AbstractThe three-dimensional ionospheric tomography (3DCIT) algorithm based on Global Navigation Satellite System (GNSS) observations have been developed into an effective tool for ionospheric monitoring in recent years. However, because the rays that come into or come out from the side of the inversion region cannot be used, the distribution of the rays in the edge and bottom part of the inversion region is scarce and the electron density cannot be effectively improved in the inversion process. We present a three-dimensional tomography algorithm with side rays (3DCIT-SR) applying the side rays to the inversion. The partial slant total electron content (STEC) of side rays in the inversion region is obtained based on the NeQuick2 model and GNSS-STEC. The simulation experiment results show that the algorithm can effectively improve the distribution of GNSS rays in the inversion region. Meanwhile, the iteration accuracy has also been significantly improved. After the same number of iterations, the iterative results of 3DCIT-SR are closer to the truth than 3DCIT, in particular, the inversion of the edge regions is improved noticeably. The GNSS data of the International GNSS Service (IGS) stations in Europe are used to perform real data experiments, and the inversion results show that the electron density profiles of 3DCIT-SR are closer to the ionosonde measurements. The accuracy improvement of 3DCIT-SR is up to 56.3% while the improvement is more obvious during the magnetic storm compared to the case of a calm ionospheric state .
When a radio signal passes through the ionosphere, it expe-riences group delay and phase advance due to the presence of free electrons. By means of this phenomenon, the STEC along the line of sight (LOS) between a satellite and receiver can be obtained by GNSS observations (Burrell et al. 2009; Rideout and Coster 2006). Ionospheric tomography is an effective tool to reconstruct the distribution of the iono-spheric electron density based on GNSS-STEC observations in various directions. The ionospheric tomography technique was proposed by Austen et al. (1986) for the first time, and
after that, many authors have conducted many theoretical (Markkanen et al. 1995; Kunitsyn et al. 1997; Raymond et al. 1994) and experimental studies (Macalalad et al. 1993; Huang et al. 1999; Pryse et al. 1997) on two-dimensional ionospheric tomography algorithms based on low earth orbit satellites. In 1997, Rius et al. (1997) obtained the global three-dimensional tomography image of a 200–650 km height range using the GPS/MET (Global Positioning Sys-tem/ Meteorology) occultation data of 28 stations and com-bining the data of 160 Global Positioning System (GPS) continuous stations around the world. The advent of GNSS provides high temporal resolution and high precision data for ionospheric tomography, making ionospheric tomography studies gradually to develop from two-dimensional tomog-raphy to three-dimensional tomography (Howe et al. 1998; Bust et al. 2000; Hansen et al. 1997; Hernández Pajares et al. 2000; Hernández-Pajares et al. 1998; Wen et al. 2007). How-ever, due to the sparsity of GNSS stations and the limited elevation angles of LOS, three-dimensional tomography is a typical ill-posed problem.
Generally, there are many categories of algorithms for solving this problem. Austen et al. (1988) first proposed the classical algorithm of algebraic reconstruction tech-nique (ART), and later, Raymund et al. (1990) proposed an improved version of multiplicative algebraic reconstruction technique (MART) that avoids a negative electron density in the inversion results. These iterative algorithms iteratively improve the background and make it gradually approach the actual situation (Das and Shukla 2011; Gerzen and Minkwitz 2016; Ssessanga et al. 2015; Prol et al. 2017; Wen et al. 2008). In recent years, various algorithms have been pro-posed and investigated, such as simultaneous iteration recon-struction technique (SIRT) (Pryse et al. 1993), constrained simultaneous algebraic reconstruction technique (C-SART) (Hobiger et al. 2008), and two-step algorithm (TSA) (Wen et al. 2012).
The singular value decomposition (SVD) method, or trun-cated singular value decomposition (TSVD) method, is the most commonly used non-iterative tomography algorithm. Its advantage is that it does not need prior ionospheric infor-mation, but the design matrix is so large that the decom-position results are unstable. The regularization method transforms the ill-posed problem into a posed problem by appending the additional external information (Nygrén et al. 1997; Wang et al. 2016). Fehmers et al. (1998) used the regularization method to reconstruct the ionosphere at Euro-pean mid-latitudes in the spring of 1995. Lee et al. (2007) performed a high spatial ionospheric resolution Tikhonov and Total Variation (TV) method. Ma et al. (2005) proposed the tomography algorithm by a residual minimization train-ing neural network (RMTNN) for the first time, and Hirooka and Hattori (2016) and Hirooka et al. (2011a) validated the reconstruction performance of RMTNN using numerical simulations based on both sufficiently sampled and sparse data. The Bayesian approach employs a Gaussian random field using a covariance matrix obtained from NeQuick2 or Chapman profile prior information to constrain the iono-spheric inversion process (Mitchell and Spencer 2003; Markkanen et al. 1995; Norberg et al. 2015). The constrained least-squares with boundary conditions is developed by See-mala et al. (2014), which does not require an ionospheric model as the initial guess. Chen et al. (2016) investigated the medium-scale traveling ionospheric disturbances based on this approach. Saito et al. (2017) reconstructed the real-time three-dimensional ionospheric electron density distribution over Japan using variable grid sizes and introduced the boundary condition to fix the electron density at the top and bottom edges of the ionosphere. In addition, Farzaneh and Forootan (2018) used the empirical orthogonal function and the spherical Slepian base functions to describe the vertical and horizontal distribution of electron density, respectively.
