Key points

  • PCA enables analysis of the spatiotemporal variability and asymmetry of the ionosphere

  • TEC spatiotemporal variations over different phases of the solar cycle

  • Newly found diurnal cycle and characteristic epochs of the variability of TEC

1 Introduction

The Earth’s ionosphere, ionized mainly by solar and cosmic radiation, has complicated process dynamics. Total electron content (TEC) is one of the critical physical parameters indicating the state of the ionosphere. Its spatial and temporal variations reflect the behavior of the electron concentration. Modeling and analyzing the variations in the TEC is motivated not only by a purely scientific interest in the research of the upper atmosphere of the Earth but also by the necessity of solving some applied problems concerning the radio propagation, on which many technical systems are based, such as radio communications, satellite positioning and geodetic radio systems (Afraimovich et al. 2008; Hernández-Pajares et al. 2014).

Works investigating the spatiotemporal structure and variations of the ionosphere have been recently reviewed by Laštovička (2013) and Qian et al. (2013). Chapman’s theory, which describes the formation of ionized layers with a simple mathematical model based on the fact that energetic photons from the Sun split atmospheric atoms and molecules into electrons and positive ions, represents major characteristics of variations in the different layers of the ionosphere (Chapman 1931). Any departure from this theory in the ionosphere is considered as an anomaly. Three major anomalies include the winter maximum (seasonal anomaly), the equinoctial maxima (semiannual anomaly), and the annual anomaly (Rishbeth et al. 2000; Zou et al. 2000; Zhao et al. 2007). Variations of the ionosphere manifest significant spatiotemporal characteristics. For example, the winter anomaly falls off in amplitude and area with declining solar activity, and it disappears at night. The semiannual anomaly varies with longitude and local time, and it is asymmetrical between the two hemispheres. The annual anomaly shows that, in the world as a whole, the F2-layer peak electron content NmF2 is, on average, greater in December than in June, either at day or at night. Although a variety of theories have been proposed to interpret these three phenomena (Rishbeth and Setty 1961; Torr and Torr 1973; Araujo-Pradere et al. 2005; Hocke 2008; Zhang et al. 2013), to name but a few, many phenomena regarding the spatiotemporal variability of the ionosphere are yet not to be explained. One of the difficulties in capturing and explaining the variability of the ionosphere is due to traditional observation techniques having limited samples in the temporal and spatial domains (Afraimovich et al. 2008; Zhang et al. 2013;Verhulst and Stankov 2015).

The International GNSS (Global Navigation Satellite Systems) Services (IGS) coordinates a global GNSS observation network that produces near real-time TEC of the global ionosphere with unprecedentedly high spatial and temporal resolution (Schaer et al. 1998; Afraimovich et al. 2008; Dow et al. 2009; Hernández-Pajares et al. 2009; García-Rigo et al. 2014). First, GNSS measurements of two or multiple frequencies are collected from a number of global GNSS observation stations, and they are used for calculating ionospheric TEC along slant signal paths (Hernández-Pajares et al. 2009). The slant TEC along signal paths is then transformed into vertical TEC using a mapping function (Schaer et al. 1998). These vertical TEC values are modelled using a variety of models, such as spherical harmonic expansion or bi-cubic splines. Finally, a global ionosphere map (GIM) is generated by calculating TEC values on a spherical grid using a corresponding model (Dow et al. 2009). The series of GNSS TEC maps provides a large-scale observation source of the ionosphere for understanding the Sun–Earth connection and thermosphere–ionosphere coupling (Hocke 2008). Of the scientific groups and institutes that are presently dedicated to ionospheric studies using GNSS observations, four are involved in producing global TEC maps: the Center for Orbit Determination in Europe (CODE) at the Astronomical Institute of the University of Bern, Switzerland (http://www.aiub.unibe.ch/igs.html); the European Space Operations Centre of the European Space Agency (ESA-ESOC), Darmstadt, Germany; the Jet Propulsion Laboratory (JPL) (http://iono.jpl.nasa.gov), Pasadena, California; and the group of Ionospheric determination and navigation based on Satellite and Terrestrial systems in the Technical University of Catalonia (UPC-IonSAT) (http://ionsat.upc.es), Barcelona, Spain. Global Ionosphere Maps, which contain global TEC values and the associated root mean square (RMS) errors, are distributed globally in a common IONosphere map EXchange (IONEX) format (Schaer et al. 1998).

The long-term IGS GIM dataset since June 1\(\mathrm{st}\), 1998, has allowed one to model and analyze the variability of the ionosphere, such as periodic oscillations, anomalies, and asymmetries in temporal and spatial domains, which are still under investigation (Araujo-Pradere et al. 2005; Astafyeva et al. 2007; Hocke 2008; Hernández-Pajares et al. 2009; Liu et al. 2009; Zhang et al. 2013; García-Rigo et al. 2014). Lean et al. (2011a, b) represented global TEC with four periodic oscillations and a secular trend using IGS TEC maps, and they constructed a linear model of ionosphere climatology, which is capable of capturing more than 98% of the variability in the daily averaged global TEC. Zhang et al. (2013) presented ionospheric longitudinal progression and symmetry with respect to zero magnetic declination by analyzing a 12-year GNSS TEC dataset in North America using the empirical orthogonal function (EOF) technique. Chen et al. (2015) constructed a regional empirical model of the TEC climatology of North America using ionospheric characteristic temporal–spatial variations represented by the first four modes of the EOF decomposition.

