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 An experimental method, using simultaneous vertical and slant observations, has been derived for estimating the effective shell height for electron content measurement by GPS reception. At a latitude of approximately 53° north, Lancashire is well placed as an observing site since GPS satellite orbits are inclined at 55° to the Earth's equatorial plane and, as a result, many GPS tracks pass almost directly overhead, giving true zenithal measurements. This paper focuses on the question of oblique-to-zenithal correction and related matters. In particular, plasmaspheric effective height and satellite bias corrections are determined by measuring the total electron content (TEC) from pairs of satellites to a single ground station, each pair giving simultaneous observations of oblique and zenithal TEC. Additional bias corrections are determined by extracting the TEC using pairs of satellites with the same elevation at the same time. The Chapman Production Function Model and the Sheffield University Plasmasphere and Ionosphere Model are both used to determine a theoretical value for the plasmaspheric effective height. The results indicate that the plasmaspheric effective height used in the oblique-to-zenithal thin shell conversion is considerably greater than the commonly adopted value of 350 km. It is suggested, on the basis of all the available evidence, that a value between 600 and 1200 km is preferred. Assuming a lower value could produce an error of 15 to 30% or more in the electron content.
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 The Global Positioning System [Hoffmann-Wellenhof et al., 1997] is a constellation of satellites intended to provide precise positional information at points on and above the Earth's surface. At the time of the present study there were 28 satellites in 12-hour orbits (four in each of six orbital planes, plus four spares), the orbits being inclined at 55° to the equator. The orbits are circular at altitude 20,200 km. Between four and eight satellites are visible above 15° elevation at any time, making it possible to obtain a position fix from any location on the Earth's surface. The satellites carry precision time and frequency standards and transmit digitally modulated L band signals (L1 and L2) at 1227.60 and 1575.42 MHz. Position fixing makes use of the propagation time delays, on which the presence of the plasmasphere also has a small effect that is slightly different on the two frequencies.
 Because of the differential time delay on L1 and L2, transmissions from the GPS satellites may be applied to measuring the electron content (the integrated electron density along the propagation path) through the ionosphere and protonosphere between the satellite and a receiver at the ground. Considerable effort has been put into this method, but in practice the utility of the technique is greatly handicapped because no one measurement is absolute. Each satellite of the constellation imposes an individual phase shift between the signals that are to be compared at the ground, and this represents a “bias” or “offset” which needs to be determined as well as the electron content. A number of satellites may be observed at any one time, which is a potential advantage, but these are all in different parts of the sky, and the electron content varies both spatially and temporally; further, no two satellites impose the same bias (unless by chance). However the problem is considered, one ends up with more variables than observations!
 To solve the problem, some assumptions have to be made. First, the ionosphere and protonosphere (which we shall collectively call the plasmasphere) may be assumed to be spatially uniform. Then a solution may be obtained by comparing observations made at different angles to the zenith and using the fact that the bias is fixed for a given satellite whereas the electron content depends on the zenith angle at which the ray traverses the plasmasphere.
 In Figure 1 the plasmasphere is simplified to a thin layer at height hs above the Earth's surface. This is the “shell” model, and its height is the “effective height” of the plasmasphere. We now suppose that we have two readings of electron content: Rz in the zenith and Rs at zenith angle χ from a single satellite having bias B, the true zenithal content being I. Then by definition,
and by simple geometry,
where β is the zenith angle at which the oblique ray crosses the shell. Thus
from which I may be determined if β is known. If the shell height is hs, β is related to the zenith angle at the ground χ by
REbeing the radius of the Earth.
 This is, of course, a simplified approach, but the principle that it illustrates appears to be embodied within most, if not all, of the reduction methods in use, since the effective height of the shell, hs, is usually quoted.