Although much research has been conducted and consid-erable progress has been made, problems such as the sparse
and uneven distribution of observation stations, especially limited angles of LOS, still exist, and the ill condition of the tomography algorithm is still prominent (Yao et al. 2015). The STEC observations used in the three-dimensional tomographic reconstruction represent the electron content of the entire ray path. Therefore, the rays that pass through or off to the side of the inversion area cannot be used because we have no way of knowing the partial STEC values of these rays in the inver-sion region. This leads to the insufficient utilization and high elevation angles of rays. The high elevation angles of these rays make it difficult to extract efficient vertical distribution information from STEC observations for tomographic recon-struction while also leading to almost no rays passing through the voxels near the edge part. The problem is more serious when the height of the inversion region is larger. Although the ionosonde and occultation data can provide information about the vertical structure of the ionosphere, the data are scarce and it is difficult to meet the needs of real-time ionospheric monitoring at a high temporal resolution.
We propose a modified tomography algorithm in which the side rays are used together in iterative tomography with non-side rays that pass through the top of the inversion region. The partial STEC of side rays in the inversion region is obtained by GNSS observations with the use of ionospheric empiri-cal model NeQuick2. In addition, the correction along GNSS ray is assigned to voxels according to the product of intercept of ray in voxel and electron density instead of according to the square of intercept. The algorithm can effectively increase the voxels penetration rate, which is the percentage of voxels penetrated by GNSS rays in all voxels. The algorithm can also increase the rays utilization rate, which is the percentage of rays used in 3DCIT in all GNSS rays, and thereby improving the inversion accuracy of the edge and bottom region. The features and effectiveness are validated by simulation experi-ments and GNSS data of IGS stations in Europe.
Ionospheric tomography with side rays
The ionospheric delay of the GNSS signal is inversely pro-portional to the square of its frequency. According to this, the total electron content in the signal propagation path can be calculated using the GNSS dual-frequency observations. In brief, the STEC from the carrier phase smoothing pseudorange is given as:
where f1 and f2 are the frequencies of the GNSS signals; p̃2and p̃1 are the carrier phase smoothing pseudorange values of the two frequency signals; ∆bk indicates the differential
(1)STEC =f 21f 22
40.3(f 21− f 2
2)(p̃2 − p̃1 + Δbk + Δbs)
GPS Solutions (2018) 22:107
1 3
Page 3 of 18 107
code bias of receivers, and ∆bs represents the differential code bias of satellites.
STEC measurements calculated from GNSS data repre-sent the total electron content of a whole ray between the satellite and receiver, so only the rays that pass into the inversion region from the top and their receivers are in the inversion area can be used in tomography. As shown in Fig. 1, the bottom line is the ground, the upper line is the top boundary of the ionosphere, the green area is the inver-sion area, and the surrounding area is the support area. The three black solid lines of B, C and D in the figure are the rays that can be used for conventional tomographic reconstruc-tion, and all of the elevation angles of the ray can be used for conventional tomographic reconstruction are between B and D. The two dashed lines of A and E are the rays pass-ing from the side of the inversion area. Since the red part of dashed line E is out of the inversion region but still, in the ionosphere, the blue part cannot be obtained from GNSS data, and this type of rays cannot be used in conventional ionospheric tomography algorithms. The three-dimensional ionospheric tomography algorithm with side rays was first presented in 2016 and applied to GNSS water vapor tomog-raphy (Yao et al. 2016). In this approach, the interested inversion region is embedded into a larger region, which is termed support area, and the right part of Fig. 1 shows how the range of the support area is determined: ray F is the LOS where the receiver is located at the edge of the tomographic region, its elevation angle is the cut-off angle, and the range of the support area is calculated by the elevation angle α and the height of the tomographic region H. Therefore, all the rays whose receivers are in the inversion region and whose elevation angles are larger than α completely pass through the inversion region and the support region. If the ratio of the total electron content of the blue part to the whole ray can be calculated by some prior information, the partial STEC (PSTEC) of the blue part can be obtained using STEC measurements calculated from GNSS data, and side rays
can finally be employed with non-side rays in ionospheric tomography.
As shown in Fig. 1, taking ray E as an example, the blue part passing through the side boundary is laying inside the inversion region, while the red part is outside the inversion region but still inside the support region. The STECGNSS of the whole ray can be calculated by GNSS data. At the same time, the STECNeQuick2 and PSTECNeQuick2 of the ray can be obtained directly from NeQuick2, as the spatial position of the ray can be determined according to the receiver and satellite positions. Therefore, the PSTEC of the side ray in the inversion region can be expressed as:
The ionospheric empirical model NeQuick2 is a time-dependent three-dimensional ionospheric electron density model (Nava et al. 2008). The performance of NeQuick2 has been validated (Nigussie et al. 2013; Oladipo and Schüler 2012; Hirooka et al. 2011a) and many studies have been performed using this model (Jin and Li 2018; Ezquer et al. 2017; Schüler and Oladipo 2014; Hirooka et al. 2011b; Nava et al. 2011). By integrating the electron density, NeQuick2 can also be used to obtain STEC between two positions. The NeQuick2 model is used as a priori information to calculate the proportion of the partial ray in the inversion region to the entire ray. Therefore, the side rays with measured informa-tion can be used in the inversion. The side rays with lower elevation angles can significantly improve the distribution of rays and make voxels in the bottom and edge part penetrated by rays.
