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 Measurements from the Global Positioning System (GPS) satellites provide a valuable source of information about the ionosphere in the form of integrations of electron concentration. The slant total electron content (TEC) through the ionosphere can be estimated for specific satellite-to-ground paths using the two GPS frequencies and knowledge of the dispersive properties of the ionosphere. However, these TEC values are in error because of the interfrequency biases of the satellites and receivers. In order to assess the accuracy of TEC in the ionospheric images, the determination of interfrequency biases must be studied. This paper addresses the determination of the magnitudes of these biases for individual GPS satellites paired with GPS receivers in Europe using the ionospheric imaging tool Multi-instrument Data Analysis System (MIDAS). This is done so that the accuracy of the TEC in the ionospheric images can be assessed. A simulation study was undertaken to verify the approach, then experimental results were compared with independent values of the biases calculated by the Center for Orbit Determination in Europe. Experimental results reveal that changes in the biases can be related directly to documented changes in receiver hardware. They allow an estimate of the receiver biases and hence the error in TEC estimation using GPS data.
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 Dual frequency GPS data can be used for ionospheric investigation because the ionosphere is a dispersive medium causing a phase advance and a group delay to radio signals. In fact, provided that both frequencies are monitored simultaneously the data can be used in two different ways to obtain information about the total electron content (TEC) between any satellite and receiver. The first way is to look at the difference between the signal phases and provided that the receiver does not lose lock on either signal this yields the change in slant TEC. The second method uses the codes on the two signals to find the difference in propagation speeds of the two signals and hence the slant TEC. Once noise and multipath have been considered, this method might at first appear to provide absolute TEC. However, it suffers from significant errors known as the interfrequency biases (IFBs). These occur largely because of the unknown differences in time for the two signals to propagate through the satellite and receiver hardware; hence they are classified as either satellite interfrequency biases or receiver interfrequency biases. Frequently these IFBs are determined in pairs, such that each satellite plus receiver IFB constitutes the difference between an apparent and the actual ionospheric delay and hence forms most of the error in slant TEC determination.
 The topic of IFB determination has been addressed before, notably by Mannucci et al. , Coco et al. , and Sardón and Zarraoa . The reason to revisit it here in the context of ionospheric imaging is that the magnitudes of these biases and the accuracy to which they can be determined is very important for ionospheric measurement and imaging, since they are a source of error for ionospheric measurements based upon differential code records.
 The majority of ionospheric research using GPS data uses a mapping technique to approximate the ionosphere as a thin shell of ionization at a nominal height of the ionosphere, usually between 350 and 450 km. However, Birch et al.  have demonstrated that an inappropriate choice of shell height is a major source of error in TEC determination. Another approach that avoids this shell approximation and is therefore far less susceptible to height-induced errors is ionospheric imaging, for example Multi-instrument Data Analysis System (MIDAS). Other ionospheric imaging methods include work carried out using multiple layers such as Hernández-Pajares et al. , and Bust et al.  who use several data sources as input to tomographic algorithms.
 Early work on tomographic imaging of the ionosphere showed that ionospheric imaging does not require precalibrated TEC data and can make use of relative measurements along connected phase arcs [Andreeva et al., 1992]. Ionospheric imaging using data from GPS satellites is a rather new technique but it is already a promising method for studying the large-scale distributions of ionospheric features. For example, Meggs et al.  have used independent measurements from incoherent scatter radar (ISR) to demonstrate that GPS imaging can reveal the depletion known as the main trough. Results from the USA by Yin et al. , have shown that the disturbed ionosphere can be imaged during the great storm of July 2000, revealing dramatic uplifts in the peak height. These case studies with colocated ISR observations are very useful for verification of isolated ionospheric features that are seen in the images. This paper has a different focus; it is concerned with making an estimate of the accuracy of vertical TEC maps obtained from ionospheric images over Europe.
