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Cycle Analysis Method of Tree Ring and Solar Activity Based on Variational Mode Decomposition and Hilbert Transform

Cycle Analysis Method of Tree Ring and Solar Activity Based on Variational Mode Decomposition and... Hindawi Advances in Meteorology Volume 2019, Article ID 1715673, 8 pages https://doi.org/10.1155/2019/1715673 Research Article Cycle Analysis Method of Tree Ring and Solar Activity Based on Variational Mode Decomposition and Hilbert Transform Guohui Li , Mengtao Zheng, and Hong Yang School of Electronic Engineering, Xi’an University of Posts and Telecommunications, Xi’an, Shaanxi 710121, China Correspondence should be addressed to Guohui Li; lghcd@163.com and Hong Yang; uestcyhong@163.com Received 31 October 2018; Accepted 20 December 2018; Published 8 January 2019 Academic Editor: Anthony R. Lupo Copyright © 2019 Guohui Li et al. *is 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. According to the correlation of tree ring and solar activity, the cycle analysis method based on variational mode decomposition (VMD) and Hilbert transform is proposed. Firstly, the tree ring width of cypress during 1700 to 1955 beside the Huangdi Tomb and the long-term sunspot number during 1700 to 1955, respectively, are decomposed by VMD into a series of intrinsic mode functions (IMFs). Secondly, Hilbert transformation on the decomposed IMF component is performed. *en, the marginal spectra are given and analyzed. Finally, their quasiperiodic properties are obtained as follows: the tree ring width has the quasiperiodicity of 2 to 7a, 10.8a, and 25a; the sunspot number has the quasiperiodicity of 8.3a, 9.9a, 11.1a, 52.2a, and 101.2a. *e result obtained by analyzing that quasiperiodicity shows that the main periods of tree ring width and the sunspot number in the same period are basically consistent, and tree ring width has other cycles. *is shows that sunspot activity is an important factor affecting tree ring growth, and tree ring width is influenced by other external environments. a common change cycle. In 1979, Jr et al. [9] validated the 1. Introduction phase locking degree between the change in the arid western *e influence of solar activity on climate has attracted much United States and the Hale sunspot cycle since 1700 and attention as early as the seventeenth century. Solar activity is suggested that the drought rhythm is in some manner an important controlling factor of the earth’s climate. It plays controlled by long-term solar variability directly or in- a leading role in the earth’s climate change [1]. Many research directly related to the solar magnetic effect. In 1998, Zhou et al. [10] measured the correlation coefficient between the studies[2–4]haveshownthatsolaractivityiscloselyrelatedto natural environment changes such as temperature and var- ring index and the sunspot cycle length with tree ring data ious dust storms, floods, and droughts. *erefore, it affects over 598 years and confirmed that there was a negative people’s production and life. If the prevention is not proper, it maximum correlation coefficient between sunspot cycle may bring great loss to the normal life of human beings. length and climate. In 2002, Rigozo et al. [11] analyzed the Sunspot [5] is an important parameter of solar activity. Tree sunspot number and tree ring width time series of Concadia ring [6, 7] has some characteristics such as accurate age inBrazil from 1837to1996and verified the influence ofsolar determination, strong continuity, high resolution rate, and activity on the growth of tree ring. In 2004, Miyahara et al. being easy to acquire the duplicate. It records the processes of [12] studied the content of radioactive carbon in the tree ring climate and environmental change and is an important global from 1645 to 1715. *e results showed that sunspot could climate change research data [8]. *erefore, the correlation maintain periodic polarity reversal during the minimum extension period. In 2017, Yin et al. [13] used EMD to between tree ring and sunspot activity has important sig- nificance for people’s life. analyze the stalagmite and tree ring time series of Holocene Many scholars [9–13] have researched the correlation and discussed the possible effects of solar activity on climate between solar activity and tree ring by using a lot of data. change at the millennium scale. To research the correlation between solar activity and tree Climate signal is a nonlinear and nonstationary signal. ring growth, the main way is to calculate whether there is Since both sunspot number and tree ring width belong to the 2 Advances in Meteorology climate signal [14], it is not possible to directly analyze the 2.3. Research Method periodic characteristics by using the traditional power 2.3.1. Variational Mode Decomposition. VMD is an effective spectrum estimation method or the wavelet analysis method signal decomposition method. Its overall framework is a based on the prior basis function. *e application of con- variational problem [26–28]. VMD is different from the volution in the calculation process of wavelet will lead to EMD of circular filtering. Each IMF is assumed to be a finite mutation at the end point, and the wavelet does not have self- bandwidth with a different center frequency. Its goal is to adaptability [15]. If power spectrum analysis and wavelet minimize the sum of the estimated bandwidths for each analysis are usedto calculate theperiodic