Chapter 7.

Relativistic Effects

7.1 Introduction

This chapter details an investigation into relativistic effects that could cause clock errors

and therefore position errors in clock coasting mode. The chapter begins with a description of

relativistic corrections that are already taken into account for the GPS satellites. Though these

corrections are known and implemented [ICD-GPS-200, 1991], no similar corrections are made

for GPS receivers that are in motion. Deines derived a set of missing relativity terms for GPS

receivers [Deines, 1992], and these are considered in great detail here. A MATLAB simulation

was used to predict the relativity effects based on the derivation of Deines, and flight data were

used to determine if these effects were present.

7.2 Relativistic Corrections for the GPS Satellite

The first effect considered stems from a Lorentz transformation for inertial reference

frames. This effect from special relativity is derived from the postulate that the speed of light is

constant in all inertial reference frames [Lorentz et. al., 1923]. The correction accounts for time

dilation, i.e. moving clocks beat slower than clocks at rest. A standard example is as follows

[Ashby and Spilker, 1996]. Consider a train moving at velocity v along the x axis. A light pulse

is emitted from one side of the train and reflects against a mirror hung on the opposite wall. The

light pulse is then received and the round trip time recorded. If the train car has width w (see

Fig. 7.1), then the round trip time according to an observer on the train is:

110
 Mirror

Light Pulse Emit/Receive

Light Pulse Experiment as Observed on Board the Train

Mirror (at time of reflection)

v
w

Emit Pulse Receive Pulse

Light Pulse Experiment as Viewed by a Stationary Observer

Figure 7.1 Illustration of Time Dilation Using Light Pulses

111
 2w
ttrain ' (7.1)
c

where c is the speed of light. Now consider an observer that is stationary with respect to the train

tracks. The train has moved a distance vttrack between emit and receive times according to this

observer's clock. Thus, the total time elapsed in the stationary frame is given by the following

relation:

1 2
w2 % v ttrack
ttrack ' 2 2 (7.2)
c

This yields:

2w
ttrack '
v2 (7.3)
c 1 & 2
c

Implicit in this derivation is the assumption that the speed of light is the same in both reference

frames. This is one of Einstein's postulates and was verified by the Michelson-Morley

experiment [Krane, 1983]. Thus, the moving clock beats slower than the stationary clock as

measured by a stationary observer:

ttrack 1
'
ttrain v2
(7.4)
1 &
c2

Using a binomial expansion and omitting the higher order terms:

112
 v2
ttrain . 1 & ttrack (7.5)
2c 2

This represents one clock correction term that must be applied for GPS satellites. The clocks in

orbit experience time dilation and must therefore be adjusted in order to maintain agreement with

clocks on the surface of the Earth. The orbital velocity of the GPS satellite is determined from

Kepler's Third Law [Pratt and Bostian, 1986]:

4 B2 3
T2 ' a (7.6)
G Me

where: a = 26561.75 km, the semimajor axis of the GPS orbit

GMe = 3.986005 x 1014 m3/s2, the Earth gravitational constant
T is the orbital period (seconds)

From this we find that T = 43,082 s. Assuming circular orbits the satellite velocity is:

2Ba
v '
T (7.7)
' 3873.8 (m/s)

Now the clock drift due to time dilation is calculated as:

v2
& ' & 0.83 @ 10&10 (s/s) (7.8)
2c 2

In one day this drift would cause a clock offset of -7.2 µs.

113
 Another correction must be made to account for the height of the GPS satellites above the

surface of the Earth. With an orbital altitude of about 20,200 km the satellite clock experiences a

gravitational potential that is significantly different from that experienced by a clock located on

the surface of the Earth. This causes a gravitational frequency shift in the GPS carrier as the

signal travels from the satellite to an antenna on or near the surface of the Earth [Spilker, 1978].

The result is that the satellite clock appears to run faster to an observer on the Earth than it would

to an observer at the satellite.

If the Earth is considered to be spherically symmetric, the gravitational potential N(r) can

be approximated as [Nelson, 1990]:

G Me
N(r) ' & (7.9)
r

where r is the radial distance from the center of the Earth. The gravitational effect is thus

[Hofmann-Wellenhof, 1994]:

)N N( Re % h ) & N( Re )
' ' &1.67 @ 10&10 & (& 6.95 @ 10&10 )
c 2
c 2
(7.10)
' 5.28 @ 10&10 (s/s)

where: Re = 6,378 km, the equatorial radius of the Earth

h = 20,184 km, the altitude of the GPS satellite

This clock drift accumulates to 45.6 µs after one day.

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 The combination of gravitational frequency shift and time dilation results in a satellite

clock drift for observers on the Earth [Deines, 1992]:

T
)N v2
m
J . 1 % & dt (7.11)
c2 2c 2

where: J is proper time of the clock carried by the GPS satellite

t is coordinate time in the inertial reference frame
)N is the gravitational term
v is the satellite velocity
c is the speed of light

The combined effect is a drift of 4.45 x 10-10 s/s which would cause a clock offset of 38.4 µs after

one day. Therefore, the satellite clocks are tuned so that the observed frequency on Earth is

10.23 MHz:

fobs & ftr

' 4.45 @ 10&10 (7.12)
fobs

where: fobs = 10.23 MHz, the clock frequency as observed on Earth

ftr = the clock frequency as observed at the satellite

Thus, the clocks are tuned to 10.22999999545 MHz before launch [Spilker, 1978]. This may

seem like a minor correction, but a 1 µs clock error is roughly equal to 300 m in terms of

distance. A 1 µs error would build up in 38 min if the relativistic effects of Eq. 7.11 were not

corrected.

