Acta Geophysica
vol. 55, no. 4, pp. 509-523
DOI 10.2478/s11600-007-0029-z
Effects of ionospheric horizontal gradients
on differential GPS
Mardina ABDULLAH1, Hal J. STRANGEWAYS2 and David M.A. WALSH2
1
Department of Electrical, Electronic and Systems Engineering, Faculty of Engineering
Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia
e-mail: [email protected]
2
School of Electronic and Electrical Engineering, The University of Leeds, Leeds, U.K.
e-mail: [email protected]
Abstract
This paper outlines the effect of horizontal ionospheric gradients on transionospheric path propagation such as for the case of GPS signals. The total electron content (TEC) is a function of time of day, and is much influenced by solar
activity and also the receiving station location. To make the model applicable for
long baselines, for which the ionosphere is not generally well correlated between
receiving stations, the ionospheric gradients should be taken into account. In this
work the signal path is determined using a modified ray-tracing technique together
with a homing-in method. Results show that horizontal gradients can have a significant effect on GPS positioning for both single station positioning and differential
GPS. For differential GPS, the ionospheric delay can, however, be either increased
or decreased compared with the case of no gradient, depending on the gradient direction.
Key words: ionosphere, ionospheric gradients, GPS, ray-tracing, TEC.
1. INTRODUCTION
Transionospheric L-band ray paths can be deviated azimuthally by electron density
variations with both latitude and longitude known as horizontal gradients. These gradients may be classified as North-South (latitudinal) and East-West (longitudinal).
Both are generally associated with diurnal variations of the electron density. NorthSouth gradients are likely to dominate over East-West gradients around noon and
midnight, whereas East-West gradients are likely to dominate near dawn and dusk.
© 2007 Institute of Geophysics, Polish Academy of Sciences
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The seasonal and space weather also affect the gradient magnitude. Communication
and navigation systems relying on transionospheric propagation must be able to compensate for the effects of sharp changes (e.g., 90% reduction) in TEC (total electron
content) associated with the ionospheric trough and also storm-time disturbance effects at mid-latitude (Foster 2000) and high-latitude. Strong horizontal structures are
often present, particularly in the equatorial and polar regions (Rios et al. 1999). Large
gradients may have an impact on ambiguity resolution when processing GPS (Global
Positioning System) data for precise measurements because the difference in ionospheric delay can itself influence the ambiguity resolution. A real-time ionospheric
tomographic model was shown to be precise enough for the successful resolution of
the widelane ambiguities for distances up to 1000 km (Hernandez-Pajares et al. 2000).
An analytic ray-tracing method was developed by Cannon and Norman (1997) to
approximate the horizontal gradients along the ray path for HF sky wave communications. However, an important shortcoming of this method is that gradients cannot be
introduced perpendicular to the ray path. The possibility of using ray-tracing with a
homing-in method for transionospheric paths, such as a GPS signal based on this application to an HF sky wave propagation path, was also demonstrated by Strangeways
(2000). In this work the horizontal gradients were included by introducing a linear
gradient of the electron density and/or the height of electron density maximum, with
latitude and/or longitude. In one example, a realistic profile for the ionosphere was
formed by three Chapman layers. Modelling has also been done for a realistic model
of the latitudinal and longitudinal variation of the ionospheric electron density at all
heights which shows an excess delay of about 10 m at 5 degrees elevation for the case
of transequatorial propagation corresponding to a TEC of 70x1015 el. m−2 (Ioannides
and Strangeways 2002). Experimental work has been done to study these gradients
applying Leitinger’s method of differential Doppler measurement to signals from the
NNSS (Navy Navigational Satellite System) satellites (Rios et al. 1999). They estimated gradient values in TEC of around 1015 el. m−2 per degree of latitude for moderate solar activity conditions and a range of (4-100)x1015 el. m−2 was observed for high
solar activity during geomagnetic disturbances. A current generally assumed value of
spatial decorrelation for GPS paths through the ionosphere is about 2 mm/km, which
is equivalent to an ionosphere induced error of 2 cm for differential Doppler determination with a 10 km baseline reference station. However, a larger gradient of 55
mm/km which would lead to a 55 cm induced error for a 10 km baseline has also been
observed (Christie et al. 1999). Further, Wanninger recorded a 5 m ionosphere induced error over a 100 km baseline during the solar maximum period (Christie et al.