characteristicsofthe IMF. *e algorithm can be divided into the structure and sunspot number and tree ring, the calculation results are not solution of the variational problem. *e detailed description ideal [16]. Empirical mode decomposition (EMD) [17, 18] is as follows [29]: provides a new idea for processing the nonlinear and non- stationary signal. When EMD is applied to other fields [19], it (1) +e Structure of Variational Problem. *e VMD algo- is found that mode decomposition may not be unique in rithm converts the signal decomposition into an iterative physics, and sometimes there are mode mixing and end- solution process of the variational problem. *e original point flying wings. *e introduction of ensemble empirical signal is decomposed into K mode functions, u (t) (k � mode decomposition (EEMD) [20] and variational mode k 1,2, . . . , K), such that the sum of the estimated bandwidths decomposition(VMD)[21]hasimprovedthemodemixingto for each mode function is minimized. Its constrained vari- some extent. Especially, VMD transforms signal de- ational model is constructed as follows: composition into the nonrecursive variational mode. Its � � number of components is also less than that of EMD and � �2 ⎧ ⎪ ⎧ ⎨ ⎫ ⎬ � j � ⎪ −jω t � � � � ⎪ min 􏽘 z 􏼔􏼒δ(t) + 􏼓u (t)􏼕e , EEMD, and it shows better noise robustness [22]. *e res- � � ⎪ t k ⎩ ⎭ � � πt u , ω 2 ⎪ { } { } k k k�1 olution of the Hilbert spectrum in the time-frequency domain (1) is much higher than that of the wavelet spectrum, so the result ⎪ K can accurately reflect the original physical characteristics of s.t. 􏽘 u � f. the system. *e marginal spectrum obtained by integral k k�1 transformation can truly reflect whether the frequency exists in the signal and avoid false periodic peaks. Zhong et al. [23] verified the superiority of the marginal spectrum in appli- (2) +e Solution of Variational Problem. (a) *e above cation through the measured vibration signal. constrained variational problem can be changed into a In view of the characteristics of VMD and Hilbert nonbinding variational problem by introducing a quadratic transform, the cyclical analysis method based on VMD and penalty factor C and Lagrange multipliers, where C guar- Hilbert transform is proposed. It is adopted to research the anteesthereconstructionaccuracyofthesignaland maintains cyclical change in the tree ring in the Huangdi Tomb and the rigor of the constraint. *e augmented Lagrange is sunspot number. *is will provide further evidence for the denoted as follows: research of the relationship between the solar activity and � � � �2 � j � climate change. −jω t � � � � L 􏼈u 􏼉, 􏼈ω 􏼉, θ􏼁 � C 􏽘 z 􏼔􏼒δ(t) + 􏼓u (t)􏼕e � � k k t k � � πt k�1 2. Data and Research Method � � � � � � � � 2.1. Tree Ring Data. Tree ring width data are derived from � � � � + f(t)− 􏽘 u (t) � k � � � the study in [24]. *e sample sequence is composed of tree � � k�1 ring width of the cypress in the Xuanyuan Huangdi Tomb (altitude 980m, slope aspect northwest, and slope 40 ) and + 􏼪θ(t), f(t)− 􏽘 u (t)􏼫. the cypress inLiu’s Tomb (altitude 930m, slopeaspect south, k ° k�1 and slope 15 ) 7km away from the Liyuan Park sampled by (2) the Meteorological Bureau of Shaanxi Province in Huan- ° ° gling Country (35 31′N, 109 12′E) in 1975. *e cypress (b) *e alternate direction multiplier method (ADMM) sample is sampled on the upper part of the forest belt, and is used to solve the saddle points of the above variational the section is taken at 30m above the ground. *e sequence n+1 n+1 n+1 problem, and then u , ω , and θ are updated alter- k k length is 256 years (from 1700 to 1955) as shown in Figure 1. nately (n represents the number of iterations), which are given by 2.2. Sunspot Data. In this paper, the annual average of 􏽢 􏽢 f(ω)− 􏽐 u 􏽢 (ω) + (θ(ω)/2) n+1 k�1 k sunspot [25] is from the Solar Influences Data Analysis u 􏽢 (ω) � , (3) k 2 1 + 2C ω− ω Center (http://www.sidc.be/silso/datafiles) during 1700 to 1955, as shown in Figure 2. *is sequence only has a length 􏼌 􏼌 􏼌 􏼌2 􏼌 􏼌 of 256 years, so it is difficult to accurately research the 􏽒 ω u 􏽢 (ω) dω 􏼌 􏼌 n+1 k ω � , (4) 􏼌 􏼌 periods of more than 100 years. *erefore, the periods of less k ∞ 2 􏼌 􏼌 􏼌 􏼌 􏽒 􏼌u 􏽢 (ω)􏼌 dω than 100 years are only researched. Advances in Meteorology 3 1700 1750 1800 1850 1900 1950 Time (years) Figure 1: Tree ring width in the Huangdi Tomb. 1700 1750 1800 1850 1900 1950 Time (years) Figure 2: Sunspot number. Step 3. Judge whether or not it meets the convergence n+1 n+1 n+1 􏽢 􏽢 ⎡ ⎣ 􏽢 ⎤ ⎦ θ (ω) � θ (ω) + τ f(ω)− 􏽘 u 􏽢 (ω) , (5) condition (6). Repeat the above steps to update parameters k�1 until the convergence stop condition is satisfied. n+1 where u 􏽢 (ω) is equal to the Wiener filtering results of the n+1 􏽢 Step 4. *e corresponding mode subsequences are obtained current remaining amount f(ω)− u 􏽢 (ω). ω is the i k i≠k according to the given mode number. center of gravity of the power spectrum of the current IMF. Inverse Fourier transform on u 􏽢 (ω) is performed to obtain the real part 􏼈u (t)􏼉. (c) Given a discriminant accuracy e > 0, the convergence 2.3.2. Hilbert Transform. Hilbert transform has the advan- condition of the stop iteration is as follows: tages of good time-frequency resolution and self-adaptation. � � In the analysis of the nonlinear and nonstationary signal, it � �2 n+1 n � � 􏽢 􏽢 �u − u � k k can effectively avoid the high-frequency interference result. 