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 The dual-frequency signals of GPS can be used to measure the total electron content (TEC). The differential instrumental biases inherent in GPS satellite and receivers are considered as the main sources of error, and they must be removed for an accurate estimation of TEC. We aim at developing an effective method to solve the difficulties involved in the TEC measurement; there are only a few usable ground receivers, especially in lower-latitude areas near the geomagnetic equator where large ionospheric variability exists. For this purpose a new parameter estimation method based on a residual minimization training neural network is applied to determination of the GPS receiver biases. The alternative method is realized by making use of the excellent features of neural networks to approximate a wide range of mapping functions, for which the network training is carried out by minimizing squared residuals of integral equation. To determine receiver biases (unknown parameters), we used additional “neural networks,” each of which consists of only one neuron without an input channel. It is assumed that satellite biases have already been determined by applying the least squares method to the GEONET data gathered by a large number of receivers. Various cases of observation data for different seasons, different local times, and different geographic locations of the receiver as well as the cases of model data are analyzed, and it is confirmed that the method is very effective for a small number of receivers located in the lower-latitude areas.
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 The Global Positioning System (GPS) is a useful tool for the measurement of the ionospheric total electron content (TEC). A GPS satellite broadcasts dual-frequency signals (f1 = 1575.42 MHz and f2 = 1227.60 MHz) to allow users to derive the ionospheric delays. The measured differential delays, however, contain not only the delay due to the ionospheric TEC, but also the delay generated by interior electronic circuits of the GPS satellite transmitters and ground receivers. The latter delay is referred to as instrumental bias (or interfrequency bias). Therefore, for the purpose of accurate measurement of TEC, we have to remove these instrumental biases. Several methods have been proposed for determination of the instrumental biases. Among them most methods assume a rather smooth spatial and temporal variation of the ionospheric behavior over the observation area during several hours of observation. Lanyi and Roth  proposed a method in which the vertical TEC is modeled by a quadratic function of latitude and longitude of two-dimensional position under consideration. In this method, GPS observation data obtained during a night in a midlatitude area from a single GPS receiver are used to estimate the coefficients of a two-dimensional quadratic model (by using the least squares method), and at the same time, satellite and receiver biases are derived. Coco et al.  used this method to investigate the general tendency of the variability of the GPS instrumental biases. To improve the accuracy of the bias estimation, Wilson et al. [1992, 1995] extended single-site technique proposed by Lanyi and Roth  to a multisite one by modeling a global vertical TEC by two-dimensional spherical harmonics. They applied this technique to obtain the diurnal and semidiurnal average of the ionospheric TEC map of the Northern Hemisphere by using data from a global network of GPS stations. An approach based on the Kalman filtering technique was proposed by Sardon et al. , which uses a second-order polynomial approximation for TEC over each station in a local reference frame. In this method instrumental biases and TEC are estimated by using stochastic parameters in the Kalman filter. These methods assume a smoothly varying ionosphere over the observation area during several hours so that the TEC can be modeled with a lower-order polynomial.
 The GPS Earth Observation Network (GEONET) [Miyazaki et al., 1997] installed by the Geographical Survey Institute of Japan (GIS) has densely located receivers covering Japan (more than 1000 receivers) and allows derivation of TEC with high spatial resolution. Otsuka et al.  developed a technique to remove instrumental biases with a least squares fitting procedure and construct two-dimensional maps of the absolute TEC over Japan by using GPS data of GEONET. Although this method can be derived the TEC maps over Japan with considerably high spatial and temporal resolutions, the error becomes more noticeable with lowering latitude of the observation areas because of the assumption that the hourly average of vertical TEC is uniform within an area covered by a receiver. The area covered by a receiver approximately corresponds to the surrounding 1000 km, so this method should be applied carefully to data collected the lower latitudes where large ionospheric variability exists. Differently from the method of Otsuka et al. , Ma and Maruyama  developed a method in which both the biases and TECs are determined at the same time by using GPS data from GEONET collected within one day. This method is based on the assumption that TEC is uniform in a small area (2 by 2 in latitude and longitude, respectively), and the TECs and biases of the satellites and receivers are determined by using the least squares fitting technique. Both methods are applicable only in those regions where a network of densely located receivers, such as GEONET, has been established. Thus a simple method to estimate a single receiver bias was proposed by Ma and Maruyama , in which the satellite biases determined from GEONET were used. However, because the estimation error tends to increase at lower latitudes; the simple method should be applied at lower latitudes (<30°N) where a great latitudinal TEC gradient and the equatorial anomaly exist. It is, therefore, still difficult to determine the receiver bias when a receiver is located in a low-latitude area.