115
 The assumption of circular orbits is somewhat erroneous. GPS orbits typically have an

eccentricity of less than 0.01 [Ashby and Spilker, 1996], but even the slightest eccentricity causes

the satellite to change altitude periodically during orbit. This results in the need for an additional

relativity term, separate from the terms of Eq. 7.11 [Nelson, 1991]:

2 G Me a
)tper ' & e sin( E ) (7.13)
c2

where: )tper is the periodic relativistic error in seconds

e is the orbit eccentricity
E is the eccentric anomaly

For an eccentricity of 0.01, the maximum effect is ±22.9 ns, or ±6.86 m. The correction is added

to the clock polynomial based on the broadcast coefficients for that satellite [ICD-GPS-200,

1991]:

)tSV ' af 0 % af 1 ( t & t oc ) % af 2 ( t & t oc )2 % )tper (7.14)

where: )tSV is the SV PRN code phase time offset in seconds

t is GPS system time
toc is the clock data reference time
af 0 , af 1 , af 2 are polynomial coefficients broadcast by the satellite

Thus, the complete satellite clock correction is as follows. The terms of Eq. 7.11 are

corrected by setting the clocks to 10.22999999545 MHz. The polynomial coefficients

( af 0 , af 1 , af 2 ) are used to correct for satellite clock bias, drift, and acceleration with respect to

GPS time. The additional term, )tper, accounts for GPS orbit eccentricity in the application of

116
Eq. 7.11. Prescriptions from special and general relativity are applied to correct known

relativistic errors for the satellites.

Some scientists have observed small GPS pseudorange errors known as the "bowing

effect" which are independent of what type of receiver is used [Klepczynski, 1986], and have

theorized that these errors may be uncorrected relativistic effects [Deines, 1992]. Nelson (1991)

has concluded that the relativity corrections as applied for GPS time transfer are not in need of

modification, but that ephemeris errors could be the source of systematic errors. However, the

large relativistic effects are clearly accounted for in the satellite clock corrections. The remainder

of this chapter deals with relativistic corrections for GPS receivers on dynamic platforms. These

effects are unaccounted for in GPS receiver algorithms.

117
7.3 GPS Receivers on Moving Platforms

The goal of this work is to analyze the effects that Deines claims are not accounted for in

GPS algorithms. This claim has not been verified experimentally, which is the purpose of this

study. In order to determine a flight profile that would maximize the predicted effects, an

analysis was done which shows that two of the terms Deines claims are missing would be

observable in the double difference (Eq. 3.16). Methods of verifying the presence of the

relativistic effects based on errors in the double differences will be presented in Sec. 7.4.

In this section we seek to develop a method of testing for the presence of relativistic

effects predicted by Deines. We begin by introducing the MATLAB simulation used to compute

the predicted effects, and continue by using this simulation to check for agreement with examples

given by Deines [Deines, 1992]. The flight profile and satellite positions were used in a

MATLAB simulation to evaluate the relativistic effects. A prediction of relativistic effects for a

GPS receiver undergoing acceleration in a turn is given, followed by a calculation of relativistic

effects in differential GPS. Here, the relativistic effects are calculated both for the aircraft and

the ground station, given a straight and level flight profile. The resulting errors that would be

evident in the double differences are presented, from which a flight test experiment is devised.

118
7.3.1 Predicted Relativity Terms for GPS Receivers

The missing relativity terms that are predicted by Deines come from the following

equation [Deines, 1992]:

T T
)N v2 A@D V@ v V2
m m
J . 1% & dt % % & dt (7.15)
c2 2c 2 c2 c2 2c 2

where: J is proper time

t is coordinate time
T is the integration time
)N is a gravitational term
V is the receiver velocity vector
V is the receiver speed
v is the satellite velocity vector
v is the satellite speed
A is the receiver acceleration
D is a vector from the receiver to the satellite
c is the speed of light

The first integral includes the satellite clock corrections that are currently implemented

(Eq. 7.11). The terms in the second integral are the relativity terms that Deines claims are

unaccounted for in GPS algorithms. During the remainder of this chapter we will be concerned

primarily with the terms under the second integral of Eq. 7.15. These are represented here to

avoid later confusion:

T
A@D V@ v V2
m
Missing Terms ' % & dt (7.16)
c2 c2 2c 2

119
The third term in Eq. 7.16 is a time dilation term which is analogous to the time dilation term

applied for the GPS satellite as shown in the first integral of Eq. 7.15. The time dilation term for

the receiver does not depend on any satellite parameters. The other two terms presented by

Deines vary depending on which satellite is considered, because v and D are different for each

satellite. To numerically calculate these terms, a GPS simulation can be used to provide satellite

positions and a flight profile can be generated in MATLAB. First, consider the test cases

presented by Deines which were used to illustrate that the predicted relativistic effects are

significant.

In the examples given by Deines, a receiver is on board an aircraft at the equator. A GPS

satellite passes directly overhead at an altitude of 20,200 km in an inclined orbit of 55E. The

aircraft moves at a speed of 900 m/s in each of the four principal directions. Satellite positions

from a FORTRAN based GPS simulation were used along with a simulated flight profile to

numerically calculate the relativistic effects in MATLAB. Each case was then compared to the

results given by Deines to check for agreement.

In the first example the aircraft flies east at a constant altitude and therefore has an

acceleration as it follows the curvature of the Earth. The flight profile was generated in

MATLAB with a time increment of one second. A GPS simulation provided satellite positions

at one second intervals, and a satellite that is just passing overhead was chosen for the

computation. The simulated flight began at a point directly under the satellite and continued

eastward along the equator. For this case Deines predicted a clock drift of 8.9 x 10-11 s/s which is

120
a combination of +6.56 x 10-11 s/s for the A @ D term and +2.34 x 10-11 s/s for the 2 V@ v & V
2

c2 2c 2
term.

The results of the MATLAB simulation for each term are shown in Figs. 7.2-7.3. The

simulation agreed with Deines in magnitude but differed in sign for the A @ D term. This is due
c2

to the fact that A points toward the center of the Earth while D is a vector from the receiver to the

satellite. Thus, the vectors are parallel but the directions are opposite which yields the negative

result. The drift was not constant in the MATLAB simulations because the geometry between

the receiver and the satellite changed slightly during the 100 second run time. The 2 V@ v & V
2

2c 2

term is in agreement with Deines. The results of the MATLAB simulations were compared to

Deines' results for each of the four cases. Table 7.1 shows the clock drifts presented by Deines,

and Table 7.2 shows the results of the MATLAB simulations. Though the numbers are different

the discrepancies between the two are minor. The simulation is accurate in the sense that it

agrees with Deines' examples.

