To study the imprints of the solar–ENSO–geomagnetic activity on the Indian subcontinent, we have applied singular spectral analysis (SSA) and wavelet analysis to the tree-ring temperature variability record from the Western Himalayas. Other data used in the present study are the solar sunspot number (SSN), geomagnetic indices (aa index), and the Southern Oscillation Index (SOI) for the common time period of 1876–2000. Both SSA and wavelet spectral analyses reveal the presence of 5–7-year short-term ENSO variations and the 11-year solar cycle, indicating the possible combined influences of solar–geomagnetic activities and ENSO on the Indian temperature. Another prominent signal corresponding to 33-year periodicity in the tree-ring record suggests the Sun-temperature variability link probably induced by changes in the basic state of the Earth's atmosphere. In order to complement the above findings, we performed a wavelet analysis of SSA reconstructed time series, which agrees well with our earlier results and increases the signal-to-noise ratio, thereby showing the strong influence of solar–geomagnetic activity and ENSO throughout the entire period. The solar flares are considered responsible for causing the atmospheric circulation patterns. The net effect of solar–geomagnetic processes on the temperature record might suggest counteracting influences on shorter (about 5–6-year) and longer (about 11–12-year) timescales. The present analyses suggest that the influence of solar activities on the Indian temperature variability operates in part indirectly through coupling of ENSO on multilateral timescales. The analyses, hence, provide credible evidence of teleconnections of tropical Pacific climatic variability and Indian climate ranging from inter-annual to decadal timescales and also suggest the possible role of exogenic triggering in reorganizing the global Earth–ocean–atmospheric systems.

Several recent studies of solar–geomagnetic effects on climate have been examined on both global as well as on regional scales (Lean and Rind, 2008; Benestaed and Schmidt, 2009; Meehl et al., 2009; Kiladis and Diaz, 1989; Pant and Rupa Kumar, 1997; Gray et al., 1992; Wiles et al., 1998; Friis and Svensmark, 1997; Rigozo et al., 2005; Feng et al., 2003; Tiwari and Srilakshmi, 2009; Chowdary et al., 2006, 2014; Appenzeller et al., 1998; Proctor et al., 2002; Tsonis et al., 2005; De Freitas and Mclean, 2013). The Sun's long-term magnetic variability caused by the sunspots is considered to be one of the primary drivers of climatic changes. The short-term magnetic variability is due to the disturbances in the Earth's magnetic fields caused by the solar activities and is indicated by the geomagnetic indices. The Sun's magnetic variability modulates the magnetic and particulate fluxes in the heliosphere. This determines the interplanetary conditions and imposes significant electromagnetic forces and effects upon the planetary atmosphere. All these effects are due to the changing solar-magnetic fields, which are relevant for planetary climates, including the climate of the Earth. The Sun–Earth relationship varies on different timescales ranging from days to years, bringing a drastic influence on the climatic patterns. The ultimate cause of solar variability, on timescales from decadal to centennial to millennial or even longer scales, has its origin in the solar dynamo mechanism. During the solar maxima, huge amounts of solar energy particles are released, thereby causing the geomagnetic disturbances. The 11-year solar cycle acts as an important driving force for variations in the space weather, ultimately giving rise to climatic changes. It is, therefore, imperative to understand the origin of space climate by analyzing the different proxies of solar-magnetic variabilities. Another important phenomenon is El Niño–Southern Oscillation (ENSO), which is associated with droughts, floods, and intense rainfall in different parts of the world. The strong coupling and interactions between the tropical ocean and the atmosphere play a major role in the development of the global climatic system. The El Niño events generally recur approximately every 3–5 years, with large events spaced around 3–7 years apart. The ENSO phenomena have shown a huge impact on the Asian monsoon (Cole et al., 1993), Indian monsoon (Chowdary et al., 2006, 2014), as well as globally (Horel and Wallance, 1981; Barnett, 1989; Yasunari, 1985; Nicolson, 1997). In particular, the El Niño, solar, geomagnetic activities are the major affecting forces on the decadal and interdecadal temperature variability on global and regional scales in a direct/indirect way (El-Borie et al., 2010; Gray et al., 2010). Recent studies (Frohlich and Lean, 2004; Steinhilber et al., 2009) indicate the possible influence of solar activity on Earth's temperature/climate on multi-decadal timescales. The 11-year solar cyclic variations observed from the several temperature climate records also suggest the impact of solar irradiance variability on terrestrial temperature (Budyko, 1969; Friis and Lassen, 1991; Friis and Svensmark, 1997; Kasatkina et al., 2007). The bi-decadal (22-year) cycle, called the Hale cycle, is related to the reversal of the solar-magnetic field direction (Lean et al., 1995; Kasatkina et al., 2007). The 33-year cycle (Bruckener cycle) is also caused by the solar origin, but it is a very rare cycle (Kasatkina et al., 2007). The 2–7-year ENSO cyclic pattern and its possible coupling process are the major driving forces for the temperature variability (Gray et al., 1992; Wiles et al., 1998; Mokhov et al., 2000; Rigozo et al., 2007, Kothawale et al., 2010). El-Borie and Al-Thoyaib (2006) and El-Borie et al. (2007, 2010) have indicated in their studies that the global temperature should lag the geomagnetic activity with a maximum correlation when the temperature lags by 6 years. Mendoza et al. (1991) reported on possible connections between solar activity and El Niños, while Reid and Gage (1988) and Reid (1991) reported on the similarities between the 11-year running means of monthly sunspot numbers and global sea surface temperature. These findings suggest that there is a possibility of strong coupling between temperature–ENSO and solar–geomagnetic signals.

