavelet variance of scale is 85 years, and it means the strongest oscillation appears in an 85 year period, and it is the uppermost period of precipitation series throughout 286 years. In addition, the second and third periods are 35 years and 21 years, respectively. The wavelet power spectrum of precipitation data series of 286 years in Beijing area were shown in Figure 8 to detect the statistically signiﬁcance of main periodicities of 85 years, 35 years and 21 years. In Figure 8, the dashed line represents the threshold values of 95% conﬁdence level and obviously the spectral density of 33–200 years (values of frequency are from 0.005 to 0.03) surpass the dashed line (95% conﬁdence level), which proves that the periodicities of 85 years and 35 years are statistically signiﬁcant. Meanwhile, the spectral density of 20 years (value of frequency is from 0.05) surpasses the dashed line, and the main period of 21 years is also statistically signiﬁcant. In previous studies shown in Table I, 6–7 years, for the short series around 50 years, 7 years, 14 years and 26 years were found to control the whole series and for longer precipitation data series (117 years and 153 years), 21–22 years and 35–36 years were found to be the main periods in Beijing area. In fact, the results for longer time series above took a consistency with the part of results in this study (35 years and 21 years as the main period events). However, because of the different length of data series acquired for this study and others, a slightly different but signiﬁcant conclusion could be reached. From a view of longer data series of nearly 300 years, 85-year periodic event of precipitation is identiﬁed as the strongest main period controlling the whole data series in Beijing area. This has never been detected in previous Figure 8. Wavelet power spectrum of 286 years precipitation data series in Beijing area Hydrol. Process. 27, 2934–2943 (2013) 2940 M. LI ET AL. Table I. Documented results of wavelet transform analysis in Beijing (BJ) and North China Plain (NCP) Period 1880–1996 1841–1994 1950–2000 1958–2008 1724–2009 Records Methods for WT Site Main periods References 117 153 51 51 286 Mexican hat Mexican hat Mexican hat Morlet Morlet NCP BJ NCP BJ BJ 6–7 years, 21–22 years,35–36 years 35 years 15 years 7 years, 14 years, 26 years 21 years, 35 years, 85 years Kuang et al. (2000) Xie et al. (2000) Niu et al. (2004) Li et al. (2010) this study work on rainfall data series. Moreover, in the CWT analysis, a time scale of 284 years showed with an alternating positive and negative phase in Figure 4. Although it is not clear because of its non-closed contour of phase, and much longer data series is needed to make a validation, it is certain that 85-year period is the strongest one as the peak value in the ﬁgure of wavelet variance (Figure 7). Trend analysis results of wavelet components Decomposition of precipitation data. The Mallat’s algorithm is used to decompose the standardized data according to Equation (9). The whole process includes some successive ﬁltering steps. First, the standardized signal is decomposed into an approximation and accompanying detail, and then the iterating algorithm mentioned above with successive approximation is decomposed in turn, so that the standardized signal is divided into many lower resolution components (Mallat, 1989). Figure 9 presents the time series of seven components, which include 2-year mode (D1), 4-year mode (D2), 8-year mode (D3), 16-year mode (D4), 32-year mode (D5), 64-year mode (D6), 128-year mode (D7) and the approximation acquired from Equation (8) for standardized precipitation data series. The D transform coefﬁcients elaborate precipitation variations and interannual scale at different periods of the whole long-term data set. Table II is the results of M–K test of different D transform and total mean square errors (MSEs). The third column of Table II represents the D combination values of M–K test which surpass the signiﬁcant level a = 0.05(1.96) from the whole 128 models with different D components. The M–K values of A + D5 model is 5.51 which indicates the increasing trend of precipitation data series is most signiﬁcant and also the A + D2 + D3 + D5 + D7 model is the nearest D components compared to the trend of standardized series. As found by continuous Morlet wavelet transform and wavelet variation, 35-year period is the one of main periods Figure 9. Time series of (D1) 2 year mode, (D2) 4 year mode, (D3) 8 year mode, (D4) 16 year mode, (D5) 32 year mode, (D6) 64 year mode, (D7) 128 year mode and approximation mode in precipitation series Copyright © 2012 John Wiley & Sons, Ltd. Hydrol. Process. 