As noted, the A @ D term showed a difference in sign due to the interpretation of which
c2

way A and D point. Deines states that D is a vector from the receiver to the satellite. Clearly the

centripetal acceleration of the aircraft points toward the center of the Earth. Thus, the A @ D
c2
terms from the MATLAB simulation are negative whereas Deines showed these terms to be

positive.

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Figure 7.2 First Term of Clock Drift from MATLAB Simulation for an Aircraft Flying
East Along the Equator at 900 m/s with a GPS Satellite Directly Overhead

122
Figure 7.3 Second Term of Clock Drift from MATLAB Simulation for an Aircraft
Flying East Along the Equator at 900 m/s with a GPS Satellite Directly
Overhead

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 Table 7.1 Relativistic Effects Predicted by Deines for Test Cases at the Equator

From A @ D (s/s) 2 V@ v & V 2 (s/s)

Vinertial (m/s)
Deines c2 2c 2

EAST 1365 +6.56 x 10-11 +2.34 x 10-11

NORTH 1013 +3.61 x 10-11 +3.01 x 10-11

SOUTH 1013 +3.61 x 10-11 -4.15 x 10-11

WEST 435 +0.67 x 10-11 -1.18 x 10-11

Table 7.2 Relativistic Effects from MATLAB for Test Cases at the Equator

From A @ D (s/s) 2 V@ v & V 2 (s/s)

Vinertial (m/s)
MATLAB c2 2c 2

EAST 1365 -6.56 x 10-11 +2.34 x 10-11

NORTH 1013 -3.63 x 10-11 +3.76 x 10-11

SOUTH 1013 -3.63 x 10-11 -2.60 x 10-11

WEST 435 -0.67 x 10-11 -1.18 x 10-11

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 In the North and South Cases for the second term, Deines used 1013 N̂ ( Ŝ ) rather than

900 N̂ ( Ŝ ) + 465 Ê . This caused the 2 V@ v & V terms to be in disagreement for those two
2

2c 2

cases. However, the simulation is largely consistent with the work presented by Deines and can

be used to predict relativistic effects for more general cases.

7.3.2 Example of an Aircraft Accelerating During a Turn

As an example, consider what happens when an aircraft is banking. The receiver

acceleration vector becomes large, for instance 19.6 m/s2 for a 2g turn. This represents a

2-3 order of magnitude increase over the centripetal acceleration of an airplane traveling straight

and level to the east at 100 m/s along the equator. In that case *A* is only 0.05 m/s2. The

relativistic effects predicted by Eq. 7.16 for a 2g turn are shown in Fig. 7.4.

In Fig. 7.3, the relativity terms for nine satellites in view are shown. Recall from the

discussion of Eq. 7.16 that two of the terms proposed by Deines are different for each satellite.

This results in an accumulated effect that depends on which satellite is used. During a 2g turn

lasting 16 s, eight of the nine predicted relativistic effects reach 10 ns or more. For two satellites

the error is as large as 40 ns, and for one satellite the error is nearly 60 ns. A 40 ns timing error

represents a 12 meter range error to the satellite. However, errors of this magnitude are not

experienced in practice. This indicates that some discretion should be shown in interpreting the

acceleration vector. For the remainder of this chapter A will be taken as the centripetal

125
Figure 7.4 Predicted Relativistic Effects for an Aircraft in a Simulated 2g Turn — Each
Satellite in View Produces a Different Effect

126
acceleration pointing towards the rotation axis of the Earth. Based on this analysis, the flight

profile to be considered for a flight test is straight and level. The next step is to determine the

relativistic effects in DGPS for such a case, and determine if the effects proposed by Deines are

large enough to measure.

7.3.3 Relativistic Effects in Differential GPS

A ground station is now included to show the impact of the relativistic effects on

Differential GPS. In this case, both the ground station and airborne receivers are moving in the

Earth Centered Inertial frame and therefore experience relativistic effects according to Deines.

As an example consider a ground station at 0E Latitude and 0E Longitude. The ground

station velocity is 465 m/s in the ECI frame. The aircraft starts at the ground station and flies

east at 100 m/s, resulting in a velocity of 565 m/s in the ECI frame. The difference between air

and ground clock offsets is manifest in the carrier phase single difference (Eq. 3.14). This

difference between the clock offsets is shown in Fig. 7.5 for a 100 second run time for each

satellite in view. Again the satellite positions are taken from a FORTRAN simulation, and there

are nine satellites in view. The result of differencing the relativistic effects between the ground

and the air is a clock drift between the receivers that shows up in the single difference for a

particular satellite. Again we note that the effect depends on which satellite is used because v

and D are different for each satellite.

127
Figure 7.5 Predicted Relativistic Effects for Straight and Level Flight as Observed in
the Single Differences — Each Satellite Produces a Different Effect

128
 By choosing the highest elevation satellite as a reference, the double difference

(Eq. 3.16) can be formed. Because the effects predicted by Deines are satellite dependent, a

residual relativity effect is predicted for the double differences. In other words, the relativistic

effects are not common for all channels, unlike temperature effects which were shown in

Chapter 6 to affect each single difference by the same amount. This means that temperature

effects would be removed from the double difference, but not uncommon errors like the

relativistic effects proposed by Deines. Fig. 7.6 shows that significant errors are predicted to

build up in the double difference over time. Here we have nine satellites in view, so there are

eight double differences. Two of the double differences show errors that grow to about -0.34 ns,

or -0.1 m over a short flight of only 100 s.

The results shown in Fig. 7.6 indicate that significant errors are predicted by Deines.

