Eleven years of global total electron content (TEC) data derived from the assimilated thermosphere–ionosphere electrodynamics general circulation model are analyzed using empirical orthogonal function (EOF) decomposition and the corresponding principal component analysis (PCA) technique. For the daily averaged TEC field, the first EOF explains more than 89 % and the first four EOFs explain more than 98 % of the total variance of the TEC field, indicating an effective data compression and clear separation of different physical processes. The effectiveness of the PCA technique for TEC is nearly insensitive to the horizontal resolution and the length of the data records. When the PCA is applied to global TEC including local-time variations, the rich spatial and temporal variations of field can be represented by the first three EOFs that explain 88 % of the total variance. The spectral analysis of the time series of the EOF coefficients reveals how different mechanisms such as solar flux variation, change in the orbital declination, nonlinear mode coupling and geomagnetic activity are separated and expressed in different EOFs. This work demonstrates the usefulness of using the PCA technique to assimilate and monitor the global TEC field.

The ionosphere is highly variable and has a complex system of drivers including variable solar radiation plus geomagnetic activity from the upper atmosphere and momentum and energy fluxes associated with neutral wind dynamics from the lower atmosphere. While magnetospheric forcing dominates the variability at high latitudes in the ionosphere, photochemistry and neutral dynamics play dominant roles in the ionospheric structure and variability at mid- and low latitudes. One critical quantity describing the ionosphere and its variability is the total electron content (TEC). A variety of in situ and remote-sensing techniques has been employed to study the Earth's ionosphere in terms of the electron density. The method and platform used for measurement determine resolution in time and space, with the measurement often being distributed unevenly. Unless assimilated general circulation models are used, no one method effectively allows for the sampling of large areas of the ionosphere with high and uniform resolution in both time and space. Since ionospheric electronic densities respond to a complex set of highly variable driving mechanisms, the global characterization of the response to solar variability posts a significant challenge.

The X-ray and ultraviolet solar irradiance which creates the ionosphere varies on all timescales: with an 11-year solar cycle associated with the 22-year magnetic cycle of the solar dynamo, with a quasi-27-day period due to active regions rotating with the sun, and on the order of minutes to hours as eruptions occur on the disk of the sun (Lean, 1987; Tobiska, 1993). The ionosphere varies on all these timescales in response to the solar inputs, while the geographic relationship of the Earth's orbit, rotation and seasonal tilt creates the solar zenith angle dependence that yields the observed diurnal, seasonal and annual variations in ionospheric density. This is further complicated by the tilt of the magnetic field of Earth, the magnetospheric inputs that drive the ionosphere at high latitudes, and the neutral dynamics generated by solar and magnetospheric forcing, indicating significant longitudinal and hemispherical asymmetries in space. Thus, the multi-scale variability and complexity in both time and space induced by different kinds of physical processes are the main characteristics of the ionosphere.

In this paper, we perform a set of multi-year assimilated runs of the thermosphere–ionosphere electrodynamics general circulation model (TIEGCM) (Roble et al., 1988; Richmond et al., 1992) driven by a lower boundary condition of tidal forcing derived from the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument (Talaat and Lieberman, 1999; Talaat et al., 2001). To effectively decompose and understand the complicated temporal and spatial variations of the ionosphere, we apply the empirical orthogonal function (EOF) and the associated principal component analysis (PCA) to the TEC fields derived from the TIEGCM runs. Section 2 describes the EOF and PCA notations and their application to different TEC fields. Section 3 shows how the EOF and PCA are able to reveal major physical processes associated with different EOFs derived from data fields. Section 4 provides a final summary and conclusions.

Explained variance

First four EOFs for the 11-year daily averaged global TEC as functions of longitude and geographic latitude calculated by the spatial and temporal settings given in Fig. 1.

Same as Fig. 1 except for a 6-year time period (2190 days) from 1999 to 2004.

Same as Fig. 2 except for a 6-year time period (2190 days) from 1999 to 2004.

