A solar eclipse is the feature that the Moon’s shadow projects on the Earth, leading a short shelter from the Sun. This provides us a good opportunity to investigate the ionospheric response to the solar eclipse. The solar eclipse-induced ionospheric electron density anomalies have been observed and discussed by many previous studies (e.g., Liu et al. 1998*; *Farges et al. 2001*; *Goncharenko et al. 2018). These anomalies are roughly divided to two periods, the eclipse and the post-eclipse. At the start of solar eclipse, the plasma density in the ionospheric *E* and *F1* regions will gradually decrease due to the loss of solar ionization rate. At the same time, the upward plasma diffusion at the ionospheric *F2* region becomes weaker. The plasma in the plasmasphere will downward supply the ionosphere by the gravity effect, which is similar to the nighttime ionosphere. This plasma downward diffusion from the plasmasphere will moderate the plasma reduction on the ionospheric *F2* region during the solar eclipse period (Le et al. 2009). After the moon shadow passes by, the ionospheric plasma will be increased due to the restoration of solar ionization rate. The plasma density after the solar eclipse was even higher than the normal ionospheric day, observed by previous studies (Huang et al. 1999; Jakowski et al. 2008; Cherniak and Zakharenkova, 2018). Furthermore, this enhancement of plasma density will continue few hours after the solar eclipse (Cherniak and Zakharenkova, 2018).

The ionospheric plasma density variations triggered by the total solar eclipse event on 21 August 2017 over the central United States have been investigated by many studies using the ground-based global positioning system (GPS) total electron content (TEC) observations, showing the TEC depletion, the large-scale traveling ionospheric disturbances (LSTIDs), the wave-like electron density structures, and the TEC enhancement after the eclipse (Coster et al. 2017; Nayak and Yiğit 2018; Zhang et al. 2017; Cherniak and Zakharenkova, 2018; Goncharenko et al. 2018; Sun et al. 2018). However, the TEC is the integrated observation of ionospheric electron density along the ray path from a GPS satellite to a ground-based GPS receiver. It is hard to study the eclipse-induced electron density structure variation, especially in the vertical direction, by directly using the TEC observations. In this paper, a GPS tomography method (Seemala et al. 2014; Chen et al. 2016; Saito et al. 2017) is used to investigate the three-dimensional ionospheric electron density variations during the 2017 August solar eclipse event. Furthermore, a procedure of Fourier analysis (Kuo et al. 1993; Liu et al. 1998; 2007) is conducted to derive power spectra, vertical phase, and group velocities of the eclipse-triggered electron density variations.

### GNSS ionospheric tomography

Around the mid-1980s, Austen et al. (1986, 1988) first suggested and applied the tomographic methodology to reconstruct the two-dimensional structure of the ionospheric electron density by using the line-of-sight TEC observations from Naval Navigational Satellite System (NNSS). The accuracy of ionospheric tomography is limited by the background ionospheric model, the incomplete ray geometry, and the measurement error, resulting in an ill-posed inverse problem. Many ionospheric tomography methods were published trying to reconstruct the ionospheric electron density structures (e.g., Austen et al. 1988*; *Raymund et al. 1994*; *Pryse 2003).

Recently, a new GPS ionospheric tomography algorithm has been developed by employing the dense ground-based GPS networks as well as the multiple Global Navigation Satellite System (GNSS) for solving this ill-posed inverse problem and further to reconstruct the three-dimensional structure of ionospheric electron density. In order to reduce the effect of background ionospheric or plasmaspheric electron density models as the initial condition on the reconstruction accuracy, a constrained least-square method is used in this GPS ionospheric tomography without using any initial model guess (Seemala et al. 2014). Later, Chen et al. (2016) applied this constrained GPS tomography method and successfully reconstructed the ionospheric electron density perturbation at the scale size of ~ 200 km in wavelength around Japan region, the medium-scale traveling ionospheric disturbance (MSTID), showing the capability of this GPS tomography for the investigation of disturbed ionosphere. Using the TEC observations provided by Japan GPS Earth Observation Network (GEONET), a near real-time ionospheric tomography system has been developed to continuously monitors the three-dimensional electron density variations over Japan every 15 min (Saito et al. 2017).