1999).
Travelling ionospheric disturbances (TIDs) or travelling wave packets (TWPs)
can also affect the precise GPS positioning. TIDs commonly occur in winter and autumn, and mostly during the daytime (Afraimovich et al. 2002). Their wavelengths are
about 50-500 km and they travel at speeds of 5 to 10 km per minute. They can be
categorised as large, medium and small scale TIDs with occurrence periods of about
30 minutes to several hours, 20 to 30 minutes and less than 20 minutes, respectively
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(Hargreaves 1992, Rieger and Leitinger 2002). The quasi-sinusoidal variation in electron density with distance that they introduce is a major limitation for extending real
time kinematics in GPS positioning beyond baselines of few tens of kilometres (Chen
et al. 2003, Hernandez-Pajares et al. 2006, Orus et al. 2003, Wanninger 2004). Hernandez-Pajares et al. (2006) showed that MSTIDs’ velocity was mostly equatorward
during daytime in local winter (~100-400 m/s) and westward during nighttime in local
summer (~50-200 m/s), related to the solar terminator and also modulated by the solar
cycle. They also suggested a technique to estimate TIDs in real-time that could be
used for precise navigation. Strangeways et al. (2003) reported that a large quasisinusoidal variation seen simultaneously for several GPS satellites using differential
GPS over a baseline of 40 km was most likely due to a MSTIDs since it exhibited the
expected quasi-sinusoidal variation and had a typical period of about 20 minutes.
Even though the measurements used in this paper were recorded during a geomagnetic
undisturbed day (Kp index < 5), MSTIDs could still be seen because they can in fact
be observed at any time during any season and any phase of solar cycle although their
occurrences do maximize at certain times and particularly during the solar maximum
(Warnant et al. 2002).
This paper presents the effect of horizontal ionospheric electron density gradients
on GPS signals using precise ray-tracing calculations. Total electron content (TEC) is
a function of time of day, and is much influenced by solar activity and the receiving
station location. To perform differential GPS for long baselines, e.g., > 100 km, over
which the ionosphere is not generally so well correlated, the ionospheric gradients
should be taken into account. Previously, the effect on differential GPS of the different
elevation angles for the same satellite at the reference and mobile receivers has been
considered by Abdullah et al. (2003). Gradients generally exist in the ionospheric
electron density, especially when ionospheric disturbances such as geomagnetic
storms occur (Bounsanto 1999, Coster et al. 2001, Foster 1998, Ho et al. 2002). Here,
in the following sections the effect of horizontal gradient on standalone reference and
mobile receiving stations will be investigated. In this work the signal path is determined using a modified ray-tracing technique together with homing-in method. The
ionospheric gradients are added in the signal path. The difference in delay between the
paths with gradients is compared with the path without a gradient. Both the latitudinal
and longitudinal gradients are considered. The gradient’s effect is considered at receiving stations located at mid-latitude and equatorial region. The correlation of gradient’s effect and baseline length for differential GPS will also be discussed.
