Three-dimensional ionospheric tomography is effective for investigations of the dynamics of ionospheric phenomena. However, it is an ill-posed problem in the context of sparse data, and accurate electron density reconstruction is difficult. The Residual Minimization Training Neural Network (RMTNN) tomographic approach, a multilayer neural network trained by minimizing an objective function, allows reconstruction of sparse data. In this study, we validate the reconstruction performance of RMTNN using numerical simulations based on both sufficiently sampled and sparse data. First, we use a simple plasma-bubble model representing the disturbed ionosphere and evaluate the reconstruction performance based on 40 GPS receivers in Japan. We subsequently apply our approach to a sparse data set obtained from 24 receivers in Indonesia. The reconstructed images from the disturbed and sparse data are consistent with the model data, except below 200 km altitude. To improve this performance and limit any discrepancies, we used information on the electron density in the lower ionosphere. The results suggest the restricted RMTNN-tomography-assisted approach is very promising for investigations of ionospheric electron density distributions, including studies of irregular structures in different regions. In particular, RMTNN constrained by low-Earth-orbit satellite data is effective in improving the reconstruction accuracy.
 Understanding the behavior of ionospheric electron densities has recently become important for both academic purposes and practical applications. For investigation of the dynamics of ionospheric phenomena, three-dimensional ionospheric tomography is effective. The ionospheric total electron content (TEC) observed by a ground-based receiver is an integrated value of the ionospheric electron density along the raypath between the satellite and the receiver. The electron density distribution can be reconstructed based on a number of TEC data points. However, it is an ill-posed problem, and accurate reconstruction is difficult because of the small number of data points and the lack of a horizontal raypath. Especially, the lack of a horizontal raypath made reconstruction of vertical electron density distribution extremely difficult. Previous studies have proposed various algorithms for ionospheric tomography.
 Two-dimensional tomography was proposed by Austen et al. , who applied the algebraic reconstruction technique (ART) to reconstruct simulated data. Two-dimensional tomography has since been studied by several researchers [Raymund et al., 1990; Kunitake et al., 1995; Mitchell et al., 1997]. However, these approaches can only obtain two-dimensional ionospheric distributions within a cross-section defined by the satellite orbit and the receivers on the ground. Therefore, the reconstruction area and time interval are limited.
 To obtain the vertical electron density profile, radio occultation observations have been performed using GPS satellites and a low-Earth-orbit (LEO) satellite. In essence, the vertical electron density profile is estimated using the Abel inversion technique based on the assumption of a spherically symmetric electron density distribution around the Earth [Hajj et al., 1994; Hajj and Romans, 1998; Tsai et al., 2001; Liu et al., 2010]. Although this technique is useful in investigating the comprehensive altitudinal electron density distribution, the assumption is not always accurate for the actual ionosphere, because the actual ionospheric electron density distribution varies both vertically and horizontally. To overcome this problem, Garcia-Fernandez et al.  proposed a method that considers the horizontal gradient in the Abel inversion by reference to the International Reference Ionosphere (IRI) model. Recently, many researchers have investigated the vertical profile and the global ionospheric electron density distribution using occultation observations by FORMOSAT-3/COSMIC [Lei et al., 2007; Hsiao et al., 2009, 2010]. However, these radio occultation observations are difficult to use for investigations of fine structure in the ionosphere. Therefore, in addition to occultation observations, three-dimensional tomography, based on data from ground-based receivers, is required to understand the detailed dynamics of the ionosphere. Saito et al.  and Lee et al.  proposed ground GPS-receiver-based three-dimensional tomography, and they showed the significant promise of the technique for reconstruction of local ionospheric distributions with large data set from GEONET (Japan). Similarly, Mitchell and Spencer  demonstrated the ground-based tomography for European region.