 Another large error source in previous studies stems from the assumption of a thin shell, where electrons lie in an infinitely thin ionosphere located at a constant height above the Earth. There are two major drawbacks to estimation of TEC by using the thin-shell model. First, the height of the ionospheric layer is commonly chosen at a fixed height (e.g., 400 km altitude), whereas the actual ionospheric layer is changeable depending on season, latitude, local time, etc. The height of the ionospheric layer should be precisely estimated before an accurate measurement of the TEC values. Second, at the low latitudes (in the vicinity of the geomagnetic equator) the spatial changes in TECs are too drastic to be expressed by the thin-shell model. Methods based on the thin-shell model are therefore not very applicable to analyses of the ionosphere near the geomagnetic equator.
 In order to cope with the above problems we propose an alternative method for determination of the receiver biases by using RMTNN [Liaqat et al., 2003]. In this method we reconstruct an approximate three-dimensional electron distribution as a computer tomographic image [Ma et al., 2000] and determine the biases of the ground receivers from the observation data by making use of the excellent capability of the multilayer neural network to approximate an arbitrary function. We successfully applied this method which does not rely on the assumption of a thin shell, to both the model data and the observation data analyses.
 The neural network parameter estimation method is described in section 2, where, we summarize the basics and features of a multilayer neural network. Numerical experiments for model problem are presented in section 3. In section 4, we evaluate the proposed method on the real observation data. Section 5 has a summary of the paper and mentions future work.
2. Neural Network Parameter Estimation Method
 A multilayer neural network is regarded as a nonlinear mathematical function, which maps a set of input variables into a set of output variables. The process of determining the values of the parameters (weights assigned to interconnections between neurons) is called learning or training [Bishop, 1994; Rojas, 1996]. Under a certain mathematical condition, a multilayer neural network can approximate any function with arbitrary accuracy by training it appropriately [Funahashi, 1989; White, 1990]. As the network training process contains smoothing and interpolating functions, a smooth inverse transformation of data with noise can be performed without special consideration of regularization. Those features make multilayer neural networks extremely useful and interesting. Throughout the present study, we define an object function composed of a sum of squared residuals of an integral equation instead of the conventionally used object function composed of a sum of squared differences between the output data and the corresponding teacher data. By employing an appropriate numerical line integral for the above integral equation, we can construct a residual minimization training neural network, which can be used for determination of GPS receiver biases as unknown parameters for cases with a small number of receivers.
 As measured observation data are sums of line integrals of ionospheric electron density along ray paths (TEC) and biases of satellite and ground receivers, we solve a general three-dimensional CT problem by assuming that electron density distribution of the ionosphere does not change during the time span when all the data are observed. To check the effectiveness of the new method we used a data set where satellite biases have already been determined by using the least squares method proposed by Ma and Maruyama . Although reconstruction of the spatial electron distribution in the ionosphere (an ionospheric CT reconstruction) and determination of the satellite biases are also possible with the new method, a large amount of the observation data is necessary for accurate analyses. In this paper, therefore, we restrict ourselves only to determination of the receiver biases.
2.1. Formulation of GPS Receiver Bias Determination Problem
 The slant TEC along the ray path connecting a GPS satellite and a receiver can be obtained directly from the GPS measurements by comparing the group delays (P1 and P2) and the phase delays of the L1 and L2 signal. We use a general tomographic model to determine the ionosphere electron density and instrumental biases for geometry shown in Figure 1. The slant TEC can be considered as integrated value of the ionospheric electron density including instrumental biases
where ij is the integrated value of the ionospheric electron density along the ray path from the satellite j to the ground receiver i, Ne(, t) is the electron density at the observation time t and three-dimensional position . i and j are the position vectors of the receiver i and satellite. j, Bi and Bj are the biases of the receiver and the satellite. The electron density Ne(, t) and instrumental biases are obtained by solving the above integral equation. For simplicity we assume that the ionospheric electron density do not change during a short time span and the integral equation is reduced to
 In the numerical calculation equation (2) is discretized with numerical integration and the above equation can be rewritten as
p and q denote the ray path and the sampling point for the numerical integration, respectively, and α denotes the weight for the integration.
