Abstract
 Top of page
 Abstract
 1. Introduction
 2. Ionospheric Tomography by Using RMTNN
 3. Numerical Experiment Using Artificial Model Data
 4. Reconstruction of the Density Distribution From Actual Observation Data
 5. Conclusions
 Acknowledgments
 References
 Supporting Information
[1] In this paper we present a new method based on a Residual Minimization Training Neural Network (RMTNN) to reconstruct the threedimensional electron density distribution of the local ionosphere with high spatial resolution (about 50 km × 50 km in east/west and 30 km in altitude) using GPS and ionosonde observation data. In this method we reconstruct an approximate threedimensional electron density distribution as a computer tomographic image by making use of the excellent capability of a multilayer neural network to approximate an arbitrary function. For this application the network training is carried out by minimizing the squared residuals of an integral equation. We combine several additional techniques with the new method, i.e., input space discretization, use of ionosonde observation data to improve the vertical resolution, automatic estimation of the biases of the satellite and the ground receivers by using the parameter estimation method, and estimation of plasmasphere contributions to the total electron content on the basis of an assumption of diffusive equilibrium with constant scale height. Numerical experiments for the actual positions of the GPS satellites and the ground receivers are used to validate the reliability of the method. We also applied the method to the analysis of real observation data and compared the results with ionosonde observations which were not used for the network training.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Ionospheric Tomography by Using RMTNN
 3. Numerical Experiment Using Artificial Model Data
 4. Reconstruction of the Density Distribution From Actual Observation Data
 5. Conclusions
 Acknowledgments
 References
 Supporting Information
[2] Reconstruction of ionospheric electron density profiles based on the measurement of satellite radio signals has become an indispensable technique for various purposes ranging from academic to practical applications. As the ionospheric total electron content (TEC) is an integrated value of the ionospheric electron density along a ray path of the radio signal from a satellite to a receiver, reconstruction of the electron density profile from a set of TEC values is a form of computerized tomography (CT), which has been intensively studied for applications in various fields [Herman, 1980; Natterer, 1986].
[3] The tomographic technique for reconstructing the ionospheric electron density profiles was studied by Austen et al. [1988], in which the algebraic reconstruction technique (ART) was applied to the analysis of model TEC data generated by a computer simulation. The aim of the analysis was to recover a twodimensional image of the electron density profile of the model ionosphere by assuming a polar orbiting satellite at an altitude of 1000 km with three or five ground stations. Following their work, twodimensional ionospheric tomography has been studied extensively from both the theoretical and observational aspects [Raymund et al., 1990; Fremouw et al., 1992; Kunitake et al., 1995]. In these studies the observation data are radio signals from the Navy Navigational Satellite System (NNSS) received at several ground stations, and the NNSS satellites move along polar orbits at a height of about 1100 km. Although these studies are useful for understanding the basic ionospheric structure, there are two significant limitations: first, only a twodimensional structure of the ionosphere within a cross section defined by the satellite orbit and the ground receiver array can be obtained; second, the time and place of observation are limited, so the electron density profile cannot necessarily be obtained at any time at any place around the world. Recently, several authors have investigated vertical ionospheric profiles by means of the Abel inversion technique. This technique is based on radio occultation measurements with the Global Positioning System (GPS) and a low Earth orbit (LEO) satellite by assuming a spherically symmetric electron density distribution around the Earth [Hajj and Romans, 1998; Tsai et al., 2001]. This assumption leads to considerable inaccuracy in estimates of the reconstructed electron density because the actual electron density varies not only vertically but also horizontally. Considering horizontal gradients in Abel inversion, GarciaFernandez et al. [2003a] proposed an approach based on a separability hypothesis by considering horizontal gradients in the Abel inversion by the assist of the IRI model and obtained clear improvement in comparison with the conventional Abel inversion method. Instead of using LEO GPS observations, GarciaFernandez et al. [2003b] first proposed a method to estimate electron density by combining the ionosonde and ground GPS data, in which vertical profiles of electron density is derived from ionosonde measurements by using NeQuick model.