27, 2934–2943 (2013) VARIATION ANALYSIS OF PRECIPITATION IN BEIJING AREA Table II. Mann–Kendall (M–K) test signiﬁcant results of different D transform and total mean square errors between data and different combinations of different subseries of data No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Model combinations M–K test MSE R Original time series 1.98 - - A + D3 A + D4 A + D5 A + D7 A + D2 + D5 A + D3 + D4 A + D3 + D5 A + D3 + D7 A + D4 + D5 A + D4 + D7 A + D5 + D7 A + D2 + D3 + D5 A + D2 + D5 + D6 A + D2 + D5 + D7 A + D3 + D4 + D5 A + D3 + D4 + D7 A + D3 + D5 + D7 A + D4 + D5 + D7 A + D2 + D3 + D5 + D7 A + D3 + D4 + D5 + D7 3.48 4.58 5.51 4.94 2.46 3.09 3.13 3.1 3.4 3.74 4.57 2.07 2.05 2.23 2.41 2.73 2.94 2.9 2 2.32 1.29 1.81 1.16 1.03 0.37 0.79 0.2 0.74 1.36 1.22 0.97 0.46 0.67 0.24 0.79 0.44 0.52 1.02 0.12 0.63 0.75 0.78 0.88 0.62 0.82 0.76 0.84 0.85 0.87 0.82 0.91 0.8 0.95 0.89 0.86 0.84 0.91 0.91 0.9 0.9 MSE, total mean square error; R correlation between data and D models. for the whole long-term precipitation data set. Thus, we ﬁnd that the study of series by DWT reveals which component is responsible for the whole precipitation data set. In all the 128 D combinations, total 20 D combinations 2941 are beyond the signiﬁcance positive level, and 13 combinations include D5 periodic component. Thus, we consider that D5 (32-year periodicity) periodic component is the effective one on the data and is responsible for producing a real trend found in the data. Comparison of trends In this part, we discuss the relation between the results of sequential M–K test from different DWT coefﬁcient of D combinations and standardized precipitation time series calculated from observed data set in Beijing area. The fourth and ﬁfth columns of Table II show total MSE and correlation coefﬁcients R for these correlations. It is found that MSE value of A + D2 + D3 + D5 + D7 model is 0.12 which means this model is the nearest to the M–K trend values of standardized data and it is parallel to M–K test statistics. Moreover, models of A + D3 + D5, A + D2 + D3 + D5, A + D2 + D5 + D7 which all include D5 component show the best results (less MSE values and higher R values). When every periodic component is separately focused, the value of R between D5 and standardized data reaches 0.88. All these analysis above reveal that D5 is the dominant periodic component of precipitation data series in Beijing area. Moreover, it is expected from Figure 9 that the approximation mode which is the residuals as all periodicities are removed from data series shows potential trend of the whole data set. Thus, we may conclude that A + D5 mode is responsible for the real whole precipitation data series (periodicity and trend) As shown in Figure 10, there are also good match to the values of sequential M–K test on models of A + D3 + D5, Figure 10. Sequential values of the Mann–Kendall test on original values and D transform values of Beijing precipitation series. Solid lines represent the original series and the dotted lines mean D transform series Copyright © 2012 John Wiley & Sons, Ltd. Hydrol. Process. 27, 2934–2943 (2013) 2942 M. LI ET AL. A + D2 + D5 + D7 and A + D2 + D3 + D5 + D7 and standardized data. It exhibits a new opinion with values on the wavelet coefﬁcient computed by D transform. In this ﬁgure, the sequential values of M–K test, both observed data and D components, are illustrated by the respective solid and dotted lines. Only are best match periodic components (least MSE and highest R) shown in Figure 10 for precipitation series in Beijing meteorological station. The sequential M–K test values of models A + D3 + D5, A + D2 + D5 + D7 and A + D2 + D3 + D5 + D7 are rather similar to that of observed data. The sequential M–K values of A + D5 are harmonious with one of the data. These ﬁgures together with Table II present that decadal periodic changes (30–35 years) are responsible for forming ﬂuctuating trend at Beijing. Table II and Figure 10 could clearly indicate that the A + D5 component dominated the whole long-term precipitation