2.2. Numerical Scheme and Algorithm for Determination of GPS Receiver Biases
Figure 2 shows a schematic diagram of the data flow of the new method. The whole system of the GPS receiver biases determination is composed of three modules: M1, M2, and M3. These are a module (M1) of the main feed-forward network, a module (M2) composed of additional “neural networks” consisting of only one neuron corresponding to numbers of GPS receiver, and a module (M3) to calculate the object function and its gradient for sake of updating the weights. In M1, an input pattern (xn, yn, zn) (a position along the observation path) is entered and the density at the position Ne(xn, yn, zn) is obtained. The symbols Noffi is the “network” composed of a single neuron corresponding bias, Bi of the GPS receiver i. The incremental value for updating the weights of the main neural network and the outputs of single neurons in M2 are denoted as Δ and ΔB's in the figure. It should be noted that in the single neuron network Noffi the output value is directly updated instead of a weight different from the usual neural network. Bj is a value taken from the analyses of Ma and Maruyama . As there are not supervisor data for Ne(xn, yn, zn) and we cannot define an object function for the usual neural network back-propagation training algorithm [Rumelhart et al., 1986], we modified the algorithm so that the target values of the line integrals along the measuring paths coincide with the measured data (the supervisor data). The object function is, therefore, defined as
where p is a measurement slant TEC value obtained from dual-frequency GPS radio signals, and IpNN is a line integral along the pth path obtained numerically by using the output data of the neural network as
where Q is the total number of sampling points of the numerical integral on a single path. We employ the trapezoidal rule with 20 sampling points for numerical integral calculation, therefore, Q is equal to 20, and αq is the weight of the integration of the point (xq, yq, zq). For the above object function the increment of the weight in the updating process of the error back-propagation method of the main network NN is derived as
 In the above equation it should be noted that the weights are updated every time a single line integral is calculated, which means that the weights are updated once every Q mapping calculations. Considering that the error function is defined as a sum squared residuals of a line integral (equation (4)), we call this scheme the “quasi-online updating scheme.” In the case of the density reconstruction problem, however, the output data should be always positive and it may be convenient to employ activation functions whose output data are always positive and upper unbounded. For this purpose, we employ the following skimmer-shaped activation function [Ma et al., 2000],
 To estimate the GPS receiver biases (as the unknown parameters) we use additional neural networks that give corresponding GPS receiver biases Bi as output values. These additional networks consist of only one neuron without an input channel. The output values of these networks are incorporated with the solution IpNN of the main network to evaluate the object function, therefore, the weight updating process of the error back-propagation method for the additional networks is given as
where is learning rate of these additional networks which can be chosen to have a different value than in the main network η.
 We summarize the GPS biases determination process as follows:
 1. Initialize all weights and offsets (w, Bi) with random values, and set learning rate η and to small positive values.
 2. Give input data (position xn, yn, zn along the measurement path) and teacher data (measurement slant TEC).
 3. Calculate output and evaluate the error function using equation (4).
 5. Return to step (2) until the stopping criterion is satisfied.
3. Numerical Experiment for Artificial Model Data
 To examine the effectiveness of the new method, we applied it to a model problem, in which the electron density distribution from 100 km to 1000 km in altitude is generated by using the International Reference Ionosphere model (IRI-95) [Bilitza, 1990]. The electron density distribution above 1000 km (plasmasphere) naturally affects the observed TECs because the GPS satellites are moving on six circular orbits at an altitude of 20200 km. In order to estimate the contribution of the plasmasphere to the TEC, we employ a simple diffusive equilibrium approach [Angerami and Carpenter, 1966] of the plasmasphere and give the electron density distribution from 1000 km to 20200 km in altitude. In this model the electron density distribution n(h) is simply expressed by the exponential function as
where n0 is the electron density at the altitude of h0 = 1000 km and Hs is the scale height of the density decay in the plasmasphere. Though the scale height Hs varies with the magnitude of the mean values of the ion masses and the ion temperatures we fixed Hs to be the value at the altitude of 2000 km as our aim is just to demonstrate the validity of the new method. Consequently the contribution from the plasmasphere amounts to ∼10% of the total observed TEC during the day time and 20–30% during the night.
 Actual positions of the GPS satellites and the ground receivers realized during a particular observation are used for all of the model calculations whereas the bias values of the satellites and the receivers are assigned artificially which we call hereafter as the “true” bias values. We defined the range of ionosphere to be within 100 to 1000 km in altitude; i.e., the electron density under 100 km and over 1000km are assumed to zero. To evaluate the error of the analysis we define the average GPS receiver bias error Eb as
where I, Bi, and BiNN are the total number of receivers, “true” and evaluated bias values of the Ith receiver, respectively. The unit of a bias value is TECU (1TECU = 1.0 × 1016e/m2).