􏽘 < e. (6) � � � � � � u 􏽢 *e marginal spectrum [30] is obtained by integral trans- � � k�1 formationof the Hilbertspectrum. *eexistence ofenergy at *e specific process of the VMD algorithm is summa- a certain frequency f in the marginal spectrum indicates rized as follows: that there must be a vibration wave at that frequency. *e marginal spectrum is the essential difference between the 1 1 Step 1. Initialize 􏼈u 􏼉, 􏼈ω 􏼉, θ , and n. marginal spectrum, the power spectrum, and the Hilbert k k spectrum. *e marginal spectrum is a major breakthrough n+1 n+1 n+1 Step 2. Update the value of 􏼈u 􏽢 􏼉, 􏼈ω 􏼉, and θ in signal analysis, and its calculation process is detailed according to equations (3)–(5). in [23]. Tree ring width (mm) Sunspot number 4 Advances in Meteorology 3.2. Multiscale Analysis of the Sunspot Number. *e multi- 2.3.3. +e Proposed Periodic Analysis Method. *e cycle analysis method of tree ring and solar activity based on the scale decomposition of sunspot number is carried out by VMD, and the result is shown in Figure 5. It can seen from VMD and Hilbert transform is proposed. Figure 5 that the IMF components of 5 different time scales (1) *e original signal is decomposed into a series of and trend components have been obtained by the multiscale IMFs by VMD decomposition. Each IMF component has a regular fluc- (2) *e Hilbert transformand variance contribution rate tuation around the zero mean, so it is a stationary signal. It are calculated for the IMF components, and the can be seen from the decomposition results that sunspot marginal spectrum is given number has multiple time scales. *e IMF5 component has (3) According to the marginal spectrum, the quasipe- the largest amplitude and the most regular fluctuation, in riodicity of each modality is extracted which basically the maximum or minimum value keeps appearing once every 10 years. *e amplitudes of the other components are sequentially weakened. *eir frequency 3. Data Analysis increases, and the period decreases gradually. 3.1. Multiscale Analysis of the Tree Ring Width. *e multi- Hilbert transform is performed on the IMF components decomposed by VMD. *e marginal spectrum of sunspot scale decomposition of tree ring width is carried out by VMD, and the result is shown in Figure 3. It can be seen number is obtained and normalized, as shown in Figure 6. from Figure 3 that the IMF components of 9 different time *e normalized marginal spectrum peaks of the sunspot scales and trend components have been obtained by the number are all over 0.2. *e corresponding frequencies of multiscale decomposition. Each IMF component has a the main period peak are 0.0096, 0.0192, 0.0923, 0.1006, and regular fluctuation around the zero mean, so it is a stationary 0.1198 from small to large. *eir cycles are 102.1a, 52.2a, signal. It can be seen from the decomposition results that 10.8a, 9.9a, and 8.6a, respectively. *e variance contribution rate, correlation coefficient tree rings have multiple time scales. *e IMF1 component has the minimum frequency, namely, the maximum cycle. calculation, and sorting condition of IMF components of the sunspot number are shown in Table 2. According to Table 2, *e IMF2 component has the largest amplitude and the most regular fluctuation, in which basically the maximum or 10.8a has the largest variance contribution rate up to 54.53%, minimum value keeps appearing once every 10 years. IMF3 and the correlation with the original data is also as high as and IMF2 are similar in fluctuation, and the latter 4 com- 0.74. Itis significant correlation. *e periodic components of ponents have smaller amplitude. Generally, IMF1–IMF9 8.8a and 9.9a might belong to the same cycle of 10.8a. Al- frequencies are increasingly larger, while the cycles are though 52.2a and 102.1a are only slightly related to the decreasing in a proper sequence. *e amplitude of vibration sunspot number original signal, the variance contribution remained basically unchanged from 1940s to 1950s, which rates are 6.54% and 10.15%. It indicates that the low- indicates that the ten-year period is the dormant period of frequency components are also very important and can- not be ignored. local tree growth. Hilbert transform is performed on the IMF components decomposed by VMD. *e marginal spectrum of tree ring 4. Discussion width is obtained. In order to highlight the sharpness of the periodic peak, the marginal spectrum shown in Figure 4 is *e tree ring width of Huangdi Mausoleum in Shaanxi normalized. *e peak of the marginal spectrum corresponds Province and sunspot number have nonlinear and non- to the average period of each IMF component. *e re- stationary characteristics, so the selection of the method is ciprocal of abscissa frequency at the peak point is the cycle. very important for the periodic analysis. In this paper, the *e normalized marginal spectrum peaks are all over 0.2. It VMD and Hilbert transform are adopted to make analysis of shows that the periodic components of each mode are real. tree ring width and sunspot number in the same period, *e corresponding frequencies of the main period peak are which obtained the multiscale time-series change rule. 0.039, 0.102, 0.137, 0.174, 0.205, 0.283, 0.346, 0.410, and *e tree ring width of the Huangdi Tomb in Shaanxi 0.447 from small to large. *eir cycles are 25.6a, 9.8a, 7.3a, Province is decomposed into 9 IMF components, and there 5.8a, 4.9a, 3.5a, 2.9a, 2.4a, and 2.2a, respectively. exists quasiperiodicity of about 25.6a, 10a, and 2–7a. *e *e variance contribution rate, correlation coefficient