Modified iterative tomography algorithm
The line integral of the ionospheric electron density along the ray path is defined as
in which STEC is the total electron content of the slant ray path, Ne is the ionospheric electron density at position r⃗ and time t, and l is the ray propagation path between a satel-lite and a receiver. The inversion region can be divided into small voxels, and the TEC measurements of GNSS data can be formulated as:
where M is the number of STEC measurements, N is the number of voxels of the inversion region, A is the design matrix, x is the vector consisting of all the unknown electron
(2)PSTEC = STECGNSS ×PSTECNeQuick2
STECNeQuick2
(3)STEC = ∫l
Ne(
r⃗, t)
ds
(4)STECM×1 = AM×N ⋅ xN×1 + �M×1
support region
support region
AB
C D E
Finversion
region
Fig. 1 Schematic diagram of support region determination and calcu-lation of PSTEC of side rays. For the sake of clarity, only one GNSS receiver is shown in the figure. The bottom line is the ground, the upper line is the top boundary of the ionosphere, the green area is the inversion area, and the surrounding area is the support area
GPS Solutions (2018) 22:107
1 3
107 Page 4 of 18
densities in all the voxels, and � is an error column vector of STEC measurement noises.
In the SIRT algorithm, the correction for a voxel is the average of corrections of all GNSS rays, which could be repressed as,
(5)
xs+1j
= xsj+
1
R
�R
i=1� ⋅ Δ ⋅W
W =A2i,j
∑M
m=1A2i,m
Fig. 2 Schematic diagram of correction assignment along the GNSS ray. The electron density values are obtained from NeQuick2 of 10°E altitudinal–latitudinal slice at 12:00 UT on March 17, 2015. The white line represents the GNSS ray; j and k represent the voxels num-bers and the double-headed white arrows represent the intercepts of GNSS ray in jth and kth voxel
Fig. 3 Geographical distribution of the GNSS stations and ionosonde station. The blue and green dots are GNSS stations and the red trian-gle is the ionosonde station. The GNSS stations represented by green dots are used as the external data checks in the real data experiment
Fig. 4 Three-dimensional distribution of non-side rays crossing the inversion region voxels
Fig. 5 Three-dimensional distribution of side rays crossing the inver-sion region voxels
GPS Solutions (2018) 22:107
1 3
Page 5 of 18 107
For s = 0, 1, 2,…, the term xs+1j
is the electron density
of the jth voxel after s + 1 iterations; R is the number of rays which traverse the jth voxel; λ denotes a relaxation parameter in (0,1); Δ is the total correction of the ith ray path; Ai,j is the intercept of the ith ray in jth voxel; W is the weight of the jth voxel for the ith ray total correction assignment.
It is known from (5) that the total correction along the ith ray path is distributed to voxels by the square of intercepts without taking electron density of voxels in consideration. As is shown in Fig. 2, the electron den-sity corrections for jth and kth voxel would be the same if their intercepts are equal in a traditional way. However, the contributions of voxels to the total correction varies a lot because the electron density magnitude varies greatly with height. The correction assignment is obviously unrea-sonable since the electron density of the jth voxel is about 11 × 1011 el/m3 while kth voxel is only about 2 × 1011 el/m3 but the correction to the two voxels are equal if the intercepts on the two voxels are the same. This will result in inadequate correction at high electron density height, while providing excessive correction at low electron den-sity height.
In 3DCIT-SR approach, the total correction is assigned to voxels according to the product of intercept and electron density:
In this approach, the voxels at high electron density height get more correction, while the voxels at low electron density height get less correction. The modified algorithm makes voxels get corrections corresponding to their elec-tron density magnitude. This is important for the inversion since non-side rays and side rays are employed together in the reconstruction.
Simulation experiment
The GNSS data of the IGS stations in Europe on December 22, 2015, at 14 UT are used in this simulation experiment to simulate the spatial positions of rays. The inversion area and distribution of GNSS stations are shown in Fig. 3. The inver-sion region covered 40°–60°N in latitude and 0°–20°E in longitude, and the height ranged from 100 to 1000 km. The spatial resolution is 1° in the latitude and longitude direc-tions, and 50 km in the altitude direction. The simulation experiment steps are as follows:
1. The range of the support region is calculated according to Fig. 1.
(6)W =Ai,j ⋅ x
kj
∑M
m=1Ai,m ⋅ xk
m
Fig. 6 Voxels penetration rate of 24 time periods. The green and red bars represent the voxel penetration rates of the 3DCIT and 3DCIT-SR algorithms, respectively
Fig. 7 Rays utilization rate of 24 time periods. The green and red bars represent the 3DCIT and 3DCIT-SR algorithms, respectively
Fig. 8 Comparison of the iteration accuracies of two algorithms
GPS Solutions (2018) 22:107
1 3
107 Page 6 of 18
2. The construction matrix A is determined with the inver-sion region, support region and ray positions, which are obtained by receiver locations and satellite positions in the GNSS data.
3. The STECs and PSTECs could be calculated from NeQuick2 and then random noise esimu is added. The
maximum noise used is 10% of the average of the simu-lated STECsimu (STECs and PSTECs) values.
4. The background is obtained by adding random noise to the electron densities of the voxels of NeQuick2. The maximum noise used is 20% of electron densities since
(7)STECsimu = STECNeQuick + esimu
Fig. 9 Electron density longi-tude plane distribution of the truth, background, 3DCIT inver-sion result, 3DCIT-SR inversion result, 3DCIT inversion error and 3DCIT-SR inversion error
GPS Solutions (2018) 22:107
1 3
Page 7 of 18 107
the accuracy of NeQuick2 is usually worse than GNSS STECs observation in actual circumstances.