 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 for 0400 to 0415 JST, 22 December 2001 and three GPS receivers located in Okinawa (Figure 3). It is found that the training error of a four-layered neural network decreases rapidly and the resulting average error of the receiver biases Eb is 0.22 TECU. The result of the analysis is summarized in Table 1. It took about 60 s of CPU time to train the network two thousand times, where 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. On the basis of these results we employ a four-layered neural network with 2, 12, 12, and 1 neurons in each layer for the following analyses.
 We produced nine sets of model observation data corresponding to 15–17 June, 20–22 September, and 21–23 December, for three GPS receivers located in Okinawa (Figure 3). The average error of the receiver biases determined for the above 9 days are summarized in Table 2. The result shows that the bias determination is carried out with sufficiently high accuracy as average error less than 0.3 TECU even for the case with such a small number of receivers.
 We applied our method to actual GPS observation data obtained from GEONET in Japan, and compared the results with the least squares method and the “simple method” for a single receiver given by Ma and Maruyama . We analyze the GPS observation data from three different receivers located around 27°N, 128°E, Okinawa, Japan and investigated the possibility of the receiver bias determination in the low-latitude areas. We assumed that the local ionospheric electron density does not change within 15 min. It should be remarked that we used a data set for which satellite biases have already been determined by applying the least squares method to GEONET data gathered from large number of receivers.
 The GPS satellites broadcast a dual-frequency signal (f1 = 1575.42 MHz and f2 = 1227.60 MHz). The derivation of slant TEC from GPS observation data using both group delays (P1 and P2) and carrier phase (L1 and L2) was proposed by Ma and Maruyama . In brief, the slant path I can be obtained from the group delays and carrier phase (L1 and L2) of two frequency signals
where λ1 and λ2 are the wavelengths corresponding to frequency f1 and f2, respectively. Although the accuracy of the slant TEC derived from the phase measurement Il is higher than Ip, it only provides the relative change in TEC, since the ambiguity is included in the phase measurement. A precise slant TEC value is derived by using both of those group delays Ip and carrier phase Il by introducing a baseline BL for the differential phase related Il [Mannucci et al., 1998; Horvath and Essex, 2000]
here BL is calculated as the following
where N is the number of measurements from a receiver to a satellite and α is the elevation angle. It should be noted that the square sine of the satellite's elevation αn is included as a weighting factor because the pseudorange with low elevation angle is apt to be affected by the multipath effect and the reliability decreases. Consequently, the contribution to the baseline determination is greatly depleted from slant paths with low elevations. When making the above calculations of BL, a data processing step is included to identify possible cycle slip in either L1 and L2 phase measurements [Blewitt, 1990]. This process is then carried out for satellite receiver pair. In order to determine the GPS satellite and receiver biases separately, the receiver located at 34.16°N, 135.22°E was selected as reference receiver.
 To investigate the effects of the season, local time, and geographic location of the GPS receiver, we carried out two cases of receiver biases determination experiments as follows.
 We used observation data obtained during several days around solstices and autumnal equinox (15–17 June, 20–22 September, and 21–23 December 2001) by three GPS receivers located in Okinawa (Figure 3) at the local time of observation of 0400 to 0415 JST. For assessing the accuracy of the analysis we define the projection error (slant TEC error) Ep and the average GPS receiver bias error Eb as
where P, I, BiG, and BiNN are the number of the projection data, the number of receivers, the bias value determined by the least square method from a large amount of the GEONET observation data, and the bias value by the new method, in respective order, where BiG is considered to represent a reliable bias value.
 We trained the network two thousands times and the result is summarized in Table 3. The result shows that almost all the average GPS bias errors Eb are less than 1.5 TECU. The day-to-day variation of the determined receiver biases is shown in Figure 4 as the standard deviation of the receiver biases to the 9-day mean (RMS). It is seen that most of the day-to-day variation are less than 3 TECU and the largest value is about 4 TECU.