2. METHODOLOGY
Horizontal gradients can be included either by introducing linear gradients or by modelling the real latitudinal and longitudinal variation of the ionospheric electron density
at all heights. The effects of such gradients have also been evaluated for differential
GPS. In this work, only linear gradients are considered and their effect has been studied for various baseline distances and orientations to gain a better understanding of
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their effect compared to the case of no gradients. This was done by introducing linear
gradients of the ionospheric electron density as a function of co-latitude and longitude
as follows:
N e (h, θ , φ ) = N e,0 (h) ⎣⎡1 + C (θ − θ n ) ⎦⎤ ⎣⎡1 + D(φ − φn ) ⎦⎤ ,
(1)
where Ne,0 (h) is the electron density at latitude θn and longitude φn at height h, θ and φ
are the co-latitude and longitude, respectively, at any point on the ray path. Employing
linear gradients enables the quantitative effect of a given gradient on the transionospheric ray path to be determined in terms of the gradient magnitude. This cannot easily be done if a realistic gradient model, in which the gradient changes with latitude,
longitude and altitude, is employed. Such calculations for realistic 3-D gradients in the
equatorial anomaly region (where they are likely to have the greatest effect) have,
however, been performed using a realistic 3-D IRI (International Reference Ionosphere Model, Bilitza 2001) by Strangeways and Nagarajoo (2005). The 3-D ionosphere models such as IRI are, however, difficult to implement in ray-tracing since,
for high accuracy, an analytical model of the ionosphere for which analytical expressions can also be derived for all the spatial gradients is required.
In eq. (1), θn and φn can be taken to be the point where the ray path intersects the
maximum electron density or some other reference point. Parameters C and D represent the fractional increase or decrease in electron density per radian away from θn and
φn. Since θ is equivalent to co-latitude, a positive value of C implies that the electron
density Ne (h, θ, φ) is decreasing with increasing latitude (e.g., in Fig. 3b). Since both
electron density and its gradient could influence the electron density at the reference
point, it was important to normalize the latitude and longitude in order to see the effect
of the gradients without addition of a general increase or decrease of electron density
over the whole propagation path. Shown in Fig. 1 is one of the electron density profiles used in this work, determined from the IRI model, which has a TEC of 78 TECU
(1 TECU = 1016 el. m−2) corresponding to local winter at mid-latitude local noon.
To achieve the most accurate results for GPS range finding and to evaluate the
gradient effect, a numerical ray-tracing method was used. The numerical Jones 3-D
ray-tracing was modified including the accurate home-in procedure (Strangeways
2000) and a modeling method for the electron density profile using exponential layers
to ensure continuous spatial derivatives for precise transionospheric propagation paths
(Ioannides and Strangeways 2000). The homing-in method enabled the ray-tracing to
be performed for the path from the exact location of the satellite to the exact location
of the receiver. This obviates employing a “trial and error” method to determine the
path between transmitter and receiver which is particularly difficult and timeconsuming when there are two variables to be determined (initial elevation and
azimuth angles). To accomplish the homing-in method, the downhill simplex method
using the Nelder-Mead algorithm (Nelder and Mead 1965) was utilized which is
a convenient and robust method for function minimization and works particularly well
when the number of variables does not exceed 5 or 6 (Strangeways 2000). This ensures the ray path is homed-in precisely at the receiver with very great accuracy even
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Fig. 1. Ionospheric profile made up of 40 exponential layers.
to a few milimetres, providing the ray-tracing is sufficiently accurate and the homingin tolerance is set sufficiently small. Used with this homing-in method, the ray-tracing
program has the capability to trace GPS signals from a particular receiver location to a
particular satellite location (or vice versa) taking into account the Earth’s geomagnetic
field, the magneto-ionic mode and ionospheric horizontal gradients and can also determine the path length of the group and phase for both GPS frequencies and how this
differs from the LOS path (path curvature effect).
Previously, the modified ray-tracing program was utilized for differential GPS
without gradients (Strangeways et al. 2003). Here, in this work, the modified
ray-tracing program incorporated linear gradients with values for C, D, θn and φn as
additional input parameters. The effect of gradients was studied at both single and
mobile stations as used in differential GPS. The receiving stations were located at
mid-latitude and also in the equatorial region for which a different profile for
local summer with a TEC of 67 TECU was employed. Only the high GPS frequency
(L1 = 1557.42 MHz) was considered in this work. Calculation was performed for both
the carrier phase and group path.
