In this paper, SRF data (F10.7 cm data) during 1947–2015 are obtained from the National Oceanic and Atmospheric Administration (NOAA) Data Center (http://www.esrl.noaa.gov/psd/data/correlation/solar.data). The F10.7 cm data are expressed in solar flux units (sfu) where 1 sfu = 10^–22 W m⁻²·Hz⁻¹. The winter SRF is obtained by the average of the data in December and the subsequent January and February. The Solar Sunspot Number (SSN) data for 1770–2014 used in this paper is collected from the Solar Influences Data Analysis Center (SIDC), which is the solar physics research department of the Royal Observatory of Belgium (http://sidc.oma.be/sunspot-data). The Total Solar Irradiance (TSI) reconstruction data during 1610–2013 is downloaded from Laboratory Atmospheric Space Physics of University of Colorado Boulder (Coddington et al., 2015; Kopp et al., 2016) (http://lasp.colorado.edu/home/sorce/data/tsi-data/). The monthly mean data of atmospheric circulation are the reanalysis data from the National Centre for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) in the United States (Kalnay et al., 1996).

The snow indices used in this study are winter and spring snow depth (WSD/SSD) and snow day number (WSDN/SSDN) from 1951 to 2015 (Song et al., 2011). All of these four indices are daily observational data over the Tibetan Plateau from the China Meteorological Information Center. In order to eliminate data discontinuity, the original data have been chosen carefully and interpolated strictly to obtain continuous monthly data over 51 gauge stations of 1961–2015 (Song et al., 2011). The distribution of the observational stations over the Tibetan Plateau is shown in Figure 1, in which black dots indicate gauge stations. The shaded area indicates that the elevation is above 3,000 m. The monthly snow data are the sum of daily snow data in 1 month. SSD and SSDN are obtained by calculating the average of monthly snow depth and snow day number from March to May at each station. Similarly, WSD and WSDN can be defined as the average of snow depth and snow day number from December to January and February of the following year at each station. Thus, we acquire the seasonal snow data of all stations over the plateau.

Summer precipitation data over 160 observation stations in China can be obtained from the National Meteorological Information Center, China Meteorological Administration (http://10.1.64.154/). The strong ENSO events data involving strong El Niño and La Niña events are obtained from the National Climate Center of China, which uses the Nino indices data published by the National Oceanic and Atmospheric Administration to calculate the Nino Z SSTA Index, which is the composite Nino index of the area weighted average for Nino1+2, Nino3, and Nino4 zones in the tropical Pacific Ocean (https://www.esrl.noaa.gov/psd/gcos_wgsp/Timeseries/). The climatological mean of all data sets used in this paper is the 30-year average from 1981 to 2010.

Monte Carlo significance test method: For different data sets with different time scales, it is necessary to use different significance test methods. For example, the Student’s t-test method is adopted to test the significance levels of correlation coefficients of raw data sets. However, for data sets after running mean, Monte Carlo method should be used to test the significance levels (Yan et al., 2003; Zhao and Han, 2005; Zhou et al., 2012), because of the decrease in both data number and degree of freedom.

The detailed steps in computing the critical values of correlation coefficients with the Monte Carlo method are as follows:

(1) Produce two random sample sequences, then compute the correlation coefficient of the two sample sequences after running mean; finally, get 5,000 correlation coefficients after calculating 5,000 times repeatedly.

See Zhou and Zheng (1999) for the method to calculate the degree of freedom of new sample series after running mean. The raw data set of winter snow depth over the Tibetan Plateau used in this paper is a discrete time series, 55 years in total, i.e., { x ( nΔ ) }   ( n = 0 ,   1 ,   2 ,   … ,   55 ) , in which △ is 1 year. After filtering of the 11-year running mean, the new data set has been changed into { x ( nΔ ) } ( n = 6 ,   7 ,   8 ,   … ,   50 ) , and △ is 1 year.

So, the bandwidth of the original data set is f w = f n - f k = 1 2 × 1 - 1 55 × 1 = 0.482 . The bandwidth of the new data set after filtering of the 11-year running mean is f p = f n - f l = 1 12 - 1 50 = 0 . 083 - 0 . 02 = 0 . 063 . Then, 1 χ = f p f w = 0.063 0.48 = 0.13125 .

So, the degree of freedom of the new data set is equal to that of the original sequence multiplied by 1 Χ , i.e., it is equal to 55*0.13125 = 7.22 ∼ 7.

Therefore, the degree of freedom of the new winter snow depth data set for 55 years after the 11-year running mean is changed to approximately 7.

Composite Mean Difference (CMD) method: The CMD used in this paper is an intuitive way to obtain the spatial pattern by calculating the differences between the strong and weak solar activity composites. CMD is often used to deduce the spatial pattern of the solar cycle response (Camp and Tung, 2007). In this study, we divide all the data into two groups according to the strong and weak solar activity years [e.g., normalized data values larger than zero are ascribed to HSAY, and those smaller than zero are ascribed to low solar activity years (LSAY)] and then calculate the difference between the two groups to obtain a spatial pattern of solar response. The Student’s t-test is used to examine the significance level.

1) Continuous Wavelet Transform (CWT)

Geophysical science often needs to decompose one signal into wavelets. The fourier transform is used to decompose a signal into infinite number of terms, which lose most time-localization information. The continuous wavelet transform (CWT) could decompose a time series into time-frequency space and could be used for feature extraction purposes. The CWT method is often used for analyzing localized intermittent oscillations in a time series, especially the time series that are not normally distributed.