Consider that a -0.1 m error is predicted in two of the double differences after only 100 s, which

means that a longer flight would produce even larger effects. Analyzing Eq. 7.16 further, we find

that the first two terms are largely dependent on the velocity of the aircraft and the time in flight.
2
Vinertial V@ v
The magnitude of the centripetal acceleration is *A* ' , and the term is clearly
R c2

dependent on receiver velocity. The implication is that a flight of 200 s at 50 m/s would produce

very similar results to the accumulated relativity errors shown in Fig. 7.6. This means that a

small aircraft traveling at moderate speeds (50-60 m/s) can be used to produce measurable

effects, if they exist. Fig. 7.6 illustrates a 10 km flight which produces double difference errors

as large as 0.1 m. Therefore, a 40 km flight should produce errors roughly four times as large.

129
Figure 7.6 Predicted Relativistic Effects for Straight and Level Flight as Observed in
the Double Differences — the Highest Elevation Satellite is the Reference

130
Errors of that magnitude (0.4 m) in the double differences would be measurable. Thus, the

results shown in Fig. 7.6 are the basis of the flight test experiment that was devised to test the

theory of Deines.

131
7.4 Flight Test Experiment

7.4.1 Flight Test — Introduction

The purpose of this flight test was to measure relativistic effects as predicted by Deines in

Eq. 7.16. The test is based on the results shown in Fig. 7.6, and the flight profile is straight and

level flight over a distance of about 40 km. The relativistic effects predicted by Deines would

build up in the double differences during the flight. Thus, we seek to measure an accumulated

effect at the conclusion of the flight. Two methods will be used to verify the presence of the

relativistic effects.

First, the double difference errors can be converted into position errors, and this can be

compared with experimental position error. This requires knowledge of true position at the

beginning and end of the flight test. The true position at the beginning of the flight can be used

to initialize an L1 carrier phase ambiguity resolved solution. This solution is carried through the

flight, and is compared to the true final position to check for agreement with the position errors

predicted by Eq. 7.16.

Second, a computational method called the QR factorization can be used in the presence

of redundant measurements to check for inconsistency among the measurements. An observable

known as the parity vector describes the level of inconsistency. It has been shown that the errors

predicted by Deines are different for each satellite. Thus, the double difference errors can be

used to predict the magnitude of the parity vector in the presence of the predicted relativistic

132
effects, and can be compared with the parity vector from the ambiguity resolved solution at the

end of the flight.

The ideal test would have been to fly straight and level at a constant velocity, with initial

and final positions known to within 1-2 cm. This is not practical, however, so this profile was

approximated as closely as possible. An accurate survey was made on the ramp at University

Airport (UNI) in Albany, Ohio, followed by a direct 40 km flight to Rhodes Airport near

Jackson, Ohio, where another survey was taken. This second survey served as both the end of

the first flight test and the beginning of the second experiment which included the return flight.

Thus, two data sets were collected. It is the return flight to UNI that is considered extensively

here, because six satellites were available continuously during the flight. Six satellites were

available during most of the first flight as well, but a few bad measurements were recorded for

one satellite and therefore made the second data set more useful.

A description of the flight test is presented, followed by an extensive outline of how the

predicted relativistic effects were calculated. The comparison between experimental and

theoretical results is then made both in terms of position error and the parity vector.

133
7.4.2 Flight Test Description

On October 7, 1996, a flight test was conducted in Ohio University's Piper Saratoga

(N8238C) Flying Laboratory. One passenger seat was removed and replaced by a palette which

was bolted into place. A rack containing an Ashtech Z-12 GPS receiver was secured into place

on the palette with a rachet tie-down. Two 12 volt batteries were connected to the rack, which

can be set up internally as either a series (24 V) or parallel (12 V) connection. Before the flight,

the batteries were connected in parallel to allow swapping of batteries without losing power to

the receiver. A dual-frequency GPS antenna was secured on the top of the airplane and

connected to the receiver via an antenna cable that fed directly into the cabin.

In the Avionics Hangar at University Airport (UNI), the Ground Station receiver was

connected to an antenna in the crow's nest. The receiver remained in the GPS lab for the duration

of the two flight tests. The Saratoga was pulled out to the front row of parking spaces on the

ramp at UNI, and faced the runway. At approximately 15:27 GMT (11:27 AM local) both

receivers began collecting data at an interval of once per second.

To obtain an accurate survey, a static data collection of about one hour was conducted.

At approximately 16:31 GMT the engine was started and preparations were made for taxi and

departure. The pilot took off at 16:39 GMT and climbed to about 1300 ft (~ 500 ft AGL) and

initiated a left turn to a magnetic heading of 235 degrees. The destination was Rhodes Airport

which is roughly 21 nmi (40 km) southwest of UNI. At 16:45 GMT the pilot executed a climb to

1700 ft to avoid a tower which was marked on the Ohio Aeronautical Chart at 1243 ft. At

134
16:51 GMT the pilot landed the airplane at Rhodes Airport. After taxiing to the parking area at

Rhodes, the airplane was situated for another static data collection beginning at 16:54 GMT.

Again data were collected for about one hour.

At 17:56 GMT the engine was started and takeoff for the return trip to UNI occurred at

18:01 GMT. The altitude for the return flight was approximately 2300 ft. Landing at UNI was

delayed slightly due to inbound traffic. The pilot entered the pattern on the downwind leg and

touchdown on Runway 25 took place at 18:16 GMT. At 18:20 GMT the airplane was parked and

was again collecting static data. This final data collection concluded at 19:28 GMT.

7.4.3 Calculation of Expected Relativistic Effects

To numerically calculate the terms in Eq. 7.16, the ephemerides from the collected data

were used to determine satellite positions during the flight test. Also, the beginning position at

Rhodes Airport and the ending position at UNI were used to construct a straight and level flight

profile between these two points. This was implemented as a constant rate of change in latitude

and longitude with a constant ellipsoidal height of 500 m. The baseline was about 39.5 km and

the duration of the simulated flight profile was 720 s yielding an approximate speed of 55 m/s for

the aircraft, consistent with the ground speed of the Piper Saratoga during the flight test.

Using the aircraft position from the flight simulation and the satellite positions based on

the received ephemerides, numerical first and second derivatives were taken to approximate the

135
velocity and acceleration vectors for the aircraft and also the velocity vector for the satellite.