[4] From the practical viewpoint to determine the ionospheric correction for TEC value by the pseudorange calculation, Hansen et al. [1997] proposed to apply the conventional CT method based on the Radon transformation. In this analysis an ionospheric delay corresponding to the “true” TEC value along a ray path is derived from the dualfrequency signal (f1 = 1575.42 MHz and f2 = 1227.60 MHz) transmitted by a GPS satellite. By using the electron density profile reconstructed from the tomographic inversion, a better ionospheric correction has been calculated in comparison with the usual correction derived by assuming the thinlayer ionospheric model where all the electrons are concentrated in a thin shell located at a fixed height. Further studies of the tomographic approach using two or threelayer models confirmed that this is appropriate for TEC estimation in comparison with the thinlayer model, especially when a large TEC gradient exists [Juan et al., 1997; HernandezPajares et al., 1999]. However, it is still very difficult to reconstruct an accurate vertical electron density profile by simply applying the above methods because there are no horizontal ray paths from a satellite to a ground receiver. To cope with this problem, HernandezPajares et al. [1998] developed a method to reconstruct the threedimensional electron density distribution of the ionosphere on a global scale by using a combination of ground receivers and LEO GPS data. In this method the tomographic problem is solved by means of Kalman filtering with a filter updating time of 1 hour in a Sunfixed reference frame and with a resolution of 10 × 10 deg in latitude/longitude and 100 km in height. Despite the capability of this tomographic method for globalscale analysis, it is difficult to apply the method to local tomographic profile reconstruction with high spatial and temporal resolutions because the ray path passing through the ionosphere from a LEO receiver to a GPS satellite is often very long (more than 1000 km); moreover, sudden smallscale variations in the ionosphere are not represented with resolution of 10 × 10 deg (about 1000 km) in latitude/longitude.
[5] To solve the above problems it is necessary to develop a locally applicable, highresolution, threedimensional ionospheric CT method which uses radio signal data measured within a short period of less than an hour. For this purpose, we propose a new ionospheric CT method and validate it in this paper. This method is based on the Residual Minimization Training Neural Network (RMTNN) [Liaqat et al., 2003] and use data from the GPS Earth Observation Network (GEONET) [Miyazaki et al., 1997] installed by the Geographical Survey Institute of Japan (GIS). By making full use of the spatially dense (more than 1000 receivers) GEONET data, several authors have derived TEC values with high spatial resolution on the basis of the thinlayer model [Otsuka et al., 2002; Ma and Maruyama, 2003]. We combine several additional techniques with the new method, i.e., input space discretization, use of ionosonde observation data to improve vertical resolution, automatic estimation of the biases of satellite and ground receivers by using the RMTNN [Ma et al., 2004], and estimation of plasmasphere contributions to the TEC on the basis of the assumption of diffusive equilibrium with constant scale height. These techniques are described in the next section.
[6] The method is realized by making use of the excellent capability of multilayer neural network to approximate an arbitrary function [Funahashi, 1989; White, 1990], for which the network training is carried out by minimizing the squared residuals of an integral equation. Numerical experiments on the actual positions of the GPS satellites and the ground receivers are used to validate the reliability of the method, where a local electron density of the ionosphere and plasmasphere are taken from the Global Core Plasma Model (GCPM) [Gallagher and Craven, 2000] for the model problem. We also applied the method to actual observation data analysis.
[7] In the next section we describe in detail the ionospheric CT reconstruction method that uses the RMTNN. Numerical experiments for model problems are presented in section 3. In section 4 we evaluate the proposed method with real observation data. Section 5 has a summary of the paper and mentions future work.