 The methods discussed above require an initial ionospheric model and/or a large amount of data for computation, but model-free reconstruction is essential for investigations of the disturbed ionosphere. In fact, our analysis is focused on the disturbed and/or irregular ionospheric electron distribution. Moreover, for applications in regions where only small numbers of ground-based GPS receivers are available, satisfactory reconstruction based on sparse data is a necessity. At tropical latitudes, equatorial plasma depletions and traveling ionospheric disturbances have been observed by ground-based instruments such as GPS receivers, radars, and airglow cameras. In addition, many anomalous ionospheric phenomena, possibly associated with large earthquakes, have been reported [Calais et al., 2003; Heki et al., 2006; Liu et al., 2006, 2009; Otsuka et al., 2006]. For instance, Sumatra island (Indonesia) is one of the most seismically active regions in the world, where large earthquakes occur frequently. However, at these low-latitude regions the number of available GPS receivers is insufficient for application of these reconstruction techniques using only ground-based receivers.
 In this paper, the Residual Minimization Training Neural Network (RMTNN) tomographic approach is selected [Ma et al., 2005a; Takeda and Ma, 2007], using TEC data, including location and altitude, derived by ground-based GPS receivers and ionosondes. This approach can be applied to reconstruction based on sparse data and a multilayer neural network. They proposed the new ionospheric tomography method and demonstrated the effectiveness of the resulting reconstruction using GEONET data from Japan under a quiet ionosphere. Although they found that the RMTNN method is promising, they did not consider reconstruction in practical situations where ionospheric disturbances are present, and/or when the region of interest is sampled by a small number of ground-based GPS receivers. We validate the performance of RMTNN tomography under both disturbed and sparse data conditions.
2. Ionospheric Tomography Using the RMTNN Method
 RMTNN is a multilayer neural network trained by minimizing an object function composed of an appropriately prepared residual of equations [Ma et al., 2000; Liaqat et al., 2003]. Using this method, we reconstruct three-dimensional ionospheric electron density distributions from TEC data as a computer tomographic image, using the RMTNN's ability to approximate an arbitrary function.
2.1. Basic Equations
 The slant TEC (STEC) [Mannucci et al., 1998] along a raypath between a GPS satellite and a ground-based receiver is defined as the integrated value of the ionospheric and plasmaspheric electron densities, including instrumental bias, as follows:
where Iij is the STEC, is the electron density, ri and rj are the position of the ith ground-based receiver and the jth satellite, and Bi and Bj are the instrumental bias of the ith ground-based receiver and the jth satellite. To determine , a neural network is constructed; is the neural network's input parameter and its output. In the neural network system, is given by a geographical latitude, longitude and altitude. For evaluation of the residuals, the equation is separated the ionospheric and the plasmaspheric part and discretized as
where q and α denote a sampling point and the corresponding weight for the numerical integration, respectively; Q is the total number of sampling points along a raypath in the ionosphere; and P is the contribution of the plasmaspheric electron density to the STEC I. We define altitudes from 100 to 700 km as the ionosphere and altitudes above 700 km as the plasmasphere, as modeled by the simple diffusive-equilibrium model [Angerami and Thomas, 1964]. Based on this model, the plasmaspheric contribution to the STEC value along the raypath is represented by Pij. In this model, we assume the altitude of the top of the ionosphere is 700 km and the scale height of density decay in plasmasphere is 480 km, respectively. To estimate the electron density , we take the squares of the residuals of the integral equation (2) as the neural network's objective function. The objective function E1 is given as
In order to estimate the , we minimize the objective function E1 in the learning process of the neural network. Although the instrumental biases Bi and Bj can be estimated during reconstruction by the RMTNN method using an additional single neuron [Ma et al., 2005b], we assumed that these biases are removed before processing.
