Ionospheric tomography based on the total electron content (TEC) data along the ray path from Global Navigation Satellite Systems (GNSS) satellites to ground receivers is a typical ill-posed inverse problem. The regularization method is an effective method to solve this problem, which incorporates prior constraints to approximate the real ionospheric variations. When two or more prior constraints are used, the corresponding multiple regularization parameters are introduced in the cost functional. Assuming that the ionospheric spatial variations can be separable in the horizontal and vertical directions, different prior constraints are used in each direction, and the dual-parameter regularization algorithm is established to reconstruct the three-dimensional ionospheric electron density in the present paper. To make the reconstruction results comprehensively reflect the observation information and background (prior) information, it is crucial to determine the optimal regularization parameters. The linear model function method is used to choose these regularization parameters. Both an ideal test and a real test show that this regularization algorithm can effectively improve the background model output.

The ionosphere is an important part of the earth's environment, significantly influencing the propagation of electromagnetic waves through reflection and absorption. It is generally accepted that radio waves up to 10 GHz can be affected by the ionosphere to some extent when they propagate through the ionosphere.

The ionosphere has extremely complex temporal and spatial variations. Nowadays, the amount of ionospheric measurements steadily increases, and their accuracies continually improve. However, when ionospheric data are collected into certain temporal–spatial bins, some bins have rather sparse measurements, or even have no measurements.

The Global Navigation Satellite Systems (GNSS) ground beacon receiver has the advantages of being low cost, having wide distribution, and having operational simplicity. It can provide a great deal of ionospheric total electron content (TEC) data along the ray path. Moreover, the three-dimensional electron density can be reconstructed through the ionospheric tomography technique by using these TEC data (e.g., Austen et al., 1988; Fougere, 1995; Na and Lee, 1991; Raymund et al., 1990), which can greatly enrich the ionospheric data resource. Due to the limitations of the receiver-satellite geometry, the TEC observation in the horizontal direction is limited and the measurement is incomplete, and the ionospheric tomography is a typical ill-posed problem.

The common methods used in ionospheric tomography are iterative algorithms (e.g., Andreeva, 1990; Hobiger et al., 2008; Wen et al., 2012), singular value decomposition algorithms (Hajj et al., 1994), Bayesian approaches (Markkanen et al., 1995; Norberg et al., 2015), regularization methods (Fehmers et al., 1998; Lee et al., 2007; Nygrén et al., 1997), data assimilation approaches (e.g., Bust et al., 2004; Pi et al., 2003; Schunk et al., 2004; Wang et al., 2004), basis functions methods (Garcia and Crespon, 2008; Mitchell and Spencer, 2003), artificial neutral network methods (Ma et al., 2005), multisource data fusion algorithms (e.g., Alizadeh et al., 2011; Dettmering et al., 2011; Yue et al., 2012), and constrained least-squares algorithms (Seemala et al., 2014). However, it needs to be noted that ill-posedness is still a crucial problem in ionospheric tomography algorithms (Yao et al., 2015).

The regularization method is an effective method to solve this ill-posed problem by incorporating some prior constraints to approximate the real ionospheric electron density variations. The classical Tikhonov regularization method uses a single constraint to treat ill-posed problems, and naturally has a single regularization parameter. The regularization parameter is applied to balance the weights between background information and real measurements, and different regularization parameters can lead to different reconstruction results. Many methods have been proposed to determine the regularization parameter, such as unbiased predictive risk estimation (UPRE; Mallows, 1973), generalized cross-validation (GCV; Golub et al., 1979), the L-curve method (Hansen and O'Leary, 1993), and the damped Morozov discrepancy principle (Kunisch, 1993). Chen et al. (2008) have analyzed the superiority of the multiparameter regularization over the single-parameter regularization. When two or more prior constraints are imposed on the cost functional, the reconstruction accuracy may be improved further, and multiple regularization parameters are introduced accordingly. The question of how to optimally determine multiple regularization parameters is an important research avenue in the study of regularization algorithms. The recently proposed model function method is a simple and practical way to choose multiple regularization parameters.

In this paper, the spatial variations of the ionosphere are separated into the horizontal and vertical directions, and a dual-parameter regularization algorithm is established by incorporating different constraints in each direction. The linear model function method is adopted to determine the optimal regularization parameters, and the ideal test and real test are carried out to validate the effectiveness of this algorithm.

The reconstruction area covers 35–50

The dual-frequency GNSS receiver can provide continuous phase and
pseudorange observations with a sample interval of 30 s. Ionospheric
TEC can be derived by using phase observations to smooth pseudorange
observations. The elevation cutoff angle is 15

Geographical position of the GNSS ground receivers (circle) and MHJ ground ionosonde (square).

