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[1] The plasmasphere (also denoted as the protonosphere) is a large toroidal domain of light ionized particles situated above the ionosphere and confined by the Earth's magnetic field. While plasmaspheric charge densities are considerably less than those of the ionosphere, the large extent of the plasmasphere can produce significant charge column densities, or total electron content (TEC), for lines-of-sight passing through the plasmasphere. A method for Self-Calibration of Range Errors (SCORE) has been developed previously both to determine combined bias calibration values for GPS receivers and satellites and to calculate absolute TEC values for the ionosphere. An enhanced SCORE process, described here, retains the “self-calibration” feature of the original SCORE process, by not requiring any measurements beyond those performed by the GPS receiver system being calibrated. The enhanced SCORE process also determines a characteristic plasmasphere amplitude parameter, thus providing an autonomous determination of both the ionospheric and plasmaspheric TEC. Case studies for a near-equatorial site are presented, with model parameters derived from 1998 data being applied to determine ionospheric and plasmaspheric TEC for measurements made in 1999.

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[2] The plasmasphere is a large toroidal domain of light ionized particles situated above the ionosphere and confined by the Earth's magnetic field. Its principal constituent ion species is hydrogen, so that it also is denoted as the protonosphere. While plasmaspheric charge densities are considerably less than those of the ionosphere, the large extent of the plasmasphere can produce significant charge column densities, or total electron content (TEC), for lines-of-sight passing through the plasmasphere. The high altitudes of the Global Positioning System (GPS) satellites guarantee that lines-of-sight for equatorial GPS users will pass through the plasmasphere, while equatorward lines-of-sight for midlatitude GPS users also will be subject to a varying degree of plasmaspheric effects.

[3] A method for Self-Calibration of Range Errors (SCORE) has been developed previously both to determine combined bias calibration values for GPS receivers and satellites and to calculate absolute TEC values for the ionosphere [Bishop et al., 1994]. The assumptions underlying this method are applicable primarily at middle latitudes, but the method has been applied in polar and equatorial regions with some success. Use of SCORE with ionospheric model simulations for a midlatitude site indicated that the plasmasphere can have a measurable effect on the bias values and ionospheric TEC determinations, but the latitudinal asymmetry of the plasmasphere, as seen from midlatitudes, can be used advantageously to exclude the plasmaspheric effects and obtain highly accurate bias and TEC results [Lunt et al., 1999a]. This study also indicated that SCORE would be a valid bias calibration method over a larger geographic domain if the plasmasphere contribution could be removed from TEC measurements along the lines-of-sight to the GPS satellites (“slant TEC”) before the standard SCORE algorithm is applied. This latter technique has been investigated and is described here.

2. Extension of SCORE

[4] The original SCORE algorithm implemented a minimization condition for differences of associated measurements of the equivalent vertical TEC performed for different satellites at different times, but with common latitudes and local times for the ionospheric penetration points (IPP) associated with the individual measurements. In mathematical form, the quantity to be minimized is

where

M_{αi} (M_{βj})

is the measured slant TEC, for GPS satellite α (β) and data sample index i (j) (including biases),

θ

is the IPP latitude for the measurement,

τ

is the IPP local time for the measurement,

B_{α}

is the satellite bias,

B_{R}

is the receiver bias.

The function f(ε), whose argument is the elevation angle for the line-of sight, is a simple geometrical conversion factor from slant TEC to equivalent vertical TEC, based on the zenith angle of the line-of-sight at the ionospheric penetration point.

In this formula, R_{e} is the radius of the Earth (6371 km) and H_{i} is the altitude of the assumed ionospheric shell (350 km).

[5] The quantity W_{αiβj} is a weight function dependent on the parameters for the individual measurements.

where Δθ and Δτ define (empirically determined) scale lengths for latitude and local time, respectively, in comparing measurements that are not precisely co-located in the chosen ionospheric coordinates, T_{αi} and T_{βj} are the Universal Times for the measurements (accounting for day transitions), ΔT is the associated scaling value, and the exponent η defines the degree of elevation weighting, allowing a trade-off between increased sky coverage and decreased accuracy of the slant TEC conversion factor, f(ε), at lower elevations. A strict elevation threshold (normally 35 degrees, but reduced to 30 degrees for the case studies described here) also can be specified.

