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Keywords:

  • GPS;
  • plasmasphere;
  • TEC

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Kalman Filter Formulation
  5. 3. Carpenter-Anderson Plasmasphere Model
  6. 4. Numerical Results
  7. 5. Preliminary Validation
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[1] We extend a Kalman filter technique for GPS total electron content (TEC) estimation by explicitly accounting for the contribution to the line-of-sight TEC from the plasmasphere. The plasmaspheric contribution is determined by integrating the electron density predicted by the Carpenter-Anderson model along GPS raypaths and allowing the Kalman filter to scale the results to fit the observations. The filter also estimates the coefficients of a local fit to the ionospheric TEC and the receiver and satellite instrumental biases. We compare algorithm results with and without the plasmasphere term for three GPS receivers located at different latitudes. We validate the approach by comparing ionospheric TEC estimates with Advanced Research Projects Agency Long-range Tracking and Instrumentation Radar measurements along coincident lines of sight to calibration objects in low Earth orbit. We find the technique is effective at separating the ionospheric and plasmaspheric contributions to the TEC by exploiting differences in the spatial distributions of electron density, and the time scales on which they vary, in these two regions.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Kalman Filter Formulation
  5. 3. Carpenter-Anderson Plasmasphere Model
  6. 4. Numerical Results
  7. 5. Preliminary Validation
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[2] The ionosphere is a rarefied region of the Earth's upper atmosphere which is partially ionized by solar radiation and extends approximately from 70 to 1000 km in altitude. The bulk of the ionospheric plasma is typically concentrated in a relatively thin altitude layer between about 200 and 400 km. The plasmasphere is the high-altitude extension of the ionosphere which consists primarily of electrons and H+ ions together with smaller concentrations of O+ and He+ ions. In the plasmasphere, the plasma is distributed along geomagnetic field lines forming a roughly toroidal region that extends to an equatorial distance of several Earth radii. The outer boundary of the plasmasphere is termed the plasmapause, which is characterized by a rapid drop in plasma density (more than an order of magnitude). The position of the plasmapause changes in response to magnetospheric convection currents near this boundary which entrain plasma and erode the outer plasmasphere. Free electrons in both the ionosphere and plasmasphere impart a group delay on space-to-ground radio transmissions, which can adversely affect the performance of technological systems that depend on them, such as navigation, communication, and deep space tracking radars. While the concentration of free electrons is much larger in the ionosphere than in the plasmasphere, the significantly longer GPS path lengths through the latter can result in group delay because of free electrons in both regions being comparable under certain circumstances. This is particularly true at night during solar minimum conditions.

[3] The GPS currently consists of a constellation of 31 active satellites which broadcast coded radio signals at the L1 (1575 MHz) and L2 (1228 MHz) frequencies. Because of the dispersive nature of the ionosphere and plasmasphere, a dual-frequency GPS receiver can measure the total electron content (TEC) along the signal path, provided that instrumental biases associated with the receiver and satellites can be accurately estimated and removed. A number of techniques have been developed for this purpose [Sardon et al., 1994; Rideout and Coster, 2006; Komjathy and Langley, 1996; Carrano and Groves, 2006; Anghel et al., 2008]. These techniques employ a thin slab approximation for the ionosphere and exploit the fact that to first order, the measured slant TEC depends systematically on the satellite elevation, while the instrumental biases do not. The contribution to the measured slant TEC from the plasmasphere, however, varies with azimuth and elevation in a manner that violates the thin slab approximation, and this can lead to inaccuracies in the estimation of the instrumental biases. A consequence of this violation is that estimates for the instrumental biases depend implicitly on the plasmaspheric density, which is undesirable since an instrumental bias should not depend on the geophysical environment being monitored. If the instrumental biases are inaccurately estimated, then the total unbiased GPS TEC will be inaccurately estimated as well.

[4] A number of researchers have concluded that the plasmasphere makes an important contribution to GPS measurements of TEC [Balan et al., 2002;Lunt et al., 1999; Bishop et al., 1999], and in many applications it can be useful to separate the ionospheric and plasmaspheric contributions to the total TEC. For example, when using GPS measurements to drive data-assimilative models of the ionosphere, it may be appropriate to remove the plasmaspheric contribution [Bishop et al., 2009]. It is also helpful to remove the plasmaspheric contribution from GPS TEC measurements when comparing them to TEC measurements made by other sensors that do not sample the plasmaspheric contribution, such as TOPEX/POSEIDON, ionosonde, or radar observations of calibration objects in low Earth orbit (LEO). On the other hand, when applying ranging corrections to radar measurements of deep space objects residing outside the plasmapause, the total TEC in both the ionosphere and plasmasphere is required. Mazzella et al. [2002] developed the first technique for estimating the unbiased ionospheric and plasmaspheric contributions to the GPS TEC. In this paper, we present an alternative approach using a Kalman filter and the Carpenter and Anderson [1992] plasmasphere model. Since this paper was first written, Anghel et al. [2009] have reported on a variation of this approach which uses the plasmasphere model by Gallagher et al. [1988] rather than the Carpenter-Anderson model.