2) Cross Wavelet Transform (XWT)

CWT could be used to examine whether two time series are linked in some way. Therefore, two CWTs could be used to construct the Cross Wavelet Transform (XWT) so as to expose their common power and relative phase in time-frequency space.

The cross wavelet spectrum of two time series X and Y with wavelet transform W n X and W n Y is described as | W n X Y ( S ) | = | W n X ( S ) · W n ∗ Y ( S ) | (Torrence and Compo, 1998), where W n ∗ Y is the complex conjugate of W n Y and W n X Y is the local relative phase between time series X j and Y j . Statistical significance is estimated against a red noise model (Torrence and Webster, 1998). Thus, XWT denotes their common power and relative phase in time-frequency space.

3) Wavelet Coherence (WTC)

In this paper, we used another tool to further identify the significant coherence between two CWTs. Even though the common power is low, Wavelet Coherence (WTC) can exhibit how confidence levels against red noise backgrounds are calculated. Different from the cross wavelet power, coherence detects the intensity of the covariance in time-frequency space with a measure of the common power of two time series.

A red noise to determine the 95% statistical confidence level of the coherence is identified by Monte Carlo method (Torrence and Webster, 1999; Jevrejeva et al., 2003).

Figure 6 shows the correlations of SSDN and summer precipitation in China for normal years, HSAY, and LSAY, respectively. HSAY (LSAY) is defined as the year of normalized value of SRF being more (less) than zero. It is seen that there are few highly correlated stations between SSDN and gauge precipitation in China (Figure 6A), but in HSAY (Figure 6B), the number of stations with significant correlation is more than that of normal years. Most parts of Northeast China and North China, as well as the south of the Yangtze River, have significant negative correlation coefficients, which indicates that when the solar activity is strong and SSDN is larger, these areas get less precipitation and are relatively dry. Between the Yellow River and the Yangtze River, significant positive correlation coefficients are identified, especially in the south of the Hetao area. It means that when the Sun is active and SSDN is larger, the precipitation in these areas increases. LSAY (Figure 6C) are accompanied with weakening correlations. However, there are still some stations with significant correlation coefficients in North China, south of the Yangtze River and Southwest China, while the patterns are opposite to those in HSAY. Therefore, solar activity could regulate the correlations between SSDN and summer precipitation in China.

Based on the above analyses, it can be deduced that the correlation between SSDN and summer precipitation in China is regulated by solar activity. However, the data we used could be contaminated by strong ENSO events. As we know, ENSO events would also have strong influence on summer precipitation in China (Zhang et al., 1999; Wu et al., 2003). Therefore, analyses need to be done for HSAY and LSAY years with strong ENSO events (i.e., strong El Niño and La Niña events) excluded.

In order to filter out the impact of ENSO events on summer precipitation in China, the intensity grade of ENSO events issued by the National Climate Center is adopted to calculate partial correlation coefficients of SSDN and summer precipitation in China, as shown in Figure 7.

Figure 7 is similar to Figure 6, but it excludes the impact of ENSO events. It is apparent that without the contamination of ENSO events, the correlations between SSDN and summer precipitation in China in Figure 7A have the same pattern as Figure 6A. In particular, in Shandong Peninsula, Southeast of the Tibetan Plateau, Guangdong province and south of Yunnan province, there are significant negative correlations, while there are remarkable positive correlation coefficients in the Daba Mountain area. In HSAY (Figure 7B), the correlation coefficients become more significant than those in normal years (Figure 7A). A number of remarkable negative correlation coefficients are seen in Northeast China, North China, Southwest China, Guizhou, and Guangxi areas, which means that, in HSAY, more spring snow over the Tibetan Plateau could cause less precipitation in these areas. There are significant positive coefficients in the south of Hetao Basin, around Shaanxi, the north of Sichuan, Zhejiang, and Fujian provinces, which indicates that, in HSAY, more spring snow could lead to more precipitation in these areas. In LSAY (Figure 7C), in Shandong province and east of Henan province, i.e., the north of lower reach of Huaihe River and the south of Yellow River, southeast of the Tibetan Plateau, and the Southeast China, there are obvious negative anomalies, indicating that more spring snow could cause less precipitation in these areas. Meanwhile, the positive anomalies are not obvious in Figure 7C. Therefore, in HSAY and LSAY, without the impact of ENSO events, there are obviously different correlation patterns between SSDN and summer precipitation in China, which testifies that the modulation of solar activity to correlation between snow and summer precipitation is valid without the influence from ENSO.

In terms of summer precipitation in China, in order to distinguish the direct influence of solar activity from the indirect influence of snow cover over the Tibetan Plateau, which is caused by solar activity, the composites of summer precipitation in peak years and valley years are investigated, as shown in Figure 8A and Figure 8B, respectively. Compared with Figure 7, the anomaly patterns of summer precipitation in Figure 8 are obviously different. Here, wintertime solar radio flux peak years are selected as the peak years of solar activity namely 1957/58, 1957/68, 1979/80, 1990/91 and 2001/02, respectively; and wintertime solar radio flux valley years are selected as the valley years of solar activity namely 1963/64, 1975/76, 1986/87, 1995/96 and 2008/09. In peak years, around the middle and south of North China, there are negative anomalies, indicating the decreased precipitation there, while in Guangxi, Guizhou, south of Xizang and west of Yunnan, positive anomalies are found with increased precipitation (Figure 8A). In the valley years, less stations can pass the significance test. When SSDN increases, the precipitation in most parts of North China, Shandong Peninsula and a few stations of the Yangtze River Basin increases significantly, and other observation stations can not pass the significance test. The results indicate that, to a certain extent, solar activity could regulate the relationship between snow over the Tibetan Plateau and summer precipitation in China.