Same as Fig. 1 except for a spatial resolution of 36

Same as Fig. 2 except for a spatial resolution of 36

The empirical orthogonal function (EOF) and the corresponding principal
component analysis (PCA) are the standard statistical techniques for
analyzing atmospheric data since being introduced by E. N. Lorenz in 1956
(Wilks, 2006). The PCA technique decomposes a given spatial–temporal field
such as TEC into a set of base functions called EOFs that is internally
determined from the data sets. Let

Explained variance

Equation (

We ran the TIEGCM with realistic solar inputs and tidal forcing over the
11-year time period from 1990 to 2010. The horizontal resolution of the TEC
field is

Since the EOFs are determined internally from the data, it is worthwhile
examining in this application how sensitive the derived

We have already indicated that it is the high correlation of the
spatial–temporal variations of the field that leads to a few EOFs being able
to explain the majority of the variance. Such a high correlation in data is
also reflected by the fact that the first few EOFs capture relatively
smooth or large-scale structures of the field as shown in Figs. 2 and 4 for
the first three EOFs. Since the large-scale structures can also be
effectively represented by a lower-resolution field, one would expect a
similar PCA result for the same field with a lower spatial resolution. In
Figs. 5 and 6, we show the same PCA as in Figs. 1 and 2 except for a lower
horizontal resolution of

Next, we examine the TEC field that includes the local-time variations. The
PCA is applied to the

First six EOFs for the 11-year global TEC as functions of longitude and geographic latitude calculated by the spatial and temporal settings given in Fig. 7.

Time series of the principal components of the PCA for the first
four EOFs shown in Fig. 2:

Daily mean time series of the longitudinally averaged TEC at five different latitudes: 0

One major advantage of the PCA technique is the data compression, so the
physical field is effectively projected onto a few modes that include the
majority of the variance of the original field. We have already shown that the four EOFs in Fig. 2 and the six EOFs in Fig. 8 contain 98.2 and 93.6 % of
the total variance, respectively. Based on their spatial patterns, we have
also indicated in the above analysis that there exist clear physical
mechanisms that drive the different EOFs solely derived from the data. To
illustrate the roles played by different physical mechanisms in extracting
EOFs, we show in Fig. 9 the time series of the decomposition coefficients

To further demonstrate the benefit of the PCA technique in the current
application, we show in Fig. 10 the time series of the longitudinally
averaged TEC in the unit of TECU (1 TECU

Differences in TEC time series between those shown in Fig. 10 and
the ones calculated based on highly truncated expansion Eq. (

Fourier power spectra of the first four PCs shown in Fig. 9
(panel

Time series of the principal components of the PCA for the first
EOFs shown in Fig. 8:

Scatterplots between the daily averaged

In Fig. 12, we show the Fourier power spectra of the time series shown in Figs. 9 and 10. There are four vertical dashed lines denoting frequencies corresponding to 2-year, annual, semiannual and 27-day periods of oscillations. Now, it is more clearly and quantitatively shown in the figure that the major spectral features among different EOFs are different, whereas those among the longitudinally averaged TEC at different latitudes are nearly the same. The first EOF spectrum peaks at all four frequencies, whereas the second EOF only shows one striking peak in the annual period that is dominant over the entire spectral domain. For the third and fourth EOFs, only two peaks at semiannual and 27-day periods are noticeable. Since different spectral characters often correspond to different driving mechanisms, Fig. 12 suggests that the PCA technique is able to differentiate directly between physical mechanisms and the data.

Two-hour resolution time series of the longitudinally averaged
TEC at five different latitudes: 0

Figure 13 shows the time series of PCA for the case of Fig. 8, which includes
local-time variation

Fourier power spectra of the first four PCs shown in Fig. 13
(panel

In Fig. 16, we show the Fourier power spectra for the time series shown in
Figs. 13 and 15. The six vertical lines denote the peak frequencies that
correspond to the following periods: annual, semiannual, 27-day, 1-day,
0.5-day and 0.25-day. Panel

In this study, we apply the EOF and PCA to the global TEC data derived from TIEGCM forced under the realistic solar inputs from above the SABER-observed tidal waves from below. We demonstrate the effectiveness of the EOF decomposition of the ionospheric variations in both time and space. It is shown that for the daily averaged TEC field, the first EOF explains more than 89 % and the first four EOFs explain more than 98 % of the total variance of the TEC field. When PCA is coupled with the spectral analysis of the time series of the EOF coefficients, it is also shown that the EOF analysis is not only a data compression technique but also a powerful tool to objectively reveal the relative importance of individual physical mechanisms (such as solar flux variation, change in the orbital declination, nonlinear mode coupling and geomagnetic activity) that are responsible for the total TEC variance.

The TIEGCM output TEC fields that generated all the figures in this paper are available from Xun Zhu (xun.zhu@jhuapl.edu) upon request.

This research was supported by NASA Living With a Star Program under grants NNX09AJ61G and NNX13AF91G and the Heliophysics Supporting Research program under grant NNX16AG68G to the Johns Hopkins University Applied Physics Laboratory. Comments on the paper by two anonymous reviewers are greatly appreciated. The topical editor, K. Shiokawa, thanks two anonymous referees for help in evaluating this paper.