2. Ionospheric Tomography by Using RMTNN
 Top of page
 Abstract
 1. Introduction
 2. Ionospheric Tomography by Using RMTNN
 3. Numerical Experiment Using Artificial Model Data
 4. Reconstruction of the Density Distribution From Actual Observation Data
 5. Conclusions
 Acknowledgments
 References
 Supporting Information
[8] RMTNN is a multilayer neural network which is trained by minimizing an object function composed of an appropriately prepared residual of an equation. As a multilayer neural network is a “universal approximator” represented by an analytical equation [Funahashi, 1989; Hornik, 1989], a residual of any equation is expressed analytically by regarding a neural network as an approximate solution of the equation. By choosing an appropriate combination of the squared residuals of various equations and minimizing it, a solution for a complicated system is obtained comparatively easily as the trained neural network (the RMTNN). In this method the role of the multilayer neural network is very important, but we will not describe the details of the neural network as it has already been described by many authors [e.g., Bishop, 1994; Rojas, 1996]. As the interpolation and the smoothing functions are inherently included in the multilayer neural network [Poggio and Girosi, 1990], the difficulty in CT image reconstruction that arises from an imperfect set of projection data is relaxed considerably by using the neural network. Comparison of the results with those by the filtered backprojection, the representative CT image reconstruction method, shows that the RMTNN method for the CT analysis is very effective for such an imperfect set of projection data [Ma et al., 2000].
2.1. Derivation of Slant TEC
[9] The derivation of slant TEC from GPS observations is commonly made using both group delays (Φ_{1} and Φ_{2}) and carrier phase advances (L1 and L2) by many authors. In brief, the slant TEC from the group delays I_{ϕ} and the carrier phase difference of the two signals I_{l} are given as
where λ_{1} and λ_{2} are the wavelengths corresponding to frequency f_{1} and f_{2}, respectively. Although the accuracy of the slant TEC derived from the phase measurement (I_{l}) is higher than the other (derived from I_{ϕ}), it only provides the relative change in TEC because of the ambiguity is included in the phase measurement. An accurate slant TEC value is derived by using both of those group delays I_{ϕ} and carrier phase advances I_{l} by introducing a baseline B_{L} for the differential phaserelated I_{l} [Mannucci et al., 1998; Horvath and Essex, 2000]
where B_{L} is calculated as the following:
where P is the number of measurements from a receiver to a satellite and θ is the elevation angle. It should be noted that the square sine of the satellite's elevation θ_{n} is included as a weighting factor because the pseudorange for the path with low elevation angle is apt to be affected by the multipath effect and the reliability decreases. When making the above calculations of B_{L}, a dataprocessing step is included to identify possible cycle slip in either L_{1} and L_{2} phase measurements [Blewitt, 1990]. This process is then carried out for satellitereceiver pair.
2.2. Basic Equations and Computational Domain
[10] The slant TEC I_{i}^{j}(t) along the ray path between a GPS satellite j and a ground receiver i is the integrated value of the ionospheric and the plasmaspheric electron density, including the instrumental biases of the transmitter in the satellite B^{j} and the ground receiver B_{i} given as
where N(, t) is the electron density at the observation time t, and I and J are the respective total numbers of ground receivers and satellites used for the measurement. The computational domain of the above integral equations of N(, t) is divided into lower region (the ionospheric region) spanning from 100 km to 700 km in altitude and an upper region (the plasmaspheric region) above 700 km, and the ionospheric region is the target region of the tomographic analysis (Figure 1). The reason of choosing 700 km as the upper boundary of ionospheric region is described in section 3. The CT analysis of the ionospheric region is comparatively easy because this region is located at far lower altitudes than the orbits of the satellites (20,200 km in altitude) and densely located ray paths are available for the analysis.
2.3. Object Function of the Residuals of the Integral Equations
[11] To determine the function N(, t), which represents the electron density at a position and a time t, we consider a neural network with input channels for and t and an output channel for N(, t). If observation data for the slant TEC are given, this function is derived as a trained neural network by minimizing a set of squared residuals of integral equations (5). To evaluate the residuals of the integral equations (5), these equations are discretized as
where q and α denote a sampling point (from 100 km to 700 km in altitude) and the corresponding weight for the numerical integration, respectively, Q is the total number of the sampling points on a path, and P_{i}^{j} is the contribution of the plasmaspheric electron density to the slant TEC I_{i}^{j}. To estimate the electron density N(, t), we take the squares of the residuals of the above integral equations E_{ri} as the object function of the RMTNN system (Figure 2),
The RMTNN system is composed of a main feedforward neural network, NN, and additional “neural networks” consisting of single neuron, NB, each corresponding to the bias of a GPS satellites or a ground receiver.
