Electron density height profiles from GPS receiver data



[1] A successful ionospheric electron density model assimilates relevant data for determination of its driving parameters. We process about two hours worth of two-frequency GPS ground-receiver data from five satellites in order to separate the least squares solution of up to four driving parameters of the electron density profile (EDP) model from the solution of four relative (between satellites) differential hardware biases. The EDP model we use is part of the Ionosphere and Troposphere Raytrace Model (ITRAY), which is our upgrade of the Raytrace/Ionospheric Conductivity and Electron Density model (ICED)-Bent-Gallagher (RIBG) model. We use the Westford GPS receiver in Massachusetts to update the EDP model, which we then use to predict EDP distributions at nearby Millstone Hill for comparison with ground truth incoherent-scatter radar and Digisonde data. We find that a simple procedure often works well when the model time dependence is correct, but we find that forcing the ionospheric model to more closely fit GPS data actually degrades ionospheric specification in the F layer region. We suggest a data diversification remedy.

1. Introduction

[2] This paper investigates the capability of a system for determining ionospheric specification from two-frequency GPS receivers on the ground. Specification consists of the distribution of electron density in latitude, longitude, and height. Satisfactory system performance requires accuracy and timely distribution of the specification. We previously developed a method of processing GPS receiver data [Reilly and Singh, 2001] that uses the electron density model in the Ionosphere and Troposphere Raytrace Model (ITRAY), our upgrade of the Raytrace/ICED-Bent-Gallagher (RIBG) model [Reilly, 1993]. In this method, least squares analysis provides both a single effective sunspot number driver of the height profile model and hardware differential biases. The data consist of 2 hours of shifted differential phase pseudorange (SDPP) (L1–L2) data in total electron content units (TECU) (one TECU = 1016 electrons per cubic meter) on 5 satellite receiver paths. The data interval is 30 s. Two hours of data provide enough ionospheric variation to obtain ionospheric model driving parameter and relative hardware differential bias (RDB) solutions. SDPP data traditionally serve as a proxy for group path length without the effects of multipath and without the free space range contribution. We showed evidence from sounder measurements that ITRAY, updated by this method, could predict foF2 within a few MHz. In this paper, we attempt to exploit the flexibility of the EDP model in ITRAY by varying four of its driving parameters independently in order to obtain a better fit of the GPS data. The four driving parameters are: (1) SF2, which determines the value of the maximum plasma frequency foF2; (2) SM3, which determines the value of M3000 (multiplying factor of foF2 that gives the maximum usable frequency (MUF) on a 3000 km path), which determines hmF2 (height at maximum plasma frequency) in the ionospheric model; (3) SWDTH, which determines the width of the F2 Chapman layer in the model; and (4) CFAC, which affects how fast the EDP drops off with height above hmF2. A fifth driving parameter of the model is Kp, the 3-hourly planetary magnetic activity index, which we determine separately from external geomagnetic data. The conditions SF2 = SM3 = SWDTH and CFAC = 0.86 were previously used to constrain these parameters.

[3] The theory for fitting the model to the data is similar to before [Reilly and Singh, 2001] except that we now work with differences of SDDP data between the satellites, or relative SDDP data, in order to remove the influence of receiver clock drift. In our previous paper, we assumed that hardware differential biases between each satellite and the receiver would remain constant over the 2-hour period. Now that we work with relative SDDP data between satellites, the assumption of constant RDBs is substantially more reliable. Hence the number of unknown model parameters for five satellites is now eight: four driving parameters of the ionospheric model and four RDBs. We obtained nearly identical answers from this and the previous method for many cases, thus confirming the method. A discrepancy found in a few cases was apparently due to receiver clock drift.

[4] The method we choose to obtain the multiparameter fit of relative SDDP data is the Levenberg-Marquardt (LM) method, as implemented and explained in the work of Press et al. [1992]. We adapt this implementation to our problem as follows. First, obtain the solution for the single sunspot number driver in the above default condition. We later refer to this as the F solution. This takes less than 1 min on a typical personal computer (PC) for a wide range of initial guesses of starting sunspot number. The F solution could be the starting guess for the LM method, which finds model parameter solutions that minimize the value of a chi-square merit function (chisq), given by

equation image

where xi is an index parameter that runs over the satellite pairs and the time, yi represents a particular one of N relative SDDP data points, and y(xi, a) represents the model calculation of this datum as a function of the 8-element model parameter vector a. We assume the data to have equal weights. Unfortunately, the F solution guess sometimes converges to a local minimum of chisq, which is different from the global minimum. We need the starting guess close enough to the global minimum solution in order to avoid this. Hence we start with a grid of driving parameters centered on the preceding F solution guess. This grid presently consists of a range of values for each of SF2, SM3, SWDTH, and CFAC. For each value in this grid we calculate RDB values that minimize chisq. The overall minimum chisq grid point is used as the new starting guess for the LM method. We refer to this later as the LMG solution. Convergence of LMG takes about 50–150 times longer than F since several tens of iterations and many more ionospheric model driving parameter interpolations are involved.