5. The inversion region is reconstructed by conventional 3DCIT and 3DCIT-SR.
6. Using NeQuick2 as the truth, the accuracies of two tomography algorithms are tested by the root mean square error.
(8)� =
√
√
√
√
√
1
J
J∑
j
(
NeNeQuick2
j− Nerecon
j
)
2
Fig. 10 Electron density latitude plane distribution of the truth, background, 3DCIT inversion result, 3DCIT-SR inversion result, 3DCIT inversion error and 3DCIT-SR inversion error
GPS Solutions (2018) 22:107
1 3
107 Page 8 of 18
Figures 4 and 5 show the three-dimensional distribu-tions of rays crossing the tomographic areas of 3DCIT and 3DCIT-SR, respectively. The green lines are non-side rays, the red lines are side rays, and the depth of the blue color represents the number of times the voxels are crossed by rays. In Fig. 4, more signals pass through the tomographic center, while fewer signals cross the edge and bottom of the
tomographic region; in particular, the voxels at low altitudes and fringe region are not passed by rays. However, as we can see in Fig. 5, after adding the side rays in the inversion, most of the voxels at low altitudes and fringe region are crossed by red lines.
Figures 6 and 7 display the comparison of voxels penetra-tion rate (percentage of voxels penetrated by GNSS rays in
Fig. 11 Electron density altitude plane distribution of the truth, background, 3DCIT inversion result, 3DCIT-SR inversion result, 3DCIT inversion error and 3DCIT-SR inversion error
GPS Solutions (2018) 22:107
1 3
Page 9 of 18 107
all voxels) and rays utilization rate (percentage of rays used in 3DCIT in all GNSS rays) of the two approaches, respec-tively. The voxels penetration rate increased from 60% to more than 80% once the side rays are introduced. It can be seen from Fig. 6 that the ray utilization rate without the side rays is approximately 30–50%, mainly caused by the side rays that cannot be used. With the addition of the side rays, the utilization rate of the rays increased to 97%, which has been greatly improved.
Figure 8 displays the RMSE (Root Mean Square Error) of the electron density values of 3DCIT and 3DCIT-SR result for 30 iterations. Because of the higher penetration rate and the higher ray utilization rate, the rays in the iterations can more effectively improve the inversion region, and the accu-racy of the reconstructed electron density of 3DCIT-SR is better than 3DCIT after the first iteration. With the increase in the number of iterations, the RMSE of the 3DCIT-SR algorithm decreases more quickly than the 3DCIT algo-rithm. The convergence accuracies of the two inversion algorithms are approximately 0.078 and 0.046 × 1011 el/m3. The results of the simulation experiment indicate that side
rays play an important role in improving the accuracy and effectiveness of tomography algorithms.
To further analyze the inversion performance of the two tomography methods, the true values and background, the inversion results of 3DCIT and 3DCIT-SR after 30 itera-tions, and errors are given in Figs. 9, 10 and 11.
The longitude planes of the truth, background, and the inversion results and errors of 3DCIT and 3DCIT-SR are given in Fig. 9. There is a significant difference between the truth and background: every longitude plane of the truth is very smooth, and the background electron density distribution of the detail in the longitude planes is not smooth. Compared with the 3DCIT inversion results, the inversion results of 3DCIT-SR are closer to the truth, and the inversion errors are smaller. The inversion errors of 3DCIT range is [− 0.4, 0.4] × 1011 el/m3, while the 3DCIT-SR error range is [− 0.2, 0.15] × 1011 el/m3. As the inver-sion error longitude planes of the 3DCIT show, the errors of the fringe planes (0°E, 20°E) are larger than the inside planes, which means the effect of tomographic improve-ment is limited in these voxels since the distribution of rays is scarce. There is no significant difference between the errors at the edge and inside for 3DCIT-SR, and they are all smaller than respective 3DCIT errors. The addition of side rays has improved the accuracy both in the edge and in the interior of the inversion region.
Similarly, Fig. 10 shows the latitude plane results. It can be seen from the 3DCIT error latitude plane that the errors of 40°N are obviously larger than those of other latitudes because of the lower ray penetration rate and higher elec-tron density. However, there is no similar phenomenon in the 3DCIT-SR tomography results, and the error of each latitude plane is smaller, indicating that side rays passing through the latitude planes have effectively improved the inversion region.
The altitude plane results are summarized in Fig. 11. Although the inversion results of the 3DCIT tomography algorithm have improved compared to the initial values, the height surfaces are still not smooth, especially at heights of 250 and 350 km. The performance of 3DCIT-SR is better, as the altitude planes of the inversion results become very smooth, and the overall trend is more consist-ent with the truth.
Real data experiment
The GNSS data of the IGS stations in Europe on March 17 (Great Magnetic Strom), June 22 (Summer Solstice), September 23 (Equinox), December 22 (Winter Solstice) of 2015, are used in this real data experiment. The measured data under a magnetic storm are used to verify the inversion accuracy with the ionosphere experiencing the disturbances
Fig. 12 Geomagnetic activity indices of real experiment days. The four rows from top to bottom are Kp and Dst indices of March 17, June 22, September 23 and December 22. The left column is the Kp index, and the right column is the Dst index
GPS Solutions (2018) 22:107
1 3
107 Page 10 of 18
(March 17). Three time periods (00:00, 06:00 and 12:00 UT) of each of the 4 days are chosen to test the algorithm inversion accuracy performance. The NeQuick2 model is used to calculate the ionospheric inversion background and PSTEC. The geographic distribution of stations is shown in Fig. 3 in which the data of the stations represented by blue dots are used in tomography, while the two stations represented by green dots (LEIJ and REDU) are employed to
verify the inversion precision. The ionosonde station PQ052 represented by a red triangle is used to compare the electron density profiles with the inversion results. The differential code bias (DCBs) is the hardware delay that occurs between two different observations, which can affect the accuracy of GNSS STEC (Han et al, 2018; Themens et al, 2013). The satellite and the receivers DCBs are estimated and corrected
Fig. 13 Inversion results of 3DCIT and 3DCIT-SR and the difference between them of 00:00 UT, 06:00 UT and 12:00 UT on March 17, 2015. The three columns from left to right are the 3DCIT inversion results, 3DCIT-SR inversion results, and the difference between the inversion results of 3DCIT and 3DCIT-SR. The three rows from top to bottom are the inversion results at 00:00 UT, 06:00 UT and 12:00 UT
GPS Solutions (2018) 22:107
1 3
Page 11 of 18 107
in advance, and the details of estimation method can be found in Jin et al. (2012).