2.4. Combined Use of Ionosonde Data
[12] The weakness of the ionospheric CT reconstruction method using GPS signals is that satisfactory vertical resolution is very difficult to attain because there are few ray paths close to the horizontal directions. To cope with this problem and improve the vertical resolution, we use information on the peak electron density, N_{m}F_{2} and the corresponding height h_{m}F_{2} obtained from ionosonde measurements. The neural network is trained with these data through conventional supervised training and the object function E_{is} is given as
where S denotes the total number of ionosonde stations, N_{s} is the output value of the neural network, and N_{s}^{ion} is the electron density value obtained from ionosonde measurements. By combining the object function for the ionosonde data E_{is} with that for the residuals of the integral equation E_{ri}, the overall object function E is derived as
where g is a penalty coefficient. In the training process, E_{is} is evaluated only at the observation point of the ionosonde and the E_{ri} values are evaluated by numerically integrating the values at the points on the ray paths. Note that this process is a special case of weak constraint data assimilation [Liaqat et al., 2001], though the solution is mainly governed by the very constraint imposed by the integral equations in the ionospheric CT reconstruction method.
2.5. Input Space Discretization
[13] Before the numerical computation, we discretize the temporal and the spatial domain into finitesized meshes. In this paper we restrict ourselves to the threedimensional CT of the ionospheric electron profile at a “fixed” time; therefore the temporal domain is composed of only one mesh with small time span, during which a sufficiently large number of ray paths are available, but it is permissible to assume that the ionospheric electron density profile is unchanged. Hereafter, we omit t from the above equations because we analyze the ionospheric density profile for a fixed time. This time span is taken as 15 min throughout the model and actual data analysis in this paper. As for the spatial domain, the discretization is carried out so that on average more than one ray path crosses through a twodimensional crosssectional area of each mesh. Though by the continuous mapping function realized by a neural network it seems unnecessary to discretize the computational domain, continuous treatment of the input variables cause overfitting and make the system unstable because of the finite number of constraints (ray paths). To avoid these defects, input space discretization is especially effective. For this reason, we assume that the electron density is constant within an area of 0.5 × 0.5 deg (about 50 km × 50 km) in latitude/longitude and 30 km in altitude as shown in Figure 1.
2.6. Contribution of the Plasmaspheric Electron Density to the Observed TEC Values
[14] To estimate the contribution of the plasmaspheric region to the observed TEC value, we employ a simple diffusive equilibrium model proposed by Angerami and Thomas [1964]. According to this model, we assume the electron density decreases exponentially with altitude, so the electron density distribution n(h) from 700 km to 20,200 km in altitude is expressed by an exponential function
where n(h_{0}) is the electron density at the top of the ionospheric region (i.e., at the altitude of h_{0} = 700 km) and H_{s} is the scale height of the density decay in the plasmasphere. Although the electron density of the plasmasphere is distributed along magnetic field lines and the scale height H_{s} varies with the solar activity, season, local time, and so on, we fix H_{s}, for simplicity, to the value at the height of 2000 km [Reinisch et al., 2001; Leitinger et al., 2002]. This is acceptable because our aim is to reconstruct the electron density profile of the ionospheric region, where the contribution of the plasmaspheric region to the TEC value is considered to be very small. Under this assumption the plasmaspheric contribution to the TEC value of the path (i, j), P_{i}^{j}, is given as
where h_{sat} is the altitude of each GPS satellite and θ is the inclination of the ray path with respect to the vertical direction.
2.7. Procedure to Update Weights of the Neural Networks
[15] To calculate the ionospheric electron density profile from the observed TEC values and the density values measured at particular points by the ionosonde, we train the RMTNN system of Figure 2 by optimizing the object function (9). After the training is over, the main neural network NN becomes an approximate density function N(), and the bias values of the transmitters on the satellites and the ground receivers are also obtained. As described in section 2.3, in the weightupdating process the different parts of the object function, E_{is} and E_{ri}, are evaluated whether the input data (the evaluation position) are ionosonde positions or the positions on the ray paths for the TEC data. When the input data are the ionosonde positions, the process is the conventional supervised training, where E_{is} is minimized. For this training process we employ online mode weight updating because of the high efficiency. In the training for the TEC data, though, some explanation may be necessary as provided in the following.