This paper analyzes the dynamics and variability of the ionosphere in temporal and spatial domains using the GIM TEC dataset and the principal component analysis (PCA) technique. PCA is able to identify spatial structures that have significant contributions to the total variability, together with their time evolution, without the need to propose any particular a priori functional model. Natali and Meza (2010, 2011) demonstrated that the PCA is an efficient tool for analyzing the annual, semiannual, and seasonal effects in the global TEC. However, existing studies assumptively use local time (LT) 12 (noon) and 22 (nighttime) as the characteristic epochs, and they only utilize the dataset of a particular solar condition. For example, 1-year datasets for 2000 and 2006 represent high and low solar activity phases, respectively (Natali and Meza 2010, 2011). Artificially selected samples of limited time slices may lead to partial or even biased knowledge, although it is acknowledged that these works are valuable in demonstrating the effectiveness of the PCA method.

This study utilizes the PCA technique and the GIM TEC dataset of 1999–2013 to analyze the time-varying aspect of the variability of the global ionosphere for different local times throughout the period of more than one solar cycle. This paper concentrates on the statistical analysis of spatial and temporal variations of the ionosphere at the global scale using the numerical analysis technique of PCA. The analysis is conducted at two different time scales: (1) 15 years as a whole and (2) each of the individual years. Particular characteristics associated with different local times and with the geomagnetic region are highlighted. Finally, the variability of the ionosphere throughout the various solar conditions is discussed. The proposed method and findings will provide a numerical reference for theoretical and computational studies of the ionosphere.

2 Methodology

2.1 Principal component analysis

The PCA technique transforms a set of correlated variables into a number of uncorrelated variables based on a new orthonormal base of minimum dimension. This method was proposed by (Pearson 1901), and it was further developed in subsequent works. From the mathematical point of view, the orthonormal base of the decomposition consists of eigenvectors of the variance–covariance matrix formed of an initial dataset. The PCA method determines the shape of the base functions from the dataset itself, and hence, it is more interesting than other techniques such as Fourier analysis when the phenomena under study are not necessarily a superposition of determinate function components. The algebraic essentials of the PCA are presented briefly in this section, and more details can be found in (Preisendorfer 1988; Storch and Zwiers 1999), for example.

Let us represent the dataset of global TEC as a matrix \(X=\{X(p,t)\}_{M\times N}\), where the subscript p indicates a grid point of GIM, the subscript t denotes the corresponding epoch time, M is the length of the time series, and N is the number of grid points of an epoch. As described in Sect. 2.2, the GIM TEC dataset in the IONEX format is reorganized according to the local time of reference. The TEC values of different local times are grouped as individual matrices X, and each of these matrices is processed separately.

As a first step in the PCA decomposition, a centering operation is performed so that the temporal means of the TEC values of each grid point p are removed to form a zero-mean series:

$$\begin{aligned} \check{X} =\{ {\check{X} \left( {p,t} \right) } \}_{M\times N} =\{ {X\left( {p,t} \right) - \overline{X(p)}} \}_{M\times N} \end{aligned}$$
(1)

where \(\check{X}\) is the centered dataset, \(\overline{X(p)} =\frac{1}{M}\mathop \sum \nolimits _{t=1}^M X\left( {p,t} \right) \).

Generally, the idea of PCA is to represent the original centered dataset \(\check{X}\) within a set of orthogonal bases as follows:

$$\begin{aligned} \check{X} \left( {p,t} \right) =\mathop \sum \nolimits _{k=1}^N a_k \left( t \right) e_k \left( p \right) \end{aligned}$$
(2)

where \(e_k \left( p \right) , k=1,\ldots ,N\) is a set of orthogonal bases of decomposition, and \(a_k \left( t \right) \) is the principal component (or time function) of the corresponding base function \(e_k\).

The orthogonal base functions and associated principal components (PCs) are empirically resolved from the variance–covariance matrix of the dataset \(\check{X}\), and it is constructed as follows:

$$\begin{aligned} S\left( {p,p{\prime }} \right) =\mathop \sum \nolimits _{t=1}^M \check{X} \left( {p,t} \right) \check{X}\left( {p{\prime },t} \right) \end{aligned}$$
(3)

where \(S\left( {p,p{\prime }} \right) \) is the covariance matrix of the dataset \(\check{X}\), and \(p{\prime }\) is one of the grid points.

