Multiresolution Analysis of the Relationship of Solar Activity, Global Temperatures, and Global Warming

Sunspot number is an important parameter for presenting the intensity of solar activity. Based on the sunspot number series, which has been replaced by a new improved version since 2015, we confirm that the sunspot number has significant variations at 11-year and 112-year periods. .e sunspot number has also increased from 1700 to 2016 with 0.08 annual increments on the basis of wavelet analysis and least-square fitting. We further confirm that global temperatures are remarkable in 22-year and 64-year cycles. .e result of wavelet transform coherence (WTC) analysis suggests that solar activity has a positive lag effect on global temperatures in the period band of 22 years with a 3-year lag. However, the linearly increasing global temperature has hampered WTC analysis since 1960. Aiming to solve this problem, we apply wavelet decomposition and cross correlation to determine whether the aforementioned lag effect in the period band of 22 years has a 2-year lag rather than a 3-year lag. We find that the 22year magnetic field solar cycle plays a greater role in global climate change than the 11-year sunspot cycle. In addition, we notice that the solar activity is not a representation of the driving force of the upward trend of global temperature after the industrial age. .e Granger causality test results demonstrate that the phenomenon of the global warming is caused by excessive CO2 emissions.


Introduction
An increase in global temperature in the last 50 years has been observed and is thus considerably discussed [1][2][3].A growing number of studies suggest that the increase in greenhouse gas concentrations in the air caused by anthropogenic effects is responsible for the climatic warming [4][5][6].e prevailing view of the Intergovernmental Panel on Climate Change (IPCC) suggests that more than 90% of the causes of the significant increase in earth's average temperature over the past 50 years can be attributed to greenhouse gases emitted by human activities, and thus, the impact of climate change can be ignored [7][8][9][10].However, a body of evidence also suggests that solar activity may account for a substantial amount of global temperature variability [11][12][13][14][15][16][17].Eddy [18] proposed the existence of the "Little Ice Age" and the "Medieval Climatic Optimum," which incidentally coincide with Maunder's solar minimum and the grand solar maximum, respectively.Ice core data on several centuries of climate change show that the current global warming belongs to the scope of climate fluctuations [19].erefore, the effect of solar activity on the current climate variability cannot be neglected.Valev [20] applied statistical analysis to investigate the correlation between global surface air temperatures and sunspots.However, the method was performed in the time domain rather than the time-frequency space, in which the correlation of some frequency components of two signals can be examined.By using yearly data, Souza Echer et al. [21] performed wavelet multiscale analysis to decompose the sunspot and global temperature time series into five levels and examine their correlation in five different frequency bands.However, their study was limited by the lack of cross validation of higherresolution data, such as monthly data.Moreover, on July 1, 2015, the sunspot number series was replaced by a new improved version (version 2.0) to include several corrections of past inhomogeneities in the time series.us, revisiting the basis of investigating the effects of solar activity and human activity on global climate change is necessary.
In this study, we employ wavelet analysis to detect the period in the sunspot number time series and the global air surface temperature time series by using yearly and monthly data.en, based on the monthly data, wavelet transform coherence (WTC) analysis is conducted to investigate the correlation between the two time-series pro les in the timefrequency space.
e most signi cant frequencies for the sunspot number and the global temperature are determined and intercorrelated.Furthermore, wavelet decomposition and the Granger causality test are used to analyze whether global warming is caused by solar activity.).e details of the new sunspot number series are given by Clette and Lefevre [22] and are not reported here.Meanwhile, the global temperature timeseries pro les are obtained from the Goddard Institute for Space Studies surface temperature analysis (GISTEMP) (http://data.giss.nasa.gov/gistemp).e GISTEMP analysis provides a measure of the changing global surface temperature with monthly resolution for the period since 1880, when a reasonably global distribution of meteorological stations was established.e input data that the GISTEMP team uses for the analysis, collected by many national meteorological services around the world, are the adjusted data of the Global Historical Climatology Network (GHCN) version 3, United States Historical Climatology Network As shown in Figure 1(a), the sunspot numbers clearly have signi cant cyclical patterns with time-varying amplitudes, and they exhibit a dominant period every 128 months (approximately 11 years).Meanwhile, as shown in Figure 1(b), the global temperature variabilities are somewhat steady before 1960.However, from 1960 until the end of the analyzed period, a clear upward trend is observed.e upward trend is usually associated with excessive greenhouse gas emissions.