Several studies have been carried out to understand the climatic changes of India in the past millennium using various proxy records, e.g., ice cores, lake sediments, glacier fluctuations, and peat deposits. There is a lack of high-precision and high-resolution palaeo-climatic information for longer timescales from the Indian subcontinent. Tree-ring data are a promising proxy to retrieve high-resolution past climatic changes from several geographical regions of India (Bhattacharyya et al., 1988, 1992, 2006; Hughes, 1992; Bhattacharyya and Yadav, 1996; Borgaonkar et al., 1996; Chaudhary et al., 1999; Yadav et al., 1999; Bhattacharyya and Chaudhary, 2003; Shah et al., 2007). It has been noted that tree-ring-based climatic reconstructions in India generally do not exceed 400-year records except at some sites in the northwestern Himalaya. Thus, a long record of tree-ring data is needed to extend available climate reconstruction further back to determine climatic variability on sub-decadal, decadal, and century scales. However, non-availability of older living trees at most of the sites is hindering the preparation of a long tree chronology. In a previous study (Tiwari and Srilakshmi, 2009), we have studied the periodicities and non-stationary modes in the tree-ring temperature data from the same region (AD 1200–2000). To reveal significant connections among the solar–geomagnetic–ENSO “triad” phenomena on tree-ring width in detail for the period from 1876 to 2000, we have applied here singular spectral analysis (SSA) and the wavelet spectral analysis for sunspot data, geomagnetic data (aa index), the Troup Southern Oscillation Index (SOI), and the Western Himalayan tree-ring data. Here our main objective is to employ wavelet-based analysis in SSA reconstructed time series to find evidence of the possible linkages, if any, among ENSO–solar–geomagnetic activity in the Indian temperature records.

The data analyzed here include the time series of the (1) smoothed sunspot
number for solar activity, (2) geomagnetic activity data (aa indices),
(3) the Troup Southern Oscillation Index (SOI) for the study of the El
Niño–Southern Oscillation, called ENSO, and (4) the Western Himalayan
temperature variability record. All the data sets have been analyzed for the
common period of 125 years spanning over 1876–2000. The monthly sunspot
number data have been obtained from the Sunspot Index Data Center
(

To analyze the temporal series and to find the climatic structure, we have here three methods: principal component analysis (PCA), singular spectral analysis (SSA), and wavelet analysis.

As a preliminary analysis, we have applied principal component analysis (PCA) to the data sets for the reduction and extraction of dimensionality of the data and to rate the amount of variation present in the original data set. The purpose of applying PCA is to identify patterns in the given time series. The new components thereby obtained by the PCA analysis are termed PC1, PC2, PC3, and so on (for the first, second, and third principal components), and are uncorrelated. The different PCs capture part of the variance and are ranked depending on their corresponding percentage variance.

The singular spectrum analysis (SSA) method is designed to extract as much information as possible from a short, noisy time series without any prior knowledge about the dynamics underlying the series (Broomhead and King, 1986; Vautard and Ghil, 1989; Alonso et al., 2005; Golyandina et al., 2001). The method is a form of principal component analysis (PCA) applied to lag-correlation structures of the time series. The basic SSA decomposes an original time series into a new series that consists of trend, periodic or quasi-periodic, and white noises according to singular value decomposition (SVD), and provides the reconstructed components (RCs). The basic steps involved in SSA are decomposition (involves embedding and singular value decomposition, SVD) and reconstruction (involves grouping and diagonal averaging). Embedding decomposes the original time series into the trajectory matrix; SVD turns the trajectory matrix into the decomposed trajectory matrices. The reconstruction stage involves grouping to make subgroups of the decomposed trajectory matrices and diagonal averaging to reconstruct the new time series from the subgroups.

