Two different analysis techniques for mapping ionospheric total electron content (TEC) are compared. The first technique approximates the ionospheric electron concentration as a thin shell at a fixed altitude. In this case, slant TEC observations are converted into vertical TEC values using a mapping function and interpolated across a grid. Other slant TEC values are then calculated from the vertical TEC grid using another mapping function. The second technique applies an advanced tomographic algorithm to invert the slant TEC observations into a time-evolving three-dimensional grid of electron concentration. Either slant or vertical TEC can then be extracted from the electron concentration images without the need for a mapping function. Results based on both simulated and experimental data are presented. The results indicate that the inversion offers improvements over a thin shell in the mapping of TEC at middle latitudes.
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 The Global Positioning System (GPS) has been used as an ionospheric research tool for almost 2 decades [Yunck et al., 2000] and offers near-global coverage combined with continuous operation. Networks of fixed ground-based receivers have been established for many different scientific applications, and daily observations from these receivers are available via the Internet. Davies and Hartmann  discuss the opportunities presented by GPS to study the ionosphere and compare the use of GPS with polar-orbiting satellites such as the U.S. Navy Navigation Satellite System (NNSS).
 The slant total electron content (sTEC) is defined as the integrated value of electron concentration along the line of sight from the transmitter to the receiver. Transionospheric radio signals suffer a phase advance and a group delay, both of which are dependent on the signal frequency. Since GPS satellites transmit on two frequencies, L1 (1575.42 MHz) and L2 (1227.6 MHz), TEC can be inferred by measuring either the phase advance or the group delay of the L1 signal with respect to the L2 signal [Lanyi and Roth, 1988]. This permits spatial maps of ionospheric TEC to be generated. An important application of such TEC maps is in the estimation of the ionospheric error in single-frequency GPS navigation solutions [Harris et al., 2001].
 Local TEC maps can be produced using a single receiver station [Coco et al., 1991; Ciraolo and Spalla, 1997]. A simple way to map TEC over wide geographical areas is to approximate the whole ionosphere, which extends from about 80 to over 1000 km, by a thin shell at a fixed altitude [Mannucci et al., 1998]. A problem with this approach is that the ionosphere is a highly variable medium, both temporally and spatially, and the vertical distribution of ionization cannot be represented accurately at all times and locations by a thin shell.
 An alternative approach is to invert the TEC observations to yield the spatial distribution of electron concentration represented in three-dimensional voxels of constant electron concentration. Slant or vertical TEC can then be calculated along any direction by integration through the grid along any required path without the need to interpolate across a shell or apply any mapping functions to convert between slant and vertical TEC. This technique is applied here using the inversion algorithm of Spencer and Mitchell , known as the Multi-instrument Data Analysis System (MIDAS). MIDAS carries out a full mathematical inversion in three spatial dimensions and one time dimension using an algorithm extended from the least squares linear matrix inversion applied by Fremouw et al.  to two-dimensional ionospheric tomography.
 In the case of two-dimensional inversions (tomography) the ionization distribution is assumed to be stationary during the time of a low-Earth-orbit satellite pass. Initially it seemed that GPS satellites could not be used for ionospheric imaging because their transit time is many hours, too long to image a moving ionosphere. Nevertheless, multiple GPS satellites are in view at any time, and hence a three-dimensional inversion is conceptually possible. However, this snapshot set of measurements results in a sparseness of data, whereas a time-dependent inversion allows a great increase in the quantity and angular coverage of measurements that can be used in each inversion. For this reason a four-dimensional determination of the ionization distribution has been implemented. The algorithm of Fremouw et al.  is extended into a time-dependent inversion by incorporating a priori information about the evolution of the electron concentration during a specified period of time, typically 1 hour, assuming that the change in electron concentration within a voxel with time is linear. Details of the MIDAS inversion technique are given by Mitchell and Spencer . In the present paper, TEC mapping using the MIDAS four-dimensional inversion is compared with the thin shell approach for both simulations and experimental case studies.
2. Simulation Study
 A simulation study was carried out using actual GPS receiver locations across Europe selected from the International GPS Service (IGS) network. Figure 1 shows the locations of these receivers. A particular day, 30 September 2000, was selected, and model ionospheres were simulated every 2 min using the international reference ionosphere (IRI) model [Bilitza, 1990]. A dynamic representation of the midlatitude trough was superimposed across the IRI model for the nighttime simulations. The temporal and spatial position of the trough was defined using a formula taken from Collis and Haggstrom . All GPS satellites actually in view from the receivers were used for the simulation, subject to an elevation limit of 15°. TEC observations were simulated by integration between each GPS satellite and receiver. The interfrequency biases, multipath, and tropospheric sources of error were neglected.