[5] The next section calculates the results of analysis of Westford GPS data for 6 dates in 2000 and 2001. The Westford receiver is located at 42.613°N–71.493°E. We compare updated ITRAY model predictions of EDPs with available EDP data from the Millstone Hill incoherent scatter radar data and Digisonde data over the 2-hour GPS data processing period. Millstone Hill is located at 42.620°N, –71.492°E. This is a severe test of the ionospheric model since available incoherent scatter radar (ISR) data concentrate in the late afternoon to evening hours, when the ionosphere varies rapidly. We use a single set of driving parameter solutions for the predictions, thus relying only on the time dependence within the ionospheric model.

2. GPS-Updated ITRAY Comparisons With Ground Truth Data

[6] The first cases considered are for 25 September and 24 October 2000, when GPS data were processed in the intervals 22.00–23.99 UT and 19.75–22.24 UT, respectively, in order to update (find driving parameters of) ITRAY. ITRAY was then used to generate EDPs for comparison with Millstone Hill data. Results are shown in Figure 1 for the indicated times. LMG, which denotes the multiparameter method discussed previously, denotes the first update method. F denotes the simpler solution method in Figure 1, also discussed above. We find that, although LMG gives a better fit to GPS data (smaller chisq; see Table 1 below) than F does, Figure 1 indicates that the F update method gives a better ITRAY prediction of Millstone Hill ISR and Digisonde data, at least near the peak density. This is a recurring theme and we will discuss it later. The Digisonde parameter available to us from the Internet is foF2 and the corresponding density appears as the small square on the horizontal axis. For 25 September we see that the F update predictions do not vary as rapidly as the data in the 2-hour period for GPS data processing. Hence the slower ITRAY model time dependence underpredicts ISR at the beginning of the 2-hour interval and overpredicts it at the end of the interval. However, the data and model time dependencies agree quite well for the case of 24 October.

Figure 1.

Updated ITRAY predictions (LMG and F) of ISR and Digisonde data: 25 September and 24 October 2000.

Table 1. Summary of Westford GPS Data Analysis
Start TimeMethodSF2SM3SWDTHCFACKpGPS Fit Error, TECUISR Predicted Error, 106/cm3
25 Sep. 2000, 22.00 UTLMG75.2850.5910.001.564.02.100.484
24 Oct. 2000, 19.75 UTLMG174.99121.36180.000.682.00.840.306
17 Apr. 2001, 21.75 UTLMG109.11250.9710.
18 Apr. 2001, 21.75 UTLMG14.09200.610.
11 Apr. 2000, 18.50 UTLMG224.7758.5110.
12 Apr. 2000, 18.50 UTLMG130.56−83.05180.001.38561.51.940.243

[7] Figure 2 shows the results of a similar analysis for 17 and 18 April 2001. For these cases, Digisonde parameters are not available, and so we rely on the ISR data as ground truth. For 17 April, F again predicts the ISR data better than LMG, but the agreement again suffers from the much greater time variation of the ISR EDP, as compared to the model time dependence. The ITRAY model, which is based on the International Union of Radio Science (URSI) climatological map data, consequently underpredicts the ISR data in the first half of the GPS processing interval and overpredicts the ISR data in the second half of the GPS processing interval. However, the F-updated ITRAY results for 18 April exhibit satisfactory agreement with the ISR data, where the rates of change of model and data are about the same. These results are thus similar to Figure 1.

Figure 2.

Updated ITRAY predictions (LMG and F) of ISR data: 17 and 18 April 2001.

[8] Figure 3 shows the results of a similar analysis for 11 and 12 April 2000, for which we have both Digisonde and ISR data at Millstone Hill. For 11 April, the agreement of F-updated or LMG-updated ITRAY predictions with ISR data is only fair. The ISR EDP changes rapidly between the first and second times, much more so than either the ITRAY predictions or the Digisonde data. The F predictions are more in agreement with the Digisonde data and adequately agree with the maximum height and shape of the ISR profile, although not with its maximum density. For 12 April, we again note rather anomalous time variation of the ISR profiles, thus guaranteeing that the F- or LMG-updated ITRAY prediction will not impressively fit ISR for every one of the times. The prediction of Digisonde data is better.

Figure 3.

Updated ITRAY predictions (LMG and F) of ISR and Digisonde data: 11 and 12 April 2000.