Similar to the Japan GEONET, there are ~ 2000 ground-based GNSS receivers located at North America (shown in Fig. 1), which provide lots of ionospheric TEC observations and are suitable to employ the GPS tomography for reconstructing the three-dimensional electron density structure. In this study, we applied the GPS tomography algorithm by the method of Saito et al. (2017) over the North America region during the total solar eclipse period on 21 August 2017. Furthermore, not only the GPS signals, but also the GLObal NAvigation Satellite System (GLONASS) signals are used to calculate the ionospheric TEC values along the ray paths from the satellites to the ground receivers for the first time. The increase of line-of-sight TEC observations can further reduce the effect of incomplete observational geometry on the electron density reconstruction accuracy. An elevation mask angle of 30° is set for the GNSS TEC observations used in this study.

The three-dimensional space of tomography is from − 125°E to − 71°E in longitude, 24°N to 52°N in latitude, and 80 km to 20,000 km in altitude. The resolutions are defined differently for trying to balance the observational path number in each grid and shown as follows. The longitudinal resolutions are set to 3° within − 112°E to − 91°E range and 2° outside (shown in Fig. 1). The latitudinal resolutions are set to 1° within 30°N to 44°N range and 2° outside (shown in Fig. 1). The vertical resolutions are 20 km within 80 km to 600 km range, 50 km within 650 km to 900 km range, 100 km within 1000 km to 2000 km range, and 5000 km within 5000 to 20,000 km range, respectively. Therefore, the total number of grids is 25,344 in the tomography space. A constraint term is applied in the cost function as shown as:

$$J\left(x\right)={\Vert b-Ax\Vert }^{2}+{\lambda \Vert Wx\Vert }^{2},$$

(1)

where *J(x)* is the cost function, showing the combination of least-square fitting (first term) and the constraint condition (second term) at the right-hand side. The parameters of *x,* and *A* indicate the matrixes of the electron concentration and the length of path in each grid, respectively. *b* matrix is the observed TEC values along the ray paths from the satellites to the ground GNSS receivers. The parameter, \(\lambda \), is determined empirically to balance the least-square term and the constraint condition (Saito et al. 2017). *W* is the constraint matrix, which is a zero matrix but with the value of − 1 at the six neighbor grids of \({x}_{ij}\) (north, south, east, west, up, and down sides) and 6 at the center grid of \({x}_{i}\) (see the appendix in Chen et al., 2016). Therefore, the definition of constraint condition in the cost function is the total electron density differences of each grid from its six neighbor grids, showing as:

$$Wx={\sum }_{i=1}^{N}{\sum }_{j=1}^{6}{C}_{ij}\left({x}_{i}-{x}_{ij}\right)$$

(2)

where *N* is the total number grids, 25,344 in this case. *C* is the constraint parameter, which is determined as a function of latitude, longitude and altitude based on the empirical electron density model, NeQuick (DiGiovanni and Radicella, 1990; Radicella and Zhang, 1995). It is the weighting of constraint conditions in different altitudes (Seemala et al. 2014; Saito et al. 2017). A relatively small constraint parameter is applied around the high electron density regions, such as ionospheric *F* region and low-latitude regions, to allow the spacial variety in the electron density. Therefore, the electron density in each grid \(x\), in Eq. (1) can be solved as,

$$x={({A}^{T}A+\lambda {W}^{T}W)}^{-1}{A}^{T}b.$$

(3)

### Tomography results

Figure 2 shows the time variations of the reconstructed electron densities by the 3D GNSS ionospheric tomography method at 260 km altitude plane (a–d), 42^{o}N latitude plane (e–h), and − 100^{o}E longitude plane (i–l), respectively. At 1720 UT, the ionospheric electron density depletion started from the north-west of American where is around the location of the solar eclipse maximum obscuration point (Fig. 2a). From the altitude–longitude plane along the latitude of 42^{o}N (Fig. 2e), it also shows the lower electron densities appeared at the western American than the eastern. Then, the maximum obscuration point moved to the location around the latitude of 42^{o}N and the longitude of − 100^{o}E at 1755 UT (Fig. 2b). The maximum electron density depletion appeared at the north-west of obscuration region (around the longitude of − 110^{o}E and the latitude of 46^{o}N), which also can be seen in the fixed altitude and latitude planes (Fig. 2b, f). Besides, the start time of eclipse at the location of (42^{o}N, − 70^{o}E) was ~ 1730 UT. It shows that the electron density depletion (as compared to Fig. 2e) in Fig. 2f at the east edge, − 70^{o}E longitude, was affected by the decrease of solar radiation. At 1830 UT, the maximum electron density depletion moved to the longitude of − 100^{o}E, following the movement of solar eclipse maximum obscuration (Fig. 2c). After the solar eclipse, 1945 UT, the solar ionization rate returned to the normal level and triggered the increases of ionospheric electron density at all the American region (Fig. 2d). These electron density reductions and increases are also shown in the fixed latitude (Fig. 2e–h) and longitude (Fig. 2i–l) planes during and after the solar eclipse.