[5] The algorithm we present is an extension of a Kalman filter technique for GPS TEC estimation called WinTEC, which was developed by the Space Weather Prediction Center [Anghel et al., 2008]. In the WinTEC approach, biased measurements of slant TEC are modeled according to their elevation-dependent paths through a thin ionospheric slab. The distribution of vertical TEC above the receiver is modeled as a first-order polynomial in a geomagnetic coordinate system. Biases for the receiver and each of the satellites appear in the model as additive terms, with no elevation angle dependence. The Kalman filter assimilates the biased slant TEC measurements at each time step and yields estimates of the combined satellite plus receiver biases, the ionospheric fit parameters, and the error covariances. In the extension of WinTEC presented here, which we will refer to as WinTEC-P, a term representing the integrated density through the Carpenter-Anderson plasmasphere model has been included. A scaling factor for this term is estimated by the Kalman filter along with the other components of the state vector.

2. Kalman Filter Formulation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Kalman Filter Formulation
  5. 3. Carpenter-Anderson Plasmasphere Model
  6. 4. Numerical Results
  7. 5. Preliminary Validation
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[6] The ionosphere and plasmasphere impart a group delay and carrier phase advance which are, to first order, equal in magnitude, proportional to the number of free electrons encountered along the signal path, and inversely proportional to the square of the signal frequency. Expressed in units of meters, the pseudoranges P1 and P2 and carrier phases Φ1 and Φ2 on the L1 and L2 frequencies, respectively, may be combined to produce two independent estimates of the total TEC along the propagation path:

  • equation image

The TEC estimate obtained from the pseudoranges, TECP, is absolute except for the instrumental biases, which will be removed later. TECP is very sensitive to multipath, however, and therefore noisy. The TEC estimate obtained from the phases, TECΦ, is much less susceptible to multipath but contains an unknown offset due to phase cycle ambiguity. After correcting the phase measurements for cycle slips, we form a new TEC estimate which is equal to the less noisy phase estimate, leveled according to the pseudoranges:

  • equation image

The bracket notation in (2) indicates an elevation-weighted average over phase-connected arcs (between cycle slips). We estimate the phase-leveling error as the elevation-weighted standard deviation over these phase-connected arcs, and we discard the data from the entire arc if this value exceeds 5 total electron content units (1 TECU = 1016 el m−2) in an effort to avoid the effects of multipath. We refer to TECRS as the line-of-sight (LOS) phase-leveled TEC observable for a given receiver-satellite pair (RS). The measurements from each of the satellites are assimilated by the Kalman filter during each Kalman update. In the results presented here, these updates occur every 60 s. We assimilate measurements only for satellites above 20° in elevation in order to further reduce the effects of multipath.

[7] Employing a thin slab model for the ionosphere, a bilinear spatial approximation of the vertical TEC in the ionosphere, and a model for the slant TEC in the plasmasphere, we express the line-of-sight phase-leveled TEC observable, at epoch k, in a geomagnetic reference frame as

  • equation image

In equation (3), the parameters a0,Rk, a1,Rk, and a2,Rk are the coefficients of a bilinear fit to the vertical TEC in the ionosphere. The quantity P(αRSk, ɛRSk) is the slant TEC in the plasmasphere according to the Carpenter-Anderson plasmasphere model along the path from the receiver to the satellite in the azimuth αRSk and elevation ɛRSk direction. This quantity also depends implicitly on the geographic location of the receiver and, depending on the plasmasphere model chosen, geophysical parameters such as the day of year, average sunspot number, and Kp. The details on how we compute this term are presented in section 3. The parameter a3,Rk is a scaling factor for the slant plasmaspheric TEC predicted by the model. Note that in this formulation, a0,Rk represents the modeled vertical TEC in the ionosphere, and the quantity a3,RkP(α, ɛ) evaluated at ɛ = 90° represents the modeled vertical TEC in the plasmasphere above the receiver. The additive terms bRk and bSk in equation (3) represent the receiver and satellite instrumental biases, which cannot be determined separately in this formulation unless an additional assumption is made (generally, the satellite biases are assumed to have zero mean). The quantity RSk is the difference between the geomagnetic local times at the ionospheric penetration point (IPP) and the station, while the quantity RSk is the difference between the geomagnetic latitudes of the IPP and the station. Finally, the thin slab mapping function M(ɛ) relates the line-of-sight (slant) and vertical equivalent TEC values at the IPP:

  • equation image

In equation (4), Re is the radius of the Earth, and h is the height of the ionospheric thin slab, which we have taken to be 350 km. It is important to note that in equation (3) the thin slab mapping function is applied to the ionosphere terms and is not applied to the plasmasphere term.