Thus, all the terms in Eq. 7.16 were known and the clock drifts predicted by Deines were

calculated. See App. C for the MATLAB software used to calculate the predicted relativistic

effects for this flight test.

The predicted relativity errors were calculated for each satellite-receiver pair. The results

of the simulation for the ground and air receivers are shown in Tables 7.3 and 7.4, respectively.

Here, the (SV#) term in the first column refers to one of the six satellites in view during the flight

test. Recall that the relativistic effects predicted by Deines are different for each satellite.

According to Deines, the terms in Eq. 7.16 should be calculated using Earth-Centered Inertial

(ECI) coordinates, then applied for users in the Earth-Centered Earth-Fixed (ECEF) frame. The

ground receiver had a velocity and acceleration in the ECI frame due to Earth rotation, which is

why the terms in Table 7.3 are nonzero. The third term is the same for all satellites because it

depends only on receiver speed. The results show the accumulated effect after a simulated

12 minute flight from Rhodes Airport to University Airport.

It is not possible to observe errors of this magnitude without differencing techniques. To

represent the error that would be observed in the carrier phase single difference (SD), the

relativistic effects for each satellite were differenced between the ground and air receivers. The

error would also be present in the code phase single differences, but the less noisy carrier phase

measurements were used here. The single difference eliminates common satellite clock errors.

136
Table 7.3 Simulated Relativity Terms (Based on Deines) for the Ground Receiver after
12 Minutes Concurrent with the Flight from Rhodes Airport to University
Airport

G @ DG i
720 A 720
VG @ v i 720
VG
2

m m m
dt dt & dt
GROUND 0
c2 0
c2 0
2c 2 Result

SV3 0.073 m -0.913 m -0.156 m -0.996 m

SV9 -1.041 m 1.348 m -0.156 m 0.151 m

SV17 -1.165 m 2.168 m -0.156 m 0.847 m

SV23 -0.620 m 3.245 m -0.156 m 2.469 m

SV26 -0.071 m 1.908 m -0.156 m 1.682 m

SV28 -0.748 m 0.849 m -0.156 m -0.056 m

i = 3, 9, 17, 23, 26, 28

Table 7.4 Simulated Relativity Terms (Based on Deines) for Air Receiver after
12 Minutes of Straight and Level Flight from Rhodes Airport to University
Airport

A @ DA i
720 A 720
VA @ v i 720
VA
2

m m m
dt dt & dt
AIR 0
c2 0
c2 0
2c 2 Result

SV3 0.256 m -0.309 m -0.197 m -1.250 m

SV9 -1.417 m 1.801 m -0.197 m 0.187 m

SV17 -1.474 m 2.186 m -0.197 m 0.515 m

SV23 -0.749 m 3.603 m -0.197 m 2.657 m

SV26 -0.281 m 2.250 m -0.197 m 1.835 m

SV28 -0.773 m 0.641 m -0.197 m -0.328 m

i = 3, 9, 17, 23, 26, 28

137
Table 7.5 shows the results of differencing the predicted relativistic effects between the ground

and air receivers. Again (SV#) refers to one of the six satellites in view during the flight test.

Because there are other error sources that are large in comparison to the predicted

relativistic effects, further differencing was performed. Temperature effects, for instance, were

shown in Chapter 6 to cause errors on the order of 10 meters in the single differences. Therefore,

using atomic clocks to augment the receivers is not effective in trying to observe the relativistic

effects. Instead, temperature effects and other common clock errors were removed by

differencing against a reference satellite. Table 7.6 shows the predicted relativity errors for the

carrier phase double differences. A high elevation satellite was chosen as the reference, in this

case SV 17.

The double difference contains noise, multipath, and residual troposphere and ionosphere

errors [Diggle, 1994]. For an L1 ambiguity resolved carrier phase solution, these effects combine

to produce position errors on the order of a few centimeters. Noise and multipath errors are

small for carrier phase measurements, which are very clean compared to code phase

measurements. Carrier phase advance through the ionosphere is approximately the same for

receivers on a short baseline compared to the distance to the satellite. During the flight test, the

maximum baseline was 40 km and even a satellite that passes directly overhead is still 20,200 km

away. Thus, ionospheric errors are minor and diminish as the aircraft proceeds to University

Airport where the ground station was located. Path delay through the troposphere depends on the

altitude of the user, so we would expect an error to build up when the plane takes off and hold

138
Table 7.5 Simulated Relativity Terms (Based on Deines) after 12 Minute Flight from
Rhodes Airport to University Airport for the Single Differences (Ground -
Air)
SD Term 1 Term 2 Term 3 Result

SV3 -0.183 m 0.396 m -0.041 m 0.254 m

SV9 0.376 m -0.454 m -0.041 m -0.037 m

SV17 0.309 m -0.017 m -0.041 m 0.332 m

SV23 0.129 m -0.358 m -0.041 m -0.188 m

SV26 0.147 m -0.342 m -0.041 m -0.154 m

SV28 0.025 m 0.207 m -0.041 m 0.273 m

720 A G @ DG & A A @ DA
m
Term 1 = i i
dt
0
c 2

720
VG @ v i & VA @ v i
m
Term 2 = dt
0
c2

720 2 2
VG & V A
m
Term 3 = & dt
0
2c 2

i = 3, 9, 17, 23, 26, 28

139
Table 7.6 Simulated Relativity Terms (Based on Deines) After 12 Minute Flight From
Rhodes Airport to University Airport for the Double Differences — SV 17 is
the Reference
DD Term 1 Term 2 Term 3 Result

SV17 - SV3 0.492 m -0.413 m 0m 0.079 m

SV17 - SV9 -0.067 m 0.436 m 0m 0.369 m

SV17 - SV23 0.180 m 0.340 m 0m 0.521 m

SV17 - SV26 0.162 m 0.324 m 0m 0.486 m

SV17 - SV28 0.284 m -0.225 m 0m 0.060 m

720 A G @ DG & A A @ DA & A G @ DG & A A @ DA

m
Term 1 = 17 17 j j
dt
0
c 2

720
VG @ v17 & VA @ v17 & VG @ vj & VA @ vj
m
Term 2 = dt
0
c2

720 2 2 2 2
VG & V A & V G & VA
m
Term 3 = & dt Ñ 0
0
2c 2

j = 3, 9, 23, 26, 28

140
approximately constant enroute. The error would drop out as the plane and the ground station

reach a state of similar altitude during landing. Note that tropospheric path delay is more

predictable than carrier phase advance through the ionosphere, and can therefore be modeled.