The trajectory matrix is thus written as

Considering Eq. (3), let

An alternative method for studying the non-stationarity of the time series is wavelet transform. For non-stationary signals, wavelet decomposition would be the most appropriate method because the analyzing functions (the wavelet functions) are localized both in time and frequency.

During the past decades, wavelet analysis has become a popular method for the analysis of aperiodic and quasi-periodic data (Grinsted et al., 2004; Jevrejeva et al., 2003; Torrence and Compo, 1998; Torrence and Webster, 1999). In particular, it has become an important tool for studying localized variations of power within a time series. By decomposing a time series into time–frequency space, the dominant modes of variability and their variation with respect to time can be identified. The wavelet transform has various applications in geophysics, including tropical convection (Weng and Lau, 1994) and the El Niño–Southern Oscillation (Gu and Philander, 1995). We have applied the wavelet analysis to analyze the non-stationary signals, which permits the identification of the main periodicities of ENSO–sunspot–geomagnetic activity in the time series. The results give us more insight information about the evolution of these variables in frequency–time mode.

A wavelet transform requires the choice of analyzing function

In several recent analyses, a complex Morlet wavelet has been found useful
for geophysical time series analysis. The Morlet is mostly used to find areas
where there is high amplitude at certain frequencies. The complex Morlet
wavelet can be represented by a periodic sinusoidal function with a Gaussian
envelope and is excellent for a Morlet wavelet that may be defined
mathematically, as follows:

Time series data of

The cross-wavelet transform is applied to two time series to identify the
similar patterns that are difficult to assess from a continuous wavelet map.
Cross-wavelet power reveals areas with high common power. The cross-wavelet
of two time series

Wavelet coherence is another important measure to assess how coherent the
cross-wavelet spectrum transform is in time–frequency space. The wavelet
coherence of two time series is defined as (Torrence and Webster, 1999)

First four principal components (PCs 1–4) for time series:

We analyzed the data sets spanning over the period of 1876–2000 using the PCA, SSA, and wavelet spectral analyses. Figure 1 shows four time series: (1) the smoothed sunspot number representing solar activities; (2) geomagnetic (aa indices); (3) the Troup Southern Oscillation Index (SOI) for the study of ENSO, and (4) the Western Himalayan temperature variability record, which are analyzed in the present work. From visual inspection it is apparent from Fig. 1 that both WH and SOI data show an irregular and random pattern, while sunspot numbers have a quasi-cyclic character. Furthermore, the WH tree-ring record also exhibits distinct temperature variability but nonstationary behavior at different scales. This variability might be suggestive of coupled global ocean–atmospheric dynamics or some other factors, such as deforestation, anthropogenic, or a high latitudinal influence (Yadav et al., 2004).

Hence it is quite difficult to differentiate such a complex climate signal
visually, and difficult to infer any clear oscillation without the help of
powerful mathematical methods. For identification of any oscillatory
components and understanding of the climatic variations on regional and
global scales, we have applied PCA, SSA, and wavelet analysis. Figure 2 shows
the principal components (PCs) for the first four eigen triples (PC1, PC2,
PC3, PC4) for the given data sets. Figure 3 shows the power spectra of the
principal components (PCs) for the four data sets shown in Fig. 2. From
Fig. 3, it is observed that the power spectra of PCs 1–4 for the sunspot
data exhibit high power at 124, 11, and 4–2.8 years. The presence of a high
solar signal at 124 years indicates the quasi-stable oscillatory components
in the data. The power spectra of geomagnetic data also show the presence of
strong signals at 124, 10–11, and 4–2 years, suggesting a strong link of
solar–geomagnetic activity. The power spectra of WH temperature data show
strong high power at

Power spectra of the first four principal component (PCs) (PCs 1–4 shown in Fig. 2) for all the data sets with their significant periodicities at 124, 11, 4 and 2.8 years indicated in bold letters.