[8] We employ a Kalman filter algorithm with process noise to estimate the TEC in the ionosphere and plasmasphere, along with the receiver and satellite instrumental biases. The process to be estimated is modeled as

  • equation image

where xk is the state vector, Φk,k−1 is the state transition matrix, wk is the process noise (a zero-mean white Gaussian noise with the covariance matrix Q), Hk is the measurement matrix, yk is the measurement vector, and vk is the measurement noise (a zero-mean white Gaussian noise with the covariance matrix R).

[9] For the single receiver case, the Kalman state vector xk consists of the three ionospheric fit parameters that describe the vertical TEC in the ionosphere above the site, the scaling factor for the plasmaspheric TEC, and the instrumental biases for the receiver and N GPS satellites:

  • equation image

The measurement vector yk contains the line-of-sight phase-leveled TEC observations:

  • equation image

To simplify the notation, let M = N + 5. The state transition matrix, Φk,k−1, is the M × M identity matrix, and the N × M measurement matrix, Hk, is given by

  • equation image

Each component of the yk vector and each row in the Hk matrix corresponds to a particular GPS satellite. If the satellite is not in view, then its corresponding component in yk and corresponding row in Hk are set to zero. Following Anghel et al. [2008], we specify the M × M process noise covariance matrix Q and the N × N measurement covariance matrix R as follows:

  • equation image

In the specification of Q as above, we obtain the best results when only the diagonal elements corresponding to the ionospheric state variables are nonzero since the ionosphere generally varies on much shorter time scales than either the plasmasphere or the instrumental biases. Alternatively, one may specify the process noise as a Gauss-Markov process, which may provide improved control of the time scales over which the ionosphere and plasmasphere are permitted to vary in the Kalman filter solution. Details on how to perform the Kalman filter updates using equations (5)–(9) may be found in a standard text on Kalman filtering. Error estimates for the unbiased TEC are available in terms of the postfit residuals and the phase-leveling error.

[10] We note that a number of alternative formulations of this algorithm are possible. For instance, Anghel et al. [2008] describe how WinTEC can be used to process the TEC from a network of GPS receivers simultaneously using a modified form of equations (6)–(9). Anghel et al. [2009] explore the use of higher-order terms (quadratic and cubic) when fitting the vertical TEC in the ionosphere. In section 5, we employ a modified version of the algorithm which uses satellite biases provided by the Center for Orbit Determination in Europe (CODE) [Schaer et al., 1998] rather than solving for them as part of the Kalman state. This is accomplished by zeroing out the columns of the Hk matrix corresponding to the satellite biases and presubtracting the satellite biases from the line-of-sight phase-leveled TEC observations before they are assimilated into the Kalman filter. In section 4, however, we solve for the combined satellite and receiver biases explicitly because organizations such as CODE do not account for the plasmasphere when they estimate the satellite biases they provide.

3. Carpenter-Anderson Plasmasphere Model

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Kalman Filter Formulation
  5. 3. Carpenter-Anderson Plasmasphere Model
  6. 4. Numerical Results
  7. 5. Preliminary Validation
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[11] Carpenter and Anderson [1992] presented an empirical model for the electron density in the inner plasmasphere (nei) in cm−3, as a function of the McIlwain L parameter (L), day of year (d), and 13 month average sunspot number (equation image):

  • equation image

Their model employs an empirical expression for the location of the plasmapause which is given by

  • equation image

where Kpmax represents the highest value attained by the planetary magnetic index over the previous 24 h. For the width of the plasmapause, we use the estimate provided by Gallagher et al. [2000], neglecting the local time dependence:

  • equation image

We use the estimate for the electron density in the plasmaspheric trough (in cm−3) provided by Sheeley et al. [2001], again neglecting the local time dependence:

  • equation image

Finally, we employ the hyperbolic tangent “step function” used by Gallagher et al. [2000],

  • equation image

to transition between the inner plasmasphere and the trough. This results in a general expression (applicable both inside and outside the plasmapause boundary) for the electron density at altitudes beyond the ionosphere:

  • equation image

We compute the plasmaspheric slant TEC term P(αRSi, ɛRSi) in equation (3) by numerically integrating the electron density in equation (15) along the line of sight from the GPS receiver to each satellite starting from 700 km up to the GPS orbital altitude of 20,200 km. While the Carpenter-Anderson model is not recommended for use below L = 2.25 (the model is known to underestimate the electron density in the topside ionosphere), we do not expect this to cause difficultly since the Kalman filter will scale the model output to match the observations.

[12] The contribution from the plasmasphere to the TEC measured along the line of sight to a GPS satellite is a strong function of elevation and the geomagnetic latitude of the receiver. Figure 1a shows the distribution of electron density in the plasmasphere as predicted by the Carpenter-Anderson model with Kpmax = 1 and equation image = 5.8. Figures 1b–1d show how the plasmaspheric TEC varies as a function of elevation angle for stations located at different geomagnetic latitudes. Note that in general, the dependence of the plasmaspheric slant TEC on the elevation angle differs significantly from that of the ionospheric slant TEC, which varies with elevation as 1/M(ɛ) according to the thin slab model. The Kalman filter effectively exploits this difference to distinguish between the plasmaspheric and ionospheric origins of the TEC.

image

Figure 1. (a) Modeled plasmaspheric electron density in a plane containing Earth's rotation axis. The dotted line represents the base of the plasmasphere, here taken to be 700 km. The modeled plasmaspheric TEC as a function of northward viewing and southward viewing elevation angles for receivers located at (b) high (60°), (c) middle (30°), and (d) low (0°) geomagnetic latitudes. The curves are labeled according to the value assumed for Kpmax, ranging from 1 to 9.

