3.
Specific Corrections
Ionosphere (2frequency
model) 
The ionosphere
correction is often only a few meters. But in extreme cases near solar maximum, it might
reach 50 meters in the zenith, or three times that near the horizon. It is generally five
times larger during the day than at night, four times larger in November than in July, and
four times larger at solar maximum than at solar minimum in the 11.1year sunspot cycle.
f_{0} 
10.23 MHz 
f_{1} 
154 f_{0}
= 1575.42 MHz 
f_{2} 
120 f_{0}
= 1227.60 MHz 

Table
6. GPS fundamental and carrier frequencies. 
The standard
twofrequency corrections for ionospheric delay may be found in many references; for
example, (Wells, 1987) or (Klobuchar, 1996). It is presumed that we have measured the
pseudorange PR simultaneously at two different Lband carrier frequencies, L_{1}
= f_{1} and L_{2} = f_{2}. These
frequencies are multiples of the fundamental frequency f_{0}.
Numerical values for the GPS system may be found in Table 6. Note that the Russian GLONASS
system uses different frequencies.
Let PR_{1} and
PR_{2} be the two simultaneous measures of the pseudorange at L_{1} and L_{2},
respectively; and let ADR_{1} and ADR_{2} be the corresponding measures of
the accumulated delta range. Then the corrections for ionospheric delay are given by:
For GPS, the numerical
values of the constants are k_{1} = 1.545728 and k_{2}
= 2.545728. After these corrections are applied, the two observed pseudoranges will be
equal. Note that this is the only correction that affects pseudorange and accumulated
delta range differently. This is because the phase velocity of electromagnetic waves
through a resisting medium exceeds the speed of light.
Corrections for
tropospheric delay are generally about two meters in the zenith, but can be an order of
magnitude larger at 10° above the horizon. The correction is in two parts, one for the
dry atmosphere and one for the wet component. The latter is usually only about 10% of the
total correction, but has considerably more uncertainty because it depends on the water
content of the air all along the path, not just at the ground station. For greatest
accuracy, barometric pressure, temperature, and dew point data are all required. Because
these are no longer measured at the Air Force monitor stations, we obtained this data from
the National Oceanic and Atmospheric Administration’s National Weather Service at the
National Meteorological Center.
Troposphere models may
be found in several references, e.g. (Spilker, 1996) and (Navstar, 1978). We take as
inputs to the model the following quantities: r = distance of receiver from
Earth center in meters; P_{s} = pressure at receiver in kilopascals;
T_{c} = temperature at receiver in degrees Celsius; T_{d}
= dew point at receiver in degrees Celsius; is the geocentric satellite position vector, and is the geocentric
receiver position vector. Then the correction to the observed pseudorange is:
Orbital relativity (due to eccentricity) 
In Einstein’s
theory of general relativity (GR), gravitational potential changes the rate of a clock by
the factor (1  GM_{Å} / Rc^{2}) , where G =
gravitational constant, M_{Å} = mass of Earth, R = distance of clock from center of
Earth, and c = speed of light. A similar but much smaller effect caused by
the change in gravitational potential of the Sun across the orbit of a GPS satellite will
be discussed in the analysis section. For now, this will be neglected. Adopting a mean
value for R = 26,560,000 m for all GPS satellites and taking the other
constants from Table 7, their rate slowing due to Earth’s gravitational potential is
1.67x10^{10} relative to clocks at infinity. The rate slowing due to Earth’s
gravitational potential for a clock located on the Earth’s geoid, say, at the north
pole where R = 6,357,000 m, is 6.98x10^{10} relative to clocks at
infinity. Since clocks on the Earth’s geoid are used to define the international
standard second of time, GPS satellite clocks run fast relative to ground clocks by
the difference of these two factors, 5.31x10^{10}. So satellite atomic clocks are
rateadjusted downward before launch by that factor, resulting in clocks in orbit that
tick at the same average rate as ground clocks. Because this compensation is complete for
the mean altitude of GPS satellites, the GR effect on the GPS system can essentially be
ignored.
Parameter 
Value 
c 
299,792,458 m s^{1} 
GM_{Å} 
3.986008x10^{14}
m^{3} s^{2} 
GM_{2} 
4.902798x10^{12}
m^{3} s^{2} 
GM_{3} 
1.327124x10^{20}
m^{3} s^{2} 
h_{2} 
0.6090 
l_{2} 
0.0852 