[6] The SCORE method was extended for the plasmasphere by including an additional correction term for the conversion of slant TEC to equivalent vertical TEC.

[7] The quantity A_{P}P_{αi} represents the slant TEC contribution of the plasmasphere, which is represented by a geometrical functional model (P_{αi}) with fixed parameters and an amplitude (A_{P}) that is determined from the measurements together with the unknown bias values. Implicit in this formulation is the assumption that the plasmaspheric TEC contribution is nearly constant on a diurnal basis, but this assumption appears to be justified by a recent study [Lunt et al., 1999b]. An alternative assumption is discussed in the section describing future developments.

[8] The weight function also was modified by including additional factors that were dependent on the plasmasphere contribution for each of the lines-of-sight, employing an additional scaling value P_{SCL} (equal to 2.0).

The use of the plasmasphere contributions within the weighting function is especially advantageous for midlatitude applications, because it provides a simple, effective means for implementing the latitudinal plasmasphere exclusion employed by Lunt et al. [1999a].

[9] The geometrical model of the plasmasphere that forms the basis for the derivation of the functional form P_{αi} is a simple cylindrical torus whose axis is determined by the local configuration of the Earth's magnetic field. This is accomplished by calculating an effective magnetic dipole axis that produces magnetic coordinates close to those of the Corrected Geomagnetic Coordinate system (IGRF80) [Knecht and Shuman, 1985]. The two radii defining the torus initially were determined to confine the plasmasphere to (geomagnetic) latitudes equatorward of 46.5 degrees, but these values were revised by subsequent studies to be 10000 km for the toroidal radius about its axis and 5175 km for the toroidal radius about its circular core, thus confining the plasmasphere to within about 30 degrees of the geomagnetic equator, as indicated in Figure 1. Note that this plasmasphere model will still produce a TEC contribution for equatorward lines-of-sight at locations poleward of these latitudes, because of its extent in altitude. A constant lower-altitude limit of 956 km is imposed on the extent of the plasmasphere model, to maintain its distinction from the ionosphere. The outer boundary corresponds approximately to an L-shell of 2.4, for which the plasmaspheric density has decreased to about 16% of its peak value, according to the Gallagher model. This truncation is somewhat compensated by the form of the plasmaspheric slant TEC function, and by an increase in the derived quantity A_{P} over what would be strictly derived from a Chapman or Gallagher model.

[10] The purely geometrical intersections of lines-of-sight with the toroidal model determine locations, path lengths, and intersection angles that are used in the functional form for the slant TEC contribution. The factors and parameters of this functional form were derived from consideration of a Chapman model of the plasmasphere, initially determined to match the Gallagher plasmasphere model [Gallagher et al., 1988]. The geometrical factor is

where

λ

is the geomagnetic latitude of the lower plasmaspheric penetration point (PPP) for the line-of-sight,

Δλ

is the latitude scale factor for the plasmaspheric density,

h

is the altitude of the lower PPP for the line-of-sight,

H

is the scale height for the plasmaspheric density,

δs

is the path length of the line-of-sight through the torus,

ζ

is the zenith angle for the line-of-sight at the lower PPP.

The function η(λ) defines the latitudinal variation of the scale height, and is given by

where δλ is a latitude scale, which can be different from Δλ. This factor arose as a consequence of the use of altitude and latitude variables here, in contrast to the L-shell variable used for the Gallagher model.

[11] Because the plasmasphere amplitude enters into the minimization formula in the same manner as the combined satellite and receiver biases, the same solution techniques used in the earlier SCORE method can be employed. To expedite the calculations, all of the geometrical quantities are computed first for each of the individual data samples, and the minimization process is performed subsequently.