 One manner in which the accuracy of ionospheric imaging can be assessed is by the evaluation of the IFB values from the images themselves, in conjunction with the differential code delays. The idea is that integration through an ionospheric image along particular satellite-to-receiver paths can determine slant TEC and hence slant differential delay. These can be compared to the smoothed differential code delay measurement at the GPS receiver, and the difference between the two quantities is an estimate of the satellite-receiver combined IFB; it is the part of the differential delay that is not attributed to the ionosphere. If these estimates of the combined IFBs are then compared to independent IFB values, calculated using large networks of receivers and over many days, their agreement implies that the estimate of the satellite-receiver combined IFB from the ionospheric imaging is correct and hence gives credibility to the ionospheric image. It should be stressed that individual satellite or receiver IFBs are not uniquely determined by our technique. Combinations of satellite and receiver IFBs can in theory be found, but individual biases can only be found if the IFB of at least one receiver or satellite is calibrated.
 In this paper the determination of IFBs using ionospheric imaging over the European region is first assessed using a simulation study where the IFBs are set to be those calculated by the Center for Orbit Determination in Europe (CODE), an analysis centre of the IGS, and the ionosphere is simulated using the 1995 update of the International Reference Ionosphere model (IRI), Bilitza . This allows the errors in the IFB determination (and also the error in the TEC) to be found and give an indication of the expected errors in the real imaging system. Secondly, the imaging is performed for real data and the IFBs determined by ionospheric imaging are compared with those from CODE. The assumption is that the CODE values are the “truth.” Again the errors are assessed by differences between the CODE-determined IFBs and the imaging IFBs. The results are discussed in terms of the time window to determine biases and the implications for assessment of errors in the ionospheric images.
 In the first stage a simulation study was undertaken. In order to simulate dual frequency GPS observations it is necessary to model the ionospheric delay and the transmitter and receiver IFBs. To make these realistic the IRI-95 model, Bilitza et al. , was used to calculate the satellite-to-receiver differential delay and the biases were modeled using those calculated by CODE (http://www.aiub.unibe.ch/). IRI-95 is an established climatological model of the ionosphere. While it is a smoother representation of the electron density than would be expected at any one instant, it gives realistic overall estimates of electron densities and hence expected satellite-to-receiver delays. Networks of ionospheric delays were simulated for satellite-to-receiver paths across the European region.
 A particular day was selected, 15 February 2002, and the satellites' positions were obtained from the database provided by the International GPS Service (SOPAC). Since these were available at 15-min intervals they were interpolated linearly to 30-s intervals. Numerical integration between each satellite position and each receiver in view of that satellite allowed the calculation of slant TEC (b) between a specific satellite and receiver (bsr). In equation (1), A is a vector that contains measurements of the path length though each voxel (or 3-D pixel) and IRINe the electron density in each voxel given by the IRI model:
 Hence each simulated measurement is the TEC through the IRI model plus the interfrequency biases. Thus satellite and receiver IFB values of CODE (IFBs and IFBr) were added to each simulated IRI slant TEC. Thus the simulated data contained IFB values and ionospheric delay but did not include a representation of noise or multipath. The inversion of the relative differential phase changes (or changes in TEC) was performed using the MIDAS algorithm, described by Mitchell and Spencer .
 For each hour of input data the inversion results in a three-dimensional, time-dependent image showing the distribution of electron concentration for that hour. For presentation purposes it is only possible to show static images of TEC, taken from frames of the movie. In experimental studies the GPS dual frequency phase data from the database provided by the International GPS Service is used as input instead.
 With the inversion completed, the differences between the differential code measurements (i.e., those with IFBs) and the corresponding MIDAS slant TECs were found. These differences are attributed to the combined satellite and receiver IFB values. Thus, for each hour, a single bias value is found for each satellite and receiver pair. The relative values for the individual receiver and satellite interfrequency biases were estimated by a least squares fit.