Figure 12 gives the geomagnetic activity indices of exper-iment days from 00:00 UT to 12:00 UT. The left column is the Kp index, and the right column is the Dst index. The four rows from top to bottom are Kp and Dst indices of March 17, June 22, September 23 and December 22. The maximum value of Kp index of March 17 is greater than 7 and the min-imum value of Dst is less than − 70 nT. The electron density
of March 17 was seriously affected by the St. Patrickˈs Day magnetic storm. There was no strong geomagnetic activity during the other 3 days.
The inversion results of 3DCIT and 3DCIT-SR and the difference between them are shown in Figs. 13, 14, 15 and 16. It can be seen from the right columns of these figures that the two tomography method results have notable differ-ences in the edge area of most time periods. As shown in the right sub-graph in the bottom panel of Fig. 13, at 12:00
Fig. 14 Inversion results of 3DCIT and 3DCIT-SR and the difference between them of 00:00 UT, 06:00 UT and 12:00 UT on June 22, 2015. The three columns from left to right are the 3DCIT inversion results, 3DCIT-SR inversion results, and the difference between the inversion results of 3DCIT and 3DCIT-SR. The three rows from top to bottom are the inversion results at 00:00 UT, 06:00 UT and 12:00 UT
GPS Solutions (2018) 22:107
1 3
107 Page 12 of 18
UT on March 17, the difference of the 0°E plane reaches to − 6.3 × 1011 el/m3. We can see from the right sub-graph in the middle panel of Fig. 16, at 06:00:00 UT on December 22, the difference of the 0°E plane is approximately 1 × 1011 el/m3, and the 20°E plane is approximate − 0.9 × 1011 el/m3. It is noteworthy that there are differences even within the
interior inversion region in some time periods. That means the side rays have played an important role in reconstruction.
Figure 17 gives the comparison of electron density pro-files among ionosonde measurements (red curves), 3DCIT results (blue curves) and 3DCIT-SR results (green curves). For the sake of analyzing the influence of side rays on the edge part of the inversion region, the figure shows the
Fig. 15 Inversion results of 3DCIT and 3DCIT-SR and the difference between them of 00:00 UT, 06:00 UT and 12:00 UT on September 23, 2015. The three columns from left to right are the 3DCIT inversion results, 3DCIT-SR inversion results, and the difference between the inversion results of 3DCIT and 3DCIT-SR. The three rows from top to bottom are the inversion results at 00:00 UT, 06:00 UT and 12:00 UT
GPS Solutions (2018) 22:107
1 3
Page 13 of 18 107
reconstruction electron density profiles at (50°N, 20°E), which is on the edge but still near the ionosonde station location.
During the magnetic storm day of March 17, although the PSTEC estimated from NeQiuck2 is relatively inaccurate,
the voxels with PSTEC modified still give a positive con-tribution to the reconstruction result. The electron density profile of 3DCIT-SR in Fig. 17a3 is closer to the ionosonde profile than 3DCIT and the NmF2 is improved by 6.3 × 1011 el/m3. As for the 00:00 UT inversion results of September 23
Fig. 16 Inversion results of 3DCIT and 3DCIT-SR and the difference between them of 00:00 UT, 06:00 UT and 12:00 UT on December 22, 2015. The three columns from left to right are the 3DCIT inversion results, 3DCIT-SR inversion results, and the difference between the inversion results of 3DCIT and 3DCIT-SR. The three rows from top to bottom are the inversion results at 00:00 UT, 06:00 UT and 12:00 UT
GPS Solutions (2018) 22:107
1 3
107 Page 14 of 18
in Fig. 17c1, the electron density of 3DCIT-SR is lower than 3DCIT at every altitude and closer to the ionosonde meas-urements. The NmF2 of 3DCIT-SR result is approximately 1.3 × 1011 el/m3 lower than the 3DCIT result.
For further evaluation of the accuracy of the inversion results, the GNSS-STECs derived from the LEIJ and REDU GNSS station data are employed as independent data to ver-ify the reconstructed STECs of 3DCIT and 3DCIT-SR at 23:30 to 00:30, 05:30 to 06:30, and 11:30 to 12:30 UT. The reconstructed STECs of 3DCIT and 3DCIT-SR are calcu-lated by integrating the ray path with the inversion results.
Figure 18 displays the comparisons of the GNSS-STECs (red curves), 3DCIT-STECs (blue curves) and 3DCIT-SR-STECs (green curves) for the three time periods on March 17, June 22, September 23 and December 22. Only one trajectory results are shown for each time period. For the sake of convenience, the x-axis in the figure is marked with epochs instead of exact time. Although the general trends of the 3DCIT-STECs and 3DCIT-SR-STECs are consistent with GNSS-STECs, the 3DCIT-SR-STECs are closer to the GNSS-STECs, which means that 3DCIT-SR reconstructions have higher precision and better agreement with the real ionosphere state.