data series from the view of periodicity and trend, and the DWT analysis well explains the basic structure of the trend. CONCLUSIONS Non-parametric methods including the M–K test, the CWT and the DWT analysis are applied in this paper to detect the trend and periodic traits of precipitation data series of nearly 300 years in Beijing area. First, the trend of precipitation is elaborated by the M–K test (Sequential M–K test). Results show that there is an increasing trend (the value of this trend is 1.98) at the 5%-signiﬁcance level. Considering the long-term data series of about nearly 300 years, an increasing trend is found from 1786, and it is clearly signiﬁcant (beyond the 5% signiﬁcance level) from 1786 to 2003. Then, CWT and wavelet variance are used to check for signiﬁcant periodic characteristics of data series. The plots of wavelet transform coefﬁcients and ﬁgure of wavelet variance show that some periodic events affect the trend of the annual total precipitation series in Beijing area. 85-year, 35-year and 21-year periodic events are found to be the main periodic series of long-term precipitation data. Also, the results of non-parametric M–K test on different combinations of discrete wavelet components shows that D5 (32-year periodicity) is the most effective periodic component on the data and A + D5 mode is responsible for producing a real basic structure of the trend founded on the data. It is coincident with the result (35-year periodic event as one part of main periodicity) by using CWT analysis. Therefore, it is useful to explain the trend structure of the precipitation data series based on D components from DWT method. In this study, we proved that it is of importance using a longer enough data series when studying non-stationary problems in hydro-meteorological domain or climate changes. In fact, we still could not clarify if the 284-year periodic event is as one main period of the whole precipitation data series in Beijing area. Future works are Copyright © 2012 John Wiley & Sons, Ltd. needed ﬁrst to acquire further longer enough data series to clarify the 284-year periodic event and second to focus on the extreme drought or ﬂooded problems. ACKNOWLEDGEMENTS This study was supported by the External Cooperation program of the Chinese Academy of Sciences, Grant No. GJHZ1016 & GJHZ06, and the National Basic Research Program of China (2010CB428406). REFERENCES Andreo B, Jiménez P, Durán JJ, Carrasco F, Vadillo I, Mangin A. 2006. 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APPENDIX RECONSTRUCTION OF PRECIPITATION DATA SERIES The regression equations and correlation coefﬁcients of each month are constructed and calculated by using the overlapped records of 40 years during 1841–1903 from precipitation hours of ‘Clear and Rain Records’ and precipitation of observed station respectively, which Copyright © 2012 John Wiley & Sons, Ltd. covers period 1841–1843, 1845–1855, 1860–1861, 1869–1884, 1889–1890, 1892–1897 and 1899. Regression equation goes as follow, P ¼ a0 þ a1 t1 þ a2 t2 a0 represents constant values; a1 and a2 represent regression coefﬁcients; t1 means the precipitation hours of slight rainfall (slight snow); t2 means the precipitation hours of rainfall (snow). Regression coefﬁcients, correlation coefﬁcients and F-test values of reconstruction of precipitation data series for each month are shown in Table A1. The values of correlation coefﬁcients of each month are more than 0.68 except in February which value is 0.41, and F-test values are much higher than statistical signiﬁcance of F0.05 and F0.01 whose values are 3.25 and 5.24, respectively. Therefore, the restruction values of precipitation data series are reliable from the point of view of mathematical statistics. Table A1. Regression coefﬁcients, correlation coefﬁcients and F-test values of multiple regression equation for reconstruction of precipitation Month 1 2 3 4 5 6 7 8 9 10 11 12 a0 a1 a2 R F-test values 0.463 2.513 0.758 1.080 10.890 44.252 5.257 4.618 21.713 1.937 2.633 0.675 0.205 0.128 0.263 0.608 0.94 0.147 1.069 1.724 0.492 0.493 0.274 0.002 0.147 0.070 0.3 1.213 2.142 3.695 5.467 3.175 2.56 2.146 0.848 0.555 0.73 0.41 0.79 0.79 0.76 0.80 0.90 0.75 0.68 0.74 0.69 0.74 18.3 29.6 30.4 29.0 25.3 27.8 75.4 23.7 16.2 22.3 16.3 19.8 Hydrol. Process. 27, 2934–2943 (2013)

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