As the authors have shown in previous work, the tomographic approach reduces significantly the mismodeling in the electron content determination using as data the ionosphere-crossing radio signals of global navigation satellite systems (GNSS) such as the Global Positioning System (GPS). We present in this paper the results of a comprehensive study on the performance of an improved tomographic model of the ionosphere. This model is obtained from the GPS L1 and L2 carrier-phase data, and it is used to estimate undifferenced and double-differenced ionospheric corrections in real time and at very long distances between receivers (500–3000 km). The key point of the strategy is the combination of real-time geodetic and ionospheric techniques to achieve a significant improvement in the reliability of carrier-phase ambiguity resolution. This new approach includes also the use of smoothed pseudo-ranges to help in ambiguity resolution. It has been tested under difficult ionospheric conditions, during four consecutive weeks in March–April 2001, at Solar Maximum, and at latitudes ranging from −40 to +40 degrees, so as to include the equatorial region. Dual-frequency satellite altimetry from TOPEX-Poseidon has been used to verify the accuracy of the ionospheric model over the oceans in that region.
 As demonstrated in recent years, it is possible to provide ionospheric corrections from global navigation satellite systems (GNSS) such as GPS, using global ionospheric maps (GIM's) computed from GPS data [see, e.g., Wilson et al., 1995]. In the case of real-time applications such as wide-area navigation augmentation systems (WAAS, EGNOS, MSAS [see, e.g., El-Arini et al., 1995]) the ionospheric corrections have to be calculated from receiver data collected at stations typically separated by several hundreds of kilometers using only data gathered until the present epoch. This real-time limitation usually produces less precise results than when postprocessing all the data collected during a long observing session. One way to overcome this natural limitation is to run, simultaneously with the ionosphere-modeling procedure, a geodetic program to solve for a number of nuisance parameters (errors in orbits, site coordinates, tropospheric refraction model, etc.) so as to estimate very precisely the biases in the carrier-phase ionosphere-free combination, Lc. By using these procedures together, both the ionospheric corrections and also the geodetic outputs can improve significantly. Some examples of this can be found in Hernández-Pajares et al. [2000b] for ionospheric corrections, in Colombo et al.  for navigation and in Hernández-Pajares et al.  for real-time tropospheric determination.
 The purpose of this paper is to show the benefit of combining precise real-time ionospheric modeling, with a tomographic approach, a precise geodetic analysis, in a more difficult scenario than encountered in our past work. We deal here with GPS sites separated by thousands of kilometers across the equatorial region, where the ionospheric electron density presents the largest values and strongest spatial and temporal gradients, under solar maximum conditions. At such very long distances, this strategy can help improve the accuracy in the real-time global ionospheric maps that may be produced as a result of a shift of the International GPS Service (IGS) towards real-time capabilities [Zumberge et al., 1999]. These maps can help improve the accuracy for single-frequency users as well.
 This paper is organized as follows: In the first part, the technique is described. In the second part experiments showing the improvement in the availability of the precise ionospheric corrections are presented in detail. In one four-week long experiment in the tropics, in 2001, we have tested the improvement in the real-time ionospheric model, using data from GPS permanent stations from one thousand to three thousand kilometers apart during the seasonal maximum of the diurnal TEC, with high solar activity. The results have been verified over land using data from one station not included in the model, and at sea, using dual-frequency altimeter data from the TOPEX-Poseidon satellite.
2. Description of the Technique
 The free electron ionospheric distribution is approximated by a grid of voxels in which the electron density is assumed constant at a given time in an Earth Centered Inertial (ECI) system (see a typical layout in Figure 1). A grid with two layers, and 5 × 3 deg in local time and latitude represents a compromise between precision (about 1 TECU, needed to solve the carrier-phase ambiguities in real time [Hernández-Pajares et al., 1999b] and computation load (about 6000 unknowns involved).
being x(t), x(t + Δt) the unknown (column) vectors at epochs t and t + Δt; y, ŷ the observed datum (scalar) and the predicted observation with the model, respectively; K the gain vector, computed as
being a the gradient (row) vector, containing the coefficients of the model unknowns to explain observation y(t); and Cx0 the covariance matrix of the a priori unknown values x0 ≡ x(t) that incorporates the possible random process variance increase; and σ2 is the y datum variance.