Table 3. Results of Actual GPS Receiver Bias Determination
 To evaluate the performance of the new method in comparison with the “simple method” of single receiver determination proposed by Ma and Maruyama  we analyzed three cases and the results are summarized in Tables 4, 5, and 6. In these tables BG, BS, and BNN are the receiver biase determined by the least square method from the GEONET data, that by the “simple method” from a single receiver, and the receiver bias by the new method analyzed for three ground receivers. It is easily seen that the agreement of the bias by the new method BNN with that by the GEONET BG is excellent, which means that the new method is very effective even if receivers are located in low-latitude area and the number of receivers is not sufficiently large.
Table 4. Results of 16 June 2001: Ep is 1.21 TECU
Table 5. Results of 20 September 2001: Ep is 1.59 TECU
Table 6. Results of 21 December 2001: Ep is 1.19 TECU
 The observation data obtained from 22 December 2001, for receivers located at Okinawa, were analyzed to investigate the effects of hourly variations in the ionosphere from 0000 to 2400 JST local time. To clarify the errors of calculated biases caused by hourly changes in the observation data, we collected sets of hourly observation data every 15 min in each hour to calculate the biases. As the ionospheric irregularities caused by the plasma bubble are remarkable at nighttime, the data from 1800 to 2400 JST local time were not used in the analysis. The TEC observations around the geomagnetic equator are still important in the “north-south conjugate observation project” launched by CRL in the southeast Asia in 2001 [Maruyama, 2002], in which accurate determination of biases of receivers located in the low-latitude areas is indispensable. For this purpose, three test calculations were carried out by using the observation data from three GPS receivers at different geographic locations, i.e., north-south, triangle and east-west in Okinawa as shown in Figure 5. Figure 6 shows the error ENN = BNN − BG of every receiver versus local time, for three geographic locations of receivers. The mean error of three GPS receivers for each geographic location is shown in Figure 7. It is found that the biases of north-south located receivers were determined with higher accuracy than were the triangle and east-west located receivers. This result can be explained by the large north-south TEC gradient in low-latitude areas that causes more error in the bias determination in east-west located receivers compared with the contour plot of TEC over Japan (Figure 7a). Thus the result shows that the biases can be accurately determined if we install the GPS receiver in the north-south direction.
 In order to investigate the reason why similar temporal variations of bias errors are obtained for different combinations of the GPS receivers (Figure 6) we plot in a same frame the temporal variations of the mean error of bias for the north-south located receivers and the total number of observation paths (Figure 8). As the observation data with low-elevation angle are not reliable we excluded the paths with the elevation angle less than 20 degree. Figure 8 clearly shows that the bias error variation correlates inversely with the time variation of the total number of the observation paths. From this result it can be concluded that increase of observation paths improves the accuracy of the determination of receiver biases.
5. Conclusions and Future Works
 This paper presented a method of GPS receiver differential bias determination by using a residual minimization training neural network (RMTNN). The solution system of this method consists of the RMTNN, which is trained by minimizing the object function composed of a sum of squared residuals of the integral equation at collocation points and additional single neural networks representing the instrumental biases. We focused on the difficulties that there are only a few usable ground receivers, especially, in low-latitude area (near the geomagnetic equator) where large ionospheric variability exists. We successfully applied this method to the model data and the actual observation data. By comparing the results with those given by the least squares method from GEONET data (high precision) and the simple method for a single receiver, we confirmed that the new method can accurately determined the receiver biases. The special features and possible extensions of this method are as follows.
 1. As the new method does not rely on the assumption of the thin-shell ionospheric model, it works well not only for determination of biases in middle-latitude areas, but also in low latitude areas. It is easy to obtain the TEC distribution everywhere over the calculating area by using a trained neural network.
 2. When we use a neural network to construct a mapping function, the smoothing and interpolation are automatically included; therefore, a smooth inverse transforming function for the data with the experimental error or the error due to the numerical computation can be attained without any special consideration for regularization.
 3. As a neural network application for solving inverse problems, the applicable fields could be expanded considerably by employing it in combination with other processes such as retrieving unknown parameters with an additional neural network.
 4. The presented study, we focused on determining the differential biases of the GPS receiver simply and quickly. As a future works, this method can be easily extended to a general three-dimensional tomography image reconstruction if we use a large amount of the GPS observation data like GEONET data. The ionospheric electron density profiles can be reconstructed by using the neural network parameter estimation method, and the satellite bias and receiver bias can be determined simultaneously. Furthermore, by using excellent features of neural networks, difficulty of four-dimensional ionospheric tomography is relaxed considerably by considering the time evolution of ionosphere. For the electron density profile of the ionosphere, the model should be modified considering the plasmasphere contribution to the TEC.