Fig. 17 Comparisons of electron density profiles from the ionosonde measurements (red curves), 3DCIT inver-sion results (blue curves) and 3DCIT-SR inversion results (green curves)
GPS Solutions (2018) 22:107
1 3
Page 15 of 18 107
For a quantitative comparison, the RMSEs and accu-racy improvement of all trajectories of 3DCIT-STECs and 3DCIT-SR-STECs are summarized in Table 1. It can be seen from the table that the RMSEs of 3DCIT-SR are smaller than those of 3DCIT in all time periods. The accuracy improve-ment ranges from 10.0 to 56.3%. There is a definite accuracy improvement between 3DCIT and 3DCIT-SR. There are rare rays in the edge and the bottom region, and thus the voxels cannot be modified effectively. Therefore, during the mag-netic storm, the greater deviation of background would lead
to a worse reconstruction result of the 3DCIT algorithm. Since the 3DCIT-SR algorithm improves the ray distribu-tion in the edge region and the bottom region, the accuracy improvement of the inversion result of the 3DCIT-SR algo-rithm during the magnetic storm is more obvious.
Fig. 18 Comparison among the GNSS-STECs (red curves), 3DCIT-STECs (blue curves) and 3DCIT-SR-STECs (green curves) of three time periods on March 17, June 22, Septem-ber 23 and December 22. The three columns from left to right correspond to 23:30–00:30, 05:30–06:30, and 11:30–12:30 UT
GPS Solutions (2018) 22:107
1 3
107 Page 16 of 18
Conclusion and discussion
We presented a modified three-dimensional ionospheric tomography algorithm which introduces side rays into the inversion. The PSTECs of side rays is derived from GNSS-STEC and NeQuick2.
In the simulation experiment, the rates of voxel penetra-tion and ray utilization of 3DCIT-SR improved by 25 and 50%, respectively, and the iterative precision is superior to the conventional 3DCIT approach. After the same number of iterations, the inversion result of the 3DCIT-SR algorithm is closer to the truth; in particular, the edge part of the tomog-raphy region, as well as the internal part, were adequately corrected. Adding side rays to inversion makes the voxels in the edge and low altitude parts penetrated by more rays and improved more effectively.
The GNSS data in Europe are used in real data experi-ments, and the measurements of an ionosonde station and two GNSS stations are employed as independent data to evaluate the inversion results. The longitude planes of the difference between the two algorithm results show that the disparity of the two approaches is mainly in the fringe part. The electron density vertical profiles of the 3DCIT-SR results have better agreement with the ionosonde measure-ments. In addition, the 3DCIT-SR-STECs are also closer to GNSS-STEC than the 3DCIT-STECs. The statistical result of the STEC RMSE inversion shows that the accuracy improvement of 3DCIT-SR ranges from 10.0 to 56.3%. The accuracy improvement of 3DCIT-SR is more obvious during the magnetic storm time. Because the two GNSS stations
of LEIJ and REDU are located in the interior of the inver-sion area, their rays include side rays and non-side rays. The accuracy improvement of the inversion result indicates that the introduction of side rays not only improves the edge part but also does so for the electron density of the inside vox-els. The new algorithm makes the distribution of rays more even in the whole inversion region, introduces more vertical direction information of the electron density distribution, and finally improves the inversion accuracy.
The calculation of PSTEC is based on GNSS data and NeQuick2. This will inevitably introduce model errors into the reconstruction process since the empirical model NeQuick2 cannot precisely describe the real ionosphere state. A more accurate method of PSTEC determination is needed in future work. Furthermore, the weights of STECs (including STECs and PSTECs) with different elevation angles or non-side rays and side rays are not considered in this work. Future development would be the determination of the weights of STECs by their accuracy.
Acknowledgements The authors would like to thank the International Global Navigation Satellite System Service (IGS) for the data used in this work. The authors also thank the Global Ionosphere Radio Obser-vatory for the ionosonde data. The ionosonde (PQ052) data were down-loaded from ftp://ftp.ngdc.noaa.gov/ionos onde/data/.
References
Austen JR, Franke SJ, Liu CH, Yeh KC (1986) Application of com-puterized tomography techniques to ionospheric research. In: Proceedings of the international beacon satellite symposium on radio beacon contribution to the study of ionization and dynam-ics of the ionosphere and to corrections to geodesy and technical workshop, University of Oulu, Finland, pp 25–35. http://adsab s.harva rd.edu/abs/1986i bs..symp...2A
Austen JR, Franke SJ, Liu CH (1988) Ionospheric imaging using computerized tomography. Radio Sci 23(3):299–307
Burrell AG, Bonito NA, Carrano CS (2009) Total electron content processing from GPS observations to facilitate ionospheric modeling. GPS Solut 13(2):83–95
Bust GS, Coco D, Makela JJ (2000) Combined ionospheric campaign 1: ionospheric tomography and GPS total electron count (TEC) depletions. Geophys Res Lett 27(18):2849–2852