3. Simulation Study
 Examples of the input data used in the simulation are shown in Figures 1, 2, 3, and 4. Figure 1 shows the vertical TEC taken from the 3-D electron concentrations in the IRI model. Figure 2 shows the locations of the GPS receivers used in the simulation. They are a selection of the actual receiver sites of the geodetic GPS receivers in Europe. Figure 3 gives an idea of the data coverage by showing the equivalent vertical TEC at 400 km altitude between each GPS satellite and receiver. This is found by performing a geometrical correction on the slant TEC through the IRI model, without any IFBs. Continuous satellite-to-receiver arcs are clearly seen. It is these adjacent measurements along continuous satellite-to-receiver pairs that the algorithm relies on by computing the differences between adjacent measurements of the input data. Figure 4 shows the same as Figure 3, except the IFBs have been added to the slant TEC values. It should be noted that the IFBs are of no consequence to the imaging algorithm itself because they are negated by the ray differencing technique. They are calculated using the differences in slant delay through the final image and the original differential delays.
Figure 5 shows the vertical TEC through the ionospheric image reconstructed from the data of Figure 4. By visual comparison with Figure 1 it can be seen that the vertical TEC in the ionosphere is well reproduced with mean absolute error in vertical TEC of 0.4 TEC units overall.
Figure 6 shows the receiver plus satellite IFBs from CODE (i.e., the true values put into the simulation) plotted against the biases obtained from the MIDAS inversion. Each point on the graph is the combination of a satellite plus receiver IFB calculated hourly by MIDAS plotted against the corresponding 30-day mean value from CODE. All pairs of receiver and satellite interfrequency bias are plotted when a satellite is in view of a receiver for a full hour. Good correlation can be observed between the biases that were put in to the simulation and those calculated by the MIDAS algorithm. This demonstrates that MIDAS can be used to calculate the paired satellite-receiver IFB values. The correlation coefficient is +99.7% and the RMS, mean and absolute mean errors in nanoseconds are 0.917, 0.288 and 0.637. The slant TEC errors in TEC units were 2.642, 0.829 and 1.837, percentage slant TEC errors were 6.0%, 0.7% and 4.4%. The vertical TEC errors are 1.281, 0.395 and 0.908 TECU and in percentages 4.5%, 0.4% and 3.4%. Hence, for the simulation there is a very small (less than 1%) underestimate in TEC.
4. Experimental Results
 Using the receiver configuration of Figure 2, the MIDAS algorithm was used to reconstruct ionosphere images for each hour of the year 2002. First a small section of this data was used to verify the accuracy of the hourly IFBs from MIDAS for direct comparison to the simulation results. The day 15 February 2002 was again used and Figure 7 shows the receiver plus satellite IFBs obtained from the MIDAS inversion, plotted against those obtained by CODE. Good agreement in the IFB values from MIDAS and CODE was found with a correlation coefficient of +98.6%, and RMS, mean and absolute differences of 2.005, 0.354 and 1.567 ns. These correlations were lower than they had been in the simulation. This was expected, as other sources of error, such as multipath, will be included with the IFB measurements. This result also compares CODE biases calculated using a month of IFB measurements with an IFB calculated from hourly images from MIDAS. Therefore the resultant MIDAS bias will also include short-term variations of the IFB that the longer CODE average will not include.
 An improvement was carried out by comparing the 30-day average of the IFB, for each satellite-receiver pair and those from CODE. Figure 8 shows these results. In general, a good agreement is found with a correlation coefficient of +98.3%, and an RMS, mean and absolute difference of 2.028, −0.539 and 1.220 ns. It is interesting to note in Figure 8 a cluster of points where there is disagreement and these were looked at in more detail. Disregarding this cluster the statistics are improved by a small amount, with a correlation coefficient of +98.9%, and an RMS, mean and absolute error of 1.643, −0.473 and 1.158 ns. These resultant biases have an 80% larger difference than the result produced from the simulated study. This increase is partly due to using measured, instead of simulated GPS TEC data, but also due to differing ionospheric reconstruction boundaries as CODE uses global images where as MIDAS biases have been measured from European data. For these results CODE has been assumed to be correct. In the simulation study, where the CODE IFBs are added to the input ionosphere, this is valid, but for the experimental results there will be differences produced by the bias determination method, such as the number of receivers used and their locations.