sunspot number in the same period is decomposed into 5 calculation, and sorting condition of IMF components of IMF components, and there exists quasiperiodicity of about tree ring width are shown in Table 1. According to Table 1, 102.2a, 52.2a, and 11a. 9.8a and 7.3a have the largest variance contribution. *ey are *e 11a cycle obtained by multiscale analysis of sunspot real correlation. Next is the quasiperiodicity of 25.6a, and its number by VMD and Hilbert transform is the famous variance contribution rate is more than 10%. *e remaining Schwabe [31] cycle, and the variance contribution rate is as components are low-frequency quasiperiodic components high as 57.91%. It is the most significant. *e average cycle of lower than 7a. It shows the accuracy of the VMD de- IMF5 is 102.1a and close to the Gleissberg cycle [32]. *e composition. Although the correlation coefficient between average cycle of IMF4 is 52.2a. *is is called the double Hale these low-frequency components and the original signal is cycle. *e 9.9a and 8.3a cycle components might belong to less than 0.3, it is also an indispensable part of the multiscale the same cycle of 11a. Le and Wang [33] obtained cycles of information analysis of the original tree ring width. 11a, 53a, and 101a by analyzing the sunspot number through Advances in Meteorology 5 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 Time (years) Figure 3: IMF components of tree ring width. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Frequency (Hz) Figure 4: Normalized marginal spectrum of the tree ring. Table 1: Variance contribution rate, correlation coefficient calculation, and sorting condition of IMF components of tree ring width. Component IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8 IMF9 Res Quasicycles/year 25.6 9.8 7.3 5.8 4.9 3.5 2.9 2.4 2.2 Variance contribution rate (%) 10.01 15.14 15.56 8.88 6.61 5.52 3.95 7.86 5.32 21.17 Correlation coefficient 0.36 0.41 0.39 0.33 0.29 0.26 0.22 0.27 0.26 Sequencing 3 2 1 4 6 8 9 5 7 the wavelet. *e research results are more accurate than the original sequence, and the variance contribution rate and previous research results [33]. correlation are the highest. *e 25a cycle of tree ring width in the Huangdi Tomb is Moreover, Liu et al. [34] obtained an average period of calculated by VMD, and the 2–7a cycles of the tree ring are precipitation with 10a in Shaanxi Province by wavelet. Tang generally regarded as related to the El Niño-Southern Os- [35] showed the correlation between precipitation in cillation (ENSO), so there is no further discussion here. For Shaanxi Province and sunspot by EEMD. Generally, the tree the highly correlated 10a quasiperiodic-mode component, it ring reflects the variations of local precipitation [36]. Pre- is the focus of research. It is basically consistent with the cipitation is affected by sunspot. *erefore, the tree growth is Schwabe cycle. *ey are the main scale components of the indirectly influenced by sunspot. IMF9 IMF8 IMF7 IMF6 IMF5 IMF4 IMF3 IMF2 IMF1 Res Normalized marginal spectrum 6 Advances in Meteorology 1700 1750 1800 1850 1900 1950 −20 1700 1750 1800 1850 1900 1950 −20 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 Time (years) Figure 5: IMF components of sunspot number. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Frequency (Hz) Figure 6: Normalized marginal spectrum of the sunspot number. in periodic analysis for tree ring width and sunspot Table 2: Variance contribution rate, correlation coefficient cal- culation, and sorting condition of IMF components of the sunspot number in the same period, which provides a new method number. for characteristics analysis of the nonlinear climate sig- nal. *rough analytical comparisons, the proposed Component IMF1 IMF2 IMF3 IMF4 IMF5 Res method demonstrated its superiority and the following Quasicycles/year 102.1 52.2 9.9 8.3 10.8 contributions: Variance contribution 10.15 6.54 16.41 6.57 54.53 5.84 rate (%) (1) Tree ring width of the Huangdi Tomb in Shaanxi Correlation coefficient 0.35 0.29 0.47 0.28 0.74 Province from 1700 to 1955 has the quasiperiodicity Sequencing 3 5 2 4 1 of 25.6a, 10a, and 2–7a. *e sunspot number from 1700 to 1955 has the quasiperiodicity of 102.1a, 52.2a, and 11a. 5. Conclusions (2) *e major cycles of tree ring width of the Huangdi In this paper, the cycle analysis method based on VMD Tomb in Shaanxi Province and sunspot number have and Hilbert transform is proposed. *is method is used the consistency for 10a, which has provided further IMF5 IMF4 IMF3 IMF2 IMF1 Res Normalized marginal spectrum Advances in Meteorology 7 [7] B. Feng and Y. B. Han, “*e possible effects of solar activities evidence for the tree growth being driven by solar on tree ring change of 500 years of platycladus orientalis in activity. Huangdi Mausoleum,” Science China Physics, Mechanics & (3) Besides the 10a cycle of the tree ring width of the Astronomy, vol. 39, no. 5, pp. 776–784, 2009. Huangdi Tomb, there are other cycles different [8] D. H. Yang and X. X. Yang, “Study on cause of formation in from the sunspot number. 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Cycle Analysis Method of Tree Ring and Solar Activity Based on Variational Mode Decomposition and Hilbert Transform

Li, Guohui; Zheng, Mengtao; Yang, Hong
Advances in Meteorology , Volume 2019: 8 – Jan 8, 2019
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Copyright © 2019 Guohui Li 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.