3. Case Studies

[12] The initial equatorial study for the extended SCORE method utilized GPS TEC data collected at Ascension Island (7°S, 14°W) in March 1998 using a small Real-Time Monitor (RTM) system deployed by the Air Force Research Laboratory (AFRL). A sequence of geometrical model parameters was investigated, with the objective of reducing the total residual error in the equation defining the SCORE algorithm. These geometrical parameters appear (implicitly) in a nonlinear context in the minimization expression, and, consequently, are not determined by the minimization process. The results of this initial study are displayed in Table 1, and the derived ionospheric TEC pattern for one day (31 March 1998) is displayed in Figure 2.

Table 1. Parameters for the Plasmaspheric Slant TEC Model^{a}

Description

Value

a

Note: The infinite scale factor values were implemented by setting the corresponding latitude (numerator) values to zero.

Toroidal radius about its axis

10000 km

Toroidal radius about its circular core

5175 km

Lower altitude limit

955.65 km

Latitude scale factor for the plasmaspheric density (Δλ) (see Note)

∞

Scale height for the plasmaspheric density (H)

4500 km

Latitude scale factor for scale height (δλ) (see Note)

∞

Latitude of fitted magnetic north pole

68.035 deg

Longitude of fitted magnetic north pole (positive east)

233.853 deg

[13] The derived geometrical parameters were applied for an extended campaign at Ascension Island during the spring of 1999, in which an Air Force Ionospheric Measuring System was deployed by AFRL with other instruments. Results for the derived ionospheric diurnal patterns are displayed in Figure 3 and Figure 4 for two different days. In all cases, the upper plot displays the plasmasphere contribution to the slant TEC, while the lower plot displays the apparent paths of the ionospheric penetration points for each satellite, projected onto geomagnetic local time and latitude. Although the plasmaspheric slant TEC variation in geomagnetic coordinates is difficult to discern in this format, the variation of the plasmasphere TEC amplitude for the three days of Ascension Island data is evident.

[14] A better portrayal of the plasmaspheric slant TEC variation arises from the use of the plasmasphere slant TEC model, together with the derived plasmasphere amplitude, to determine the plasmaspheric slant TEC for a meridian sky track. This is displayed in Figure 5 for the 31 March 1998 case. For comparison, the slant TEC variation derived from the Gallagher model also is displayed, scaled to the same overhead slant TEC. The plasmaspheric boundary appears much more abrupt for the derived model and less variable equatorward (north) of the site.

4. Future Developments

[15] Further generalization of the plasmasphere toroidal model is planned, to include temporal variations on the timescale of about a day. Preliminary investigations of this effect for the data of 31 March 1998 indicate a significant improvement in the ionospheric vertical TEC diurnal pattern if plasmasphere variations are allowed. Further refinement of the plasmasphere TEC model already is in progress. The formula for the plasmaspheric slant TEC will be compared to slant TEC variations derived from the Sheffield University Plasmasphere Ionosphere Model (SUPIM) [Bailey and Balan, 1996], both to enhance the functional form and to define a more appropriate scope for the initial parametric values.

[16] Additional TEC data, subsequent to the spring 1999 campaign at Ascension Island, are now available from the Ionospheric Measuring System deployed there, for further performance evaluations and validation studies. Comparisons of the plasmasphere and ionosphere patterns derived for sites at various latitudes are planned, and comparisons of the derived ionospheric TEC to TOPEX measurements also are being considered.

5. Conclusions

[17] A preliminary technique for calibrating GPS receivers and measuring separate TEC contributions for the ionosphere and plasmasphere has been developed, and produces reasonable results for both components in certain conditions. Some such technique is necessary to achieve precise calibrations for obtaining accurate TEC data for equatorial regions that are affected by the plasmasphere. With further development, it may be possible for the technique described here to approach the accuracy of the SCORE method in midlatitude regions.

Acknowledgments

[18] One of the authors (A.J.M.) is grateful to Nick Lunt for performing the ionospheric model validations for SCORE that inspired this extension of the method. This research was supported by AFRL contract F19628-97-C-0078.