Because the covariance matrix \(S\left( {p,p{\prime }} \right) \) is positive, real-valued and symmetric, it has positive, real eigenvalues and eigenvectors, the number of which is N. The most important property of the eigenvectors is that they are mutually uncorrelated over dimensions (i.e., orthogonal over the space associated with the GIM TEC grid points), as mathematically expressed in Eq. (4).

$$\begin{aligned} \mathop \sum \nolimits _{p=1}^N e_j \left( p \right) e_k \left( p \right) =\left\{ \begin{array}{ll} 1&{} {j=k} \\ 0&{} {j\ne k} \\ \end{array}\right. j,k=1,2,\ldots N \end{aligned}$$
(4)

The eigenvectors \(\{ {e_j \left( p \right) ,j=1,2,\ldots N} \}\) form a set of orthogonal base functions \(\left( {e_k } \right) \) in Eq. (2) for the expansion, and the corresponding eigenvalues \(q_{j}(j=1,\ldots ,N)\) give the amount of variability in the original dataset that is accounted for by the associated eigenvector and its time function. The eigenvalues are sorted into decreasing order, and they are normalized to have the unit summation \((\tilde{q}_j =q_j{/}\sum _i {q_i })\). The associated PCs \(a_k \left( t \right) \) are the decomposition time functions of the original dataset in the dimension of a corresponding base function (\(e_k\)). It is straightforward to demonstrate that they can be computed from the original data and the eigenvectors as:

$$\begin{aligned} a_k \left( t \right) =\mathop \sum \nolimits _{p=1}^N \check{X} \left( {p,t} \right) e_k \left( p \right) \end{aligned}$$
(5)

The principal components \(a_k \left( t \right) \) are allowed to carry the variability of the original dataset with respect to the corresponding base functions, and they are also mutually orthogonal as follows:

$$\begin{aligned} \mathop \sum \nolimits _{t=1}^M a_j \left( t \right) a_k \left( t \right) =\left\{ \begin{array}{ll} q_j&{} j=k \\ 0&{} j \ne k \\ \end{array} \right. j,k=1,2,\ldots N \end{aligned}$$
(6)

where \(q_j\) is the j-th eigenvalue of the covariance matrix S.

Therefore, the set of eigenvectors are empirical orthogonal functions: “empirical” indicates that they arise from the data itself, whereas the orthogonality means that these base functions and principal components are uncorrelated over dimensions. This fact facilitates the study of the contributions of various factors to the variability of the TEC. When interpreting the data of the decomposition modes, each eigenvector represents the corresponding component of spatial structure of the ionosphere variability, and the associated principal component \(a_k \left( t \right) \) describes its temporal evolution. The combination of an eigenvector and the associated principal component is called a mode. The eigenvalue of each mode represents the degree of variability in the corresponding dimension, and it quantifies the contribution of the mode to the total variability that is given by the sum of all eigenvalues. All modes are ordered according to decreasing eigenvalues, such that the first mode contributes the most to the total variance, the next mode has the second largest contribution, and so on. Thus, the PCA decomposition identifies the most dominant spatial and temporal structures of the ionospheric variability. It is necessary to take into account the fact that only the correlation of signs at two points on a map is essential, rather than the signs of the variations themselves. Indeed, one should multiply the obtained distribution by the time-dependent amplitude. In the decomposition of centered data, the mentioned amplitude may have any sign. However, if the signs of the variations of a given mode at two points coincide (or are opposite), then this means that some process connected with this principal component leads to the correlation (or anticorrelation) of variations at these points.

2.2 Data selection and processing

The IGS IONEX format allows the exchange of the products of global TEC maps and the associated RMS referring to predefined reference epochs and to a 2D or even 3D Earth-fixed grid. In this work, we use the GIM TEC dataset of 1999–2013 provided by CODE in the IONEX format. These global ionosphere maps show a series of snapshots of the global ionospheric TEC at reference epochs of every 2 h in the Universal Time (UT) scale. In space, they present TEC values and the associated RMS in the form of a grid of 2.5\(^{\circ }\) in latitude and 5\(^{\circ }\) in longitude. The grid ranges from –180\({^\circ }\) to 180\({^\circ }\) in geographic longitude and from –87.5\({^\circ }\) to 87.5\({^\circ }\) in geographic latitude. Note that some IGS analysis centers have, recently, also provided GIM with a time resolution of 1 h or even 15 min (García-Rigo et al. 2014).

In the PCA method, it is necessary to take into account the effectiveness of data representation and known spatiotemporal variability, to correctly analyze the dynamics of the series under consideration (Maslennikova and Bochkarev 2014). By reorganizing the global TEC maps, this study filters out the extremely dominant diurnal variation, which accounts for 99% of the variability (Wan et al. 2012; Ercha et al. 2012). The original reference epochs in UT of the TEC maps are transformed to local time of the grid points, and these TEC values are regrouped into twelve sets of TEC maps according to local time, which have the reference epochs of every 2 h local time, such as LT 1, 3, 5, ..., 23. The interpolation methods in Schaer et al. (1998) are used for interpolation in the reorganized dataset. Consequently, we construct the global TEC maps of every 2 h local time for the 15-year interval 1999–2013, and the PCA is applied separately to each of the 12 global TEC series of different local times. The results are analyzed in the following section.