[16] In this training process (7) is minimized. For this training the online mode weight updating cannot be used because the object function can be used only after all the density values of the sampling points on a ray path are evaluated. For this reason we carry out the forward calculations for the sampling points on the same ray path and update the weights of the neural network every time of the calculation of the single line integral, which means that the weights are updated every Q times of the forward calculations. Considering that the error function is defined as a sum of squared residuals of a line integral (7), we therefore call it the “quasionline updating scheme” [Ma et al., 2000]. Figure 3 shows the schematic diagram of the updating method. Therefore for the object function shown in (7) the weight increment in the updating process of the quasionline method of the main network NN is derived in clusters for the pth path as
where, for simplicity, p represents the ray path (i, j), I_{p}^{NN} is a line integral along the pth path obtained numerically by using the output data of the neural network, w is a weight of the main network NN, and η is the learning rate.
[17] To determine the biases of satellite transmitters and ground receivers, a parameter estimation method proposed by Ma et al. [2004] is applied. The output values of the additional networks NB are incorporated with the solution I_{p}^{NN} of the main network NN to evaluate the object function. Therefore the weight updating process of a conventional error backpropagation method for these additional networks is given as
where P_{i} is the number of ray paths passing the ground receiver i. The biases of the satellite transmitters are similarly updated.
3. Numerical Experiment Using Artificial Model Data
 Top of page
 Abstract
 1. Introduction
 2. Ionospheric Tomography by Using RMTNN
 3. Numerical Experiment Using Artificial Model Data
 4. Reconstruction of the Density Distribution From Actual Observation Data
 5. Conclusions
 Acknowledgments
 References
 Supporting Information
[18] To examine the effectiveness of the new method, we first applied it to a model problem, in which the electron density distribution is generated by using the GCPM [Gallagher and Craven, 2000]. The GCPM is considered as the most comprehensive model available for our purpose because it includes the ionosphere, plasmasphere, magnetospheric trough, and polar cap, where the ionospheric densities are represented by the International Reference Ionosphere (IRI) model [Bilitza, 1990] at lower altitudes. In this model, electron densities are smoothly varying and available from 100 km to 20,200 km (the height of GPS satellites) in altitude. The actual positions of the GPS satellites and the ground receivers during a particular observation (the real ray path geometry) are used for the model calculation, whereas the bias values of the satellites and the receivers (which we will refer to as the “true” bias values) are assigned artificially. The values of N_{m}F_{2} and h_{m}F_{2} at the position of the actual ionosonde station located in Japan are obtained from the GCPM and used as the “observation” data of the ionosonde. To evaluate the error of the analysis, we define the average density error E_{d} as
where M is the number of sampling points used for the reconstruction after the learning process is over, and N and N^{NN} are the original (GCPM) and reconstructed density distributions, respectively. The average density error can be calculated only for the model problem where the original density distribution of the GCPM is known beforehand. The GPS satellite and receiver bias errors, E_{s} and E_{b}, are defined as
where I and J are the total number of satellites and receivers, respectively, B_{i} and B_{i}^{NN} are the “true” and determined bias values of the ith receiver, respectively, and B_{j} and B_{j}^{NN} are the “true” and determined satellite bias values of the jth satellite, respectively. The unit of a bias value is the TECU (1 TECU = 1.0 × 10^{16} e/m^{2}).
[19] In this paper, both the reconstructions from the model data and the actual observation data are carried out by dividing the computational domain of (5) (and consequently (6)) into two regions, i.e., the ionospheric region and the plasmaspheric region. To determine the appropriate upper boundary of ionospheric region (the target region of the tomographic analysis), a simple numerical experiment is carried out by varying the upper boundary from 600 km to 1000 km every 100 km in altitude. Observation data of ground GPS for the test calculation are taken from the data set of 1200–1215 JST, 22 December 2001 (0300–0315 UT, 22 December 2001) for 40 GEONET receivers located in Japan (Figure 4) and the observation data from the ionosonde station located in Kokubunji (139.51°E, 35.74°N). The results are shown in Table 2; it is found that the average density error E_{d} for the case of the altitude of 700 km is smallest, and for this reason we used 700 km as the upper boundary of the ionospheric region for the following calculations.