3. RESULTS AND DISSCUSION
Latitudinal gradients
At single station
Variations of the latitudinal gradient were first introduced using eq. (1) by varying the
parameter C without including any longitudinal gradient. Parameter C was chosen to
have values of about 1 that would give an additional electron density of ~100% above
the electron density at the reference latitude over a path extending over one radian of
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latitude. In order to determine the effect of the gradient at any receiver (e.g., at the reference station), the satellite and receiver positions were fixed for both the ray paths
with and without a gradient. The ionospheric delay or phase advance td,ref in both cases was obtained from the difference between the distance of the ray path to the receiver from the satellite Prs determined from the ray-tracing and that for propagation
over the line of sight (PLOS) at the velocity of light in vacuum. This is given as
td ,ref = Prs − PLOS .
(2)
A delay difference (δ t d) was also calculated corresponding to the difference between the delay with a gradient and with no gradient, as a function of C given as
δtd = td , ref (with grad) − td ,ref (no grad) .
(3)
Figure 2a shows the difference in group delay δ t d for a ray path at 20 degree
azimuth for a mid-latitude receiving station. This figure shows a decreasing δ t d with
increasing positive gradient indicating the group delay with no gradient is larger than
for the case with the gradient (e.g., the latitudinal gradient gives a decreased delay of
about 6 m at 20 degree elevation when the electron density increases by 50% (C = 0.5)
above the electron density at the reference point over the path length). Furthermore,
since PLOS was kept the same for both cases, this shows that there was a reduction in
the group delay for the gradient case. In this case the satellite is located at a higher
latitude than the receiving station position so that the ray path is shorter due to decreasing electron density and the path undergoes less refraction explaining the reduction in the group delay with gradient. This is depicted in Fig. 3a for the ray paths that
propagate between the receiving station and the satellite position for C = 0.5 (Fig. 3b
will be explained in a later section).
Fig. 2. Delay difference δtd between the paths with gradient and with no gradient. (a) Group
path, (b) group and phase paths. The absolute value of group |L1, group| path was plotted in (b)
for comparison to the phase path.
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1.7
x 10
12
3.5
1.65
2
β=90º, tx=39º
1.5
1.5
tx=23º
ββ=20º,
=20º, tx=23º
–0.2
0
0.2
0.5
Ref. point
1
C=0.5
1.45
α=1º, β=10º, W-E, eqt.
C= –0.5
2.5
rx=39º
1.6
1.55
12
3
Ref. point
C=0
Nc [cl. m–3]
x 10
515
1
0.5
1.4
0
1.35
-0.5
1.3
1.25
10
1.5
α=20º, mid-lat.
β=40º, tx=13º
15
20
(a)
25
30
Co-latitude [º]
35
40
-1
45
-1.5
20
(b)
30
40
50
60
70
80
90
Co-latitude [º]
Fig. 3. Electron density with latitudinal gradients (Ref. point is the point where the ray path intersects the maximum electron density, rx and tx are abbreviations for receiver and transmitter,
respectively).
The difference in delay for the phase path is in the opposite direction but with
slightly different magnitude, e.g., in Fig. 2b the phase path is advanced about 6 m and
the group path is delayed about 5 m with a latitudinal gradient given by C = 0.5 for
a path at 20 degree azimuth and 40 degree elevation. The difference in sign of the difference in group and phase paths is expected because of general difference in sign of
the group delay and phase paths. Negative values of C gave the opposite effect from
the above results so that the sign of the group delay and phase advance depends on the
direction of the gradient.