Methodology of Analysis.
e method used in this study includes the analysis of wavelets, including wavelet coherence and wavelet transform.
e continuous wavelet transform (CWT) of a time series X n (n 1, 2 , . . ., N) is de ned [23][24][25] as where s is the wavelet scale, δt is the time step, ϕ 0 is the mother function, n ′ is the reversed time, and δt/s √ is the normalization factor.Compared with traditional methods (e.g., Fourier analysis) that examine continuous and stationary periodic signals in the frequency space, CWT can detect intermittent and/or nonstationary periodicities in the time-frequency space.
After the introduction of CWT, we brie y recall the content of the wavelet coherence to establish our problem context.
e wavelet coherence of the two time-series pro les X n and Y n (n 1, 2 , . . ., N) is de ned [24][25][26] as 2 Advances in Meteorology where S is the smoothing operator, s is the wavelet scale, and R n (s) is the localized correlation coefficient.For the two CWTs, the WTC can be considered the local correlation between the two time-series profiles in the time-frequency domain.e details of the WTC analysis are given by Grinsted et al. [26] and are not reported here.
e original time series of the sunspot number and the global temperature are the wavelets transformed by the orthonormal discrete Meyer wavelet transform, which decomposes the original signal into orthogonal frequency levels and enables the operation of band-pass filtering in different frequencies, each limited by the power of 2. Wavelet decomposition represents the annual time series of the sunspot number and the global temperature until the D5 level.e corresponding band frequencies are presented in Table 1.e approximations for A5 are at the scaling level and correspond to long-term periods.

Results and Discussion
3.1.Period Analysis.Considering that generating the sunspot number variation is a highly complicated nonlinear process, we select here two groups of data to comprehensively analyze the periodicity of sunspot variations.One group is for the monthly sunspot data from 1747 to 2016, while the other group is for the yearly sunspot data from 1700 to 2016.In this analysis, we choose a complex Morlet wavelet with w 0 � 6 as the mother function.is value of w 0 offers a good balance between time and frequency localization.
Figure 2 shows the first, second, and third periods located in the period bands of 8-16, 64-128, and 16-32 years, respectively.e 11-year cycle that corresponds to the 8-to 16-year scale has passed the significance test at the 5% level against the red-noise variance test.However, the 112-year cycle for the 64-to 128-year scale and the 22-year cycle for the 16-to 32-year scale are not significant at the same test conditions.e result in Figure 2 indicates that the sunspot in 11-year cycles is significant in the entire time domain, while the 22-year and 112-year cycles are remarkable in the local time domain.In addition, we find that the intensity of the solar activity in the latter half of the twentieth century has reached the highest level since 1749, but the intensity significantly weakened since the twenty-first century.
Figure 3 shows the first, second, and third periods located in the period bands of 8-16, 64-128, and 32-64 years, respectively.e 11-year period corresponding to the 8-to 16-year scale has passed the significance test.However, the 55-year period for the 32-to 64-year scale and the 112-year period for the 64-to 128-year scale have not passed the rednoise test, which indicates that the sunspot number variability is globally significant in 11-year periods and locally notable in 55-year and 112-year periods.
e results of Figures 2 and 3 indicate that the selected different time intervals and timescales correspond to different periods, which is consistent with previous results [27][28][29][30].Moreover, the sunspot number variability is globally significant in 11-year and 112-year cycles and locally remarkable in 22-year and 55-year periods.On the basis of the yearly sunspot numbers from 1700 to 2016 and the sunspot variability cycles determined by the wavelet analysis, this study adopts 11, 22, 55, and 112 years as the period terms of the time-curve fitting equation.e least-square method is used to calculate the amplitude, phase, and trend of the sunspot time series.e curve fitting equation is where a 0 is the constant, a 1 is the linear trend, a 2 is the linear acceleration, b i is the amplitude, and T is the period.Table 2 shows the amplitude of each period and the linear trend of sunspot number variability.
e result of Table 2 indicates that the amplitude of 11 years is the maximum and the amplitude of 22 years is the minimum.
e sunspot number shows an increasing trend from 1700 to 2016 with 0.08 annual increments, and we can see that the new sunspot number displays a weak trend, which is consistent with the findings of Clette and Lefevre [22].
Moreover, we compare the original values of the sunspot number with the fitting ones.It is clear from Figure 4 that although errors have been observed between original data and fitting data, this fitting scheme describes well the general behavior of solar activity.Meanwhile, the wavelet analysis of the monthly global temperature data from 1880 to 2016 is completed.
Figure 5 shows the first, second, and third periods of global surface temperature variation located in the period bands of 64-128, 16-32, and 8-16 years, respectively.e 64year period corresponding to the 64-to 128-year scale has passed the significance test, consistent with previous results [31,32].e 22-year period corresponding to the 16-to 32year scale has also passed the significance test, consistent with previous results [33,34].However, the 11-year period for the 8-to 16-year scale has not passed the red-noise test, which means that the global surface temperature variation is globally significant in 64-year and 22-year periods and locally notable in 11-year periods.