3 Results and discussion

Applying the PCA decomposition to the 12 sets of TEC maps provides insight into the ionospheric behavior of 24 h local time for the period of more than one solar cycle that covers various solar conditions. The principal components (\(a_k \left( t \right) \)) and eigenvectors (\(e_k \left( p \right) \)) associated with specific datasets represent the variability of the ionosphere in temporal and spatial scales, respectively. As the geophysical references of the solar conditions during the period from 1999 to 2013, the Mg II index, sunspot number and solar radio flux F10.7 data show (Fig. 1) that the study period includes the recent extremely prolonged solar minimum (2007–2009) and active solar conditions (2000–2003 and 2012–2013) (Weber et al. 1998; Snow et al. 2014). In Sect. 3.1, we analyze TEC variations of different local times obtained from the whole 15-year dataset, and then, in Sect. 3.2, we study TEC variations year by year under different solar conditions.

Fig. 1
figure 1

Solar activity references of the period 1999–2013: Mg II index (top), sunspot numbers (middle) and solar radio flux F10.7 index (bottom)

Full size image
Table 1 Normalized contributions of the first four components to the total TEC variability
Full size table

3.1 Analysis using the whole 15-year TEC dataset

The 12 TEC series of local times of every 2 h for the whole 15-year period are processed separately with the PCA. Normalized relative variances (\(\tilde{q}_j ={q_j }/{\sum _i {q_i } }\)) for the first four modes of the PCA decomposition are presented in Table 1 and Fig. 2. It is shown that these variances are highly dependent on local times and that the most featured phases are dawn (LT4–6) and late morning to noon time (LT10–12), as these two phases have the lowest and highest ratios of variances between the first and second modes, respectively, which are 2.6 for LT4–6 and 27.4 for LT10–12. Figure 2 and Table 1 show that the first two modes account for more than 80% (dawn) to 90% (LT10–12) of the variations.

As shown in Table 1 and Fig. 2, in the diurnal cyclic trend, the first mode reaches the peak (90%) at LT10–12, and then, it slowly decreases to 85.6% in the afternoon. It jumps down to 76% at dusk (LT18–20), and then, it reduces gradually down to 55% at the dawn of the next day. As the Sun rises, the contribution of the first mode increases quickly and reaches the peak at LT10–12. With a similar timeline, the second mode has an opposite trend. That is, the second mode has the lowest variability at noon, and its variability increases in the afternoon and the nighttime. At dawn (LT4–6), the second mode has the largest variability (26%). After that, the second mode quickly weakens until noon. The observed variation trends and cyclic turning points of local time are highly related to solar elevation angles, and the diurnal cyclic trend describes the “solar-controlled” behavior of the ionosphere. The characteristic turning periods of local times (e.g., LT4–6 and 10–12) provide a reference for analyzing ionospheric physical processes (e.g., the ionization process and recombination). In the nighttime, from the sunset until dawn (LT4–6), the first PC has decreasing variances, whereas the high-order PCs have increasing variances. In the subsequent time, the first PC has rapidly increasing variances until later morning (LT10–12), and the other PCs have the opposite trend as the Sun rises in this period. The characteristic periods of dawn and later morning are highly related to the conditions of the ionosphere. After the sunset, the remaining effect of solar radiation decays, and the recombination processes gradually dominate the ionosphere until dawn, such that the ionosphere approaches a calm state during the night. After sunrise, the solar irradiance again controls the ionosphere to actively produce more ions and electrons. As a result, the first PC of the variability is more significant until the later morning. In the afternoon, the first PC of the variability is reduced as the ionizing radiation decreases.

Fig. 2
figure 2

Normalized relative variances of the first four principal components of the 12 sections, each of which covers 2 h of local time

Full size image

To verify the effectiveness, TEC values are calculated, respectively, using the PCA modes and GIM dataset, and the derived results are compared. Given a specific location \(\left( p \right) \) and time \(\left( t \right) \), TEC values are calculated as Eq. (7) using the first two PCA modes. TEC values of eight locations with respect to two reference epochs LT5 and 11 (as listed in Table 2) are calculated for 1999–2013 using the first two PCA modes and the GIM dataset, respectively. The correlation coefficients between the two TEC time series are about 0.9 or greater for these locations of different latitudes but the same longitude, as shown in Table 2.

$$\begin{aligned} \mathrm{T}\check{\mathrm{E}}\mathrm{C}\left( {p,t} \right) =\mathop \sum \nolimits _{k=1}^2 a_k \left( t \right) e_k \left( p \right) + \overline{X(p)} \end{aligned}$$
(7)

where \(\mathrm{T}\check{\mathrm{E}}\mathrm{C}\left( {p,t} \right) \) is the calculated TEC values using the first two PCA modes, and \(\overline{X(p)}\) is mean TEC value of the period as in Eq. (2).

Table 2 Pearson’s correlation coefficients (R) of the TEC series derived from the PCA modes and the GIM at two local times and different locations with the same longitude (\(E20{^\circ }\))
Full size table

The second verification method is to compare global mean TEC derived, respectively, with PCA modes and GIM dataset. We calculate TEC values of global grid points using the first two PCA modes to generate global ionosphere maps, and then estimate global mean TEC values of 1999–2013 from the PCA-derived global TEC maps and GIM dataset using the weighted-average equation Eq. (8) (Schaer 1999; Ercha et al. 2012). The global mean TEC values derived with the first two PCA modes and GIM have correlation coefficients of more than 0.99, as shown in Fig. 3.

$$\begin{aligned} \mathrm{T}\bar{\mathrm{E}}\mathrm{C}=\frac{1}{\sum {\cos \beta } }\sum _{(\lambda ,\beta )\in \Omega } {\cos (\beta )\mathrm{TEC}(\lambda ,\beta )} \end{aligned}$$
(8)

where \(\lambda ,\beta \) represent the longitude and latitude of the grid points, and \(\mathrm{TEC}(\lambda ,\beta )\) represents the TEC values of the global TEC map at the corresponding grid points.