 GPS signals were received at Lancaster, England (53°52′N, 2°45′W), from 21 October 1993 to 28 February 1994, though the data were of poor quality after the end of January. An Osborne Mini-Rogue receiver type ICS-4Z was used. The antenna was a covered crossed dipole with choke ring. Up to four satellites could be received at the same time, and the logging was set to 2-min resolution. The “pseudorange” was derived from comparison of the modulations on the L1 and L2 signals, and the phase difference between the L1 and L2 carriers was measured. Each of these is proportional to the electron content [Hoffmann-Wellenhof et al., 1997]. That derived from the pseudorange is absolute but noisy. That from the phase difference is less noisy but is not absolute, lacking a baseline. The data streams were combined by setting the average electron content from the phase over a pass to the average from the pseudorange. The procedure is described by Hargreaves and Burns , who also presented results from a period when the receiver was operated at a site in the auroral zone. In the present study, the initial data processing includes the bias values given by Gaposchkin and Coster , as used by Hargreaves and Burns. Satellites not included in the Gaposchkin and Coster list were given a bias of zero initially.
3. Comparison of Slant and Zenithal Observations
3.1. Height Determination: Method
 Our approach is to determine the effective height by comparing the readings from a pair of satellites observed simultaneously along slant and zenithal paths. We can achieve zenithal readings because the geographic latitude at Lancaster is close to the orbital inclination of the GPS satellites. This means that many overhead passes occur at that site.
 The geometry is now like that of Figure 1but involves two satellites, each with its own bias. If the biases for the zenithal and slant satellites are Bs and Bz,
Plotting Rsagainst Rzshould give a straight line of form y = mx+ c, with gradient m= sec β and intercept c = (Bs − Bz sec β).
giving a value for hs, the zenith angle at the ground (or the elevation E = 90° − χ) being known. It is not neccessary to know the biases.
 In this work the electron content and the bias may be expressed either as a time delay (in nanoseconds) or in total electron content units (1 TECU = 1016 m−2), the conversion being 1 TECU ≡ 0.3505 ns difference of delay time between the L1 and L2 frequencies. We shall work in units of electron content.
 From (8) the effective height of the plasmasphere (equal to ionosphere plus protonosphere) can be derived from the gradient of the regression line. This assumes that the elevation of the oblique satellite is constant for all observations, which was not strictly true. The orbital period of each satellite is slightly less than 12 hours, which causes the time of zenith passage and the elevation of the slant satellite to vary over a two-month period of observation. However, since satellite elevations varied by no more than 0.5°, errors from this cause are negligible. For example, in the comparison between satellites 14 (slant) and 15 (zenithal) the elevation of the former varied from 37.1° to 37.6°, and the time of the zenithal pass of the latter varied from 1030 to 0800 UT, during October and November 1993 (the period of observations which was adopted for this study). The method depends on the day-to-day variation of electron content and that with the time of day.
3.2. Height Determination: Results
 The observations were made between 21 October 1993 and 28 January 1994, and at that time the satellites of the GPS constellation were identified by numbers between 1 and 31. By inspection of the orbital data the following pairs were identified, in which the second measured the electron content over a slant path while the first observed within 5° of the zenith: 3/17, 3/22, 5/9, 5/20, 9/5, 9/12, 9/26, 14/15, 14/29, 15/14, 15/31, 20/5, 20/9, 20/12, 28/22, and 28/31.
 Examples of the plots of Rs against Rz are shown in Figure 2. Each of these covers the period October and November 1993. These plots are essentially linear, and the scatter is relatively small. The best estimates of the gradients are given in the caption, with their uncertainties as derived by standard linear regression analysis. Table 2 gives the results for the whole data set, including correlation coefficient, gradient derived by linear regression with its error, and effective height hs. The value of the gradient expected for a height of 400 km is included for comparison. In some cases the correlation coefficient is high, and the uncertainly in “m” is correspondingly small.
Table 2. Height Estimates From Slant/Zenithal Plotsa
Zenith Satellite PRN
Slant Satellite PRN
Mean UT, hours
Mean Azimuth, deg
Mean Elevation, deg
Number of Points
m if hs = 400
PRN, satellite identification number.