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4. Numerical Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Kalman Filter Formulation
  5. 3. Carpenter-Anderson Plasmasphere Model
  6. 4. Numerical Results
  7. 5. Preliminary Validation
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[13] In this section, we demonstrate the WinTEC-P algorithm with and without the plasmasphere term using GPS measurements collected at three ground stations located in the American sector. These GPS receivers operate as part of the Air Force Research Laboratory–Scintillation Network Decision Aid network of ionospheric monitoring instruments [Carrano and Groves, 2006]. The three-letter station codes, station names, and geographic coordinates for these GPS receivers are given in Table 1.

Table 1. Station Codes, Station Names, and Geographic Coordinates for the Three GPS Receivers Used in This Study
Station CodeStation NameLatitudeLongitude
HAYMIT Haystack Observatory, Westford, MA, USA42.61288.52
NCANorth Carolina A&T, Greensboro, NC, USA36.08280.23
ROARoatan, Honduras16.29273.41

[14] Since these stations are located at different geomagnetic latitudes, the contributions to the GPS TEC from the plasmasphere are expected to differ at these stations. In each case, we allowed the Kalman filter to assimilate data during the period 16–19 November 2007, which was characterized by relatively low geomagnetic activity (Kp less than 3+). For simplicity, we have assumed a constant value of Kpmax = 1 throughout this period. Figures 2–7 show the output of WinTEC-P (with the plasmasphere term) and WinTEC (without the plasmasphere term) for these three stations on 17–19 November 2007 after transients associated with the Kalman filter have vanished. These transients typically persist for a few hours after data from each GPS satellite is first assimilated and vanish after the Kalman filter has established a stable estimate for the satellite's instrumental bias. When a new satellite enters the GPS constellation, a new transient begins, and the Kalman filter will readjust its estimate of the instrumental bias for this satellite typically within a few hours.

image

Figure 2. WinTEC-P results as a function of geomagnetic local time for the HAY receiver on 17–19 November 2007. The standard deviation of the postfit residuals during this period was 0.86 TECU. The receiver bias (relative to the satellite biases with their mean subtracted) was estimated to be −6.3 TECU. The RMS deviation between the computed and CODE satellite biases was 0.73 TECU.

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image

Figure 3. WinTEC results as a function of geomagnetic local time for the HAY receiver on 17–19 November 2007. The standard deviation of the postfit residuals during this period was 0.96 TECU. The receiver bias (relative to the satellite biases with their mean subtracted) was estimated to be −7.0 TECU. The RMS deviation between the computed and CODE satellite biases was 0.73 TECU.

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image

Figure 4. WinTEC-P results as a function of geomagnetic local time for the NCA station on 17–19 November 2007. The standard deviation of the postfit residuals during this period was 0.99 TECU. The receiver bias (relative to the satellite biases with their mean subtracted) was estimated to be 37.4 TECU. The RMS deviation between the computed and CODE satellite biases was 0.79 TECU.

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image

Figure 5. WinTEC results as a function of geomagnetic local time for the NCA station on 17–19 November 2007. The standard deviation of the postfit residuals during this period was 1.15 TECU. The receiver bias (relative to the satellite biases with their mean subtracted) was estimated to be 37.4 TECU. The RMS deviation between the computed and CODE satellite biases was 0.78 TECU.

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image

Figure 6. WinTEC-P results as a function of geomagnetic local time for the ROA station on 17–19 November 2007. The standard deviation of the postfit residuals during this period was 1.27 TECU. The receiver bias (relative to the satellite biases with their mean subtracted) was estimated to be 39.9 TECU. The RMS difference between the computed and CODE satellite biases was 0.86 TECU.

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image

Figure 7. WinTEC results as a function of geomagnetic local time at the ROA station on 17–19 November 2007. The standard deviation of the postfit residuals during this period was 1.41 TECU. The receiver bias (relative to the satellite biases with their mean subtracted) was estimated to be 43.5 TECU. The RMS difference between the computed and CODE satellite biases was 0.80 TECU.