 The mean electron density (Ne)i,j,k of each illuminated cell i,j,k (in solar longitude, latitude and height respectively), is treated as a random walk process, and with typical process noise of , that is equivalent in our case (two layers with a total width of 1360 km) to a change of about . This is an adequate value under many different ionospheric conditions, 5 to 15 times greater than the datum a priori standard deviation, being this relationship similar to those used by Mannucci et al. . The carrier-phase data (LI = L1 − L2, in length units) are the main observations used. Previously, we have used no pseudo-range data, because of their much grater noise and multipath problems, compared to the carrier phase. For this work, we have used it only to a limited extent, in the form of phase-smoothed pseudo-range. The carrier-phase biases BI (constant in each continuous arch of carrier-phase data with the same satellite-receiver pair) are estimated simultaneously as random variables (restarted whenever there is a cycle slip). In the filter the biases BI decorrelate in real time from the electron density values Ne as long as the satellite geometry changes and the variance of both kinds of unknown became smaller, taking into account the relationship for each ray:
being α ≃ 1.05 (meters LI)/(1017 electrons · m−2).
 The filter starts assimilating data from the permanent sites several hours ahead of the intended utilization period, optimizing the unknowns management, in order to assure accurate real-time estimation with well-converged filter parameters.
 This approach is particularly suitable for detecting local features of the electron density distribution. The use of two layers of voxels in the model –instead of one as it is usually done– reduces significantly the mismodeling of the electron content [Hernández-Pajares et al., 1999a, 1999b].
 In the case of Wide Area Differential GNSS (WADGNSS) networks, from these real-time slant total electron content (STEC) corrections obtained by equation (1), it is possible to form the station-satellite double differences, ∇ΔSTEC, and to obtain a second ambiguity (the widelane) in the reference stations (see flow chart in Figure 2). At that point, these accurate double differenced ionospheric corrections can be broadcast to any roving receiver in the WADGNSS region to help users determine their positions more precisely, by fixing the carrier-phase ambiguities in their receiver data. This and other details of the technique, hereinafter called Wide Area Real Time Kinematics (WARTK), can be found, for instance, in Hernández-Pajares et al. [2000a, 2000b].
 In the case of the reference stations calculation, the long distances and the strong electron content gradients limit the performance of the technique. For this reason, we extend in this paper the algorithm to include the case of reference stations separated by thousands of kilometers, using also widelane-smoothed pseudo-range to improve the ionospheric model and, hence, help achieve real-time ambiguity determination.
 The technique can be summarized in several steps (see flow chart in Figure 2):
The input GPS data (L1, L2 carrier phases [in length units] and the corresponding pseudo-ranges P1 and P2, ephemeris information, coordinates of fixed sites, etc…) are analyzed with three programs running simultaneously in real time:
Ionospheric program: It processes data in the form of the geometry-free ionospheric combination of carrier phases (LI = L1 − L2) and of transmitter-receiver rays geometry. The outputs are the estimated unknowns of the ionospheric model explained above: the mean voxel densities and the LI biases (BI).
Geodetic program: From geometric (ionospheric-free) combination of carrier phases (Lc = (f12L1 − f22L2)/(f12 − f22) being f1 and f2 the two corresponding frequencies) the program provides several main outputs: Improved GPS ephemeris, Zenith Tropospheric Delays (ZTD) and the Lc biases of each observation (Bc).
Code smoothing program: It computes the widelane carrier-phase ambiguities (Lw = (f1L1 − f2L2)/(f1 − f2)) using the appropriate pseudo-ranges (Pw = (f1P1 + f2P2)/(f1 + f2)). These pseudo-range values are corrected by a multipath model continuously computed for each fixed site with GPS data from previous days.
In this way, and at a given epoch, we have three different ambiguities (ionospheric, ionospheric-free and widelane) estimated independently for each double differenced observation (between pairs of satellites and receivers). These three ambiguities are not independent from each other, as they are related to the L1 and L2 ambiguities (N1 and N2):
being Nw the integer widelane ambiguity, β = λ1λ2/(λ22 − λ12) and λX the X combination wavelength.