For differential GPS
The effect of horizontal gradients was then investigated for differential GPS for baseline directions of South-North (S-N) and West-East (W-E) with a baseline length of
10 km. A short baseline was chosen to see the effect of the gradient over distances for
which differential GPS correction would typically be feasible. This was done by first
finding the satellite locations at the normal GPS satellite altitude to correspond to
a given set of inclination angles of arrival at the reference station. Then the homing-in
program was used to determine the paths from the determined satellite locations to
both reference and receiving mobile stations. The latter was specified in the computation by the baseline length and by the azimuth of the mobile station from the reference
station. The horizontal gradient was incorporated for the ray-traced paths for both receiving stations. The reference point (θn and φn in eq. 1) and the gradient C in the raytracing program were set to be the same for the ray paths to both the reference and
mobile stations so that the ionosphere and gradient were similar for both points. In order to see the clear effect of latitudinal gradients for differential GPS, a path azimuth
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of 1 degree was first chosen for the W-E baseline direction so that the distance of the
satellite to both stations was nearly equal, thus reducing the effect of additional TEC
for a longer propagation path to one of the receiving stations. The difference in delay
and phase advance ∆td between the delay at the reference station td,ref and mobile station td,mob was computed as
+td = td ,ref − td ,mob .
(4)
In eq. (3) a delay difference δ t d was calculated for the same satellite and receiver locations but with and without horizontal ionospheric gradient whereas the difference in
delay (∆td) in eq. (4) is the difference in delay between the two 10 km baseline stations. This ∆td was calculated also with and without (C = 0) horizontal gradients.
Figure 4a shows ∆td with the gradient for the differential GPS for a W-E direction baseline for a mid-latitude location. The absolute value of ∆td is less than 0.5 cm
for C values less than 1. The fact that the ∆td decreases for increasing C at high elevation for the group path and vice versa at low elevations can probably be explained by
the fact that the gradients will have more of an effect compared with the altered
electron density along the path for the higher elevations, as for the latter, the range of
latitude of the propagation path is less, corresponding to a smaller change in electron
density over the path length.
The ∆td for the phase path shows a variation with increasing C in the opposite direction for all elevation angles with a slightly different magnitude to the group delay.
Since the difference can be sub cm, very precise ray-tracing is essential and requires
use of a smaller integration step of 0.1 km (for the propagation path in the ionosphere)
which results in very long simulation times, especially for the longer path at low
4.5
β=20º
4
α=1º, S-N, mid-lat.
10º
3.5
3
2.5
40º
2
1.5
1
0.5
0
89º
0
(b)
0.2
0.4
0.6
0.8
1
Value of C
Fig. 4. Differences ∆td versus latitudinal gradients C (mid-latitude): (a) W-E direction, (b) S-N
direction.
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Differences in delay [cm]
α=1°, β=20°, W-E (L1,gr)
517
α=1°, β=20°, S-N (L1,gr)
No gradient
With gradient (C=0.5)
With gradient (C=0.5)
No gradient
(a)
Baseline lenght [km]
(b)
Baseline lenght [km]
Fig. 5. Differences ∆td as a function of the station separation: (a) W-E direction, (b) S-N direction.
elevations, e.g., 10 degrees. Figure 4b shows ∆td for the S-N direction at the azimuth
of 1 degree which gives a longer path from the satellite to both stations than for the
W-E baseline. It shows a contrasting result from the W-E direction case where, at high
elevation angle paths, the altered electron density has more of an effect compared with
that of the gradient. This is because the longer propagation paths correspond to a
greater variation in electron density over the path length and thus this effect dominates
over that of the gradient.
Results also show that ∆td is increasing very linearly with increasing baseline
length for both baseline orientations, as shown in Fig. 5. The S-N direction gives the
contrasting result for the W-E direction where the no gradient case gives a larger ∆td
than that with the gradient due to the effect of the absolute value of electron density
being more pronounced than the effect of the gradient. ∆td is about 0.15 cm at 10 km
baseline length when C is 0.5 for the W-E direction and ∆td nearly reaches 5 cm for
the S-N direction for the same baseline length due to the longer path between the satellite and the receivers.