2.2. Assimilation of Ionosonde Data
 The disadvantage of using ionospheric tomography based on ground-based GPS receivers is the associated difficulty of obtaining sufficient vertical resolution because of the lack of horizontal raypaths. To solve this problem, we use information about the peak electron density (NmF2) and the corresponding height (hmF2) observed by an ionosonde station to constrain the problem. The neural network was trained using these data through the conventional supervised back-propagation algorithm [Rumelhart et al., 1986]. The object function E2 for the ionosonde data is given by
where S is the number of ionosondes, Ns is the output of the neural network for the corresponding position that gives hmF2, and Nsion is the observed NmF2 value. The objective function E2 is evaluated from electron density at hmF2 height over the ionosonde locations using the same neural network. Therefore, the overall objective function E is
where g is a balance parameter between the GPS and ionosonde data. In this paper, if the value of the objective function E is below 10−3, we decided that the neural network was convergent. The criterion of E (10−3) is empirically determined by numerical simulations. Consequently, g = 1.0 was adopted as the balance parameter in this study.
2.3. Procedure for Updating the Weight Values of the Neural Network
Figure 1 shows a schematic diagram of the data flow for ionospheric tomography by RMTNN. The objective function E1 can be used only after all density values of the sampling points along a raypath have been evaluated. Therefore, conventional online updating (where the values of the weights are updated after every forward calculation) cannot be used. For the objective function E1, the weight-updating process of the neural network is derived in clusters for the nth path as
where n represents the raypath (i, j), InNN is a line integral along the numerically obtained nth path using the output data of the neural network, w is the weight of the main neural network, and η is the learning rate. The objective function E2 can be used for conventional online updating, as follows:
The weight update for E2 is executed after updating process E1 for all GPS raypaths.
 In this study, we used data obtained during 15-min periods for the reconstruction. We assume that the electron density distribution is constant during these periods. Furthermore, the spatial resolution is 0.5° latitude × 0.5° longitude × 30 km in altitude.
3. Simulation of RMTNN Tomography
Ma et al. [2005a] proposed the application of the RMTNN method for ionospheric tomography and demonstrated the effectiveness of the resulting reconstruction using GEONET data from Japan under a quiet ionosphere. Although they found that the RMTNN method is promising, they did not consider reconstruction in practical situations where ionospheric disturbances are present, and/or when the region of interest is sampled by a small number of ground-based GPS receivers. Therefore, to evaluate the performance of RMTNN tomography in practical conditions, we performed numerical simulations assuming both disturbed and sparse conditions. First, to examine the effectiveness of the RMTNN method for a disturbed ionosphere, we applied a simple plasma-bubble model to data acquired in Japan. We subsequently checked the performance of the RMTNN method for the Sumatra region (Indonesia) as an example of a sparse case. In this region, many interesting phenomena have been reported [Heki et al., 2006; Liu et al., 2006, 2009; Otsuka et al., 2006].
 To evaluate the errors in our analysis, we define the average density error Ed as
where M is the number of sampling points used for the reconstruction and N and NNN are the original and reconstructed electron density distributions, respectively. In this simulation, the number of iterations for the RMTNN learning process is fixed at 10,000.
3.1. Case 1: Disturbed Ionosphere (Plasma Bubble) Over Japan Using 40 GPS Data Receivers
 A plasma bubble is a local ionospheric plasma depletion that occurs in the magnetic equatorial regions and is generated at the bottom of the ionospheric F region after dusk through a plasma instability. During the solar maximum, a plasma bubble generated at lower latitudes moves toward Japan [Lee et al., 2008]. Plasma bubbles are suitable for investigation of the performance of the RMTNN method because of the associated drastic changes in the ionosphere. For the simulations, the actual positions of the GPS receivers and an ionosonde station in Japan are used. The simulated STEC values are generated based on the NeQuick model [Radicella and Leitinger, 2001], which is a three-dimensional, time-dependent ionospheric electron density model developed by the Aeronomy and Radiopropagation Laboratory of the Abdus Salam International Center for Theoretical Physics (ICTP). In this simulation, we used the data set obtained during the period 03:00–03:15 UT on 22 December 2001. We used data from 40 GPS receivers (part of GEONET), as well as observations from the ionosonde station located at Kokubunji (see Figure 2). The reconstructed region covers 26°N–42°N latitude, 128°E–142°E longitude, and 100–700 km in altitude. In this case, the total number of raypaths is 6122 and the total number of voxels for reconstruction is 20,097 (33 × 29 × 21 in latitude × longitude × altitude). We use the NeQuick model to obtain the background electron density distribution. The plasma depletion covers a 1.5°-wide area from 26°N to 36°N in latitude, 100–700 km in altitude, and contains 1% of the background electron density. We compared the reconstructed electron density distribution with simulated data.