Each one of these ambiguities follows an exact relationship with the other two, that can be derived from equation (4). This allows a check on the integrity of all three ambiguities. Moreover, at this step, and for each pair of ambiguities, an additional test and computations are made following the relationships described in Hernández-Pajares et al. [2000a].
If the set of ambiguities passes the tests just described, then the ambiguities are fixed and used as additional inputs to the Ionospheric and Geodetic programs.
 The goal of this new experiment is to assess the performance of the real-time ionospheric model at greater distances, 1000–3000 km, over a wider range of latitudes and over a longer period of time, including very active ionospheric conditions.
 The data came from 12 permanent IGS sites in Asia and Australia, shown on the map in Figure 3. The TOPEX-Poseidon TEC observations (affected by an ionospheric error of 2–3 TECU ≃0.5 cm for TOPEX altimetry [see, e.g., Benada, 1997, section 4]) are also used as “truth” (see typical satellite-ground tracks in the same figure). As seen in Figure 3, the satellite data cover both ionospheric equatorial anomalies, and the distances between sites are in the range of 1000–3000 km.
 The data period spans four whole consecutive weeks, from March 6th to April 2nd of 2001 (days of the year 65 to 92), i.e. close to the Solar Maximum, and includes the Spring seasonal maximum of the diurnal TEC, and several disturbed periods (see Kp index in Figure 4).
4. Computation and Results
 As shown in the works mentioned earlier, ambiguity resolution requires the accuracy of the ∇ΔSTEC correction for the rover to be better than 2TECU (i.e, a one-sigma precision of 1 TECU). Furthermore, with distances of less than 1000 km between reference stations an ambiguity success rate higher than 90% can be obtained. This is better by about 10% than when using the smoothed pseudo-range alone to calculate the wide-lane ambiguity [see, e.g., Hernández-Pajares et al., 1999b, Figure 16]. However, at still greater distances (more than 1000 km) and with larger ionospheric variations, one should expect a lower percentage of success. In this case the smoothed pseudo-range can be useful to help fix the widelane ambiguities, to constrain the estimation of the real-time ionospheric model.
 The main problem using smoothed pseudo-range is the presence of multipath, caused by reflected signals that arrive later to the receiver than the direct transmission from the GPS satellite. For permanently installed receivers, it is possible to choose a good antenna location to mitigate this problem. And it is also possible to use the receiver data to estimate the actual antenna multipath pattern in order to correct much of this unwanted effect. Only for three of the more affected permanent receivers used (BJFS, KAYT and SHAO) has been computed a multipath model for correcting their pseudo-range.
 The precision of the real-time ionospheric corrections, based on carrier-phase data from reference stations, can be assessed in three different ways, for the vertical TEC, the STEC and the ∇ΔSTEC.
 First, they are compared to the available precise TEC estimates obtained directly from the dual-frequency altimeter on the TOPEX-Poseidon satellite [Benada, 1997] over seas close to the GPS permanent receivers (Figure 3).
 Two examples of direct comparison of vertical TEC are shown in Figure 5, for geomagnetically quiet and very active conditions (days 67 and 90, 2001, with Kp in the ranges 0–3 and 6.5–8.5 respectively, see Figure 4). In general the real-time GPS-derived TEC follows the main trends of the TOPEX-derived TEC, but with higher values. This discrepancy is compatible with the presence of a plasmaspheric electron content of several TECU between the altimeter and the GPS satellites [Lunt et al., 1999]. Remarkably, the real-time performance can be better than that of some postprocessed results, such as the GIM's distributed by UPC (see Hernández-Pajares et al. [1999a] for more details), or the GIM's available from other centers (in IONEX format, on-line at ftp://cddisa.gsfc.nasa.gov/pub/gps/products/ionex/YEAR/DOY). One of the reasons for this is the improvement due to fixing the ambiguities in the fixed-site carrier-phase data in real time (Figure 2), that provides very accurate values of ∇ΔSTEC to the real-time model. At present this is not yet done for the distributed UPC GIM's. Another difference is the poorer temporal resolution (2 hours) of the distributed global maps of the ionosphere, which have to be linearly interpolated in time for comparison purposes.