Thus, the error sources that impact the position solution cannot mask the relativistic effects

predicted in the double differences (Table 7.6), which are in some cases an order of magnitude

larger than the aforementioned error sources. Another potential error source is receiver inter-

channel biases, but these are generally calibrated by the manufacturer.

Table 7.6 shows that relativistic effects on the order of 0.5 meters can be expected for

certain double differences. These translate into position errors based on the geometry of the six

satellites under consideration. The time dilation term drops out of the double difference, as will

be shown in Sec. 7.4 and is not considered here — note that time dilation is a generally accepted

effect and has been verified experimentally [Hafele & Keating, 1972].

The first method of verifying the presence of relativistic effects is to determine the

predicted position error based on the accumulated effects after the 12 minute flight shown in

Table 7.6. The result is a horizontal position error of 0.473 meters and a vertical position error of

0.151 meters, which was determined as follows. From Diggle we have the form of the position

solution as calculated using double difference carrier phase measurements (assuming six

satellites are visible) [Diggle, 1994]:

141
 1 2 1 2 1 2
ux & ux uy & uy uz & uz
DD 12
& N 8
12
1 3 1 3 1 3
DD 13 & N 13 8 ux & ux uy & uy uz & uz
x
1 4 1 4 1 4
DD 14 & N 14 8 ' ux & ux uy & uy uz & uz y (7.17)
DD 15 & N 15 8 1 5 1 5 1 5 z
ux & ux uy & uy uz & uz
DD 16 & N 16 8 1 6 1 6 1 6
ux & ux uy & uy uz & uz

where: DDmn is the carrier phase double difference using satellites m and n
Nmn is the combined integer ambiguity for the double difference
8 is the wavelength (19 cm for L1 or 86 cm for L1 - L2)
(ux, uy, uz) is a unit vector pointing from the midpoint of the baseline
between the two receivers to a satellite
(x, y, z) is a vector representing the baseline from the ground receiver to
the airborne receiver

This equation may be rewritten as:

DD ' H $ (7.18)

where: DD contains carrier phase double difference measurements

H is a matrix of differenced unit vectors based on the satellite geometry
$ is the (x, y, z)T baseline vector

To determine position error due to errors in the double differences, DD was replaced by a

vector containing the predicted relativistic effects. Then a least squares solution for $ yielded the

position error due only to the relativistic effects. Here a snapshot of the geometry at the end of

the 12 minute run time was used along with the built-up relativistic effects. First, the geometry

matrix was formed using (ux, uy, uz) for each satellite as shown in Table 7.7. From these vectors

142
Table 7.7 Unit Vectors to Each Satellite from the Baseline Midpoint at the End of the
Simulated Flight from Rhodes Airport to University Airport
Geometry ux uy uz Azimuth Elevation

SV3 -0.857 0.414 0.308 295.8E 17.9E

SV9 0.609 -0.286 0.740 115.1E 47.7E

SV17 0.067 -0.327 0.943 168.5E 70.5E

SV23 -0.152 0.365 0.918 337.4E 66.7E

SV26 0.670 0.564 0.483 49.9E 28.9E

SV28 -0.855 -0.331 0.400 248.8E 23.6E

143
we get azimuth and elevation angles which gives an indication of how the satellites are spaced in

the sky. SV 17 was chosen as the reference satellite in this case because it was highest in

elevation. The geometry matrix was formed using the differenced unit vectors shown in

Eq. 7.17.

Second, the relativistic errors from Table 7.6 were used to form the double difference

vector of Eq. 7.17, which can now be expressed:

0.079 0.923 & 0.742 0.635

0.369 & 0.542 & 0.042 0.203 errx
0.521 ' 0.219 & 0.693 0.024 erry (7.19)
0.486 & 0.603 & 0.892 0.459 errz
0.060 0.921 0.003 0.542

where the left hand side is a vector of relativistic effects, and the 5 x 3 H matrix represents the

satellite geometry. From this, a least squares solution was formed using the generalized inverse:

$ ' ( H TH )&1H T DD (7.20)

This yielded (-0.198, -0.430, 0.151)T for the x, y, and z position errors, or 0.473 m horizontal

error and 0.151 m vertical error. The three dimensional position error is about 0.5 m.

The second method of verifying the presence of relativistic effects is to form a parity

space residual. The residual exists when redundant measurements are available and is defined as

the norm of the parity vector. In this case, two redundant measurements were available which

144
resulted in a two element parity vector. A parity vector is formed by operating on H so that it is

expressed as a product of an orthogonal matrix and an upper triangular matrix. Such a

representation is called a QR decomposition, hence H = QR and QTQ = I [Golub and Van Loan,

1989]. This allows us to partition Eq. 7.18 in order to take advantage of the redundancy. This

comes from the fact that the last m-4 rows of R contain zeros where m is the number of double

difference measurements:

$
DD ' H$

$
DD ' Q R$

Q T DD ' R$
$
(7.21)
T
Q$ U
& & & DD ' & & & $
T 0
Qp

The top partition yields the least squares solution as:

T
$ ' U &1 Q$ DD (7.22)

The bottom partition shows the following relation:

T
Qp DD ' 0 (7.23)

which is not true in general due to measurement noise, multipath, and other error sources. Thus,

the result of Eq. 7.23 is known as the parity vector [Kline, 1991]:

145
 T
p ' Qp DD (7.24)

In simple terms, the parity vector can be thought of as a measure of agreement among the

different measurements. Thus, if inconsistencies exist in DD the parity residual will be large. It

is important to note that if the double differences all have a common error, this will not show up

in parity space. However, in the case of the predicted relativistic effects we have errors that are

not common to each double difference. Therefore, disagreement among measurements should be

observable in parity space.