Fig. 3
figure 3

Global mean TEC derived using the first two PCA modes and GIM and their correlations for local times of 5 (upper) and 11 (lower)

Full size image

The diurnal cyclic trend of the first two modes has two characteristic times of LT4–6 and 10–12. Figure 4 shows the time series and spatial distribution of the first mode of these two reference epochs. In space, the eigenvector of the first mode has the same sign at the global scale, which means that the first mode is positively correlated globally. However, the spatial distributions of the first mode of the two epochs are significantly different, as shown in Fig. 4. At dawn, the ionosphere of the geomagnetic Southern Hemisphere has a higher variability than that of the Northern Hemisphere. The high geomagnetic latitudes (ML) region between South America and the Antarctic has the highest variability of the first mode, and its symmetric area with respect to the geomagnetic equator, i.e., Southern Europe and the Middle East region in the Northern Hemisphere, has the lowest variability. The observed asymmetry at both the hemispheres is agreed with the results of the coupled thermosphere–ionosphere–plasmasphere modelling in Millward et al. (1996), and it is probably related to the offset of the geomagnetic axis from the Earth spin axis and the thermospheric circulation. At LT10–12, the spatial distribution of the first mode is highly dependent on geomagnetic latitude, and it is symmetric with respect to the geomagnetic equator. The most characteristic feature of the spatial distribution at LT10–12 is that the maximum variation is distributed within the band from the geomagnetic equator (GE) to ±20\(^{\circ }\) ML, which corresponds to the area where the equatorial anomaly occurs. The degree of this variation decreases with increasing ML. Note that the geomagnetic poles, geomagnetic equator and southern and northern ML 60\(^{\circ }\) and 15\(^{\circ }\) are marked in Figs. 4, 6, 7, 8, 10, 12 and 14.

Figure 5 compares the periodic spectra of temporal variation of two epochs, which is derived by applying the discrete Fourier transform (DFT) technique to the time series of PCA modes. For local time 4–6, annual variation is the most dominant power, whereas semiannual and 27-day periods have significantly increased powers at LT10–12 (daytime). Cycles of 27 days are related to the solar rotation (Schaer 1999), and the annual/semiannual cycles may be thought to involve the thermosphere, where global circulations alter neutral composition, thereby redistributing the atomic [O] and molecular [N\(_{2}\) and O\(_{2}\)] concentrations (Lean et al. 2011b; Qian et al. 2013). The distribution of atomic and molecular concentrations alters the relative ion production and loss processes and finally affects the electron density and its variation. However, its origin, which is either solar or lower atmosphere, is under debate (Rishbeth and Müller-Wodarg 2006; Hocke 2008).

As for the second mode, the spatial structure of the variation is symmetric with respect to the geomagnetic equator, and the variation is anticorrelated between the Southern and Northern Hemispheres, as shown in Fig. 6 for LT4–6 and 10–12. The difference in the spatial distribution of different sections exists in the middle latitudes of the Southern Hemisphere. The second modes of dawn and daytime have similar periodograms, and the annual cycle is the dominant period (Fig. 5).

Fig. 4
figure 4

Temporal and spatial variations (corresponding to a\(_{k}\)(t) and e\(_{k}\)(p), respectively) in the first mode of the ionosphere of 1999–2013 for local times LT4–6 (upper) and LT10–12 (lower). a\(_{k}\)(t) and e\(_{k}\)(p) are unitless, and they represent relative variations. In the subplots of the spatial variation, the southern and northern geomagnetic poles are marked by a white diamond, the geomagnetic equator and southern and northern ML 60\(^{\circ }\) are marked by a black-star line, and the southern and northern ML 15\(^{\circ }\) are marked by a white-star line

Full size image
Fig. 5
figure 5

Comparison of periodograms at periods ranging from 10 to 1000 days of the time series of the first (upper) and second (lower) PCA modes for local time sections LT4–6 and LT10–12

Full size image
Fig. 6
figure 6

Temporal and spatial variations in the second mode of the ionosphere of 1999–2013 at local time sections LT4–6 (upper) and LT10–12 (lower)