Wavelet power spectrum of

Cross-wavelet spectrum between

Singular spectra with their SSA decomposed components and their
reconstructed time series for

To explore the stationary characteristics of these peaks obtained by the PCA, we have applied the Morlet-based wavelet transform approach (Holschneider, 1995; Foufoula-Georgiou and Kumar, 1995; Torrence and Compo, 1998; Grinsted et al., 2004). The wavelet spectrum identifies the main periodicities in the time series and helps to analyze the periodicities with respect to time. Figure 4 shows the wavelet spectrum for the (a) smoothed sunspot number for solar activity (SSN), the (b) Western Himalayan (WH) temperature variability record, (c) geomagnetic activity, and the (d) Troup Southern Oscillation Index (SOI). From the wavelet spectrum of sunspot time series (Fig. 4a), the signal near 11 years is the strongest feature and is persistent during the entire series, indicating the non-stationary behavior of the sunspot time series. The wavelet spectrum of SOI (Fig. 4c) shows strong amplitudes. However, due to the non-stationary (time-variant) character of the time series, the observed spectral peaks (power) split in the interval of 2–8 years. The wavelet power spectrum of the Western Himalayan temperature variability (Fig. 4b) reveals significant power concentration on inter-annual timescales of 3–5 years and at an 11-year solar cycle. A dominant amplitude mode is also seen in the low-frequency range at around 35–40 years (at periods 1930–1980) corresponding to AMO cycles. Our result agrees well with the results of other climate reconstructions (Mann et. al., 1995) from tree rings and other proxies. The observed variability in AMO periodicity has also been reported in other tree-ring records (Gray et al., 2004). The statistical significance of the wavelet power spectrum is tested by a Monte Carlo method (Torrence and Compo, 1998). The WH spectra depicting statistically significant powers above the 95 % significance level at around 5, 11 and 33 years suggests the possible imprint of sunspot–geomagnetic and ENSO phenomena on the tree-ring data. On shorter timescales, the wavelet power spectrum of the geomagnetic record (Fig. 4d) also reveals statistically significant power at around 2-, 4–8-, and 11-year periods.

In order to have better visualization of similar periods in two time series and for the interpretation of the results, the cross-wavelet spectrum has been applied. Figure 5 shows the cross-wavelet spectrum of the (a) SSN–WH temperature data, (b) WH data–SOI, and (c) SSN–SOI data. The contours (dark black lines) are the enclosing regions where cross-wavelet power is significantly higher, at 95 % confidence levels. The wavelet cross-spectra of WH–SSN (Fig. 5a) show a statistically significant high power over a period of 1895–1985 in a 8–16-year band. It is seen that in the WH–SOI cross-spectra (Fig. 5b), the high power is observed at a 2–4-year band and at 8–16 years as well. The SSN–SOI spectra (Fig. 5c) show a strong correlation at an 11-year solar cycle, which is stronger during 1910–1950 and 1960–2000 (Rigozo et al., 2002, 2003), suggesting the strongest El Niño and La Niña events, indicating solar modulation on ENSO (Kodera and Kuroda, 2005; Kryjov and Park, 2007). These results show a good correspondence in response to growth of the tree-ring time series during the intense solar activity. Hence the results strongly support the possible origin of these periodicities from solar and ENSO events. The interesting conclusion from Fig. 5 is that WH–sunspot connections are strong at 11 years, and ENSO–sunspot connections also exhibit strong power around 11 years; the WH–ENSO connections are spread over three bands, 2–4, 4–8, and 8–16 years, covering the solar cycle and its harmonics; the WH–geomagnetic exhibits strong connections around 2–4, 4–6, 11, and 35–40 years, indicating the influence of solar–geomagnetic activity on Indian temperature.

Singular spectral analysis (SSA) is performed for all four data sets with a
window length of 40. The SSA spectra with 40 singular values and their
corresponding reconstructed series (varying from RCs 1 to 15 in some cases)
are plotted as shown in Figs. 6 and 7. The important insights from SSA
spectra are the identification of gaps in the eigenvalue spectra. As a rule,
the pure noise series produces a slowly decreasing sequence of singular
values. The explicit plateau in the spectra represents the ordinal numbers of
paired eigen triples. Eigen triples 2–3 for the sunspot data correspond to
the 11-year period; eigen triples for 1–2, 3–5, 6–10, and 11–14 for the
WH temperature data are related to harmonics with specific periods (periods
33–35, 11, 5, 2 years); eigen triples for 2–5, 6–9, and 10–13 for the
geomagnetic data are related to periods of 11, 5, and 2 years. The eigen
triples for the SOI data represent