[20] We produced model observation data corresponding to 1200–1215 JST, 22 December 2001 (0300–0315 UT, 22 December 2001) for 40 GEONET receivers located in Japan (Figure 4) to test our reconstruction method. Total number of the ray paths P is 2128 for this case and 20 sampling points are placed on each ray path. It should be noted that total number of the ray paths varies due to the number of satellites visible in the sky at the same time. Although there are four routine ionosonde observation stations in Japan, as shown in Figure 4, we used observation data from only the ionosonde station located in Kokubunji (139.51°E, 35.74°N) for both the model and the actual data reconstruction. Observation data from the other stations are used to examine the reliability of our method in the actual data reconstruction described in the next section.
[21] In order to optimize the neural network structure, an error attained after a fixed number of iterations is investigated by varying the number of the hidden layers for the model observation data, networks with 1, 2 and 3 hidden layers are considered. The dependence of the attained average density error E_{d} for network with 3, 4, and 5 layers are shown in Table 1. It is seen that the error decreases rather rapidly up to fourlayered neural network with 2, 12, 12, 1 neurons at each layer but afterward the error decrease very slowly. Considering computation cost, we employ a 4layered neural network with 2, 12, 12, 1 neurons at each layer to the following numerical experiments of this paper. It takes about 10 min of CPU time to train the network 4000 times. The calculations are carried out on a workstation with a Xeon 2.20 GHz CPU and the programming language is ANSI standard Fortran with double precision.
Table 1. Dependence of Average Density Error on the Number of the Hidden Layers^{a}Number of Layers  Number of Units of Each Layer  E_{d} 


3  3121  0.46 
3  3201  0.39 
4  312121  0.28 
4  320201  0.26 
5  31212121  0.23 
5  32020201  0.23 
Table 2. Results of Determination of Upper Ionospheric Boundary^{a}Altitude of Upper Boundary  Average Density Error 


600 km  0.35 
700 km  0.28 
800 km  0.29 
900 km  0.31 
1000 km  0.31 
[22] The contour plots of the model and reconstructed density distributions are shown in Figure 5 for a vertical cross section with a fixed longitude of 137°E. The average density error E_{d} is 2.8 × 10^{10}e/m^{3}, which is very small compared with the typical peak electron density of 2 × 10^{12}e/m^{3}. The average error of the receiver bias E_{r} and satellite bias E_{s} are 0.12 and 0.31 TECU, respectively. A correlation diagram of the determined bias B^{NN} versus the true bias B is shown in Figure 6. The results show that the electron density reconstruction and bias determination are carried out with sufficiently high accuracy using the model data. Figure 7 compares the model vertical electron density distributions of Wakkanai (141.68°E, 45.39°N), Kokubunji (139.51°E, 35.74°N), Yamagawa (130.62°E, 31.20°N), and Okinawa (128.15°E, 26.67°N) to the corresponding distributions reconstructed by the new method. Table 3 gives the standard deviation of the electron densities in these calculations. The vertical distribution is clearly reconstructed appropriately even though we use observation data from only one ionosonde station located (Kokubunji). Similar results are obtained with model data generated for different solar activity, seasons, and local time.
Table 3. Standard Deviations of Reconstructed Vertical Density Distribution^{a}Location  Standard Deviation 


Wakanai  0.557 
Kokubunji  0.510 
Yamagawa  0.426 
Okinawa  0.525 
4. Reconstruction of the Density Distribution From Actual Observation Data
 Top of page
 Abstract
 1. Introduction
 2. Ionospheric Tomography by Using RMTNN
 3. Numerical Experiment Using Artificial Model Data
 4. Reconstruction of the Density Distribution From Actual Observation Data
 5. Conclusions
 Acknowledgments
 References
 Supporting Information
[23] In this section we discuss the application of the new method to actual GPS observation data obtained from 40 different receivers and ionosonde observation data from Kokubunji (Figure 4). Observation data obtained within 15 min every hour on 5 November 2001 are analyzed to investigate the effects of hourly variations in the ionosphere from 0000 to 2400 in JST. Some examples of the hourly variations of ionospheric density distribution at 137°E are shown in Figure 8. It should be noted that the color map scale indicating the electron density value differs for the different parts in order to clearly show the region of the maximum electron density in each part. Figures 9 and 10 show contour plots of the longitudealtitude cross section at 37°N in latitude and the longitudelatitude cross section at 300 km in altitude obtained from 0400 JST, respectively.