The difference in delay with latitudinal gradients has been calculated also for receiving stations located in the equatorial region. Figure 6a shows ∆td for the W-E direction baseline which can be seen to have a different variation with C compared to
the result for the stations located at mid-latitudes. ∆td decreases with increasing C for
almost all elevation angles for the group delay showing that the effect of the changed
electron density is more dominant than the effect of the gradients. For the S-N direction as shown in Fig. 6b, ∆td shows a similar variation with C as that found for the
mid-latitude receiving stations. The latitudinal gradient coefficient, C, cannot have too
large a value, such as 1 (100% increase or decrease in electron density above the elec-
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tron density at reference point over a radian of latitude), as this can result in a negative
electron density at large distances from the reference point. The electron densities for
different values of C are shown in Fig. 3b as a function of co-latitude as obtained from
eq. (1). This shows a negative electron density for higher latitudes for values of C
equal to or more than 1.
Results show that the ionospheric differential effect for a 10 km baseline oriented
W-E for latitudinal gradients up to C = 0.5 is of few mm for both group and phase
measurements. This is insignificant compared with the maximum allowable error of
0.25 TECU in differential slant TEC for carrier ambiguity fixing and precise GPS navigation (see, for example, Hernandez-Pajares et al. 2000). (1 TECU can contribute to
a path delay of approximately 0.16 m (≈ 0.54 ns) at the L1 GPS frequency). On the
other hand however, the S-N baseline directions show significant differential effects
(up to 20 cm for the baseline of about 50 km). Such a difference should be corrected
for precise differential GPS navigation for medium and long baseline distance. The
larger difference for the N-S baseline than the E-W results, of course, from the approximately N-S satellite to ground path and a reverse result would be expected for an
E-W or W-E path.
Fig. 6. Differences ∆td versus latitudinal gradients (equatorial): (a) W-E direction, (b) S-N direction.
Longitudinal gradients
At single station
Various longitudinal gradients have also been modelled utilizing eq. (1) by varying the
parameter D (without any latitudinal gradients). The reference station is located at the
mid-latitude. The difference in delay with gradient and with no gradient is shown in
Fig. 7 for this station for a path azimuth of 20 degrees. δ t d shows a similar variation
with gradient as for the latitudinal gradients (see Fig. 2). The difference in delay for
the phase path shows an opposite direction with slightly smaller magnitude at low elevations as illustrated in Fig. 7. For this linear longitudinal gradient model ray-tracing
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cannot be performed across the prime meridian in the Jones 3-D program because longitude is always expressed in degrees East so that there is a 360 degree discontinuity
in longitude at this meridian.
10
α =20°
20° 40°
Diff. in delay [m]
5
90°
0
90°
-5
40°
-10
β =20°
-15
solid line: group path
dotted line: phase path
-20
0
0.2
0.4
0.6
0.8
1
Value of D
Fig. 7. Differences δtd vs. longitudinal gradients.
For differential GPS
With the same approach as for the latitudinal gradients, the effect of a longitudinal
gradient was investigated for differential GPS for baseline directions of S-N and W-E
with a baseline length of 10 km for stations located at mid-latitude and in the equatorial region. In order to see the clear effect of longitudinal gradients for differential
GPS, a path azimuth of 89 degrees was chosen for the S-N direction baseline so that
the distance of the satellite to both receiving stations was nearly the same. The difference in delay with gradient magnitude is shown in Fig. 8 for the mid-latitude case for
the S-N direction baseline. For both stations, A and B, a similar variation of delay difference with gradient is seen as for the case of the latitudinal gradients for the W-E direction baseline with a rather smaller value for station A. For the W-E direction, ∆t d
for the group path decreases for all elevations for both stations as shown in Fig. 9,
even for high elevation angles.
For the equatorial region, ∆t d decreases for both S-N and W-E direction baselines
for the group path as shown in Fig. 10. The lower elevations show a sharp decrease in
∆t d. This is most probably due to the longer propagation path from the satellite to the
receiver over the equatorial region at the lower elevations that makes the altered electron density along the path have a greater effect than that of the gradient.