 The derived height is sensitive to the value of m, and the sensitivity is greatest at high elevation. The estimates of effective shell height vary considerably, with a mean of 1330 km and a standard deviation of 3964 km (largely due to pair 28/31) for the full set of results. Some of these values are plainly unrealistic. That for pair 28/31 is far too large, and others are below the ionosphere.
Figure 3 shows the location of the slant satellite path in azimuth and zenith angle. We note that those observed to the north give extreme height estimates (a possible reason for which will be suggested later). Figure 4 plots the gradient magainst elevation, the passes to the north being omitted. Figure 4 includes the estimated uncertainty in m, and it is seen that when that is taken into account, all data are consistent with an effective height of 750 km. Obviously, this is not a precise figure, but it does appear that the results fit 750 km better than 400 km. Nine of the results form a group lying between 500 and 1100 km, and the mean of that group is 763 km. These results suggest that the effective shell height of the plasmasphere is greater than the value of 350–400 km that is usually assumed. The present analysis suggests that a value of 750 km would be more appropriate.
3.3. Bias Determination
 If both the satellites used in the slant/zenithal comparison pass overhead in turn, it is possible to determine the bias of each in the following manner. We now designate the satellites as 1 and 2 and their biases as B1and B2. Following (5) and (6), when satellite 1 is overhead,
Plotting R2s against R1z gives the straight line (from equation (7))
When satellite 2 is overhead, the same procedure gives
If both lines are now plotted as R2against R1, they intersect at the point (B1, B2), which is where I = 0 (see Figure 5). This gives a method for determining the biases from the data.
 The situation that both satellites of a slant/zenithal pair passed overhead arose only for satellite pairs 14/15 and 5/9. The biases so determined are given in the upper part of Table 3. Using these values, further bias values may be derived from the regression equations involving other satellites, since for any slant/zenithal pair, the point (BZ, BS) lies on the regression line (where I = 0). This leads to the values in the lower part of Table 3, the additional uncertainty arising from the uncertainty of the gradient.
Table 3. Biases Determined From Slant/Zenithal Plotsa
 An alternative method of comparing biases is to compare the readings from pairs of satellites at the same elevation, again assuming uniformity over the horizontal distance between the penetration points. By this means, the bias of one satellite can be determined if the bias of the other is known. Figure 6 illustrates the occurrence of common elevation during a group of days in November 1993. The comparison was restricted to elevations not less than 60°, and elevation differences are not more than 0.5°. The method was applied in steps starting with satellite 14, then again starting with satellite 15, and taking the average. The results are shown in Table 4. In this process the error accumulates as the number of steps increases. The range of the error due to this comparison is 0.01 to 1.02 TECU. This is small compared to the uncertainty already present in the “seed” biases (Table 3) and is therefore negligible in practice.
Table 4. Biases Determined by the Common Elevation Methoda
 To convert slant TEC readings to zenithal electron content using the shell assumption, two parameters are needed: the effective height of the shell and the bias for the satellite concerned. Let us again suppose that the plasmasphere is uniform and constant. Then if both the height and the bias are correct, the electron content derived during the satellite track will also be constant. However, if either the height or the bias are wrong, then the derived electron content will vary as the satellite moves across the sky.
 We may apply this principle to verify the self-consistency of the values. If the bias is assumed and we vary the shell height, the resulting curve of electron content against time changes shape. If no obliquity correction is applied, effectively putting the shell at infinity, the plot of reading against time is in the shape of a “valley.” As the height is reduced, the valley flattens and changes into a “hill.” The correct height produces a curve without hills or valleys.
 Sequences of tracks were identified in which the electron content was expected to be constant or to change linearly and where we expect continuity of content between different satellites. The track sequences were then plotted (for example, Figure 7) for a range of assumed shell heights (300, 500, 1000, 2000, 2500, 3000, 5000, and 10,000 km). Inspection of Figure 7 suggests a shell height in the region of 1000 to 2000 km. Applying this approach to a selection of 15 track sequences on different days in October and November 1993 gives effective heights ranging between 500 and 2250 km (mean 1350 km) for the period from noon to evening and a range of 1000 to 4000 km (mean 2344 km) from midnight to morning. This approach is admittedly subjective, but it is interesting to note that the effective height so estimated increases from day to night, both being substantially above 400 km.