In order to investigate the ionospheric response to the total solar eclipse and compare with previous studies (Nayak and Yiğit 2018; Uma et al. 2020), the electron density variation is calculated by subtracting the reconstructed electron densities on the previous day (20 August 2017), called as reference day afterwards, from those on the eclipse day (21 August 2017). Furthermore, the difference percentage of electron density is defined as

$$\frac{{Ne}_{d}-{Ne}_{d-1}}{{Ne}_{d-1}}\times 100,$$

(4)

where *Ne* is the electron density. The index words in Eq. (4), *d* and *d*-1, indicate the eclipse day and the reference day, respectively. Figure 3 shows the electron density variations as well as the difference percentage at 1755 UT at 260 km altitude plane (a–b), 42^{o}N latitudinal plane (c–d), and − 100^{o}E longitudinal plane (e–f). Figure 3a clearly shows that the electron density depletions appeared around the lower latitude regions compared with the reference day. The electron densities variations at the 42^{o}N latitudinal plane (Fig. 3c) and the − 100^{o}E longitude plane (Fig. 3e) further show the most density depletion appeared at the west and south of maximum obscuration, respectively, and mainly around the altitude of 260 km. The difference percentage of electron density (Fig. 3d, f) shows that the more electron density depletion appeared at the lower altitude regions on the west side of − 100^{o}E longitude. The most electron density depletion is around 40% compared with the reference day.

The time series of electron density difference percentages at the fixed longitude of − 100^{o}E on 260 km altitude (Fig. 4a) and 42^{o}N latitude planes (Fig. 4b) are presented, respectively. Compared with the electron density on and before the eclipse day, the electron density depletion, denoted by the contour curve of 0%, started from the lower latitude shown in Fig. 4a region around 1700 UT. In addition, the electron density depletion of 0% contour curve in Fig. 4b first appeared at the higher altitude and then extended to the lower altitude after the start time of eclipse, around 1631 UT. After 1730 UT, the contour curves of − 20% and − 30% in Fig. 4b show that the electron density reduction is upward propagation until 1845 UT. The results further show that the electron density depletion was mainly concentrated on the altitudes lower than 600 km and reached its maximum depletion at 140 km altitude at 1825 UT, about 30 min after the time of eclipse maximum obscuration (1755 UT, denoted by the vertical white line). The maximum depletion was around − 39.9%. After 1845 UT, the electron density continuously returned to the normal level until 2030 UT. Due to the southward move of solar eclipse obscuration shown by the red curve in Fig. 4a, the electron density at lower latitude region returned to its normal level until 2100 UT.

Figure 5 shows the comparison of electron densities between the eclipse day and the reference day (a) as well as their difference percentage (b) at 260 km altitude at the fixed location of (42^{o}N, − 100^{o}E). Results show that the maximum electron density depletion on the eclipse day appeared around 15–30 min after the totality (vertical-black-dashed line in Fig. 5a) and reached around 40% compared with the reference day (Fig. 5b). The depletion rate of electron density at 260 km is around 5.33 \(\times \) 10^{10} ele/m^{3}/hr (= (2.3 \(\times \) 10^{11}–1.5 \(\times \) 10^{11})/1.5 h). Later, the electron density returned to the reference day level around 2010 UT (~ 1330 LT) with an increasing rate of 7.19 \(\times \) 10^{10} ele/m^{3}/h (= (2.7 \(\times \) 10^{11}–1.5 \(\times \) 10^{11})/1.67 h), and then kept its concentration for a few hours (Fig. 5a). These time delay and the electron density recovery features are clearly shown in the difference percentage (Fig. 5b). Noted that the increase of difference percentage after 2010 UT in Fig. 5b is due to the coupled effects of the decrease of electron density on the reference day and the preservation of electron density on the eclipse day. After 2010 UT (~ 1320 LT), because of the descending solar zenith angle, the photochemical ionization rate becomes weak and then results in the electron density decrease on the reference day. However, after the maximum obscuration on the eclipse day, the electron density can be preserved a long time by the effects of downward plasma fluxes from the topside ionosphere, like the downward plasma diffusion after sunset, and the production of plasma by the recovery of ionization in the bottom side ionosphere (Cherniak and Zakharenkova, 2018).