 The simulated observations were then used as input data for each of the two techniques, i.e., thin shell and full inversion. The shell height used in the thin shell simulations was 400 km, and the spherical surface was divided into a grid bounded by lines of constant latitude and longitude. The full inversion reconstructions covered a height range from 80 to 1180 km in 50-km increments. Both the thin-shell and full-inversion techniques were implemented with identical time-evolution terms as detailed by Mitchell and Spencer .
 Consequently, the comparison being made here is between the use of the height dimension, as in the full inversion, and the thin shell, which does not make use of height information. In each case the techniques were applied across a range of different horizontal resolutions, i.e., the latitude and longitude increments that defined the grid spacings. Two locations without GPS receivers, Milan (45.5°N, 9°E) and Hamburg (53.5°N, 10°E), were then selected from which to calculate the slant TEC to each satellite in view. These locations could represent single-frequency users requiring ionospheric corrections. The accuracy of each of the techniques to evaluate TEC was assessed by comparison with the true slant TEC (found by integration through the original model). The comparison was made by applying equation (1). The reconstructed TEC (TECc) were differenced from the simulated truth TEC (TECm) and considered over all satellite-to-ground paths (N).
Figure 2 shows a set of results for each of the two test locations. The results were divided into nighttime and daytime, and separate graphs are shown for Milan and Hamburg. On each plot, the x-axis shows the horizontal spacing between grid points in degrees. The error for the thin shell is shown as diamonds, and the error for the inversion is shown as squares. The results all show improving accuracy as the horizontal grid size is decreased. They also indicate that the four-dimensional inversion method consistently gives an overall improvement in accuracy over the thin shell.
3. An Experimental Study
 A major ionospheric storm occurred in July 2000, and the period leading up to and during the storm was selected for the experimental testing of the techniques. Three days were chosen: 11 July (day 193) and 14 July (day 196), and the main ionospheric storm day, 15 July (day 197), representing a range of ionospheric conditions at midlatitudes. The European IGS receivers shown in Figure 3 were selected, and reconstructions were carried out for all three days using both thin shell and full four-dimensional inversion methods.
 A range of shell heights can be found in the literature. For example, Mannucci et al.  assume 450 km, and Mannucci et al.  chose 350 km. A study on the choice of an effective shell height is presented by Birch et al. , in which they advocate a value in the range 750–1200 km to allow for a protonospheric contribution. In the present work the height was fixed at 400 km, as suggested by Ciraolo and Spalla , and was used previously in the simulation study. The full inversion reconstructions covered a height range from 80 to 1180 km in 50-km increments. In both methods, a horizontal grid spacing of 1° in latitude and 5° in longitude was used.
 In order to evaluate the technique experimentally it was necessary to devise a method different from that used for the simulations, since in the experimental environment there is no “truth” ionosphere with which to compare results. The approach adopted was as follows. Consider two very closely spaced receivers k and m monitoring the same GPS satellite s at the same epoch. Then, for receiver k, the pseudocode range pk measured on the L1 frequency can be expressed as
where ρ is the true geometric range from the receiver to the satellite, c is the velocity of light in a vacuum, δtk and δTs are the receiver and satellite clock offsets, respectively, Ik, Tk, and Mk are excess path lengths due to ionospheric, tropospheric, and multipath delays, respectively, and bk and bs are the receiver and satellite hardware biases. The f1 implies dependence upon the L1 frequency. Similarly, the pseudocode range measured on the L2 frequency can be expressed as
Differencing equations (2) and (3) eliminates the nondispersive (i.e., frequency independent) terms, yielding
For convenience, this can be written
where Δbk is the receiver interfrequency bias (IFB) and Δbs is the satellite IFB. A similar relationship can be expressed for receiver m, i.e.,
If receivers k and m are sufficiently close for their lines of sight to the satellite to follow similar paths, then they can be said to experience very similar ionospheric delays, i.e., ΔIk ≈ ΔIm. Since they are both monitoring the same GPS satellite, differencing equations (5) and (6) yields
Thus, if the differential delay (dual frequency) between the satellite and each receiver is found, the only difference between the measurements at each of these two receivers is the IFBs of the receivers, since the satellite IFB is common to both measurements. This assumes that differences in the multipath between the two frequencies, and between the two receivers, can be neglected. Since this difference between the IFBs of two closely spaced receivers can be calculated with some accuracy, we have used this initially as the basis for our evaluation of our techniques.
 To determine the difference in IFB (ΔIFB) correctly for widely separated receivers, it would be necessary to know the different ionospheric delay experienced by each signal. Since this is the quantity in question here, the absolute ΔIFBs cannot be calculated with certainty. However, another property of the biases, namely, their stability, can be used here to infer information relating to the accuracy of the methods. Individual receiver IFBs are known to be stable over the day on an hour-to-hour basis [Sardón and Zarraoa, 1997; Mannucci et al., 1998], so the ΔIFB can also be expected to remain stable. This expectation of stability can be exploited as a measure of the accuracy of the determination of the ionosphere. Hence each of the two techniques is applied to 1-hour data sets over a period of 1 day and is used to determine the ΔIFB values. The stability of the determined ΔIFB can be used as a measure of the relative accuracy of the two techniques in mapping the intervening ionosphere.