 From (1) and (2), assuming that I is constant during a single overpass,
This expression gives the relation between sec β (which implies the shell height through (4)) and the satellite bias B, for slant and zenithal readings of Rs and Rz. Figure 8 illustrates such relationships if the slant observation is at elevation 65°.
5. Guidance From Plasmasphere Models
 An alternative approach to the question of shell height is to use a model of the plasmasphere, comprising ionospheric and protonospheric components. No model is likely to represent the plasmaspheric profile on any particular occasion, but it should indicate an average situation, and that would be of some help. It also avoids the ubiquitous spatial and temporal irregularity of the real plasmasphere. Davies  considered the shell height problem using ionospheric models drawn from the international reference ionosphere, which are based on observations at various geographical sites and from satellites. A small arbitrary content was added to represent the protonosphere. We shall use two sorts of model, the first comprising a heavy-ion ionosphere based on the Chapman profile plus an added protonosphere, and the second comprising a mathematical model which generates both the ionosphere and the protonosphere from first principles.
5.2. Chapman Model
 This model was constructed as follows. The bottomside ionosphere is according to the Chapman formula,
where z is the reduced height (h− hm0)/H, H being the scale height with a value of 70 km (the dominant species being atomic oxygen). The height of ionospheric maximum (hm0) is taken as 300 km, with electron density Nmax3 × 1011 m−3. The solar zenith angle χ is 60°. The topside ionosphere is given by the same formula but with the scale height trebled on account of increased thermospheric temperature and the importance of ion diffusion above the peak.
 The protonosphere has an exponential form
where the scale height is now determined by atomic hydrogen and has a value 16 times that used in the topside ionosphere. The term Nbp (the electron density at the base of the protonosphere, taken as 800, 900, or 1000 km) is set to the value of the electron density in the topside ionosphere at the same height, giving a continuous transition between ionosphere and protonosphere. Figure 9 illustrates the resulting plasmaspheric profiles.
 In calculating the electron content, the summation used an increment of 10 km in the bottomside ionosphere, 20 km in the topside ionosphere, and 100 km in the protonosphere. The electron content was evaluated vertically and over a range of elevation angles. The ratio of slant to zenithal content gives a value for β, the zenith angle of the ray at the equivalent shell, from which the height of the shell is obtained from (4). The results are given in Table 5, where P is the fraction of the total electron content in the protonosphere. We note that both P and h increase with elevation angle. Over elevations of 40° to 80° the fraction of ionization in the protonospheric part of the path for the three cases (hm0 = 800, 900, and 1000 km) is 17 to 20%, 30 to 34%, and 47 to 51% of the total, values which are consistent with previous estimates based on ATS 6 data [Hargreaves, 1998]. The corresponding effective height in the three cases lies between 711 and 829 km, 887 and 1143 km, and 1136 and 1411 km, somewhat greater than those found by the method of section 3 and significantly greater than the values of 350–400 km that are commonly used.
Table 5. Chapman Production Function Derivation of Plasmaspheric Effective Heighta
Electron content is in units of 1017 m−2. Is and Io+ are the slant electron content through the whole plasmasphere and the slant electron content through the heavy-ion ionosphere, respectively. Iz is the vertical electron content through the plasmasphere. P is the fraction of the total electron content in the protonosphere.
5.3. Mathematical Models
 We use the Sheffield University Plasmasphere-Ionosphere Model (SUPIM) described by Bailey and Sellek  and Bailey et al. . It is based upon time-dependent equations of continuity, momentum, and energy balance for the atomic ions O+, H+, He+, and electrons and of continuity and momentum balance for molecular ions N2+, O2+, and NO+. The effects of the neutral air wind and of E × B drift are included. The equations are solved along centered-dipole magnetic field lines with coincident geographic and magnetic axes, from a base altitude in one hemisphere to the same base altitude in the other (conjugate) hemisphere. The model outputs data for each ion and for electrons at specific points along field lines selected by the user, the output comprising concentrations, field-aligned fluxes, field-aligned velocities, and temperatures.