Table
7. Numerical values of physical constants. 
The only exception is a
small correction, always less than 5 m, for periodic variations in satellite clock rates
caused by the small orbital eccentricity of the satellites. This may be computed to
sufficient accuracy by correcting the satellite clock readings by the amount+2290 e
sin l in ns, where the eccentricity e may be taken, and mean
anomaly l computed, from Table 2. This translates into the following
correction to the observed pseudorange in meters:
D_{3} PR = 686.5 e sin l
[9]
In Einstein’s
theory of special relativity (SR), the relative velocity between two clocks changes their
relative rates such that each sees the other tick slower; the distance between them
appears to contract; and the two effects (time dilation and space contraction) combine to
leave the apparent speed of light c unchanged as measured by observers at
either clock. These changes are expressed by the vector form of the Lorentz
transformations (Robertson & Noonan, 1968):
where is the relative velocity vector and v
is its magnitude, (t, ) are time and x, y, z
coordinates in the receiver comoving frame, (T, ) are time and X, Y,
Z coordinates in the satellite comoving frame, and the space contraction /
time dilation factor g = 1 / Ö (1  v^{2}/c^{2}).
The largest velocities
we normally need to consider in the GPS system are those of a typical satellite with
respect to the center of the Earth, about 3,900 m/s, for which v/c is 1.3x10^{5}.
This makes the g factor exceed unity by about 0.84x10^{10}.
Hence, space contraction is no more than 2.2 mm and can always be neglected for our
purposes here. Likewise, since a typical flight time for a signal from satellite to ground
is about 80 ms, time dilation during that downlink time is about 67 picoseconds (ps), also
negligible here. Of course, the slowing effect on clock rates is not negligible because it
eventually builds up a significant time difference. However, just as for the gravitational
potential effect, GPS satellite clock rates are preadjusted upward before launch to
compensate for the average value of this velocity effect, resulting in orbiting clocks
that are seen to tick at the same average rate as ground clocks. The remaining small
changes in clock rates due to orbital eccentricity causing the satellite to speed up and
slow down contribute clock corrections that are always less than 5 m for the 1993
satellite constellation. We take this correction (equation [9] above) into account
in our calculations.
So we can simplify the
Lorentz transformations considerably by taking g = 1 in the small terms,
and g = (1 + ½ v^{2}/c^{2})
in the terms factored by t:
Note especially the
appearance of the factor ½. In derivations that simply set g = 1, this
factor is lost, and the resulting equation sets are no longer symmetric between frames. In
the GPS system, the first of these equations is always further simplified to T = t,
neglecting the socalled "relativity of simultaneity" effect, which is the term
factored by ½. This is justified by arguing that all clocks will be synchronized to the
underlying ECI coordinate frame instead of to their own frame. This is also necessary
because to do otherwise would lead to contradictory requirements on the settings of
satellite clocks relative to ground clocks.
For example, consider
an arbitrary satellite, a receiver fixed at the north pole, and a set of additional
receivers moving in various directions at various speeds, yet each arriving at the north