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[15] Figures 2–7 show the unbiased total TEC, unbiased ionospheric TEC, and unbiased plasmaspheric TEC in both slant and vertical equivalent representations. The heavy solid black, red, and blue curves show the modeled (fitted) vertical total TEC (a0 + a3P(0, 90°)), modeled vertical ionospheric TEC (a0), and modeled vertical plasmaspheric TEC (a3P(0, 90°)), respectively. The combined satellite plus receiver biases are also shown. The time axis for each plot is geomagnetic local time in hours, and the unbiased TEC measurements are colored according to the geomagnetic latitude of the IPP so that meridional gradients are easily identified. After each simulation, we subtracted the mean value from the combined satellite plus receiver biases so that they may be readily compared with the satellite biases reported by CODE [Schaer et al., 1998]. This mean value has been added to the receiver bias estimated by the Kalman filter to obtain the values of the receiver bias reported here. Note that the GPS receivers described in this section are NovAtel GSV4004B models which measure the pseudoranges using the coarse acquisition code on L1 and the precise (P) code on L2. Therefore, the net satellite instrumental bias imposed on the GPS measurements is given by the difference between the P1P2 and P1C1 satellite biases reported by CODE.

[16] Figure 2 shows the output of the WinTEC-P algorithm for the HAY station, which is the most poleward of the three stations considered. The RMS deviation between the final Kalman filter estimate of the satellite biases and those provided by CODE was 0.7 TECU, which indicates that WinTEC-P is able to accurately estimate them from the measurements. The final estimate for the receiver bias was −6.3 TECU. The final estimate for the plasmaspheric scaling factor, a3, was 2.0, suggesting a somewhat denser plasmasphere than predicted by the Carpenter-Anderson model. In Figure 2b, a strong southward directed meridional gradient is evident in the vertical equivalent total TEC. The results from WinTEC-P suggest that this gradient is largely plasmaspheric in origin since it is largely absent from the vertical equivalent ionospheric TEC (Figure 2d). The modeled vertical TEC in the plasmasphere (Figure 2f) was predicted to be approximately 0.7 TECU, while the maximum slant TEC in the plasmasphere (Figure 2e) was 8.0 TECU. In Figure 3, we show the results for this station when the plasmasphere term has been neglected. In this case, the RMS deviation between the computed satellite biases and those provided by CODE was 0.7 TECU, and the receiver bias was estimated to be −7.0 TECU. Note that when the plasmaspheric term is excluded, the meridional TEC gradient in the plasmasphere is again evident in the vertical equivalent TEC (Figure 3b), and the modeled vertical total TEC is approximately 1 TECU larger (compare Figures 2h and 3d). We also note that the postfit residuals are slightly larger when the plasmasphere term is excluded.

[17] Figure 4 shows the results of the WinTEC-P algorithm for the NCA station. The RMS deviation between the computed satellite biases and those provided by CODE was 0.8 TECU. The final estimate for the receiver bias was 37.4 TECU, and the final estimate for the plasmaspheric scaling factor was 2.2. Again, a strong southward directed meridional gradient in the vertical equivalent TEC is evident (Figure 4b), nearly all of which appears to be plasmaspheric in origin, as it is largely absent in the vertical equivalent ionospheric TEC (Figure 4d). At this station, the modeled plasmaspheric contribution to the vertical TEC (Figure 4h) was estimated to be 1.4 TECU, which is approximately 50% of the modeled vertical total TEC above this station at the nighttime minimum. The maximum slant TEC in the plasmasphere (Figure 4e) was approximately 8 TECU. When the plasmasphere term is neglected (Figure 5), the meridional gradient in the plasmasphere is again evident in the vertical equivalent TEC (Figure 5b). It is interesting to note that at this midlatitude station, the receiver bias is estimated to have the same value regardless of whether the plasmasphere term is included in the model or not. Again, the postfit residuals are slightly larger when the plasmasphere term is excluded.

[18] Figure 6 shows the results of the WinTEC-P algorithm at the ROA station. The RMS deviation between the computed satellite biases and those provided by CODE was 0.86 TECU. The final estimate for the receiver bias was 39.9 TECU, and the final estimate for plasmaspheric scaling factor was 1.6. A strong southward directed meridional gradient in the vertical equivalent total TEC is once again evident (Figure 6b), and at this station much of it is plasmaspheric in origin although some is also evident in the ionosphere (Figure 6d), particularly during daylight hours. The modeled contribution to the total vertical TEC from the plasmasphere (Figure 6h) was computed to be 3.3 TECU. The maximum slant TEC in the plasmasphere (Figure 6e) was 5.8 TECU. It is interesting to note that this receiver, which is the most equatorward of the three considered, measured the largest vertical plasmaspheric TEC but the smallest maximum slant plasmaspheric TEC. This is consistent with the unscaled output from the Carpenter-Anderson model (shown in Figure 1b). When the algorithm is executed without the plasmasphere term (Figure 7), the meridional TEC gradient is again evident in the vertical equivalent TEC (Figure 7b). Without the plasmasphere term, the predicted value for the receiver bias was 3.6 TECU higher than when it was included. This suggests that much of the plasmaspheric content was attributed to the receiver bias when the plasmasphere was neglected, and as a result the modeled vertical total TEC along the signal path was approximately 3.5 TECU smaller (compare Figures 6h and 7d). Once again, the postfit residuals are slightly larger when the plasmasphere term is excluded.