[9] Table 1 summarizes our results for driving parameters of the ionospheric model of ITRAY obtained by fitting it to about 2 hours worth of data from 5 GPS satellites, as measured by the Westford GPS receiver. The first column gives the start time and date of the approximately 2-hour interval. The second column cites the methods that we have already discussed. Columns 3–6 give the driving parameter solutions for the ionospheric model. Column 7 gives the appropriate value of the magnetic activity index, available from the Internet. Column 8 gives equation image (see equation (1)) and is thus the RMS error of the ITRAY fit to each GPS data point (relative SDPP in TECU). The last column gives the RMS error of the ISR electron density prediction by ITRAY, averaged over the three times for a given date. Not shown are the four RDB values or the standard deviation errors of the model parameter solutions. We obtain these from our use of the LM method. They depend on chisq. For example, in the case of a good fit, such as the 24 October entry, the errors in SF2, SM3, SWDTH, and CFAC are about 2.5, 6.3, 222, and 0.02, respectively, and the four RDB errors are around 0.06 TECU. For a poor fit, such as the last 12 April entry, the corresponding errors in the ionospheric model driving parameters are about 13, 29, 3723, and 0.05, respectively, and the four RDB errors are around 0.2 TECU. The large errors in SWDTH indicate only a weak dependence of chisq on this parameter in our model. Some of the parameters are confined within certain limits, reflecting the underlying validity and limitations of the model, which is why, for example, values of the upper and lower limits of SWDTH, 10.0, and 180.0 show up quite frequently. The LM method allows these constraints to be incorporated in the fitting process.

3. Discussion and Conclusions

[10] It surprised us that F-updated ITRAY predictions were more effective than LMG-updated ITRAY predictions of Millstone Hill data, despite the fact that LMG analysis of GPS data enabled us to vary more model parameters for a clearly better fit of GPS data. We have observed this not only with the present data sets but also with Digisonde data from previous work. The LMG fitting process apparently deforms the EDP in height regions of the model height versus density profile that are far above the F2 peak height, with a corresponding effect in the F layer. The F procedure does better in the F region of the ionosphere by not allowing the deformation of the profile in the upper height regions. Hence our model apparently does not relate the F region to these upper height regions properly. In lieu of having a perfect model, we may compensate for this by assimilating additional data types that more specifically constrain the F region parameters. Digisonde data does this. So does GPS occultation data.

[11] F-updated ITRAY predictions often do quite well, certainly better than LMG in the F layer, despite the fact that LMG gives a better fit of GPS data. F updates take much less PC time. For example, with our moderately powerful desktop PC, it takes a few seconds to load data from a GPS receiver and about 10 s to get a converged F solution for SF2 and the RDBs if the ionospheric grids have been preloaded. An ionospheric grid in our case is a complete specification of electron density versus height profile parameters in ITRAY, for given time and driving parameters, at each point of a global latitude-longitude grid. The grid latitude spacing is 3°, and the longitude spacing is 5°. Otherwise, the additional time to generate ionospheric grids, about 20 of them for a 2-hour period, is about 20 s. These times can be reduced with further computer power and additional software efficiency, but it is already reasonable to think of this as a component of a practical system for regional and even global ionospheric electron density specification in near-real time, utilizing data from a network of GPS sounders.

[12] Convergence in the LMG method started from lowest chisq member of a grid of results centered on the F solution. We also attempted to start the LM calculation from the F solution without the grid. This often reached the same solution as the LMG solution, but sometimes it reached a different solution, associated with a local minimum of chisq as a function of the eight model driving parameters that is from the global minimum. For example, this happened in the first and fifth entries of Table 1.

[13] The issue of having to process 2 hours of GPS data, in order to separate the ionospheric variation from the constant contributions of RDBs, can be mitigated somewhat. One can find RDB solutions in this manner and then proceed to use them in future calculations for a substantial period of time since they are associated only with the more stable satellite clocks. We have indicated a method for this in our previous paper. Once the known RDB contributions are subtracted from the data, the driving parameters of the ionospheric model should be able to be determined in a smaller GPS data time interval. This would enable more frequent updates of the ionospheric model and thus make it more capable of responding to actual time variations of the ionosphere.

[14] As we have mentioned, one may obtain greater precision with the LMG method by assimilating more data types and/or by using a better ionospheric model. For example, when both GPS and Digisonde data are present, it may be possible to determine one or both of foF2 and hmF2 from Digisonde data, thereby determining SF2 and SM3 in our model, and then use the remaining driving parameters to determine EDP shape parameters. Other data sets ultimately may be used with GPS data for regional or global ionospheric electron density specification besides Digisonde data. Satellite UV airglow data, satellite Earth-occultation GPS data, and over-the-horizon (OTH) radar data are examples that come to mind because of their global coverage characteristics.


[15] Thanks to ONR for its support and to J. P. Keady for useful discussions.