pole at a common instant. Then at that common instant each receiver would place a
different requirement on the difference T  t since differs for each. These differences are of
order 1020 m, and are therefore appreciable. As an immediate consequence of neglecting
the relativity of simultaneity term when comparing receiver and satellite clocks, the
satellite clocks are generally not Einsteinsynchronized with respect to the ECF or the
receiver comoving frames. We will comment further on the implications of this in the
analysis section of this paper.
We have used the
algorithms of (McCarthy, 1992) to compute deformations of the solid Earth under the stress
of lunar and solar tides. These corrections have a maximum effect of about 0.3 meters in
the station coordinates for the cases considered here. Only McCarthy’s equation (6)
is needed for our analysis. If it were our intention to compare the station coordinates we
derive with coordinates derived from data types that have applied no Earthtide model, we
would need McCarthy’s equations (8) as well. The correction to the adopted
coordinates to include the effect solid Earth tides is:
[10]
where
D = observed station coordinate vector minus
reference station coordinate vector, 
GM_{ j}
= product of gravitational constant and mass for the Moon (j
= 2) or the Sun (j = 3), 
GM_{Å }=
product of gravitational constant and mass for the Earth, 
= unit vector from the geocenter to Moon or Sun
and the magnitude of that vector, 
= unit vector from the geocenter to the station
and the magnitude of that vector, 
h_{2}
= nominal second degree Love number, 
l_{2}
= nominal Shida number. 
Numerical values of the
constants may be found in Table 7. Since we should apply this correction to the adopted
coordinates, it will affect the computed values of the pseudorange rather than the
observed values.
Simplified models for
the vectors to the Moon and Sun, accurate to about 0.3°, were used for solid Earth tide
calculation and other lowprecision purposes. Both models used time T in
Julian centuries from J2000 as input (as defined just prior to equation [3]), and
produced ECI coordinates as output.
Solar model
Subscript 3 refers to the Sun. The rotation R_{1}
below is defined in equation [4].
L_{3} = 280.460° + 36,000.770° T 
[solar mean longitude] 
l_{3} = 357.528° + 35,999.050° T 
[solar mean anomaly] 
(v_{3}  l_{3}) =
1.915° sin l_{3} + 0.020° sin 2 l_{3} 
[solar equation of center] 
l_{3} = L_{3} + (v_{3}
 l_{3}) 
[solar true longitude] 
v_{3} = l_{3} + (v_{3}
 l_{3}) 
[solar true anomaly] 
e = 23.439 3°  0.013 00° T 
[obliquity of the ecliptic] 
e_{3} = 0.016 708 617 
[solar eccentricity] 
r_{3} = 149,597,870 (1  e_{ 3}^{2})
/ (1 + e_{ 3} cos v_{ 3}) 
[magnitude of solar radius vector, km] 
_{3} = R_{1}(e ) {r_{
3} cos l _{3}, r_{ 3} sin l _{3},
0} 
[solar radius vector, ECI coordinates, km] 
Lunar model
Subscript 2 refers to the Moon. The rotation R_{1} below is
defined in equation [4]. e is defined above in the solar model.
l_{ 2} = 134.927° + 477,198.867° T 
[lunar mean anomaly] 
F_{2} = 93.272° + 483,202.018° T 
[lunar mean argument of latitude] 
D = 297.850° + 445,267.111° T 
[mean elongation of Moon from Sun] 
W _{2} = 125.045°  1,934.136° T 
[lunar longitude of ascending node] 
l _{2} = F_{2} + W
_{2} + 22,640" sin l_{ 2}  4,586"
sin (l_{ 2}  2 D) + 2,370" sin (2 D) 
[lunar true longitude] 
b _{2} = 18461" sin F_{2}
+ 1,010" sin (l_{ 2} + F_{2}) +
1,000" sin (l_{ 2}  F_{2}) 
[lunar latitude] 
r_{ 2} = 6,378.136 3 (60.362
98  3.277 46 cos l_{ 2}  0.579 94 cos (l_{ 2}
 2 D)  0.463 57 cos (2 D) 