 For midlatitude modelling, the E × B drift may be neglected, but a pattern of neutral air thermospheric wind must be assumed since this exerts a significant force along the field lines. The model used was the revised global model of thermospheric winds by Hedin et al. . The model begins with zero ionosphere and is run for several days to achieve a steady state. Runs of 3 days were used in the present case. The computation of electron density distribution was carried out along eleven field lines having maximum altitudes in the equatorial plane of 6000, 7000, 8000, 9000, 10,000, 12,000, 14,000, 16,000, 20,000, 25,000, and 30,000 km. Figure 10 shows the resulting distribution of electron density at 0600 and 1500 UT for the autumn (day 264). Day 355, representing the winter, was also treated. Figure 10 also shows the latitude of Lancaster, the observer's zenith, and the lines representing zenith angle 25° to north and to south.
 To apply the distribution to electron content modeling, the positions of the sender in space and the receiver at the ground are specified, and the electron density is integrated along the propagation path, which is taken as a straight line. In the present case the electron content was obtained in various directions from the latitude of Lancaster, and the effective shell height was determined as before from the ratio of the oblique and zenithal values. Table 6 summarizes the results for directions to north and south for 1500 UT on day 264 (autumn equinox).
Table 6. Electron Content Over a Range of Zenith Angles in the N-S Planea
Electron Content (TECU)
Day 264 at 1500 hours. P is the fraction of the total electron content in the protonosphere. V, vertical.
 It is apparent from Figure 10 that a ray 25° to the south passes through considerably more protonospheric plasma than one 25° to the north. This is something to which Lunt et al. [1999d] have previously drawn attention. From a midlatitude site the north-south variation of ionospheric content is relatively small (except for the effect of obliquity), but because the protonosphere is largely controlled by the form of the geomagnetic field, that part of the total content varies significantly from north to south (Table 6). The model is therefore incompatible with the assumption of spatial uniformity required by the method used in section 3. Slant/zenithal comparisons made with the slant path to the north will give effective heights that are too large, while those made with the slant path to the south will be too small. If slant paths are taken to the east or to the west, the effect is reduced but not eliminated because those rays dip into the protonosphere to some extent, depending on the zenith angle.
 SUPIM is not entirely realistic because it includes only magnetic field lines corotating with the Earth. At high latitudes the tubes of force have such a large volume that they can never be filled with plasma coming up from the ionosphere by day and sinking back by night. However, this is not the whole story because at high latitudes the plasmasphere circulates in a complex pattern driven by the solar wind, providing a mechanism for transferring ionization from the dayside to the nightside directly across the pole [Sojka et al., 1993]. We may therefore doubt whether the sharp termination of the protonosphere predicted by the model is a sufficient description for observations at 53°N.
 It does, however, provide specimen values for ionospheric and protonospheric contents. Figures 11a and 11b show the daily variation of effective height using the SUPIM ionospheric content and a uniform protonosphere having (1) the zenithal value (0.16 units) and (2) the value at 65° elevation east or west (0.22 units). The plasmaspheric effective height for the equinox varies from 700 km during the day to 1200 km at night, while that for the winter solstice varies from 750 km during the day to 1600 km at night. Note that most of the daily variation is due to the heavy ion ionosphere; the protonospheric variation is relatively small (Figure 12). This is also apparent in Figure 10. The weak diurnal variation of protonospheric content is consistent with previous observations using the radio beacon experiment on the geosynchronous satellite ATS 6 [Davies, 1980]. The contribution from He+is negligible.