The QR decomposition of H was performed:

H ' QR

&0.595 &0.475 0.120 &0.182 &0.610 &1.551 0.176 &0.454

0.350 &0.077 0.519 &0.774 0.052 0 1.340 &0.615 (7.25)
' &0.141 &0.498 &0.573 &0.363 0.522 0 0 0.605
0.389 &0.717 0.323 0.470 0.102 0 0 0
&0.594 0.080 0.533 0.121 0.585 0 0 0

The bottom partition of QT was used to form the parity vector:

146
 0.079
0.369
& 0.182 & 0.774 & 0.363 0.470 0.121
p ' 0.521
& 0.610 0.052 0.522 0.102 0.585
0.486
(7.26)
0.060

&0.253
'
0.327

The length of this vector, *p* = 0.414 m, shows that the residual should be more than 40 cm after

the flight if the predicted relativistic effects were present in the data. During post processing, the

predicted position error and parity space residual can be compared directly to what is observed

when using the actual flight data.

147
7.4.4 Flight Test Results

Fig. 7.7 shows a plot of the ground track for the flight. The pilot took off on Runway 19

at Rhodes Airport and proceeded in a northeasterly heading to UNI. After entering the pattern on

the downwind leg, the pilot landed on Runway 25 at UNI. Figs. 7.8-7.9 are plots of static data

collections of 10 min duration taken before and after the flight. Fig. 7.8 shows the three

dimensional position error on the ground at Rhodes airport, while Fig. 7.9 shows the three

dimensional position error on the ground at UNI. This is determined by comparing the ambiguity

resolved L1 carrier phase solution from the FORTRAN software to the surveyed position as

calculated by the PNAV software using the static data collections from Rhodes Airport and

University Airport.

The position error at Rhodes Airport stayed mostly between 4 and 5 cm, or about a

quarter wavelength on L1. The three dimensional position error may seem a bit large, but the

baseline between the ground station and the Saratoga was 39.5 km during the static data

collection at Rhodes Airport. Certain error sources, such as ionospheric carrier phase advance,

have a tendency to decorrelate as the distance between air and ground platforms increases,

resulting in larger position errors [Diggle, 1994]. It should be noted that ionospheric modeling

was used by the PNAV software when the baseline was longer than 15 km, as was the case

during the static collection at Rhodes Airport. No such correction was applied in the FORTRAN

solution, yielding a larger position error for the 39.5 km baseline than for the 78 m baseline when

the plane was parked at UNI after the flight. This is shown in Fig. 7.9 which indicates a position

error of about 1-2 cm after the flight test. This represents a systematic error as ideally the

148
Figure 7.7 Ground Track for Flight from Rhodes Airport to University Airport

149
position error would have been zero-mean when the Saratoga was so close to the ground station.

Certain error sources may not be zero-mean, such as multipath [van Nee, 1991]. However, this

1-2 cm offset was seen during the initial static collection at UNI as well, suggesting that

multipath may not have been the cause. Still, Figs. 7.8-7.9 demonstrate a highly stable position

solution that is accurate to within a few centimeters, clearly enough accuracy to measure effects

on the order of 0.5 m.

The three-dimensional position error at the end of the flight was far less than the 0.5 m

predicted by the MATLAB simulation based on the relativity terms taken from Eq. 7.16. Thus,

the relativistic effects predicted by Deines are not supported by this flight test. Specifically, the

first two terms of Eq. 7.16 have been shown to be in error. GPS receiver algorithms should not

include corrections for these two terms.

To further verify the absence of the relativistic effects predicted by Deines, we turn to the

second method proposed earlier in Sec. 7.4.3 — examination of the parity vector. The parity

vector is plotted in Fig. 7.10 for two cases, one in which a troposphere model was used and one

in which the troposphere model was turned off. It should be noted that Fig. 7.10 is not based on

the PNAV solution from the Ashtech PRISM software package. The initial position at Rhodes

Airport was fed to the FORTRAN software to fix the L1 ambiguities. From there, the position

solution and calculation of the parity vector are done in the FORTRAN software independent of

the PNAV software. Recall that the norm of the parity vector, the parity space residual, is a

measure of consistency among the measurements (in this case double differences). The parity

150
Figure 7.8 Static Collection Showing Initial 3D Position Error for 10 Minutes at Rhodes
Airport Before the Flight to University Airport

151
Figure 7.9 Static Collection Showing Final 3D Position Error for 10 Minutes at
University Airport After the Flight from Rhodes Airport

152
space residual based on the prediction of Deines is 0.414 m. In Fig. 7.10, the parity space

residual does not exceed 0.08 m during the flight, and that is with the troposphere model turned

off. This is clear evidence that the relativistic effects predicted by Deines were not present

during the flight test.

The troposphere path delay is a particularly instructive phenomenon here because it

A@D
represents an error source that depends on which satellite is being used — much like the
c2
V@ v
and terms proposed by Deines. When the troposphere model is turned off, the parity vector
c2

changes as the aircraft gains or loses altitude. When the plane goes higher the parity space

residual goes up, and when the plane descends the parity space residual goes lower. The

correlation is evident when Fig. 7.10 is compared with Fig. 7.11, a plot of altitude during the

flight. Ultimately the plane lands at university airport and the effects of troposphere path delay

are diminished.

An important characteristic of Fig. 7.10 is that the troposphere errors drop out when the

plane lands at University Airport. This is in contrast to the predicted relativistic effects which

were predicted to accumulate during the flight to a size which would cause the parity space

residual to reach 0.4 m and not diminish. Thus, the predicted relativistic effects could not have

caused the parity space residual to grow to 0.08 m and return to 0.01 m as in Fig. 7.10. To

provide further proof, the parity space residual is also plotted (second curve in Fig. 7.10) for the

case where the troposphere model is turned on. The model clearly removes most of the

troposphere error as the parity space residual in this case stays at 0.03 m or less during the flight.