 This cluster of mismatched IFB values was found to be associated with a receiver based in Cascais, Lisbon, Portugal. On 10 July 2002, (191st day of year) between 1530 and 1600 UT, the receiver's hardware was updated. This was documented on the site log (ftp://garner.ucsd.edu/pub/docs/site_logs/casc.log.txt). It seems that the update to the hardware caused the IFB related to this receiver to alter dramatically (also documented by Sardón and Zarraoa ). A series of estimates of this receiver bias can be seen in Figure 9 for the year 2002. In this case the absolute values of the biases are not determined reliably, since the individual satellite or receiver biases cannot be found independently, but the relative change is clear. Figure 10 shows the same plot over the five days around the change in the hardware. The change of interfrequency bias over day 191 is approximately 23 ns, which is equivalent to 66.2 TEC units. The plots demonstrate that the imaging technique is able to separate out changes in individual IFB values and can detect their changes on hourly timescales. However, it should be stressed that individual satellite or receiver IFBs such as these are not uniquely determined by our technique, unlike combinations of satellite and receiver IFBs which can in theory be found.
5. Summary and Discussion
 The work here has shown a method to determine the error in TEC determination from ionospheric imaging. The method can be used in both simulation and experiment. This opens up the possibility to use ionospheric images for applications requiring estimates of the image accuracy, for example in navigation corrections and in data assimilation into physical models.
 A simulation study was carried out using a realistic representation of the ionosphere from the empirical model IRI. Satellite and receiver IFB values were simulated using values calculated by CODE. The results of the simulation study indicated that the MIDAS ionospheric imaging technique can be used to calculate IFBs values to an accuracy of RMS, mean and absolute mean values of 0.917, 0.288 and 0.637 ns. The corresponding slant TEC errors found by differencing those through the image with those from the IRI model were 2.642, 0.829 and 1.837 TEC units. In percentage terms these values were 6.0%, 0.7% and 4.4%. Vertical TEC errors were 1.281, 0.395 and 0.908 TEC units and 4.5%, 0.4% and 3.4%, respectively.
 The experimental study for the same day showed that differences between the hourly MIDAS IFBs and the long-term CODE values were RMS, mean and absolute mean values of 2.005, 0.354 and 1.597 ns. These indicate that the errors found by experiment are about double those found by simulation, assuming that the CODE bias values are correct. A large contribution to this increase in error is likely to be due to multipath, which was not accounted for in the simulation.
 On a longer timescale of one year the statistics of the differences between the monthly mean IFBs from MIDAS images and the CODE values were RMS, mean and absolute mean values of 1.643, 0.473 and 1.158 ns. The MIDAS IFB values clearly showed a sudden change in the bias for the Cascais receiver, which was traced to a documented change in the hardware at that site.
 The results are of interest for several reasons. First, they give an estimate of the errors involved in absolute TEC determination from ionospheric imaging that relies on using uncalibrated differential phase observations. The results give confidence in the determination of absolute TEC values, which is important for the subsequent use of ionospheric images. This is of interest for scientific studies in cases where bounds on the physical parameters of the ionosphere are sought [see, e.g., Yin et al., 2004]. The fact that the IFB error values are larger in the experimental study than in the simulation may indicate that the TEC errors are also larger in the experimental case. Thus an estimate of around double the 6.0%, 0.7% and 4.4% values for RMS, mean and absolute mean may be appropriate when using such measurements for assimilation work, where the errors on both the model and measurements are required.
 The results are also applicable to the navigation community who are interested in accuracies in TEC mapping for ionospheric corrections by Satellite Based Augmentation Systems. Finally, for real-time ionospheric mapping the ability to calculate IFBs on an hourly basis is very useful in situations where the values may change, such as happened in the case of the Cascais receiver. Data from newly installed hardware can be used without concerns over the site-specific IFB values. The use of precalculated IFBs for this receiver in an ionospheric imaging/mapping system would have resulted in the introduction of errors of 66 TEC units for all slant paths from that site and this could have created an artificial ionospheric feature in the maps. The use of uncalibrated differential phase data for ionospheric mapping avoids this potential problem.
 We are grateful to the International GPS Service for the use of GPS data and CODE for the use of their IFBs, and we also acknowledge the use of the IRI model. We also thank BAE SYSTEMS and the U.K. Engineering and Physical Sciences Research Council (EPSRC) for their support.