[24] The vertical resolution of ionospheric tomography is usually very limited because there are no horizontal ray paths from the satellite to the ground receiver. To confirm the reliability of the vertical resolution of the new method, we compare the N_{m}F_{2} and h_{m}F_{2} values obtained by the new method with those obtained through ionosonde observations. For this confirmation we use data observed at Wakkanai, Yamagawa, and Okinawa because the data at Kokubunji is used as training data for the tomographic reconstruction. The peak density value N_{m}F_{2} and the peak density height h_{m}F_{2} of the observation and the calculation during 24 hours on 5 November 2001 are shown in Figures 11 and 12, respectively. Table 4 shows standard deviations of reconstructed N_{m}F_{2} and h_{m}F_{2}. Those figures and table show that the agreement between the calculated and the observed values is quite good not only for Kokubunji but also for Wakkanai, Yamagawa, and Okinawa. On the basis of these results, we can safely say that an appropriate vertical electron density distribution can be reconstructed by using the new method.
Table 4. Standard Deviations of Reconstructed N_{m}F_{2} and h_{m}F_{2}^{a}Location  Standard Deviation of N_{m}F_{2}  Standard Deviation of h_{m}F_{2} 


Wakanai  1.135  15.08 
Kokubunji  0.365  14.72 
Yamagawa  1.362  22.90 
Okinawa  1.683  20.86 
5. Conclusions
 Top of page
 Abstract
 1. Introduction
 2. Ionospheric Tomography by Using RMTNN
 3. Numerical Experiment Using Artificial Model Data
 4. Reconstruction of the Density Distribution From Actual Observation Data
 5. Conclusions
 Acknowledgments
 References
 Supporting Information
[25] We have successfully applied a new method based on the residual minimization training neural network (RMTNN) to reconstruct a local ionospheric electron density profile from model data and actual observation data. In this study the solution system of the method consists of the RMTNN, which is trained by minimizing an object function composed of a sum of squared residuals from an integral equation. We successfully combined several additional techniques with the new method, i.e., input space discretization, use of ionosonde observation data to improve the vertical resolution, and automatic estimation of the biases of the satellite and ground receivers.
[26] The special features and possible extensions of this method are summarized as follows.
[27] 1. Ionospheric tomography using the RMTNN is successfully realized. The new method is constructed on the basis of the excellent capability of a multilayer neural network to approximate an arbitrary function, where the smoothing and interpolation of observation data are automatically included, to attain a smooth inverse transform for data containing experimental errors and/or errors due to the numerical computation without special consideration for regularization.
[28] 2. Although the solution of the integral equations (GPS TECs) mainly determines the reconstructed image of the ionospheric density distribution, inclusion of ionosonde data is very effective to improve vertical resolution of the electron density in the new method. As is the case with this idea, it is easy to make a combined use of other observations (e.g., topside sounder, LEO, etc.), for the training process of RMTNN. The new ionospheric tomography we have proposed is considered a special case of weak constraint data assimilation, where data from only one observation point are “assimilated” under the weak constraints imposed by the integral equation.
[29] 3. The new method can also be applied to simpler problems to obtain almost realtime (every 15 min) TEC with high precision, because the proposed method does not rely on the assumption that all the electrons are located within a thin shell at a fixed height.
[30] 4. The main source of error in the proposed method is the use of a simple exponential approximation with a fixed scale height of the plasmasphere. To evaluate the effect of the plasmasphere precisely, a more sophisticated model of the plasmasphere is needed. As future work, this method can be easily extended to fourdimensional tomography by considering the time evolution of the ionosphere. In this paper we restricted ourselves to describing the new method and we put off the studies of the physics, statistical analysis, and application to ionospheric storms for the future works.