Full size image

Tables 3 and 4 list Pearson’s correlation coefficients (R) of the first four PCA modes with the solar and geomagnetic indices, respectively, for different local times. They show strong correlation between the first mode and the solar indices with the coefficients of 0.83–0.92, whereas the correlations with the geomagnetic indices are relatively weak (the coefficients 0.1–0.25). This phenomenon shows that the solar activity represented by the indices Mg II and F10.7 is the major driver of TEC variability, accounting for 55–90% at different local times. For the second, third and fourth modes, the correlations of the solar indices have prominent daytime and nighttime dependences, and the correlations of nighttime are of more significance. Negative values of correlation coefficients may imply time delays between the solar geomagnetic driver and the response of TEC, as they are not exactly simultaneous (Ercha et al. 2012). It is worth noting that the third mode has significant temporal correlation with geomagnetic activity that is indicated by indices such as A\(_{p}\) and D\(_\mathrm{st}\) (Ahn 2002; Le et al. 2009). The correlation coefficients are 0.15–0.4 for the A\(_{p}\) index (a daily average level of geomagnetic activity) and 0.25–0.6 for the D\(_{st}\) index (which represents the intensity of the ring current), as shown in Table 4. The spatial distribution of the third mode is highly dependent on geomagnetic latitudes, and the latitude band from North ML 15\({^\circ }\) to South ML 20\({^\circ }\) is anticorrelated with the high-ML regions in both hemispheres, as shown in Fig. 7. This may imply that geomagnetic influences on TEC are highly dependent on geomagnetic latitudes. The trends in these effects may be opposite in the geomagnetic equatorial region and the high-ML areas of both hemispheres, although Lean et al. (2011b) reported that general effects of geomagnetic activity on daily averaged global TEC are relatively modest depletions, with a maximum reduction of 11 TECU (occurring in the Halloween Storms of 2003) in the 15-year period.

Table 3 Pearson’s correlation coefficients (R) of the first four PCA modes with the solar indices (Mg II and F10.7 Index) for different local times
Full size table
Table 4 Pearson’s correlation coefficients (R) of the first four PCA modes with the geomagnetic indices (Ap and Dst Index) for different local times
Full size table
Fig. 7
figure 7

Spatial variations in the third mode of local times showing high dependence on geomagnetic latitudes: LT4–6 (left) and LT10–12 (right)

Full size image

3.2 Analysis using yearly TEC series

This section analyses temporal variation and spatial structure of the TEC series of every individual year from 1999 to 2013. The 12 TEC datasets of every year that refer to every 2 h of local time are processed separately through the PCA method. It is found that the temporal–spatial variability of the ionosphere varies from year to year with solar conditions. This section selectively presents the results of four years that represent the four main phases of the solar cycle: peak solar activity (2001), declining solar activity (2005), calm solar activity (2008) and ascending solar activity (2010). Spatial and temporal variations of the first two modes are illustrated in Figs. 8, 9, 10, 11, 12, 13, 14 and 15 for the two characteristic sections LT4–6 and LT10–12 of the four selected years.

For dawn (LT4–6), the first mode mainly represents annual variation (as shown in Fig. 8), and its spatial variation (illustrated in Fig. 9) is anticorrelated and asymmetric in the Southern and Northern Hemispheres. The amplitude (in Fig. 8) multiplied by its eigenvector (in Fig. 9) shows that annual TEC variation in the first mode has maximal values in local summer (June in the Northern Hemisphere and November–January in the Southern Hemisphere) and minimal values in local winter in both the hemispheres. The degree of the asymmetry increases with solar activity. The maximum variation is distributed in the high-ML regions in both Southern and Northern Hemispheres. As the solar cycle comes to the maximum, the area of higher variation spreads from high ML toward lower ML in both hemispheres, and the variation at mid-ML, such as the Pacific Ocean and Atlantic Ocean regions, becomes higher compared to variations of the same areas in 2005 and 2008. When solar activity is at its peak (2001) and ascending (2010), the rate of change of TEC in September–October is much higher than in the other phases of the solar cycle, such as in 2005 and 2008, as shown in Fig. 8.

Fig. 8
figure 8

Temporal variations in the first mode for LT4–6 for the four selected years that represent the four major phases of the solar cycle

Full size image
Fig. 9
figure 9

Spatial variations in the first mode of the ionosphere for the four selected years for the local time section 4–6. In the four subplots, the colormap scales are unified for comparison

Full size image

For LT10–12, the first mode represents temporal variation consisting of a main semiannual variation and secondary annual variation, as shown in Fig. 10. Relative rate factors of the semiannual and annual variations in the principal component of the first mode vary with different phases of the solar cycle. In space, as shown in Fig.11, during the declining, calm and ascending periods of the solar cycle, the middle- and high-ML areas of the Northern Hemisphere have a spatial distribution of the first mode that is opposite to that of the other areas of the globe. When solar activity increasingly approaches the maxima, the variations in the northern middle- and high-ML areas become gradually coherent with the other regions. In general, the power distribution decreases with increasing ML in both hemispheres, which implies positive effects of solar irradiance on TEC variations. In high solar activity (2001), it shows that the semiannual anomaly of TEC (where TEC is greater at the equinox than at the solstice, Meza et al. 2012) exists globally, and it is two times larger at equatorial low geomagnetic latitudes than at high latitudes in the daytime.