 A summary of this comparison for the first two weeks of this study, with lower geomagnetic activity, confirms these conclusions. Figure 6 shows the bias and standard deviation of the difference TEC[GPS]-TEC[TOPEX] plotted as a function of latitude, and subtracting the TOPEX topside electron content predicted by the Parameterized Ionospheric Model (PIM [Daniell et al., 1995]). The two plots of Figure 6 correspond to the two daily TOPEX passes near local noon (top plot) and local night (bottom plot), with TOPEX observations relatively close to GPS stations. It can be seen the compatibility of the real-time TEC determination with TOPEX data, in particular in the diurnal passes with an accuracy better than the postprocessed solutions that do not fix the carrier-phase ambiguities. Similar results are obtained for the last two weeks with higher geomagnetic activity.
 A second comparison can be made between the ground-site observed unambiguous STEC value (estimated in postprocessed mode) with its real-time prediction, for a receiver that has not been used in the computation of the model. This is the case of the IGS station NTUS, separated by approximately 600 km from the nearest station, SAMP (see map in Figure 3). The variation of STEC can be predicted to better than 5 TECU in a typical case, with a RMS of 2.8 TECU (3% of the mean TEC, Figure 7). This result is compatible with the vertical TEC comparison with TOPEX.
 Finally, in a third study of the performance of this approach, the RMS of the error in ∇ΔSTEC prediction has been calculated using as “truth” the ambiguities fixed in a postprocessed solution, to determine the corresponding real-time success rate in widelane ambiguity resolution (see flow chart in Figure 2). Figure 8 shows the results for a baseline of 2354 km (COCO-KARR), under the influence of the southern Appleton anomaly. The ∇ΔSTEC RMS is typically about 1TECU, and depends on the success rate, which is greater than the 90% during the four consecutive weeks of the period studied, with a mean rate of 97%. That period includes days of high geomagnetic activity, as shown in Figure 4. These results are quite representative, as it can be seen in Table 1, for long baselines, and for both northern and southern tropical ionosphere.
Table 1. RMS of the ∇ΔSTEC Real-Time Determination, and Associated Percentage of Success for the Double-Differenced Widelane Ambiguity, for a Set of Representative Baselinesa
The distance, in kilometers, and the number of observations are also indicated.
5. Summary and Conclusions
 The Wide Area Real Time Kinematic (WARTK) algorithm performs well when used to compute precise real-time ionospheric corrections in WADGPS-like networks. In several experiments previously reported in other publications, we have shown how the WARTK procedure can provide ∇ΔSTEC better than 2.7cm (1/4 TECU) for the reference and rover receivers, in middle and high latitudes, and under different ionospheric conditions [Colombo et al., 1999, 2000; Hernández-Pajares et al., 2000a, 2000b, 2001].
 In the present work, we investigate the improvement in real-time STEC determination for the reference GPS stations in one of the worst possible scenarios: Very long baselines (1000–3000 km), at low latitudes, and near-Solar Maximum conditions, during four consecutive weeks, including periods of high geomagnetic activity (March–April 2001). The vertical TEC prediction for the TOPEX-Poseidon satellite made with this method is larger than the TEC actually observed with the TOPEX-Poseidon dual frequency altimeter. This discrepancy is compatible with the existence of a significant plasmaspheric electron content between the TOPEX and the GPS satellites. This result is more compatible with TOPEX than those based on the postprocessed global-scale models (GIM's), obtained without fixing ambiguities. The standard deviation of the discrepancies with the TOPEX vertical TEC is at the level of 1–3 TECU. This result is compatible with the agreement observed between measured and predicted STEC for GPS receivers not involved in the computations. Finally, in this study the RMS of the difference between the ∇ΔSTEC and the postprocessed “truth” is about 1 TECU, and the success rate in ambiguity resolution is greater than 90%. Similar results have been achieved for pairs of stations separated by more than 2000 km, and experiencing the strong gradients of the southern equatorial anomaly.
 The authors are grateful to the International GPS Service and cooperating organizations, for making publicly available the IGS data sets. The maps have been generated with the software package GMT [Wessel and Smith, 1995]. Some geodetic calculations were made using the GIPSY software [Webb and Zumberge, 1997]. This work has been partially supported by the Spanish projects TIC-2000-0104-P4-03 and TIC-2001-2356-C02-02.
 Arthur Richmond thanks M. Bakry El-Arini and another reviewer for their assistance in evaluating this paper.