Chen CH, Saito A, Lin CH, Yamamoto M, Suzuki S, Seemala GK (2016) Medium-scale traveling ionospheric disturbances by three-dimensional ionospheric GPS tomography. Earth Planets Space 68(32):1–9
Das SK, Shukla AK (2011) Two-dimensional ionospheric tomography over the low-latitude Indian region: an intercomparison of ART and MART algorithms. Radio Sci. https ://doi.org/10.1029/2010R S0043 50
Ezquer RG, Scidá LA, Orué YM, Lescano GE, Alazo-Cuartas K, Cabrera MA, Radicella SM (2017) NeQuick 2 total electron content predictions for middle latitudes of North American region during a deep solar minimum. J Atmos Sol Terr Phy 154(2):55–66
Farzaneh S, Forootan E (2018) Reconstructing regional ionospheric electron density: a combined spherical slepian function and
Table 1 Inversion RMSE (TECU)
Time period (UT) 3DCIT 3DCIT-SR Accuracy improvement (%)
empirical orthogonal function approach. Surv Geophys 39(2):289–309. https ://doi.org/10.1007/s1071 2-017-9446-y
Fehmers GC, Kamp L, Sluijter FW, Spoelstra T (1998) A model-independent algorithm for ionospheric tomography: 2. Experi-mental results. Radio Sci 33(1):165–173
Gerzen T, Minkwitz D (2016) Simultaneous multiplicative column-normalized method (SMART) for 3-D ionosphere tomography in comparison to other algebraic methods. Ann Geophys Germ 34(1):97–115. https ://doi.org/10.5194/angeo -34-97-2016
Han D, Kim D, Kee C (2018) Improving performance of GPS satel-lite DCB estimation for regional GPS networks using long-term stability. GPS Solut 22(1):13
Hansen AJ, Walter T, Enge P (1997) Ionospheric correction using tomography. In: Proc. ION GPS-97, Institute of Navigation, Kan-sas City, MO, USA, September 16–19, pp 249–260
Hernández-Pajares M, Juan JM, Sanz J, Sol Eacute JG (1998) Global observation of the ionospheric electronic response to solar events using ground and LEO GPS data. J Geophys Res-Space 103(A9):20789–20796
Hernández-Pajares M, Juan JM, Sanz J, Colombo OL (2000) Applica-tion of ionospheric tomography to real-time GPS carrier-phase ambiguities resolution, at scales of 400–1000 km and with high geomagnetic activity. Geophys Res Lett 27(13):2009–2012
Hirooka S, Hattori K (2016) Validation of Ionospheric Tomography Using Residual Minimization Training Neural Network. Electr Commun Jpn 99(4):50–57
Hirooka S, Hattori K, Takeda T (2011a) Numerical validations of neural-network-based ionospheric tomography for disturbed iono-spheric conditions and sparse data. Radio Sci 46(5):RS0F05. https ://doi.org/10.1029/2011R S0047 60
Hirooka S, Hattori K, Nishihashi M, Takeda T (2011b) Neural net-work based tomographic approach to detect earthquake-related ionospheric anomalies. Nat Hazard Earth Sys 11(8):2341–2353
Hobiger T, Kondo T, Koyama Y (2008) Constrained simultaneous algebraic reconstruction technique (C-SART)—a new and sim-ple algorithm applied to ionospheric tomography. Earth Planets Space 60(7):727–735
Howe BM, Runciman K, Secan JA (1998) Tomography of the iono-sphere: four-dimensional simulations. Radio Sci 33(1):109–128
Huang CR, Liu CH, Yeh KC, Lin KH, Tsai WH, Yeh HC, Liu JY (1999) A study of tomographically reconstructed ionospheric images dur-ing a solar eclipse. J Geophys Res Space 104(A1):79–94
Jin S, Li D (2018) 3-D ionospheric tomography from dense GNSS observations based on an improved two-step iterative algorithm. Adv Space Res 62(4):809–820
Jin R, Jin S, Feng G (2012) M_DCB: Matlab code for estimating GNSS satellite and receiver differential code biases. GPS Solut 16(4):541–548. https ://doi.org/10.1007/s1029 1-012-0279-3
Kunitsyn VE, Andreeva ES, Razinkov OG (1997) Possibilities of the near-space environment radio tomography. Radio Sci 32(5):1953–1963
Lee JK, Kamalabadi F, Makela JJ (2007) Localized three-dimen-sional ionospheric tomography with GPS ground receiver meas-urements. Radio Sci 42(04):1–15
Ma XF, Maruyama T, Ma G, Takeda T (2005) Three-dimensional ionospheric tomography using observation data of GPS ground receivers and ionosonde by neural network. J Geophys Res Space 110(A5):A5308. https ://doi.org/10.1029/2004J A0107 97
Macalalad FV, Jr JH, Apostol JV, Preston FW (1993) Experimen-tal ionospheric tomography with ionosonde input and EISCAT verification. Ann Geophys Ger 11(11–12):1064–1074
Markkanen M, Lehtinen M, Nygrén T, Pirttilä J, Henelius P, Vilenius E, Tereshchenko ED, Khudukon BZ (1995) Bayesian approach to satellite radiotomography with applications in the Scandina-vian sector. ISME J 8(11):2280–2289
Mitchell CN, Spencer PSJ (2003) A three-dimensional time-depend-ent algorithm for ionospheric imaging using GPS. Ann Geophys Italy 46(4):687–696
Nava B, Coisson P, Radicella SM (2008) A new version of the NeQuick ionosphere electron density model. J Atmos Solar Terr Phys 70(15):1856–1862
Nava B, Radicella SM, Azpilicueta F (2011) Data ingestion into NeQuick 2. Radio Sci, 46(6):RS0D17. https ://doi.org/10.1029/2010R S0046 35, 2011
Nigussie M, Radicella SM, Damtie B, Nava B, Yizengaw E, Groves K (2013) Validation of the NeQuick 2 and IRI-2007 models in East-African equatorial region. J Atmos Sol Terr Phy 102(9):26–33
Norberg J, Roininen L, Vierinen J, Amm O, Mckay-Bukowski D, Lehtinen M (2015) Ionospheric tomography in Bayesian frame-work with Gaussian Markov random field priors. Radio Sci 50(2):138–152
Nygrén T, Markkanen M, Lehtinen M, Tereshchenko ED, Khudukon BZ (1997) Stochastic inversion in ionospheric radiotomography. Radio Sci 32(6):2359–2372. https ://doi.org/10.1029/97RS0 2915