The different reconstruction results can be obtained when different regularization constraints are incorporated. Here, the ionospheric spatial variations are assumed to be separable in the vertical and horizontal directions. In the vertical direction, the Gaussian correlation constraint is used, and the correlation distance is derived from the statistical results of Yue et al. (2007a). The correlation distance increases exponentially with altitude. It is about 20 km at the ionospheric E layer and F layer, and is about 500 km at the height of 2000 km. In the horizontal direction, two different regularization terms are imposed (denoted as regularization method I and regularization method II) as follows.

The electron density in the horizontal direction is constrained by a multipoint finite-difference approximation of the two-order Laplace operator (Hobiger et al., 2008). For a grid on a certain layer, the constraint operator is

The constraint operator for a grid at the northern, southern, western, and eastern boundary of the layer is

and the constraint operator for a grid at the northwest, northeast, southwest, and southeast corner of the layer is

Then the smoothness constraint matrix

The electron density constraint in the horizontal direction is taken as
Gaussian correlation, and the correlation distance is also derived from the
statistical results of Yue et al. (2007a); that is, the horizontal
correlation distance is about 16

The linear model function method in the framework of the damped Morozov discrepancy principle is used to determine the regularization parameters (Wang, 2012), and its basic idea is constructing a linear function to locally approximate the original function at each iteration step, greatly reducing the calculation time. In the following, we use this method to determine the optimal regularization parameters in the cost functional (2) as an example.

According to the damped Morozov discrepancy principle, the cost functional
(2) can be rewritten as

The flow chart of determining the regularization parameters by using the linear model function method.

In the ideal test, the real position between GNSS satellites and ground
receivers on 8 March 2013 is used to establish the observation matrix

The reconstruction results using regularization method I under background I conditions. From top to bottom are the longitude–latitude slices at different altitudes, altitude–latitude slices at different longitudes, and longitude–altitude slices at different latitudes. From left to right are the background electron density, true electron density, reconstructed electron density, reconstructed absolute error, reconstructed relative error, and whether the satellite-receiver ray propagates through the corresponding grid or not.

The average relative reconstruction error and standard deviation using regularization method I under the background I situation. The red solid line is the average relative error of the background electron density. The blue solid line is the standard deviation of the background relative error. The red dashed line is the average relative error of the reconstructed electron density. The blue dashed line is the standard deviation of reconstructed relative error.

Figure 3 shows the reconstruction results using regularization method I under
the background I condition, and the background electron density is lower
than the simulated true electron density. From top to bottom are the
longitude–latitude slices at different altitudes (270 km, 290 km, 310 km,
and 330 km), the altitude–latitude slices at different longitudes
(287

The reconstruction results using regularization method I under background II conditions. From top to bottom are the longitude–latitude slices at different altitudes, altitude–latitude slices at different longitudes, and longitude–altitude slices at different latitudes. From left to right are the background electron density, true electron density, reconstructed electron density, reconstructed absolute error, reconstructed relative error, and whether the satellite-receiver ray propagates through the corresponding grid or not.

The average relative reconstruction error and standard deviation using regularization method I under the background II situation. The red solid line is the average relative error of the background electron density. The blue solid line is the standard deviation of the background relative error. The red dashed line is the average relative error of the reconstructed electron density. The blue dashed line is the standard deviation of the reconstructed relative error.

Figure 4 shows the average relative error of the reconstructed electron density (red dashed line) and standard deviation of the reconstructed relative error (blue dashed line) on 8 March, and the average relative error of the background electron density (red solid line) and standard deviation of the background relative error (blue solid line) are also superimposed for reference. After using the regularization method I, the average relative reconstruction error is significantly reduced compared with that of the background model, but the standard deviation of the relative reconstruction error increases during some periods.

The reconstruction results using regularization method II under background I conditions. From top to bottom are the latitude-longitude slices at different altitudes, altitude–latitude slices at different longitudes, and longitude–altitude slices at different longitudes. From left to right are the background electron density, true electron density, reconstructed electron density, reconstructed absolute error, reconstructed relative error, and whether the satellite-receiver ray propagates through the corresponding grid or not.

The comparisons between the average relative error and standard deviation of the reconstructed relative error using regularization method II and the average relative error and standard deviation of the relative error using the single-parameter regularization method under background I conditions.

Figure 5 is the same as Fig. 3, but for the reconstruction results under
background II conditions. The background electron density is larger than the
simulated true electron density. The regularization parameters are 0.0034
and 0.1502, and the maximum absolute error of the reconstructed electron density
is about

The reconstruction results using regularization method II under the
background I situation are shown in Fig. 7, and the background electron
density is lower than the simulated true electron density. The
regularization parameters are 0.0012 and

The reconstruction results using regularization method II under background II conditions. From top to bottom are the longitude–latitude slices at different altitudes, altitude–latitude slices at different longitudes, and longitude–altitude slices at different latitudes. From left to right are the background electron density, true electron density, reconstructed electron density, reconstructed absolute error, reconstructed relative error, and whether the satellite-receiver ray propagates through the corresponding grid or not.