Wavelet Coherence Analysis.
Considering that the traditional correlation analysis can be performed in the time domain only, we use WTC to find regions in the timefrequency domain where the sunspot time series and the

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global temperature time series are highly correlated but do not necessarily have high power values (Figure 6).e thick contour represents the 5% signi cance level against red noise, and it is the boundary line of the cone in uence in which wavelet power caused by discontinuity at the edge has dropped to e −2 of the edge value.e relative phasing relationship is shown as arrows (i.e., in-phase to the right, antiphase to the left, and sunspot number variability leading to global temperature variation of 90 °downward).As shown in Figure 6, the most prominent regions with signi cant correlations are the period scales every 256 months (approximately 22 years), and the correlation in the 11-year period band is low, which is consistent with previous research results [20,21].e arrows that point southeastward represent the sunspot number variability that leads to global temperature variation, and the uctuation range of the time-varying relative phase angle is not large in the 256month scale.us, we adopt the average value of the timevarying relative phase angle to express the phase relationship between the sunspot number and the global temperature.
e mean relative phase angle is 50.1 °, which is equivalent to a 3-year lag.However, when the global temperature has started to increase in the 1960s, the high correlation between the sunspot number and the global temperature diminished. is phenomenon indicates that the linearly increasing global temperature will hamper the WTC.erefore, we use wavelet decomposition to gain deep insights into the correlation between the sunspot and global temperature, and we investigate in more detail the levels D3 (8-16 years), D4 (16-32 years), and A5 (>64 years).
e levels encompass 11-year, 22-year, and long-term trends.Figure 7 presents the D3, D4, and A5 decomposition levels of the sunspot number and global temperature., 1920-1990).Meanwhile, we find that the D3 level of GISS temperature calculated by Souza Echer et al. [21] showed a weak 11-year cycle over 1925-1960; however, the 11-year periodical variation is quite normal in this study, which may be attributed to the difference between GHCN version 3 (this represents a change from the prior use of unadjusted version 2 data) and GHCN version 2.Moreover, the cross correlation is taken as an indicator to individually evaluate the performances of different spectral bands [35,36].As shown in Figure 8, the cross correlation between the sunspot and global temperature in the D3 band is low and ranges from −0.24 to 0.18.e cross correlation between the two time-series profiles in the D4 band shows significant correlation coefficients with values from −0.6 to 0.59, in which the maximum is 0.59 with a 2-year lag rather than a 3-year lag, which is not consistent with the wavelet-coherent result in Figure 6. e results in Figure 8 indicate that the effect of the 22-year solar cycle is more significant on global temperature than the 11-year cycle, and the linearly increasing global temperature has indeed hampered the WTC.In general, the 22-year period of the solar activity, which is called the magnetic Hale cycle, is related to the change in the sun's magnetic field polarity.us, the 22-year magnetic field solar cycle plays a greater role than the 11-year sunspot cycle in global climate change.
e approximation level (A5) shows the long-term trends.After 1970, the trend is observed in the opposite direction (Figure 7), which indicates that solar activity does not represent the driving force of the upward trend in global temperature (i.e., the attribution is mainly to the increase in greenhouse gases).
Considering that global warming is usually attributed to the rising concentrations of carbon dioxide (CO 2 ), establishing the possible connection is also necessary.
Figure 9 presents the annual emissions of CO 2 from 1880 to 2014 provided by the Carbon Dioxide Information Analysis Center database (http://cdiac.ornal.gov/trends/emis/tre-glob.html).
e CO 2 emissions have slowly increased before 1950.After 1950, a sharp growing trend is observed, and this phenomenon may hamper the period correlation detection between the sunspot and global temperature.
To further detect the relationship of sunspots, CO 2 emissions, and global temperatures, we employ the Granger causality test to investigate their dependence on one another [37,38].
Lag length selection is conducted by using the Bayesian information criterion.A p value >0.1 indicates that there is no Granger causality between the two time-series profiles.
e results in Table 3 suggest that global warming is mainly caused by CO 2 emissions rather than sunspot number variability.