 The calculation of the ΔIFBs was first tested by examining the mean IFB differences, ΔIFB, for pairs of close receivers. Differential time and phase data for each individual satellite to receiver phase arc were calibrated together using a least squares fit. This was done to take advantage of the phase variation (smoothly varying but uncalibrated) and the absolute time delay (noisy but useful for calibration). Two examples of such calibrations are shown in Figure 4.
 For closely spaced receivers, the P2–P1 differencing method is a very simple way to determine the difference in ΔIFB between two receivers, ignoring the ionospheric differences. It was found that for closely spaced receivers all three methods yielded daily mean ΔIFBs within approximately 10 cm of each other. This is entirely expected, as there is little difference between the ionospheric delays along these particular pairs of same-satellite to different-receiver paths.
 In the inversion techniques the IFBs were calculated from the difference between the raw measurements and the imaged ionosphere. To test the stability of the biases, the standard deviations of the receiver ΔIFBs for a group of close midlatitude receivers were computed from thin shell and full inversion data sets (Figure 5). As expected, good agreement can be observed between the inversion and thin shell ΔIFBs stabilities.
 In Figure 6, standard deviations of receiver IFBs for three widely separated receivers, brus, onsa, and kir0, are presented. In all cases, the ΔIFBs computed from the full inversion method display lower variability than those computed from a thin shell. Using our evaluation criterion, this implies that the inversion method has led to an improvement in the TEC correction in comparison with the thin shell method. The standard deviations of the inversion ΔIFBs can be seen to increase slightly as the geomagnetic conditions become more disturbed (i.e., as 15 July is approached). These standard deviations of the ΔIFBs will be influenced by the error in mapping the absolute difference in TEC between the two receivers in question. While the gradients in TEC over Europe may have been significant on 15 July, the absolute values were not as elevated as they were over the United States [see, e.g., Mitchell and Spencer, 2002], and consequently the absolute errors in TEC mapping have not greatly increased. It should also be noted that the data from onsa reveal evidence of severe multipath, which may have biased the results involving this receiver.
 A group of low-latitude receivers, mate, aqui, and lamp, is shown in Figure 7. The ΔIFB for the inversion data from lamp-mate and lamp-aqui shows an improvement when using the inversion. However, it can be seen that the mate-aqui pair display low variability (less than 20 cm) for both thin shell and inversion, implying that the ionosphere is well represented for this pair using either method. These receivers are situated close to each other, so it is reasonable to expect good agreement between them.
 It is important to note that the only information that we are using from these results is the relative improvement in using the full inversion over the thin shell. Since the IFB for each receiver may be very different, it is not possible to compare the results between different receiver pairs to infer information about the accuracy of the ionospheric mapping. In the case of the middle- to high-latitude receiver pairs the inversion always gave a lower standard deviation than the shell. In the case of the middle- to low-latitude receiver pairs the inversion gave either a lower standard deviation than the shell or a similar result.
5. Conclusions and Discussion
 Two different analysis techniques for evaluating slant TEC have been compared. The first method was the conventional thin shell approach, and the second was a new four-dimensional imaging technique. Simulations using the IRI model enabled a quantitative comparison between the two approaches. Results were calculated across 24 one-hourly inversions, and five different horizontal resolutions were investigated. In all cases these showed that the full inversion could evaluate better estimates of slant TEC than the thin shell. Shortcomings of the simulations were because we did not account for error due to the troposphere, multipath, and interfrequency biases, but it should be noted that the omission of these factors would not be expected to favor either technique.
 The two techniques were applied to experimental data collected across Europe before and during the major storm of July 2000. A method was devised to compare the full inversion and the thin shell approaches experimentally, based on the expectation of stability of the receiver interfrequency biases. In this experimental study we have relied on the assumption that the interfrequency bias differences are stable over a 24-hour period. This means that effects due, for example, to diurnal temperature changes should be common to site pairs. Since the inversion generally gave a smaller standard deviation, the implication is that on an hour-by-hour basis, the full four-dimensional inversion provides a more accurate estimate of ionospheric delay and hence is better than the thin shell for ionospheric mapping. These first results are very encouraging for the inversion method.
 The case studies presented in this paper have indicated that four-dimensional inversion of GPS data is a feasible technique for ionospheric imaging at midlatitudes. Furthermore, it can offer advantages over a thin shell approach, overcoming errors associated with mapping functions. There is still a requirement to establish the routine reliability of the inversion technique over all geomagnetic activities and geographic locations. Future work will address this and quantify the potential benefit of this work to navigation position corrections.