The comparisons between the average relative error and standard deviation of the relative error using regularization method II and the average relative error and standard deviation of the relative error using the single-parameter regularization method under background II conditions.

Figure 8 shows the average relative error of the reconstructed electron
density (red dashed line) and standard deviation of the reconstructed
relative error (blue dashed line) on 8 March. For comparison, the average
relative error (red solid line) and standard deviation of the relative error
(blue solid line) using the single-parameter regularization method are also
shown. The cost functional of the single-parameter regularization method
should be minimized to derive the approximate solution of the ionospheric
electron density:

Figure 9 is the same as Fig. 7, but for the reconstruction results under
the background II conditions. The regularization parameters are 0.0012 and

The comparisons between the reconstructed electron density
profiles near the MHJ station and the measured electron density profiles
from ground ionosonde under the relatively quiet geomagnetic activity
condition; LT

The comparisons between the reconstructed electron density
profiles near the MHJ station and the measured electron density profiles
from ground ionosonde under the relatively active geomagnetic activity
condition; LT

Regularization method II can transfer the observation information in one place to a nearby place by the action of the background error covariance. The regularization method I transfers the observation information under the action of the Laplace operator, and its influence area is relatively limited. Overall, when the observation data are sparse, the regularization method II has a better performance. In the following, only the reconstruction results using regularization method II are shown in the real measurements test.

In the real measurements cases, the effectiveness of the regularization method II is tested under the quiet and active geomagnetic activity conditions. The ionosonde measurement from MHJ station is used as the independent validation data. These data were obtained from GIRO and were manually scaled by the SAO software, and they can provide accurate electron peak density and peak height. Due to the topside ionospheric electron density profiles derived by the extrapolation method, the electron density data below the height of 500 km are used to validate the reconstruction result. In the real reconstruction, it is assumed that the electron density in each grid does not vary within a 15 min interval.

The geomagnetic activity on 8 March was relatively quiet. The reconstruction results are shown in Fig. 11, in which the red line is the reconstructed electron density profile, the blue line is the background electron density profile, and the green line is the observed electron density profile from ground ionosonde. There are no ionosonde data at 01:00 and 12:00 UT. Overall, the reconstructed peak electron density using the regularization method II is much closer to the ionosonde measurements compared with that from the background model, and the reconstructed electron density peak height is basically similar to the background value. This is mainly because the satellite-receiver geometry has limitations, and the TEC data contain much more ionospheric structure information in the horizontal direction than in the vertical direction. Only GNSS TEC data have limited influences on the changes in altitudinal resolution in the background model.

On 17 March the geomagnetic activity was relatively active, and the spatial correlation distance used here is the same as that under the quiet geomagnetic activity condition. The reconstruction results are shown in Fig. 12, and there is no validation data from the MHJ ionosonde at 06:00 UT. Compared with the background model, the regularization method can distinctly improve the accuracy of the electron peak density, but these measurements are still quite different from the ionosonde measurements, which may be related to the accuracy of background error covariance. Due to the limitations of spatial and temporal resolution of ionospheric observation data and the complexity of the ionospheric variations, the background error covariance is not totally known and needs to be studied further, especially under the ionospheric disturbed condition. Moreover, the observation error is artificially assumed in the present paper, and so the real spatial and temporal variations of the measurement error also need some more researches.

Ionospheric tomography based on GNSS TEC is a typical ill-posed inverse problem, and the regularization method is used to solve this problem by incorporating prior constraints to approximate the real electron density distribution. Because of the complexity of the spatial and temporal variations of the ionosphere, a single regularization term may not obtain the high-accuracy reconstruction results. Multiple constraints sometimes need to be incorporated, and multiple regularization parameters are introduced accordingly. Here, the ionospheric variations are separated in the horizontal and vertical direction. The cost functional of the dual-parameter regularization method is established, and the regularization parameters are determined by the linear model function method. This regularization algorithm is tested by the ideal cases and real cases, and the results show that it can significantly improve the background model outputs. Moreover, to improve the reconstruction accuracy further, the measurement error and the background error covariance should be studied in detail.

The GNSS observation data and precise ephemeris data can
be obtained from

The authors would like to thank the IGS for providing the observation data and precise ephemeris data, and GIRO for the ionosondes data. This research was supported by the National Natural Science Foundation of China (no. 41575026, and no. 41804149).

The topical editor, Keisuke Hosokawa, thanks Gopi Krishna Seemala and two anonymous referees for help in evaluating this paper.