Conclusion
e results of the wavelet analysis in this study indicate that the sunspot number variability is globally significant in 11year and 112-year periods and locally remarkable in 22-year and 55-year cycles, while global temperatures are globally notable in 22-year and 64-year cycles and locally remarkable in 11-year periods.Meanwhile, the least-square curve fitting indicates that the amplitude of 11 years is the maximum and the amplitude of 22 years is the minimum.
e sunspot number has increased from 1700 to 2016 with 0.08 annual  Advances in Meteorology increments.Furthermore, we apply WTC to con rm whether the solar activity before 1960 has a positive lag e ect on global temperatures in the 22-year period band with a 3year lag.e linearly increasing global temperature has hampered WTC.en, wavelet decomposition is used to gain deeper insights into the correlation between the sunspot and global temperature.e cross correlation indicates that the e ect of the 22-year solar cycle with a 2-year lag on global temperature is more signi cant than that of the 11-year cycle.
us, the 22-year magnetic eld solar cycle plays a greater role than the 11-year sunspot cycle in global climate change.Additionally, we notice that solar activity cannot explain the upward trend of global temperatures after 1970.Finally, the result of the Granger causality test demonstrates that global warming is mainly caused by CO 2 emissions.
Data Availability e data used to support the ndings of this study are available from the corresponding author upon request.Advances in Meteorology 7

(
USHCN) data, and SCAR (Scienti c Committee on Antarctic Research) data from Antarctic stations.e time-series pro les used in the present study represent monthly deviations in relation to the 1951-1980 average with a scale of 0.01 °C and cover the time interval between January 1880 and December 2016.

Figure 1 :
Figure 1: (a) Monthly average of the sunspot number and (b) global surface temperature time-series pro les.

Figure 3 :
Figure 3: Wavelet power spectrum of the annual sunspots from 1700 to 2016.

Figure 4 :
Figure 4: A comparison of the original values of the sunspot number with the tting ones.

Figure 2 :
Figure 2: Wavelet power spectrum of monthly sunspots from 1749 to 2016.

Figure 9 :
Figure 9: Annual values of carbon dioxide emissions from 1880 to 2014.

Table 1 :
Frequencies corresponding to the scales of the Meyer wavelet function with a 1-year sampling period.

Table 2 :
Parameters of signi cant periods of the sunspot number time series.