Oladipo OA, Schüler T (2012) GNSS single frequency ionospheric range delay corrections: NeQuick data ingestion technique. Adv Space Res 50(9):1204–1212
Prol FS, Camargo PO, Muella MTAH (2017) Numerical simulations to assess ART and MART performance for ionospheric tomog-raphy of Chapman profiles. An Acad Bras Cienc 89(3):1531–1542. https ://doi.org/10.1590/0001-37652 01720 17011 6
Pryse SE, Kersley L, Rice DL, Russell CD, Walker IK (1993) Tomo-graphic imaging of the ionospheric mid-latitude trough. Ann Geophys 11(3):144–149
Pryse SE, Kersley L, Williams MJ, Walker IK, Willson CA (1997) Tomographic imaging of the polar-cap ionosphere over sval-bard. J Atmos Sol Terr Phy 59(15):1953–1959
Raymond TD, Franke SJ, Yeh KC (1994) Ionospheric tomography: its limitations and reconstruction methods. J Atmospheric Sol Terr Phys 56(5):637–653
Raymund TD, Austen JR, Franke SJ, Liu CH, Klobuchar JA, Stalker J (1990) Application of computerized tomography to the inves-tigation of ionospheric structures. Radio Sci 25(05):771–789
Rideout W, Coster A (2006) Automated GPS processing for global total electron content data. GPS Solut 10(3):219–228
Rius A, Ruffini G, Cucurull L (1997) Improving the vertical resolu-tion of ionospheric tomography with GPS occultations. Geo-phys Res Lett 24(18):2291–2294
Saito S, Suzuki S, Yamamoto M, Saito A, Chen CH (2017) Real-time ionosphere monitoring by three-dimensional tomography over Japan. Navig J Inst Navig 64(4):495–504
Schüler T, Oladipo OA (2014) Single-frequency single-site VTEC retrieval using the NeQuick2 ray tracer for obliquity factor determination. GPS Solut 18(1):115–122
Seemala GK, Yamamoto M, Saito A, Chen CH (2014) Three-dimen-sional GPS ionospheric tomography over Japan using con-strained least squares. J Geophys Res Space 119(4):3044–3052
Ssessanga N, Kim YH, Kim E (2015) Vertical structure of medium-scale traveling ionospheric disturbances. Geophys Res Lett 42(21):9156–9165
Themens DR, Jayachandran PT, Langley RB, MacDougall JW, Nicolls MJ (2013) Determining receiver biases in GPS-derived total electron content in the auroral oval and polar cap region using ionosonde measurements. GPS Solut 17(3):357–369
Wang S, Huang S, Xiang J, Fang H, Feng J, Wang Y (2016) Three-dimensional ionospheric tomography reconstruction using the model function approach in Tikhonov regularization. J Geo-phys Res Space 121(12):104–112. https ://doi.org/10.1002/2016J A0234 87 (115)
Wen D, Yuan Y, Ou J, Huo X, Zhang K (2007) Three-dimensional ionospheric tomography by an improved algebraic reconstruc-tion technique. GPS Solut 11(4):251–258
Wen D, Yuan Y, Ou J, Zhang K, Liu K (2008) A hybrid reconstruc-tion algorithm for 3-D ionospheric tomography. IEEE Trans Geosci Remote Sens 46(6):1733–1739
Wen D, Wang Y, Norman R (2012) A new two-step algorithm for ionospheric tomography solution. GPS Solut 16(1):89–94. https ://doi.org/10.1007/s1029 1-011-0211-2
Yao Y, Kong J, Tang J (2015) A new ionosphere tomography algorithm with two-grid virtual observations constraints and three-dimensional velocity profile. IEEE Trans Geosci Remote 53(5):2373–2383. https ://doi.org/10.1109/TGRS.2014.23597 62
Yao YB, Zhao QZ, Zhang B (2016) A method to improve the utilization of GNSS observation for water vapor tomography. Ann Geophys Ger 34(1):143–152. https ://doi.org/10.5194/angeo -34-143-2016
Yibin Yao is currently a Professor with the School of Geodesy and Geomatics, Wuhan University. He received the Ph.D. degree from Wuhan University, Wuhan, China, 2004. He is also with the Collaborative Innovation Center for Geospatial Technology, Wuhan. His main research inter-ests include GNSS, ionospheric atmospheric meteorological studies, theory, and method of surveying data processing.
Changzhi Zhai is currently work-ing toward the Ph.D. degree in geodesy and surveying engineer-ing in the School of Geodesy and Geomatics, Wuhan University, Wuhan, China. He received the M.Sc. degree from Wuhan Uni-versity, Wuhan, China, in 2017. His research interests include Global Navigation Satellite Sys-tem (GNSS) data processing and 3-D ionospheric tomography based on GNSS observations.
Jian Kong received the B.Sc. degree from Shandong Univer-sity of Science and Technology, Qingdao, China, in 2009 and a Ph.D. degree from Wuhan Uni-versity, Wuhan, China, in 2014. His main research interests include GNSS data processing, global ionospheric modeling using multisource geodesy observations, 3-D ionospheric tomography based on GNSS observations, and ionospheric anomalies analysis under abnor-mal conditions.
Qingzhi Zhao is working in Col-lege of Geomatics, Xi’an Uni-versity of Science and Technol-ogy, Xi’an, China. He received the Ph.D. degree from Wuhan University, Wuhan, China, in 2017. His research interests include Global Navigation Satel-lite System (GNSS) data pro-cessing and 3-D tropospheric tomography based on GNSS observations.
Cunjie Zhao is currently working toward the Ph.D. degree in geod-esy and surveying engineering in the School of Geodesy and Geo-matics, Wuhan University, Wuhan, China. He received the M.Sc. degree from Wuhan Uni-versity, Wuhan, China, in 2015.His research interests include Global Navigation Satellite Sys-tem (GNSS) data processing and 3-D ionospheric tomography based on GNSS observations.