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10.1155/2019/1715673
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Hindawi Advances in Meteorology Volume 2019, Article ID 1715673, 8 pages https://doi.org/10.1155/2019/1715673 Research Article Cycle Analysis Method of Tree Ring and Solar Activity Based on Variational Mode Decomposition and Hilbert Transform Guohui Li , Mengtao Zheng, and Hong Yang School of Electronic Engineering, Xi’an University of Posts and Telecommunications, Xi’an, Shaanxi 710121, China Correspondence should be addressed to Guohui Li; lghcd@163.com and Hong Yang; uestcyhong@163.com Received 31 October 2018; Accepted 20 December 2018; Published 8 January 2019 Academic Editor: Anthony R. Lupo Copyright © 2019 Guohui Li et al. *is 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. According to the correlation of tree ring and solar activity, the cycle analysis method based on variational mode decomposition (VMD) and Hilbert transform is proposed. Firstly, the tree ring width of cypress during 1700 to 1955 beside the Huangdi Tomb and the long-term sunspot number during 1700 to 1955, respectively, are decomposed by VMD into a series of intrinsic mode functions (IMFs). Secondly, Hilbert transformation on the decomposed IMF component is performed. *en, the marginal spectra are given and analyzed. Finally, their quasiperiodic properties are obtained as follows: the tree ring width has the quasiperiodicity of 2 to 7a, 10.8a, and 25a; the sunspot number has the quasiperiodicity of 8.3a, 9.9a, 11.1a, 52.2a, and 101.2a. *e result obtained by analyzing that quasiperiodicity shows that the main periods of tree ring width and the sunspot number in the same period are basically consistent, and tree ring width has other cycles. *is shows that sunspot activity is an important factor affecting tree ring growth, and tree ring width is influenced by other external environments. a common change cycle. In 1979, Jr et al. [9] validated the 1. Introduction phase locking degree between the change in the arid western *e influence of solar activity on climate has attracted much United States and the Hale sunspot cycle since 1700 and attention as early as the seventeenth century. Solar activity is suggested that the drought rhythm is in some manner an important controlling factor of the earth’s climate. It plays controlled by long-term solar variability directly or in- a leading role in the earth’s climate change [1]. Many research directly related to the solar magnetic effect. In 1998, Zhou et al. [10] measured the correlation coefficient between the studies[2–4]haveshownthatsolaractivityiscloselyrelatedto natural environment changes such as temperature and var- ring index and the sunspot cycle length with tree ring data ious dust storms, floods, and droughts. *erefore, it affects over 598 years and confirmed that there was a negative people’s production and life. If the prevention is not proper, it maximum correlation coefficient between sunspot cycle may bring great loss to the normal life of human beings. length and climate. In 2002, Rigozo et al. [11] analyzed the Sunspot [5] is an important parameter of solar activity. Tree sunspot number and tree ring width time series of Concadia ring [6, 7] has some characteristics such as accurate age inBrazil from 1837to1996and verified the influence ofsolar determination, strong continuity, high resolution rate, and activity on the growth of tree ring. In 2004, Miyahara et al. being easy to acquire the duplicate. It records the processes of [12] studied the content of radioactive carbon in the tree ring climate and environmental change and is an important global from 1645 to 1715. *e results showed that sunspot could climate change research data [8]. *erefore, the correlation maintain periodic polarity reversal during the minimum extension period. In 2017, Yin et al. [13] used EMD to between tree ring and sunspot activity has important sig- nificance for people’s life. analyze the stalagmite and tree ring time series of Holocene Many scholars [9–13] have researched the correlation and discussed the possible effects of solar activity on climate between solar activity and tree ring by using a lot of data. change at the millennium scale. To research the correlation between solar activity and tree Climate signal is a nonlinear and nonstationary signal. ring growth, the main way is to calculate whether there is Since both sunspot number and tree ring width belong to the 2 Advances in Meteorology climate signal [14], it is not possible to directly analyze the 2.3. Research Method periodic characteristics by using the traditional power 2.3.1. Variational Mode Decomposition. VMD is an effective spectrum estimation method or the wavelet analysis method signal decomposition method. Its overall framework is a based on the prior basis function. *e application of con- variational problem [26–28]. VMD is different from the volution in the calculation process of wavelet will lead to EMD of circular filtering. Each IMF is assumed to be a finite mutation at the end point, and the wavelet does not have self- bandwidth with a different center frequency. Its goal is to adaptability [15]. If power spectrum analysis and wavelet minimize the sum of the estimated bandwidths for each analysis are usedto calculate theperiodic characteristicsofthe IMF. *e algorithm can be divided into the structure and sunspot number and tree ring, the calculation results are not solution of the variational problem. *e detailed description ideal [16]. Empirical mode decomposition (EMD) [17, 18] is as follows [29]: provides a new idea for processing the nonlinear and non- stationary signal. When EMD is applied to other fields [19], it (1) +e Structure of Variational Problem. *e VMD algo- is found that mode decomposition