153
Figure 7.10 Parity Space Residual During Flight from Rhodes Airport to University
Airport

154
Figure 7.11 Altitude (Ellipsoidal Height) During Flight from Rhodes Airport to
University Airport

155
 There is no indication of relativistic effects causing inconsistencies among the carrier

phase measurements in Fig. 7.10. The parity space residual did not approach the 0.4 m level

predicted. Thus, we can say once more that the relativistic effects predicted by Deines were not

present in the flight data.

To show how sensitive the parity space residual is to disagreement among the

measurements, a cycle slip was artificially injected in one of the carrier phase measurements (see

Fig. 7.12). This was implemented by adding one cycle (19 cm) to the SV28 carrier phase

measurement recorded by the airborne receiver, resulting in a jump of 11 cm in the parity space

residual. This was done for each of the satellites, and jumps of anywhere from 9 to 16 cm were

seen, showing the high degree of sensitivity of the parity space residual to errors on the order of

magnitude predicted by Eq. 7.16.

For more proof, the predicted relativity effects were artificially injected as shown in

Figs. 7.13-7.14. The relativity effects were implemented as linear drifts for 12 minutes, and they

were subtracted (Fig. 7.13) from the carrier phase measurements. In Fig. 7.14 the relativistic

effects were added to the carrier phase measurements. This was done in order to avoid doubling

the effects in case they existed in the actual flight data. That is, if the relativistic effects were

present the parity space residual would approximately double if the relativistic effects were

added, and go closer to zero when the effects were subtracted. Because the parity space residual

grew to about 40 cm in both cases, it is clear that the predicted relativistic effects did not occur in

the original data.

156
Figure 7.12 Parity Space Residual During Flight from Rhodes Airport to University
Airport with Artificially Injected Cycle Slip in SV 28

157
Figure 7.13 Parity Space Residual During Flight from Rhodes Airport to University
Airport with Relativistic Effects Artificially Subtracted

158
Figure 7.14 Parity Space Residual During Flight from Rhodes Airport to University
Airport with Relativistic Effects Artificially Added

159
7.5 Conclusions from Flight Test Experiment of Relativistic Effects

It has been shown that the relativistic errors predicted by Deines are not supported by

A@D V@ v
experimental evidence. In particular, the and terms of Eq. 7.16 would cause large
c2 c2

effects on the order of 0.5 m that were not observed in actual flight data. It is important that GPS

receiver manufacturers do not include corrections for these two terms in position calculation

algorithms.

The fact that these two terms were not measured during the flight leads us to wonder

where the misinterpretation exists in the derivation of Deines. Nelson (1991) suggests that the

Earth Centered Earth Fixed (ECEF), Earth Centered Inertial (ECI), and topocentric reference

frames are equivalent as long as the appropriate corrections are made. It is possible that Deines

made an error in applying a transformation between reference frames, though that question is left

to the relativity experts.

V2
Note, however, that the third term in the integral of Eq. 7.16, & , drops out of the
2c 2

double difference and is not a part of the flight test results presented in this chapter. Note in

Table 7.5 that time dilation caused a predicted error of -0.041 m during the flight that would be a

component of the error in the single differences. However, stable clocks were not used during

the flight test, and it would not be possible to verify the existence of the time dilation term based

on the data collected. However, time dilation has been previously verified experimentally

[Hafele & Keating, 1972] and would affect receivers that rely on clock-aided navigation. Note

160
that the GPS satellite clocks are corrected for time dilation as shown in the first integral of

Eq. 7.16. There should be a similar correction for receivers on dynamic platforms. Thus, the

time dilation term proposed by Deines for moving receivers is correct, but needs to be interpreted

properly.

As an example, consider a low Earth orbit satellite (LEO) at an altitude of 300 km above

the Earth's surface. By Kepler's Third Law the orbital period would be 5,431 seconds, or about

90.5 min. Assuming a circular orbit, the velocity of the LEO would be 7.726 km/s. The clock

drift due to time dilation would be -3.32 x 10-10 s/s and the gravitational term, )N, would be

0.31 x 10-10 s/s. The combined clock drift is -3.01 x 10-10 s/s, or -0.09 m/s which is equivalent to

almost half the L1 wavelength per second. In only 100 s a 9 m error would build up. It is

important to remember that double differencing would eliminate this error because it affects each

single difference by the same amount. However, if clock coasting were used during periods of

poor geometry, this error would have to be corrected.

It is intuitive that a GPS receiver at some altitude above the Earth would require a

correction for gravitational potential as well. The effect is small for users on or near the Earth,

but some applications might require a GPS receiver to be placed on a dynamic platform at a

significant altitude. As an example, a clock at 39E N latitude with an altitude of 10,000 ft

(3048 m) above the geoid (a reference surface on which ideal clocks beat at the same rate)

experiences a clock drift of -3.32 x 10-13 (s/s). This is calculated by first approximating the

effective gravity at that latitude [Ashby and Spilker, 1996]:

161
 geff(cos 8) . 9.832099 & 0.051038(cos 8)2 & 0.000779(cos 8)4 (7.27)

where: geff is the effective gravity (m/s2)

8 is the latitude

For 8 = 39E, geff is 9.801 m/s2 which is used to determine the clock drift at 10,000 ft (3048 m):

g eff &9.801 m/s 2

& h' (3048 m)
c2 c2 (7.28)
' & 3.32 @ 10&13 (s/s)

This clock drift would cause an offset of -0.1 m after 1000 s (-17 min). Thus, even though this

is a small term, it is potentially significant for clock-aided navigation. A correction for

gravitational frequency shift is applied to the satellite clocks (Eq. 7.11), and it would be

appropriate to evaluate this effect for a given airborne application to determine the size of the

error and whether or not it can be ignored for a GPS receiver.

Intuitively, it seems that the relativistic effects for a moving GPS receiver should be

satellite independent. The two terms in Eq. 7.16 that have been shown to be in error are different

for each satellite. Ultimately, the determination of appropriate relativistic corrections is beyond

the scope of this dissertation and is a question to be posed to relativity experts.

162