[19] In summary, these numerical results show that excluding the plasmasphere term results in higher estimates for the total TEC at the most poleward station and lower estimates of the total TEC at the most equatorward station. The results of Anghel et al. [2009] also suggest that neglecting the plasmaspheric contribution results in underestimation of the total TEC at equatorial latitudes and overestimation of the total TEC at middle latitudes. A. J. Mazzella et al. (GPS determinations of plasmasphere TEC, paper presented at the International Beacon Satellite Symposium, National Science Foundation, Boston, Massachusetts, 2007) and Lunt et al. [1999] also showed that by not accounting explicitly for the plasmasphere, the total TEC tends to be underestimated at equatorial latitudes and overestimated at middle latitudes in comparison with Sheffield University plasmasphere-ionosphere model simulations. Our numerical results are consistent with these studies.

5. Preliminary Validation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Kalman Filter Formulation
  5. 3. Carpenter-Anderson Plasmasphere Model
  6. 4. Numerical Results
  7. 5. Preliminary Validation
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[20] In this section, we present evidence that WinTEC-P can effectively separate the contributions to the total GPS-estimated TEC arising from the ionosphere and plasmasphere. A thorough validation is difficult to achieve for several reasons. First, the plasmaspheric contribution is frequently much smaller than the ionospheric contribution. Depending on how highly the electron density above the GPS receiver is structured, the plasmaspheric contribution may be small relative to errors associated with modeling the ionosphere. Therefore, care should be exercised when interpreting the estimated plasmaspheric contribution, if the postfit residuals are large. Second, the instrument providing truth data needs to observe along nearly the same line of sight, and at nearly the same time, as the line of sight between the GPS receiver and one or more of the GPS satellites. Third, the instrument providing truth data needs to have its own instrumental biases very carefully controlled. One instrument that meets these last two criteria is the dual-frequency (VHF/UHF) Advanced Research Projects Agency Long-range Tracking and Instrumentation Radar (ALTAIR) located at Kwajalein Atoll. We use data collected during ALTAIR tracking scans of passive calibration objects to obtain the TEC along slant lines of sight from the radar to each object. This TEC is computed by differencing the range measurements made at the UHF and VHF frequencies, as described by Caton et al. [2009]. When applying this technique to ALTAIR tracking passes of calibration objects in low Earth orbit, the slant TEC measured largely excludes the contribution from the plasmasphere. Since the range measurements from ALTAIR are very carefully and routinely calibrated, these ionospheric TEC measurements are essentially unbiased.

[21] Air Force Research Laboratory also operates an Ashtech Z-12 GPS receiver on the island of Roi-Namur (9.40°N, 167.47°E) in close proximity to the ALTAIR radar. Using ALTAIR tracking data from campaigns conducted as part of the joint U.S.-UK Wideband Ionospheric Distortion Experiment [Caton et al., 2009; Cannon et al., 2006], we identified a number of angular conjunctions between the calibration objects and the GPS satellites as viewed from Roi-Namur. Our criterion for identifying the conjunctions is that the angular separation, θ, be 5° or less between the lines of sight to the calibration object and the GPS satellite. Since the calibration objects considered are in LEO orbit, they pass over ALTAIR much more quickly than the more distant GPS satellites. In an effort to increase the number of conjunctions available, we do not insist that the conjunctions occur at precisely the same instant of time for both the calibration object and the GPS satellite. Instead, we accept the conjunction if at time TG the LOS to the GPS satellite passes within 5° of the LOS to the calibration object's location at any other time TC such that ∣TGTC∣ ≤ 5 min. During this 5 min period we assume the ionosphere to remain unchanged. Imposing these criteria resulted in six angular conjunctions being identified. The lines of sight from the GPS receiver to the GPS satellites and from ALTAIR to the calibration objects at the closest angular approach are summarized in Table 2. The GPS satellites listed in Table 2 are identified by their pseudorandom noise identification codes. The dates and times listed in Table 2 indicate the times, TG, when the LOS to the GPS satellite made its closest approach, and ΔT = TGTC indicates the number of minutes before or after this time when the LOS to the calibration object made its closest approach. The two calibration objects involved were RIGIDSPHERE-2 (LCS-4, object ID 05398), a smooth calibration sphere of 112.9 cm diameter orbiting with perigee and apogee of 743 and 834 km at 86.7° inclination, and OPS-5712 (P/L 160, object ID 02826), a smooth calibration sphere of 50.8 cm diameter orbiting with perigee and apogee of 765 and 779 km at 69.9° inclination.