may not be unique in rithm converts the signal decomposition into an iterative physics, and sometimes there are mode mixing and end- solution process of the variational problem. *e original point flying wings. *e introduction of ensemble empirical signal is decomposed into K mode functions, u (t) (k � mode decomposition (EEMD) [20] and variational mode k 1,2, . . . , K), such that the sum of the estimated bandwidths decomposition(VMD)[21]hasimprovedthemodemixingto for each mode function is minimized. Its constrained vari- some extent. Especially, VMD transforms signal de- ational model is constructed as follows: composition into the nonrecursive variational mode. Its � � number of components is also less than that of EMD and � �2 ⎧ ⎪ ⎧ ⎨ ⎫ ⎬ � j � ⎪ −jω t � � � � ⎪ min 􏽘 z 􏼔􏼒δ(t) + 􏼓u (t)􏼕e , EEMD, and it shows better noise robustness [22]. *e res- � � ⎪ t k ⎩ ⎭ � � πt u , ω 2 ⎪ { } { } k k k�1 olution of the Hilbert spectrum in the time-frequency domain (1) is much higher than that of the wavelet spectrum, so the result ⎪ K can accurately reflect the original physical characteristics of s.t. 􏽘 u � f. the system. *e marginal spectrum obtained by integral k k�1 transformation can truly reflect whether the frequency exists in the signal and avoid false periodic peaks. Zhong et al. [23] verified the superiority of the marginal spectrum in appli- (2) +e Solution of Variational Problem. (a) *e above cation through the measured vibration signal. constrained variational problem can be changed into a In view of the characteristics of VMD and Hilbert nonbinding variational problem by introducing a quadratic transform, the cyclical analysis method based on VMD and penalty factor C and Lagrange multipliers, where C guar- Hilbert transform is proposed. It is adopted to research the anteesthereconstructionaccuracyofthesignaland maintains cyclical change in the tree ring in the Huangdi Tomb and the rigor of the constraint. *e augmented Lagrange is sunspot number. *is will provide further evidence for the denoted as follows: research of the relationship between the solar activity and � � � �2 � j � climate change. −jω t � � � � L 􏼈u 􏼉, 􏼈ω 􏼉, θ􏼁 � C 􏽘 z 􏼔􏼒δ(t) + 􏼓u (t)􏼕e � � k k t k � � πt k�1 2. Data and Research Method � � � � � � � � 2.1. Tree Ring Data. Tree ring width data are derived from � � � � + f(t)− 􏽘 u (t) � k � � � the study in [24]. *e sample sequence is composed of tree � � k�1 ring width of the cypress in the Xuanyuan Huangdi Tomb (altitude 980m, slope aspect northwest, and slope 40 ) and + 􏼪θ(t), f(t)− 􏽘 u (t)􏼫. the cypress inLiu’s Tomb (altitude 930m, slopeaspect south, k ° k�1 and slope 15 ) 7km away from the Liyuan Park sampled by (2) the Meteorological Bureau of Shaanxi Province in Huan- ° ° gling Country (35 31′N, 109 12′E) in 1975. *e cypress (b) *e alternate direction multiplier method (ADMM) sample is sampled on the upper part of the forest belt, and is used to solve the saddle points of the above variational the section is taken at 30m above the ground. *e sequence n+1 n+1 n+1 problem, and then u , ω , and θ are updated alter- k k length is 256 years (from 1700 to 1955) as shown in Figure 1. nately (n represents the number of iterations), which are given by 2.2. Sunspot Data. In this paper, the annual average of 􏽢 􏽢 f(ω)− 􏽐 u 􏽢 (ω) + (θ(ω)/2) n+1 k�1 k sunspot [25] is from the Solar Influences Data Analysis u 􏽢 (ω) � , (3) k 2 1 + 2C ω− ω Center (http://www.sidc.be/silso/datafiles) during 1700 to 1955, as shown in Figure 2. *is sequence only has a length 􏼌 􏼌 􏼌 􏼌2 􏼌 􏼌 of 256 years, so it is difficult to accurately research the 􏽒 ω u 􏽢 (ω) dω 􏼌 􏼌 n+1 k ω � , (4) 􏼌 􏼌 periods of more than 100 years. *erefore, the periods of less k ∞ 2 􏼌 􏼌 􏼌 􏼌 􏽒 􏼌u 􏽢 (ω)􏼌 dω than 100 years are only researched. Advances in Meteorology 3 1700 1750 1800 1850 1900 1950 Time (years) Figure 1: Tree ring width in the Huangdi Tomb. 1700 1750 1800 1850 1900 1950 Time (years) Figure 2: Sunspot number. Step 3. Judge whether or not it meets the convergence n+1 n+1 n+1 􏽢 􏽢 ⎡ ⎣ 􏽢 ⎤ ⎦ θ (ω) � θ (ω) + τ f(ω)− 􏽘 u 􏽢 (ω) , (5) condition (6). Repeat the above steps to update parameters k�1 until the convergence stop condition is satisfied. n+1 where u 􏽢 (ω) is equal to the Wiener filtering results of the n+1 􏽢 Step 4. *e corresponding mode subsequences are obtained current remaining amount f(ω)− u 􏽢 (ω). ω is the i k i≠k according to the given mode number. center of gravity of the power spectrum of the current IMF. Inverse Fourier transform on u 􏽢 (ω) is performed to obtain the real part 􏼈u (t)􏼉. (c) Given a discriminant accuracy e > 0, the convergence 2.3.2. Hilbert Transform. Hilbert transform has the advan- condition of the stop iteration is as follows: tages of good time-frequency resolution and self-adaptation. � � In the analysis of the nonlinear and nonstationary signal, it � �2 n+1 n � � 􏽢 􏽢 �u − u � k k can effectively avoid the high-frequency interference result. 􏽘 < e. (6) � � � � � � u 􏽢 *e marginal spectrum [30] is obtained by integral trans- � � k�1 formationof the Hilbertspectrum. *eexistence ofenergy at *e specific process of the VMD algorithm is summa- a certain frequency f in the marginal spectrum indicates rized as follows: that there must be a vibration wave at that frequency. *e marginal spectrum is the essential difference between the 1 1 Step 1. Initialize 􏼈u 􏼉, 􏼈ω 􏼉, θ , and n. marginal spectrum, the power spectrum, and the Hilbert k k spectrum. *e marginal spectrum is a major breakthrough n+1 n+1 n+1 Step 2. Update the value of 􏼈u 􏽢 􏼉, 􏼈ω 􏼉, and θ in signal analysis, and its calculation process is detailed according to equations (3)–(5). in [23]. Tree ring width (mm) Sunspot number 4 Advances in Meteorology 3.2. Multiscale Analysis of the Sunspot Number. *e multi- 2.3.3. +e Proposed Periodic Analysis Method. *e cycle