Fig. 10
figure 10

Temporal variations in the first mode for LT10–12 in the four selected years

Full size image
Fig. 11
figure 11

Spatial variations in the first mode of the ionosphere for the four selected years for local time 10–12. In the four subplots, the colormap scales are unified for comparison

Full size image
Fig. 12
figure 12

Temporal variations in the second mode for LT4–6 for the four selected years

Full size image
Fig. 13
figure 13

Spatial variations in the second mode of the ionosphere for the four selected years for local time 4–6. In the four subplots, the colormap scales are unified for comparison

Full size image
Fig. 14
figure 14

Temporal variations in the second mode for LT10–12 for the four selected years

Full size image
Fig. 15
figure 15

Spatial variations in the second mode of the ionosphere for the four selected years for local time 10–12. In the four subplots, the colormap scales are unified for comparison

Full size image

As for the second mode, temporal variations at dawn in Fig.12 represent a main semiannual variation demodulated with an annual variation, and the amplitudes are positively correlated with the solar conditions, which will be illustrated later in Fig. 16. In high solar activity, the semiannual TEC anomaly is observed in mid and high geomagnetic latitudes of the Southern Hemisphere at dawn, and the larger variability is located at the band region around southern ML 60\({^\circ }\). In contrast, in the equatorial region and the Northern Hemisphere, TEC is greater at the solstice than at the equinox, as the spatial variances have negative values in Fig. 13. Under calm solar activity (2008), the variation of the equatorial low-ML region is anticorrelated with high-ML areas of both hemispheres, and the semiannual TEC anomaly is observed at dawn in the equatorial low ML region, although its amplitude is much lower than in high solar activity.

Fig. 16
figure 16

The rate factors of the annual and semiannual oscillation components in the first mode (upper) and the second mode (lower) for different years and local times. For each year, 12 bars are related to the 12 reference local times in sequence. The colorful parts indicate the rate factors of the annual oscillation (\(R_a\)), and the grey parts indicate the rate factors of the semiannual oscillation (\(R_s\)). The magenta bar corresponds to local time 12:00

Full size image

At LT10–12, the second mode shows annual variation with a TEC maximum at local summer in both hemispheres (Fig. 14). For most of the solar cycle, spatial variations of the second mode are highly dependent on geomagnetic latitudes, being anticorrelated in two parts separated by southern 15\(^{\circ }\) ML (Fig. 15). An exception at high solar activity is that the spatial variation in the northern high-ML region is anticorrelated with that in low-ML areas.

The first two modes, modulated with both annual and semiannual variations, account for an integrated relative variability of 70–90%, varying with solar conditions and local times. Annual and semiannual oscillation components of the first two modes are demodulated from the principal components using the Fourier analysis technique (Liu 2011; Liu et al. 2014b) as follows:

$$\begin{aligned} \begin{aligned} \psi (t)&=C_0 +\sum _{i=1}^N {\left[ {C_i \cos \left( {w_i t} \right) +S_i \sin \left( {w_i t} \right) } \right] }\\&= {C_0 } +\sum _{i=1}^N {\left[ {A_i \cos \left( {w_i t-\phi _i } \right) } \right] }\\ w_i&={2\pi }/{p_i}, A_i =\sqrt{C_i^2 +S_i^2 } \end{aligned} \end{aligned}$$
(9)

where \(\psi (t)\) is a time series of the sliding window, \(C_0\) is the average of the time series in the sliding window, N is the number of components (\(N=2\) for annual and semiannual components), \(w_i\) is the angular frequency of a component with a period \(p_i\), and \(C_i\) and \(S_i\) are coefficients to be estimated, which define the amplitude (\(A_i\)) and phase (\(\phi _i\)) of the corresponding components.

With the amplitude parameter (\(A_i\)) in Eq. (9), normalized rate factors R of annual and semiannual oscillations are calculated as: \(R_a =\frac{A_1 }{A_1 +A_2 }, R_s =\frac{A_2 }{A_1 +A_2}\), where \(R_a\) and \(R_s\) are the rate factors of the annual and semiannual oscillations, respectively (Liu 2014a). It is evident that, for each specific section of local time of a specific year, \(R_a +R_s =1\). Figure 16 illustrates the rate factors of the annual and semiannual oscillations in the first two modes at different years and local times. These factors are evidently dependent on the solar conditions. For example, for the first mode in daytime, the semiannual term dominates during the Solar Maximum (e.g., years 2000–2002, 2011–2012), and annual oscillation dominates in the other years. Overall, these aspects of both oscillations demonstrate distinct trends during the daytime and nighttime (from dusk to dawn) across the different phases of the solar cycle. An exception is that under active solar conditions (2000–2003), the variability of the earlier nighttime (LT18–24, represented in red bars) remains similar to that of the daytime. For example, in the upper part of Fig. 16, the red bars (LT18–24) of 2000–2003 are comparable to the yellow and magenta bars (LT10–18), while in the other years, the red bars are much higher than the yellow and magenta bars, and they are similar to the blue bars (later nighttime). It shows that extremely strong solar irradiance of the daytime has a residual effect on the ionosphere, even after sunset.

Finally, integrated annual and semiannual oscillations in the first two modes of different years and local times are calculated as Eq. (10) using these rate factors, and the results are shown in Fig. 17.

$$\begin{aligned} {\begin{array}{ll} {V_a =R_a^1 *q_1 +R_a^2 *q_2 } \\ {V_s =R_s^1 *q_1 +R_s^2 *q_2 } \\ \end{array} } \end{aligned}$$
(10)

where \(V_a\) and \(V_s\) are the normalized amplitudes of the annual and semiannual oscillations, respectively, \(R_a^i ,i=1,2\) are the rate factors of the annual oscillation in the first and second modes, respectively, \(R_s^i ,i=1,2\) are the rate factors of the semiannual oscillation in the first and second modes, respectively, and \(q_i ,i=1,2\) are the eigenvalues of the first and second modes, respectively.