 First, simulated and reconstructed distributions are shown in Figures 3a and 3b, respectively, for a vertical cross-section at a fixed latitude of 38°N (outside the plasma bubble). The reconstructed image is consistent with the original image. Next, simulated and reconstructed distributions are shown in Figures 3c and 3d, respectively, for a vertical cross-section at a fixed latitude of 34°N (inside the plasma bubble). Again, the reconstructed image is consistent with the original image. For further evaluation, horizontal cross-sections at several altitudes (100, 300, 500, and 700 km) of the original and reconstructed images are shown in Figures 4 and 5, respectively. The reconstructed images (see Figure 5) are consistent with the original images (see Figure 4), except at 100 km altitude. At lower altitudes, the raypaths are concentrated over each receiver. Therefore, the estimation errors at lower altitudes and near the edges of the reconstruction area are larger than in other regions. However, these errors are not significant enough to prevent identification of the plasma bubble. The average density error Ed is 3.30 × 1010 e/m3, which is very small compared with the typical peak electron density of 2.28 × 1012 e/m3. These results suggest that application of the RMTNN method is suitable for studies of the ionosphere, including investigation of irregular structures.
3.2. Case 2: Reconstruction From 24 GPS Data Receivers in Indonesia
 In section 3.1, we used 40 GPS receivers from GEONET for reconstruction. In this section, we investigate the performance of the RMTNN method using 24 GPS receivers in Sumatra (Indonesia), comprising 22 GPS receivers from the Sumatran GPS Array (SuGAr), 2 GPS receivers from the International GNSS Service (IGS), and an ionosonde station located at Kototabang (see Figure 6). In this simulation, we investigate a data set obtained during the period 15:00–15:15 UT on 9 September 2007. The reconstructed region covers 10°S–10°N latitude, 90°E–110°E longitude, and 100–700 km in altitude. In this case, the total number of raypaths is only 1171 and the total number of voxels for reconstruction is 35,301 (41 × 41 × 21 in latitude × longitude × altitude). We compared the reconstructed electron density distribution with the simulated distribution using NeQuick.
Figure 7 shows the model and reconstructed distributions in a longitudinal cross-section at 100°E. The reconstructed image is generally consistent with the original distribution, except at higher latitudes. The average density error Ed is 2.30 × 1010 e/m3, which is sufficiently small compared with the typical peak electron density of 7.41 × 1011 e/m3. Figure 8 shows the electron density profiles over the position at (−5°N, 100°E), (0°N, 100°E) and (5°N, 100°E). The errors of peak electron density are relatively large except at 0°N (near the Ktotabang ionosonde station) and the reconstructed profile is basically consistent with the model, except below 200 km in common. Even though it is located near ionosonde station, the estimation error above 600 km at 0°N is larger than other latitudes.
Figures 9 and 10 show horizontal cross-sections at altitudes of 100, 300, 500, and 700 km. Similarly to the vertical cross-sections, the reconstructed image is consistent with the original image, except at altitudes of 100 and 700 km. In Figure 10, the distribution at 100 km altitude is overestimated, while at 700 km it is slightly underestimated.
 Although this tendency was already shown for the plasma-bubble reconstruction, the estimation error associated with this previous reconstruction was larger than that derived here. Figure 11 shows the relationship between the distribution of penetrated voxels and altitude. At lower altitudes, the distribution of voxels that were penetrated by raypaths is concentrated over the receivers. The distribution of the effective voxels for RMTNN is scattered as a function of altitude. These distributions of raypaths at each altitude have an effect on the reconstruction accuracy at lower altitudes. In addition, the number of penetrating raypaths for each voxel at 700 km is smaller than at lower altitudes, possibly explaining the underestimation of the distribution at altitudes above 600 km.