analysis method of tree ring and solar activity based on the scale decomposition of sunspot number is carried out by VMD, and the result is shown in Figure 5. It can seen from VMD and Hilbert transform is proposed. Figure 5 that the IMF components of 5 different time scales (1) *e original signal is decomposed into a series of and trend components have been obtained by the multiscale IMFs by VMD decomposition. Each IMF component has a regular fluc- (2) *e Hilbert transformand variance contribution rate tuation around the zero mean, so it is a stationary signal. It are calculated for the IMF components, and the can be seen from the decomposition results that sunspot marginal spectrum is given number has multiple time scales. *e IMF5 component has (3) According to the marginal spectrum, the quasipe- the largest amplitude and the most regular fluctuation, in riodicity of each modality is extracted which basically the maximum or minimum value keeps appearing once every 10 years. *e amplitudes of the other components are sequentially weakened. *eir frequency 3. Data Analysis increases, and the period decreases gradually. 3.1. Multiscale Analysis of the Tree Ring Width. *e multi- Hilbert transform is performed on the IMF components decomposed by VMD. *e marginal spectrum of sunspot scale decomposition of tree ring width is carried out by VMD, and the result is shown in Figure 3. It can be seen number is obtained and normalized, as shown in Figure 6. from Figure 3 that the IMF components of 9 different time *e normalized marginal spectrum peaks of the sunspot scales and trend components have been obtained by the number are all over 0.2. *e corresponding frequencies of multiscale decomposition. Each IMF component has a the main period peak are 0.0096, 0.0192, 0.0923, 0.1006, and regular fluctuation around the zero mean, so it is a stationary 0.1198 from small to large. *eir cycles are 102.1a, 52.2a, signal. It can be seen from the decomposition results that 10.8a, 9.9a, and 8.6a, respectively. *e variance contribution rate, correlation coefficient tree rings have multiple time scales. *e IMF1 component has the minimum frequency, namely, the maximum cycle. calculation, and sorting condition of IMF components of the sunspot number are shown in Table 2. According to Table 2, *e IMF2 component has the largest amplitude and the most regular fluctuation, in which basically the maximum or 10.8a has the largest variance contribution rate up to 54.53%, minimum value keeps appearing once every 10 years. IMF3 and the correlation with the original data is also as high as and IMF2 are similar in fluctuation, and the latter 4 com- 0.74. Itis significant correlation. *e periodic components of ponents have smaller amplitude. Generally, IMF1–IMF9 8.8a and 9.9a might belong to the same cycle of 10.8a. Al- frequencies are increasingly larger, while the cycles are though 52.2a and 102.1a are only slightly related to the decreasing in a proper sequence. *e amplitude of vibration sunspot number original signal, the variance contribution remained basically unchanged from 1940s to 1950s, which rates are 6.54% and 10.15%. It indicates that the low- indicates that the ten-year period is the dormant period of frequency components are also very important and can- not be ignored. local tree growth. Hilbert transform is performed on the IMF components decomposed by VMD. *e marginal spectrum of tree ring 4. Discussion width is obtained. In order to highlight the sharpness of the periodic peak, the marginal spectrum shown in Figure 4 is *e tree ring width of Huangdi Mausoleum in Shaanxi normalized. *e peak of the marginal spectrum corresponds Province and sunspot number have nonlinear and non- to the average period of each IMF component. *e re- stationary characteristics, so the selection of the method is ciprocal of abscissa frequency at the peak point is the cycle. very important for the periodic analysis. In this paper, the *e normalized marginal spectrum peaks are all over 0.2. It VMD and Hilbert transform are adopted to make analysis of shows that the periodic components of each mode are real. tree ring width and sunspot number in the same period, *e corresponding frequencies of the main period peak are which obtained the multiscale time-series change rule. 0.039, 0.102, 0.137, 0.174, 0.205, 0.283, 0.346, 0.410, and *e tree ring width of the Huangdi Tomb in Shaanxi 0.447 from small to large. *eir cycles are 25.6a, 9.8a, 7.3a, Province is decomposed into 9 IMF components, and there 5.8a, 4.9a, 3.5a, 2.9a, 2.4a, and 2.2a, respectively. exists quasiperiodicity of about 25.6a, 10a, and 2–7a. *e *e variance contribution rate, correlation coefficient sunspot number in the same period is decomposed into 5 calculation, and sorting condition of IMF components of IMF components, and there exists quasiperiodicity of about tree ring width are shown in Table 1. According to Table 1, 102.2a, 52.2a, and 11a. 9.8a and 7.3a have the largest variance contribution. *ey are *e 11a cycle obtained by multiscale analysis of sunspot real correlation. Next is the quasiperiodicity of 25.6a, and its number by VMD and Hilbert transform is the famous variance contribution rate is more than 10%. *e remaining Schwabe [31] cycle, and the variance contribution rate is as components are low-frequency quasiperiodic components high as 57.91%. It is the most significant. *e average cycle of lower than 7a. It shows the accuracy of the VMD de- IMF5 is 102.1a and close to the Gleissberg cycle [32]. *e composition. Although the correlation coefficient between average cycle of IMF4 is 52.2a. *is is called the double Hale these low-frequency components and the original signal is cycle. *e 9.9a and 8.3a cycle components might belong to less than 0.3, it is also an indispensable part of the multiscale the same cycle of 11a. Le and Wang [33] obtained cycles of information analysis of the original tree ring width. 