 Apart from numerical estimates, the mathematical model provides two useful results of a qualitative nature. First, the predicted sensitivity of slant/zenithal comparisons to nonuniformity of the protonosphere is a warning. It may also be the reason why the experimental method of section 3 produced greater consistency when satellite passes to the north were excluded from the results. By the same token, the effective heights derived using passes to the south may be too low. Second, the model illustrates that the protonosphere (because of its large volume) varies little between day and night, and therefore day-to-day variations are likely to be smaller in the protonosphere than in the ionosphere. The consequence would be that effective heights derived using the natural variations of electron content over zenithal and slant paths would be too low rather than too high.
6. Summary and Discussion
 In this paper we have considered the following question: If the plasmasphere (i.e., ionosphere plus protonosphere) is to be represented by a shell for the purpose of obtaining absolute values of electron content from GPS data, what is the best height to use? It is a question to which there seems to be no precise answer, but some useful conclusions may nevertheless be drawn.
 The calculations based on the Chapman model showed that even if the plasmasphere is uniform, the shell height depends on the elevation of the received ray (Table 5). The models indicated heights between 750 and 1200 km (according to the amount of protonosphere included) for elevation 50°, but with up to 100 km variation at other elevations.
 According to the mathematical model, and as pointed out by Lunt et al. [1999d], the protonospheric content varies strongly with latitude, whereas the latitudinal variation of the ionospheric content is relatively weak. Ray paths to the north of the zenith are likely to be relatively deficient in protonospheric content, whereas those to the south will have more protonosphere than would a zenithal ray path. Therefore, in comparisons of zenithal and slant content, as in the experimental method of section 3, we expect those involving paths to the north to give greater heights and those involving paths to the south to give lower ones. We eliminated northward paths. The results of section 3.2, which included only southward paths, may be too low for this reason.
 The results of the various determinations are summarized in Table 7. These vary of course. The models cannot be guaranteed to be accurate representations, though they are reasonable in the light of present knowledge. The experimental method using slant/zenithal comparisons cannot be precise because of spatial variations which reduce the correlation between the day-to-day readings, leading to uncertainty in the slope. We have tried to quantify the uncertainty.
Table 7. Summary of Shell Heights Determined by Four Methods
track curvature and continuity
track curvature and continuity
 However, one conclusion can be drawn with some confidence: The range 350–400 km is too low as an effective height, and it would be better placed in the range 750–1200 km. Although the electron density in the protonosphere is much less than that in the ionosphere, it is not negligible, and its contribution to the total content is entirely above the ionospheric maximum. Davies  calculated the shell height using ionospheric profiles from the international reference ionosphere and estimated the effect of adding an exponential protonosphere having an electron content of 3 × 1016m−2. The computations were made for 25 locations where electron content has been measured and/or ionosondes have been operated. A range of latitudes in both hemispheres was covered. Most longitudes were represented, though 16 were between 205° and 310°, showing a bias toward the American sector. The resulting shell heights varied with elevation of the slant ray and with day or night, season, sunspot number, and latitude of the station. The average shell height for the ionosphere was about 615 km, rising to about 860 km when the protonosphere was included. The results from the present study are broadly consistent with these.
Figure 13shows the effect on the derived electron content of assuming a height that is too low. For example, if a height of 400 km is assumed when the correct value is 1000 km, the value obtained for the zenithal electron content will be too low by 15% to 50% depending on the zenith angle. In general, the electron content will be too low; the error will be transferred to the bias, which will be too large. Previous considerations of the accuracy of electron content data derived from GPS reception estimate an accuracy of about 1–2 TECU [Wanninger and Sardon, 1993; Jakowski and Jungstand, 1994; Davies, 1999; Mannucci et al., 1998; Bishop et al., 1985, Sardon et al., 1994]. The present study would put the uncertainty due to error in the bias determination somewhat larger than this (typically, +3 to −8 TECU), in addition to a systematic error of some 15 to 30% or more due to the effect of using too low a shell height.
 Development of SUPIM (G. J. Bailey) was supported by PPARC grant PPA/G/O/1997/00691. We thank K. Davies for useful discussions and comments on the paper. We also thank the Centre for Astrophysics and Department of Computing of the University of Central Lancashire for support during the investigation. The paper was presented at the National Radio Science Meeting, Boulder, Colorado, January 2001.