[magnitude of lunar radius vector, km] 
_{2} = R_{1}(e ) {r_{
2} cos l _{2} cos b _{2}, r_{
2} sin l _{2} cos b _{2}, r_{
2} sin b _{2}} 
[lunar radius vector, ECI coordinates, km] 
The antenna on the satellite is not located at the center of mass.
Fortunately, however, it always remains pointed straight downward, so satellite spin does
not need to be modeled. And the offset between center of mass and antenna is the same for
most satellites (see Table 8). Therefore, with quantities defined as in Equations [8],
the offset correction to the observed PR is simply the Table 8 value projected onto the
pseudorange: 
Satellite 
Offset 
SV 35, 36
& 38 
+1.8664 m. 
all others 
+0.9519 m. 

Table 8. Offsets of
satellite antenna from satellite center of mass. 


4. Sample
Calculation
UT1 
15^{h}44^{m}51.4815425^{s
} 
GMST_{t} 
3.633 080 40 rad 
GMST_{r} 
3.633 080 40 rad 
X 
+12,318,856.50 m 
Y 
10,190,204.31 m 
Z 
+21,334,249.68 m 
X’ 
+3,017.55990 m/s 
Y’ 
+2,322.31113 m/s 
Z’ 
628.77318 m/s 
x 
+6,063,015.09 m 
y 
+1,729,609.43 m 
z 
+960,842.89 m 
x’ 
126.1 m/s 
y’ 
+442.1 m/s 
z’ 
+000.0 m/s 

Table 9. UT1 time, rotation angle,
and ECI coordinates of satellite and monitor station. 
To illustrate our calculation procedure
with a numerical example, we begin with the ECF state vector of Table 4, which is for SV
32 at GPS time 1993 Aug. 23^{d} 15^{h} 45^{m} 00^{s} = JD
2,449,223.15625 exactly. If we need it for some nontabular moment, we could obtain that
by Lagrangian interpolation. Then we will compare the computed ranges and observed
pseudoranges from Kwajelain at that epoch. The ECF state vector for the monitor station
is from Table 1, where of course the velocity components in the ECF frame for a station
fixed on the Earth are all zero. Additional quantities that will be needed for the
calculations may be extracted from Table 2 through Table 8.
Our first step will be to rotate both of
these state vectors into the ECI frame. For the satellite state vector, we use [2]
to get UT1 from the GPS time, then [3] to get GMST_{t}, the Greenwich
Sidereal Time at the time of satellite transmission. For the receiver state vector, the
time of reception is found from [6]. Then we use [4] and [5] to
rotate from ECF to ECI. The extracted values and results to this point are summarized in
Table 9.
Next, we form the computed PR and ADR from
[1], and take the observed PR and ADR at frequencies L1 and L2 from Table 5. It is
somewhat arbitrary whether to apply corrections to the observed or computed values. But we
will attempt to apply them so as to bring both values as close as possible to the actual
geometric distance through space traveled by the signal. Therefore, clock corrections from
Table 3 are applied to the observed PR and ADR, as are ionosphere and troposphere delay
corrections from [7] and [8], the orbital relativity correction from [9],
and the geometric offset correction from [11]. The solid Earth tide corrections
from [10] will be used to correct the computed PR and ADR, since these change the
true geometric range.
r 
24,417,279.25 m 
eq.10 
0.00 m 
calc. 
24,417,279.25 m 


PRobs 
24,586,975.07 m 
kclock 
514,207.33 ns 
32clk 
51,794.41 ns 
net 
169,676.60 m 
D _{1}PR 
5.66 m 
D _{2}PR 
10.75 m 
D _{3}PR 
0.62 m 
D _{4}PR 
+1.00 m 
obs. 
24,417,281.44 m 