4. Improvement in Reconstruction Accuracy at Lower Altitudes
 The main cause of the overestimation at lower altitudes seems to be the unconstrained electron density at the bottom of the ionosphere. Therefore, to improve the reconstruction accuracy, information on the electron density from alternate sources in the lower ionosphere is used as an additional constraint. For RMTNN, we defined the objective function E3 as an additional constraint. Therefore, the overall objective function E is
The effect of the additional constraint is evaluated by comparison of the original and the reconstructed data. In this simulation, the number of iterations for the RMTNN learning process is fixed at 4000 and the same initial weight values for the neural network are used.
 First, information on the electron density in the lower ionosphere (at 100 km altitude) is used to constrain the NeQuick model. We selected one (0°N, 100°E), five (one point + 10°N, 90°E; 10°N, 110°E; 10°S, 90°E; 10°S, 110°E), and nine points (five points + 5°N, 95°E; 5°N, 105°E; 5°S, 95°E; 5°S, 105°E). The objective function E3 for the NeQuick model value is given by
where R is the number of points to be constrained, Nr is the output of the neural network for the corresponding position, and NrNeQuick is the value simulated by the NeQuick model.
 Second, we use information on the electron density in the E region (at 130 km altitude) and the corresponding height observed by an ionosonde station as constraints. We used the value simulated by the NeQuick model at 130 km altitude over Kototabang as ionosonde data. The objective function E3 for the information on the E region is given by
where S is the number of ionosondes, Ns is the output of the neural network for the corresponding position that gives hmE, and Nsion is the value observed by the ionosonde.
 In addition, we use information on the electron density profile obtained by the LEO satellite. For the simulations, the actual positions of the profile points obtained by FORMOSAT-3/COSMIC are used and electron densities are generated by the NeQuick model. We have assumed that only one profile dominates the reconstruction area, and the profile points are selected with an altitude spacing of 30 km (see Figure 12). The objective function E3 for the LEO profile data is given by
where L is the total number of profile points obtained from the occultation observations, Nl is the output of the neural network for the corresponding position, and Nlocc is the value obtained from the occultation observations.
 To evaluate the effectiveness of our attempts to improve the reconstruction accuracy at lower altitudes, in addition to Ed, we define the average density error Ed′ between 100 and 250 km altitude, as follows:
where M′ is the number of sampling points used for reconstruction below 250 km altitude, and N and NNN are the original and reconstructed electron density distributions, respectively.
 The result of the calculated Ed and Ed′ values are shown in Table 1. All the constraints are effective in improving the reconstruction accuracy. The use of model data at 100 km altitude resulted in a reduction in Ed and Ed′, but the five-point restriction has a relatively small effect on the overall improvement, most likely because four of the five points lie outside the region of penetrated voxels at 100 to 300 km altitude as shown in Figure 11. Similarly, the use of information on the E region resulted in a reduction in Ed and Ed′. In addition, we obtained good performance using the electron density profile from the LEO satellite, including a remarkable improvement in reconstruction accuracy across the entire region. Therefore, the use of LEO profile data is the best approach resulting from this study, even though the improvement in the region 100 to 250 km is only slightly better than obtained with one alternative electron density value near the center of the imaged region at 100 km altitude. In this paper, we defined the estimation error as the ratio of reconstructed data to model data for each altitude. Figures 13 and 14 show the error maps estimated using the conventional (hmF2 + NmF2) and newly proposed methods (conventional constraints + LEO profile data). In Figure 13, the reconstructed data at 100 km altitude are up to 85 times larger than the model data. On the other hand, although underestimated voxels exist at 100 km altitude, these maximum errors are much smaller than those resulting from the conventional method. Moreover, the electron density profiles over the position at 0°N latitude and 100°E longitude (Figure 15) show improvement for both lower altitudes and for the full altitude profiles. These results indicate the effectiveness of using LEO profile data for accuracy improvement in reconstructions based on sparse data.