The results show that the ionospheric differential effects for a 10 km baseline for
a longitudinal gradient for both W-E and S-N baseline directions are insignificant as
compared to the maximum allowable error for differential GPS processing (see
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520
above). However, since the difference in delay can reach cm values for this 10 km
baseline, the differential effect should be considered for longer baseline since it linearly increases with baseline length.
0.15
0.6
β=10º
0.1
40º
Diff. in delay [cm]
0.05
0.4
0
20º
0.2
80º
40º
0.1
-0.1
α=89º, S-N, stn. A
-0.15
(a)
β=10º
0.3
-0.05
-0.2
α=89º, S-N, stn. B
0.5
20º
0
0.2
0.4
0.6
0.8
Value of D
0
1
-0.1
(b)
80º
0
0.2
0.4
0.6
0.8
1
Value of D
Fig. 8. Differences ∆td with longitudinal gradients due North (mid-latitude): (a) Station A,
(b) Station B.
6
6
α=89º, S-N, stn. A
5
Diff. in delay [cm]
5
β=10º
β=10º
α=89º, S-N, stn. A
4
20º
4
3
3
40º
2
40º
1
2
60º
0
80º
1
–1
0
–1
0
(a)1
80º
–2
0.2
0.4
0.6
Value of D
0.8
–3
0
(b)
0.2
0.4
0.6
0.8
1
Value of D
Fig. 9. Differences ∆td versus longitudinal gradients due East (mid-latitude): (a) Station A,
(b) Station B.
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0.1
6
α=89º, S-N, eqt.
α=89º, W-E, eqt.
5
0.05
β=10º
4
0
Diff. in delay [cm]
521
3
2
-0.05
1
-0.1
β=10º
0
–1
-0.15
–2
-0.2
-0.25
–3
0
0.1
0.2
(a)
0.3
Value of D
0.4
0.5
–4
0
(b)
0.1
0.2
0.3
0.4
0.5
Value of D
Fig. 10. Differences ∆td versus longitudinal gradients (equatorial): (a) S-N direction, (b) W-E
direction.
4. CONCLUSIONS
Horizontal gradients have a significant effect on GPS positioning for both single station positioning and differential GPS. For single station in the Northern hemisphere, a
path with a positive value of C (corresponding to electron density decreasing with increasing latitude) gives decreasing delays for the group path and increasing delays for
the phase path compared with the case of no gradient. The sign of the change
depends on the respective directions of the gradient and the propagation path. As an
example, the group delay decreases about 5 m and the phase advance increases about
6 m for C equal to 0.5 (latitudinal gradient) for a path elevation angle of 40 degrees.
This difference increases at lower elevation angles and contributes an appreciable error to precise single frequency phase measurements and so should be corrected for.
For differential GPS, the difference in ionospheric delay (and phase advance) can be
either increased or decreased compared with the no-gradient case depending on the
gradient direction. However, since the difference in ionospheric delay is not that significant compared to the maximum allowable error for precise GPS positioning, it can
be neglected for a short baseline of 10 km. In the case of longer baselines, the differential delay should be considered since it increases linearly with increasing baseline
length. If the gradient can be measured or estimated, correction can also be made in
this case. This work has potential for better ionospheric correction for precise GPS applications. The gradient would need to be estimated using a 3-D ionosphere model,
real-time ionospheric measurements or a combination of both by updating the ionosphere model using real-time or quasi-real-time ionosphere measurements. This could
then result in a four dimensional ionospheric gradient model that incorporates the lati-
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522
M. ABDULLAH et al.
tude, longitude, altitude and time variations of the electron density over any propagation path to a GPS receiver.
A c k n o w l e d g e m e n t s . MA is grateful to Dr. R.T. Ioannides of Leeds University for assistance in using the precise transionospheric ray-tracing program. The authors would also like to thank the referees for their comments and suggestions for improving this paper.
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Received 4 September 2006
Accepted 13 June 2007
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