11a, 53a, and 101a by analyzing the sunspot number through Advances in Meteorology 5 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 Time (years) Figure 3: IMF components of tree ring width. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Frequency (Hz) Figure 4: Normalized marginal spectrum of the tree ring. Table 1: Variance contribution rate, correlation coefficient calculation, and sorting condition of IMF components of tree ring width. Component IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8 IMF9 Res Quasicycles/year 25.6 9.8 7.3 5.8 4.9 3.5 2.9 2.4 2.2 Variance contribution rate (%) 10.01 15.14 15.56 8.88 6.61 5.52 3.95 7.86 5.32 21.17 Correlation coefficient 0.36 0.41 0.39 0.33 0.29 0.26 0.22 0.27 0.26 Sequencing 3 2 1 4 6 8 9 5 7 the wavelet. *e research results are more accurate than the original sequence, and the variance contribution rate and previous research results [33]. correlation are the highest. *e 25a cycle of tree ring width in the Huangdi Tomb is Moreover, Liu et al. [34] obtained an average period of calculated by VMD, and the 2–7a cycles of the tree ring are precipitation with 10a in Shaanxi Province by wavelet. Tang generally regarded as related to the El Niño-Southern Os- [35] showed the correlation between precipitation in cillation (ENSO), so there is no further discussion here. For Shaanxi Province and sunspot by EEMD. Generally, the tree the highly correlated 10a quasiperiodic-mode component, it ring reflects the variations of local precipitation [36]. Pre- is the focus of research. It is basically consistent with the cipitation is affected by sunspot. *erefore, the tree growth is Schwabe cycle. *ey are the main scale components of the indirectly influenced by sunspot. IMF9 IMF8 IMF7 IMF6 IMF5 IMF4 IMF3 IMF2 IMF1 Res Normalized marginal spectrum 6 Advances in Meteorology 1700 1750 1800 1850 1900 1950 −20 1700 1750 1800 1850 1900 1950 −20 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 −50 1700 1750 1800 1850 1900 1950 Time (years) Figure 5: IMF components of sunspot number. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Frequency (Hz) Figure 6: Normalized marginal spectrum of the sunspot number. in periodic analysis for tree ring width and sunspot Table 2: Variance contribution rate, correlation coefficient cal- culation, and sorting condition of IMF components of the sunspot number in the same period, which provides a new method number. for characteristics analysis of the nonlinear climate sig- nal. *rough analytical comparisons, the proposed Component IMF1 IMF2 IMF3 IMF4 IMF5 Res method demonstrated its superiority and the following Quasicycles/year 102.1 52.2 9.9 8.3 10.8 contributions: Variance contribution 10.15 6.54 16.41 6.57 54.53 5.84 rate (%) (1) Tree ring width of the Huangdi Tomb in Shaanxi Correlation coefficient 0.35 0.29 0.47 0.28 0.74 Province from 1700 to 1955 has the quasiperiodicity Sequencing 3 5 2 4 1 of 25.6a, 10a, and 2–7a. *e sunspot number from 1700 to 1955 has the quasiperiodicity of 102.1a, 52.2a, and 11a. 5. Conclusions (2) *e major cycles of tree ring width of the Huangdi In this paper, the cycle analysis method based on VMD Tomb in Shaanxi Province and sunspot number have and Hilbert transform is proposed. *is method is used the consistency for 10a, which has provided further IMF5 IMF4 IMF3 IMF2 IMF1 Res Normalized marginal spectrum Advances in Meteorology 7 [7] B. Feng and Y. B. Han, “*e possible effects of solar activities evidence for the tree growth being driven by solar on tree ring change of 500 years of platycladus orientalis in activity. Huangdi Mausoleum,” Science China Physics, Mechanics & (3) Besides the 10a cycle of the tree ring width of the Astronomy, vol. 39, no. 5, pp. 776–784, 2009. Huangdi Tomb, there are other cycles different [8] D. H. Yang and X. X. Yang, “Study on cause of formation in from the sunspot number. It indicates that, in ad- earth’s climatic changes,” Progress in Geophysics, vol. 28, no. 4, dition to the solar activity, the tree ring growth is also pp. 1666–1677, 2013. influenced by other factors. [9] J. M. M. Jr, C. W. Stockton, and D. M. Meko, Evidence of a 22- Year Rhythm of Drought in the Western United States Related (4) *e proposed method is a cycle analysis method to the Hale Solar Cycle Since the 17th Century, Springer, which is suitable for nonlinear and nonstationary Berlin, Germany, 1979. signals. It can calculate the real and effective signal [10] K. Zhou, C. J. Rp, and C. J. Butler, “A statistical study of the period. *e proposed method is more accurate than relationship between the solar cycle length and tree-ring index the traditional spectral analysis method. values,” Journal of Atmospheric and Solar-Terrestrial Physics, vol. 60, no. 18, pp. 1711–1718, 1998. [11] N. R. Rigozo, D. J. R. Nordemann, E. Echer, A. Zanandrea, Data Availability and W. D. Gonzalez, “Solar variability effects studied by tree- *e annual average of the sunspot number time series in this ring data wavelet analysis,” Advances in Space Research, vol. 29, no. 12, pp. 1985–1988, 2002. paper comes from the Solar Influences Data Analysis Center [12] H. Miyahara, K. Masuda, Y. Muraki, H. Furuzawa, H. Menjo, of the Royal Observatory of Belgium (http://sidc.oma.be/ and T. Nakamura, “Cyclicity of solar activity during the silso/datafiles). It records the sunspot data from 1700 to the maunder minimum deduced from radiocarbon content,” present. In particular, the data analysis center significantly Solar Physics, vol. 224, no. 1-2, pp. 317–322, 2004. revised the number of sunspots on July 1, 2015. *e data in [13] Z. Q. Yin, D. Y. Liu, C. G. Pang, C. Y. Xuan, and X. N. 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