Singular spectra with their SSA decomposed components and their
reconstructed time series for

Power spectrum and wavelet power spectrum of SSA reconstructed

Squared wavelet coherence plotted for the SSA reconstructed time
series between

In general our result agrees well with earlier findings in the sense that statistically significant global cycles of coupled effects of sunspot–geomagnetic activity and ENSO are present in the land-based temperature variability record. However, there are certain striking features in the spectra that need to be emphasized regarding the Western Himalayan temperature variability: (i) inter-annual cycles in a period range of 3–8 years corresponding to ENSO in the wavelet spectra exhibit intermittent oscillatory characteristics throughout the large portion of the record (Fig. 4); (ii) the 11-year solar cycle in the cross-wavelet spectrum of SSN and SOI (Fig. 5) indicates the solar modulation in the ENSO phenomena (Kodera and Kuroda, 2005; Kryjov and Park, 2007); and (iii) the high amplitude at 11 years in the time interval 1900–1995 with a strong intensity from 1900 to 1995 shows a good correspondence to the high temperature variability for the interval of high solar–geomagnetic activity. The multi-decadal (30–40-year) periodicity identified here in the Western Himalayan tree-ring temperature record matches with North Atlantic sea surface temperature variability, implying that the temperature variability in the Western Himalayas is not a regional phenomenon but a globally teleconnected climate phenomenon associated with the global ocean–atmospheric dynamics system (Tiwari and Srilakshmi, 2009; Delworth et al., 1993; Stocker, 1994). The coupled ocean–atmosphere system appears to transport energy from the hot equatorial regions towards Himalayan territory in a cyclic manner. These results may provide constraints for modeling of climatic variability over the Indian region and ENSO phenomena associated with the redistribution of temperature variability. The solar–geomagnetic effects play a major role in abnormal heating of the land surface, thereby indirectly affecting the atmospheric temperature gradient between the land–ocean coupled systems. In the present work, the connections between solar–geomagnetic activity and ENSO on the WH time series are found to be statistically significant, especially when they are studied over contrasting epochs of, respectively, high and low solar activity. The correlation plots for the SSA-RC data sets of WH-sunspot, WH-aa index, WH-SOI, and sunspot-aa index are plotted in Fig. 10. One can notice that there is a correlation plot for the geomagnetic–sunspot activity with a maximum correlation value at a 1-year lag, suggesting the strong influence of sunspot and geomagnetic forcing on one another. The cross-correlation plot for the WH data and the SOI represents a maximum value at zero lag. The correlation plot for the WH-sunspot and WH-geomagnetic indices exhibits almost identical results, suggesting the possible impact of solar activities on the Indian temperature variability.

The net effect of solar activity on temperature record therefore appears to be the result of cooperating or counteracting influences of Earth's magnetic activity on the shorter and longer periods, depending on the indices used; scale interactions, therefore, appear to be important. Nevertheless, the link between Indian climate and solar–geomagnetic activity emerges as having strong evidence; next is the ENSO–solar activity connection.

In the present paper, we have studied and identified the periodic patterns
from the published Indian temperature variability records using the modern
spectral methods of singular spectral analysis (SSA) wavelet methods. The
application of wavelet analysis for the SSA reconstructed time series, along
with the removal of noise in the data, identifies the existence of
high-amplitude, recurrent, multi-decadal scale patterns that are present in
Indian temperature records. The power spectra of WH temperature data show a
strong high power at

Cross-correlation of SSA reconstructed time series of

The data on the Western Himalayan data was given to us by Dr. Ram Ratan Yadav of Birbal Sahni Institute of Palaeobotany, India. The data plot was given in his publication (Yadav et al., 2004).

The authors are extremely thankful to the Editor Stéphane Vannitsem and the anonymous reviewers for their professional comments, meticulous reading of the manuscript, and valuable suggestions to improve the manuscript. The authors thank Dr. Ram Ratan Yadav, Birbal Sahni Institute of Palaeobotany, India, for providing the Western Himalayan data. The authors acknowledge Prof. Francisco Javier Alonso of the University of Extremadura for using the SSA routine in a MATLAB environment. We are thankful to Dr. Alask Grinsted and his colleagues for providing the wavelet software package. The first author acknowledges the Head, University Centre for Earth & Space Sciences, University of Hyderabad, for providing the facilities to carry out this work. Rama Krishna Tiwari is grateful to DAE for a RRF (Rajaramanna Fellowship). Edited by: S. Vannitsem Reviewed by: three anonymous referees