Table 1. Average Density Error Ed and Ed′
Unit of average density error is 1010 e/m3. Ed, average density error for the entire reconstruction region; Ed′, average density error for the region between 100 and 250 km altitude.
Conventional (NmF2 and hmF2)
1 point at 100 km altitude
5 points at 100 km altitude
9 points at 100 km altitude
E region (NmE and hmE)
LEO satellite profile data
 We evaluated the performance of RMTNN tomography under more realistic conditions compared to previous studies [Ma et al., 2005a; Takeda and Ma, 2007] using numerical simulations. First, a simulation using a simple plasma-bubble model as a representation of a disturbed ionosphere shows that the RMTNN method is suitable for the detection of irregular ionospheric structures in Japan. Although estimation errors prevail at lower altitudes, RMTNN tomography is sufficiently well suited for the investigation of irregular ionospheric electron density structures. Second, we validated the performance of the method based on sparse data coverage in Indonesia. Although sparse data was used for the tomography, the reconstructed data are consistent with the model data, except below 200 km and above 600 km altitude. This discrepancy appears to reflect the lack of constraints on the electron density at lower altitudes.
 To avoid this problem, we employed information on the electron density from the NeQuick model at 100 km altitude, in the E region provided by ionosonde data, and profile data by the LEO satellite. As a result, the proposed methods show significant improvements in estimation of electron densities, not only at lower altitudes but also across the entire region. In particular, inclusion of the electron density profile from the LEO satellite is effective in improving the reconstruction accuracy from the perspective of practical use. The average density errors below 250 km and for the entire region are half those resulting from the RMTNN method without additional constraints from alternative observations of electron density. Information on the E region is not always obtained by ionosondes. In addition, although the constraints from the NeQuick model at 100 km altitude are limited, there is a possibility that it may undermine the advantage of the RMTNN method. This is the case because one of the advantages of the RMTNN method is model-free reconstruction. On the other hand, inclusion of LEO observational data is extremely effective, because it allows us to include information on both the bottom of the ionosphere and below approximately 500 km altitude. However, the accuracy of profile data obtained from occultation observations requires further validation, because the raypath through the ionosphere from a GPS to a LEO satellite is often very long, and the location where it applies is not well defined. In fact, estimation error of LEO data at low altitude is relatively high, so sufficient caution is necessary when applying to actual data [Liu et al., 2010].
 We monitored the objective function E and average density error Ed. However, for application to actual data processing, a better-defined criterion is necessary. The well-known Akaike information criterion (AIC) [Akaike, 1974] is not necessarily suitable for parameter determination of neural networks [Hagiwara et al., 2001]. Therefore, evaluation by generalized cross-validation (GCV) might be effective. Moreover, RMTNN tomography can be easily extended to four-dimensional ionospheric tomography, including time evaluation. Continuous four-dimensional tomography can provide information on dynamic variations of the ionosphere, which reveals the transport of energy from space and/or the lithosphere. In conclusion, the RMTNN method, combined with additional data (e.g., LEO profile data), is eminently suitable for investigation of the electron density distribution in the ionosphere, including studies of irregular structures in different regions.
 The authors would like to thank the Scripps Orbit and Permanent Array Center (SOPAC) for GPS data from the International GNSS Service (IGS) and from the Sumatran GPS Array (SuGAr) stations; the Geospatial Information Authority of Japan (GSI) for GPS data from GEONET; the Japanese National Institute of Information and Communications Technology for the ionosonde data; and the COSMIC Data Analysis and Archive Center (CDAAC) for FORMOSAT-3/COSMIC data. This research was supported in part by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (grant 19403002) and by the National Institute of Information and Communication Technology (Research and Development promotion funding for international joint research).