Fig. 17
figure 17

Normalized amplitudes (%) of the annual and semiannual oscillations at different local times during 1999–2013

Full size image

Figure 17 presents the normalized amplitudes of the annual and semiannual oscillations of different years and local times, which are decomposed from the first two modes of the PCA. It shows that annual and semiannual oscillations have different relative amplitudes at different phases of the solar cycle. In general, under conditions of lower solar activity (e.g., during the recent solar minima of 2007–2009), the annual oscillation has higher relative amplitudes than those of active solar conditions. Amplitudes of semiannual oscillation are higher during high solar activity than during lower solar activity. Lean et al. (2011a) also reached similar conclusions regarding the overall ionosphere by simulating annual and semiannual oscillations of daily average global TEC with an empirical regression model. Figure 17 shows that the annual oscillation in the nighttime is much higher than that of the daytime. The peak annual oscillation variation occurs at dawn (LT4–6), and it has 1.5–2 times the amplitude of the annual variation at noon (LT12–14) on average throughout the various solar conditions. In contrast, the semiannual oscillation of the daytime is much higher than that of the nighttime. The peak semiannual oscillation variation occurs in the afternoon (LT14–16) under active solar conditions (1999–2003, 2011–2013) and at LT12–14 under lower solar activity. The amplitude of the peak semiannual oscillation variation is 2–3 times higher than that of the dawn. When it is in the early stage of the active solar condition, such as in 2000 and 2011, the semiannual oscillation variation of LT14–16 has the highest amplitude, and the amplitude is even higher than that of the annual oscillation of the same period. As solar activity reaches the peak (e.g., 2001–2002 and 2012–2013), the amplitude of the semiannual oscillation decreases. Daytime solar irradiance under active solar conditions has a residual effect on semiannual oscillation of TEC even after the sunset when the solar cycle is at the maximum phase. The differences in semiannual oscillation in daytime and nighttime are consistent with that in Fig. 5, and they may be attributed to composition changes and energy transport processes (Rishbeth and Müller-Wodarg 2006; Lean et al. 2011b; Qian et al. 2013).

4 Conclusions

This study analyses the spatial and temporal variability of the ionospheric TEC under the various solar conditions using the statistical analysis technique of principal component analysis and the GIM TEC maps derived from global GNSS measurements. The dataset of global TEC maps of 1999–2013, which covers the period of more than one solar cycle, is first reorganized according to the local times. The TEC series of different local times are then processed separately with the PCA technique at two time scales: one is the whole 15 years of the period of study, and the other is every individual year. Both the spatial structure and the temporal variation of the variability of the ionosphere are analysed under the two time scales. This work presented quantitative outcomes and new observations related to the variability of the global ionosphere, thanks to the twofold diversity from previous studies: the new data representation method and the long-term dataset that covers various solar conditions of more than one solar cycle. The statistical analyses of the ionosphere variability are summarized as follows.

First, it is shown that the temporal–spatial variability of the ionosphere varies in a diurnal cyclic trend, and dawn (e.g., LT4–6) and the late morning (LT10–12) are more remarkable characteristic times. This is different from previous studies that empirically considered LT12 and 22 as characteristic epochs. The presented characteristic times and cyclic trend are highly related to the diurnal position change of the Sun that affects the dominant processes governing the production and loss of ions and electrons in the ionosphere.

Second, the PCA performs an orthogonal decomposition of the data itself, and it does not force any specific form of components. This feature provides the PCA technique with an ability to freely decompose the principal components of minimum dimensions, from which it is possible to observe the asymmetries such as the semiannual TEC anomaly. It shows that, in high solar activity, the semiannual anomaly of TEC exists globally in the daytime, and the variability is two times larger at equatorial low geomagnetic latitudes than at high latitudes. At dawn, the semiannual TEC anomaly is observed in mid and high geomagnetic latitudes of the Southern Hemisphere, with larger variabilities around southern ML 60\({^\circ }\). In calm solar activity, the semiannual TEC anomaly is observed in the equatorial low ML region at dawn, and its amplitude is much lower than at high solar activity. Similar conclusions were reached by Meza et al. (2012) using TEC data and Millward et al. (1996) using NmF2 data.

Third, the first multiple modes contribute to most of the temporal and spatial variability, which is dependent on the solar conditions and local times. The diurnal variation of TEC has been filtered out in this study. The first and second modes account for 80–90% of the total variability over a solar cycle, and the corresponding spatial and temporal variations are presented in terms of the characteristic local times and typical phases of the solar cycle. The amplitudes of the annual and semiannual oscillations demodulated from the first two modes are quantitatively analysed. It is found that the ionosphere behaves with evidently different patterns of variability for the daytime and nighttime, which is presented in this work. An exception is that, under maximum solar cycle conditions, there is a residual effect of extremely strong daytime solar irradiance on the variability of the ionosphere even after sunset (e.g., from LT18 to 24).