Table 2. Angular Conjunctions Between GPS Satellite and Calibration Object Lines of Sight From Roi-Namura
Date/TimeGPSALTAIRDifferences
PRNɛ(deg)α(deg)equation imageKpmaxTECWTECITECPNorad IDɛ(deg)α(deg)TECRΔT(min)θ(deg)TECR−TECWTECR−TECI
  • a

    ALTAIR, Advanced Research Projects Agency Long-range Tracking and Instrumentation Radar. PRN, pseudorandom noise. UT, universal time. Simulation results listed include the TEC estimates from WinTEC (TECW), the ionospheric TEC estimates from WinTEC-P (TECI), the plasmaspheric TEC estimates from WinTEC-P (TECP), and the ionospheric TEC estimates from ALTAIR (TECR). The geophysical parameters used as input to WinTEC-P are also given, and units of the TEC estimates and the RMS errors are TECU.

4 May 2006/1127:25 UT0674.4184.017.52.75.86.76.50539874.3189.19.4−4.21.43.62.8
28 May 2006/0926:56 UT0661.9187.017.52.09.810.86.20539861.8189.111.2−5.00.91.50.4
14 Sep 2006/1123:20 UT1135.0330.014.92.319.720.62.20282637.8332.422.34.33.32.61.7
15 Sep 2006/1046:22 UT1942.019.514.91.07.810.85.30282641.920.58.94.50.81.2−1.8
3 May 2008/1007:22 UT2121.0174.83.53.08.611.43.00539821.7179.910.7−4.24.82.1−0.7
3 May 2008/1007:22 UT3035.0359.03.53.06.39.13.00539834.84.09.44.74.13.20.4
RMS Error2.51.6

[22] Once the times of conjunction were identified, we used WinTEC-P with and without the plasmasphere term to estimate the TEC from the GPS measurements collected at Roi-Namur. In each case, data from the 24 h period preceding the conjunction were assimilated by the Kalman filter in order to allow transients to disappear. In section 4, we showed that the RMS error between estimates of the satellite biases provided by CODE and those computed using WinTEC and WinTEC-P were less than 1 TECU. We therefore see no compelling reason to solve for the satellite biases when estimates of the satellite biases are available from CODE. The results presented in this section have been computed using estimates of the satellite biases from CODE. We note that since the Ashtech Z-12 receiver measures the pseudoranges using the precise code (P) on both the L1 and L2 signals (via the Z-tracking technique), only the P1P2 biases from CODE are required.

[23] The WinTEC-P algorithm requires two geophysical parameters as input since they are needed by the Carpenter-Anderson model. These are the 13 month average sunspot number and the maximum value of Kp attained over the previous 24 h. The values for these parameters used in our simulations are also listed in Table 2. Since there are several different estimates of TEC involved in this validation study, for clarity, we assign them the following symbols. We refer to estimates of slant TEC made while excluding the plasmasphere term as TECW. We refer to estimates of the ionospheric and plasmaspheric contributions to the total slant TEC made while including the plasmasphere term as TECI and TECP, respectively. Finally, we refer to estimates of the slant TEC made by differencing ALTAIR two-frequency range measurements to calibration objects as TECR. Since the calibration objects considered are in LEO orbits, during angular conjunctions with a GPS LOS, TECR represents an unbiased measure of the ionospheric contribution to the total TEC along the slant path to the GPS satellite. We consider our validation successful if two conditions are met. The first condition is that the ionospheric slant TEC estimate made while accounting for the plasmasphere (TECI) should be closer to the ionospheric slant TEC estimate made by ALTAIR (TECR) than the slant TEC estimate made while neglecting the plasmasphere (TECW). As shown in Table 2, this was indeed the case for five out of the six cases considered. The RMS error between TECW and TECR for the six cases combined was 2.5 TECU, while the RMS error between TECI and TECR was 1.6 TECU. In other words, the error in estimating the ionospheric contribution to the total TEC was reduced by 36% when accounting for the plasmasphere. We note that in the cases considered, both TECW and TECI tended to systematically underestimate TECR, but TECI did so less than TECW. The second condition we expect to be satisfied by this validation is that the total GPS TEC estimated while accounting for the plasmasphere should be greater than the estimate of the ionospheric TEC made by ALTAIR (in other words, TECI + TECP > TECR should be satisfied). As shown in Table 2, this condition was satisfied by all of the six cases examined. We conclude that when the plasmasphere term is included, the estimated ionospheric TEC is in better agreement with the ionospheric TEC measured by ALTAIR. When the plasmasphere term is neglected, the contribution from the plasmasphere is partially transferred to the estimated receiver bias instead, and as a result the unbiased estimates of ionospheric TEC are too low compared to the ALTAIR measurements.

[24] We note that these results constitute only a preliminary validation of the technique. First, because of the limited number of ALTAIR tracking passes available, only six angular conjunctions were identified with the LOS to a GPS satellite. A more thorough validation is clearly needed using many more cases. Second, although we have treated the ALTAIR estimates of TEC as truth data, they are not perfect. Caton et al. [2009] estimate that range gate discretization may cause errors in the range-derived TEC from ALTAIR of up to 1.5 TECU. Third, the equatorial location of ALTAIR is perhaps not the ideal location to carry out this validation. From Figure 1 it can be observed that at equatorial latitudes the variation of plasmaspheric TEC as a function of elevation more closely resembles that of the ionospheric slant TEC, according to the thin slab approximation. Therefore, we might anticipate more difficulty separating the plasmaspheric and ionospheric contributions to the TEC at equatorial latitudes than at middle latitudes, where the variations of these contributions with elevation are more distinct.

6. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Kalman Filter Formulation
  5. 3. Carpenter-Anderson Plasmasphere Model
  6. 4. Numerical Results
  7. 5. Preliminary Validation
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[25] The estimation of receiver and satellite instrumental biases is a leading source of error when measuring TEC using the GPS. Most techniques for estimating these biases from ground-based GPS measurements employ a thin slab approximation for the ionosphere that is inappropriate for the plasmasphere, which the GPS signals also traverse. These techniques attribute a portion of the group delay imparted by free electrons in the plasmasphere to the ionosphere and instrumental biases instead. As a result, the estimated instrumental biases are implicitly a function of the plasmaspheric density.

[26] In this paper, we have extended the WinTEC Kalman filter technique for estimating TEC from ground-based GPS measurements by adding a term representing the plasmasphere to the observation equation. The extended technique provides unbiased estimates for the ionospheric and plasmaspheric contributions to the GPS TEC, as well as the satellite and receiver instrumental biases. While we used the Carpenter-Anderson plasmasphere model in this study, the approach is flexible, and other models may be used instead. We have compared the performance of WinTEC with and without the plasmaspheric term for three GPS receivers located at different latitudes in the American sector. We find that when the plasmaspheric contribution is neglected, a portion of this contribution is transferred to the estimated biases and the estimated ionospheric TEC instead. The apportionment of this transfer between the estimated biases and the estimated ionospheric TEC is determined by the structure of the plasmasphere (and hence the geomagnetic latitude of the observing station) and the specific plasmasphere model chosen. As a result, when the plasmaspheric term is neglected, the total TEC along the path from the receiver to the GPS satellites tends to be underestimated at low latitudes and overestimated at middle latitudes. This observation is in agreement with the results of other studies [Anghel et al., 2009; Lunt et al., 1999; Mazzella et al., presented paper, 2007].

[27] We performed a preliminary validation of WinTEC-P by comparing its estimates of the ionospheric contribution to the total GPS TEC with ALTAIR measurements of TEC along approximately coincident lines of sight to calibration objects in LEO orbit. The error in estimating the ionospheric contribution to the total TEC was reduced by 36% when accounting for (and removing) the plasmaspheric contribution. In the cases considered, we found the technique to be effective at separating the contributions to the TEC from the ionosphere and plasmasphere by exploiting differences in the spatial distributions of electron density, and the time scales on which they vary, in these two regions.

[28] In general, we expect the accuracy of modeling the plasmaspheric contribution to the GPS TEC to be limited by an incomplete knowledge of the state of the plasmasphere and its dynamic coupling to the ionosphere and magnetosphere. The results described here, however, suggest the plasmaspheric contribution to the TEC can be accounted for in a systematic fashion (consistent with the plasmasphere model chosen) when estimating the instrumental biases and ionospheric TEC. In addition to the cases presented here, we have successfully tested the algorithm using GPS measurements collected at equatorial latitudes during solar maximum and also during storm time conditions. As may be expected, the postfit residuals are larger in these cases. Examples of these simulations (using the Gallagher et al. [1988] plasmasphere model instead of the Carpenter-Anderson model) are given by Anghel et al. [2009]. In addition to the usefulness of the WinTEC-P approach for remote sensing the plasmasphere, we anticipate that its estimates of the instrumental biases may exhibit fewer spurious temporal fluctuations (since they should be largely decoupled from conditions in the plasmasphere). Our initial validation with ALTAIR data suggests that estimates of ionospheric TEC made while accounting for the plasmasphere are more accurate than those obtained by implicitly assuming all the measured TEC is ionospheric in origin. A more extensive validation should be performed, using a larger data set of unbiased ionospheric TEC measurements, to further quantify the improvement in accuracy obtained.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Kalman Filter Formulation
  5. 3. Carpenter-Anderson Plasmasphere Model
  6. 4. Numerical Results
  7. 5. Preliminary Validation
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[29] This work was supported by AFRL contracts FA8718-06-C-0022 and FA8718-08-C-0012. The authors would like to thank Andrew Mazzella and the anonymous reviewers, whose helpful comments have improved this paper.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Kalman Filter Formulation
  5. 3. Carpenter-Anderson Plasmasphere Model
  6. 4. Numerical Results
  7. 5. Preliminary Validation
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Kalman Filter Formulation
  5. 3. Carpenter-Anderson Plasmasphere Model
  6. 4. Numerical Results
  7. 5. Preliminary Validation
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information
FilenameFormatSizeDescription
rds5656-sup-0001-t01.txtplain text document0KTab-delimited Table 1.
rds5656-sup-0002-t02.txtplain text document1KTab-delimited Table 2.

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