(OC) 
+2.19 m 

Table 10. Corrections and adjusted
values of pseudorange and accumulated delta range, observed and computed. 
The final step is to form the (OC) from
the (Observed PR minus Computed PR). The results of computations for all these corrections
and various intermediate quantities are summarized in Table 10. The final residual
differs from this (OC) in that additional corrections to the clocks, station coordinates,
and the demodulator delays are derived from the analysis of all observations, as described
in the next section.
5. Data Analysis
Reducing our data in this way, we can
compute calculated ranges to compare with the observed PR values. The differences (OC) in
the sense of observed minus calculated can then be examined to determine how good our
modeling has been. For each solution, we choose to analyze PR or ADR values, or some
combination such as the average of the two. If we analyze ADR values, then each
uninterrupted span of data during which the ADR value is accumulated requires solving for
a single arbitrary additive constant to convert ADR into a pseudorangelike quantity.
Typically, this is one constant per passage of each satellite for each monitor station, or
one constant for several hours of 1.5second data. Because the ADR values need this
additive constant, and because of occasional cycle slips corrupting the ADR counts, we
prefer to use PR values alone. However, the higher precision of ADR measurements makes
them very useful for reducing the scatter in PR measurements over any short arc.
Our second decision before performing an
analysis run is what clock corrections to solve for. Typically, we must solve for one
arbitrary offset correction and one rate correction for each clock in the system, both for
satellite and ground clocks. Of course, one clock must be treated as the "master
clock" providing the definitions of epoch and length of the second, or time would be
indeterminate. For that purpose we adopt the Colorado Springs (CSOC) monitor station clock
as the most closely tied to the U.S. Naval Observatory master clock. So we accept the
clock corrections to the CSOC clock in Table 3 as exact, and solve for additional
corrections needed to all other clocks.
Existing atomic clocks are not generally
expected to remain wellbehaved over a fourday time span because their rates are not
stable for that long. Therefore we may choose to solve for additional clock corrections
– for example, one additional rate correction for each clock each day. Because clock
rate instability seemed to become significant in a bit less than a day, and to avoid
confusion with 12hour and 24hour signals in the data, we adopted one clock rate
correction every 16 hours for most of our analyses.
Our next decision is whether to solve for
corrections to the adopted coordinates of the monitor stations, or whether to keep those
coordinates fixed at their adopted values. For best results, corrections should be solved
for, because the adopted coordinates are not as accurate as the precision our solutions
can reach. However, sometimes other quantities of interest could become indeterminate if
solved for simultaneously with station coordinates.
Each monitor station also has two delay
corrections associated with it, mainly caused by cable lengths. The delay is different for
the L1 and L2 band signals. These two delay corrections, one for each frequency, are
called "demodulator" corrections. These corrections are not guaranteed to remain
fixed over time because demodulators and cabling are sometimes changed. But over our
fourday span we have assumed that the two corrections per demodulator do remain
unchanged. There are potentially twelve oddnumbered demodulators associated with L1 and
twelve evennumbered demodulators associated with L2 for each monitor station, totaling 60
pairs over the five monitor stations. However, ten of those pairs were inactive during
this time span, leaving 50 pairs of demodulator corrections to be solved for.
We further note that the existence of two
unknown demodulator corrections means that the two simultaneous measures of PR for L1 and
L2, which we called PR_{1} and PR_{2} in Equations [7], may differ
by a constant. But this will imitate a delay produced by the ionosphere. So to eliminate
indeterminacy in our solutions, we solve instead for one constant bias in PR for each
oddnumbered demodulator, except for one oddnumbered demodulator per monitor station for
which no correction to PR is allowed. And we solve for one constant bias in (PR_{2}  PR_{1})
for each evennumbered demodulator, except for one evennumbered demodulator overall for
which no correction to (PR_{2}  PR_{1}) is allowed. We chose
the demodulators that seemed to need the smallest corrections in trial solutions as our
reference demodulators.
Finally, we determine if there are any
special parameters we might wish to solve for to test various theses about the data, and
if we wish to use the entire fourday data span or some subset of it. We initially did not
provide for corrections to the adopted satellite orbits, since those orbits were
advertised to be good to of order 20 cm. However, we discovered that was apparently not
always the case, as we will see in our discussion of the analysis.
So a typical analysis run might have
almost 300 unknowns. However, this is only because we have so many receivers, satellites,
and clocks in the GPS system. The partial derivatives of these unknowns are generally
minimally correlated, and there are plenty of observations. So typical solutions are
welldetermined, and we seldom had an unexpected problem with matrices becoming nearly
singular.
