zotero/storage/X65X88AU/.zotero-ft-cache

Chapter 1
THE SAGNAC EFFECT IN THE GLOBAL POSITIONING SYSTEM

Neil Ashby
Department of Physics, UCB 390. University of Colorado. Boulder, CO 80309-0390 USA. n ashby@mobek.colorado.edu

Abstract

In the Global Positioning System (GPS) the reference frame used for navigation is an earth-centered, earth-ﬁxed rotating frame, the WGS-84 frame. The time reference is deﬁned in an underlying earth-centered locally inertial frame, freely falling with the earth but non-rotating, with a time unit determined by atomic clocks at rest on earth’s rotating geoid. Therefore GPS receivers must apply signiﬁcant Sagnac or Sagnac-like corrections, depending on how information is processed by the receiver. These corrections can be described either from the point of view of the local inertial frame, in which light travels with uniform speed c in all directions, or from the point of view of an earth-centered rotating frame, in which the Sagnac effect is described by terms in the fundamental scalar invariant that couple space and time. Such corrections are very important for comparing time standards world-wide.

1. Introduction
The purpose of the Global Positioning System (GPS) is accurate navigation on or near earth’s surface. GPS also provides an accurate world-wide clock synchronization and timing system. Most GPS users are interested in knowing their position on earth; the developers of GPS have therefore adopted an Earth-Centered, Earth-Fixed (ECEF) rotating reference frame as the basis for navigation. Speciﬁcally, in the WGS-84(873) frame, the model earth rotates about a ﬁxed axis with a deﬁned rotation rate, ωE = 7.2921151247 × 10−5 rad s−1.[1],[2]
In an inertial frame, a network of self-consistently synchronized clocks can be established either by transmission of electromagnetic signals that propagate

11
G. Rizzi and M.L. Ruggiero (eds.), Relativity in Rotating Frames, 11-28. © 2004 Kluwer Academic Publishers.

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N. Ashby

with the universal constant speed c (this is called Einstein synchronization), or by slow transport of portable atomic clocks. On the other hand it is wellknown[3] that in a rotating reference frame, the Sagnac effect prevents a network of self-consistently synchronized clocks from being established by such processes. This is a signiﬁcant issue in using timing signals to determine position in the GPS. The Sagnac effect can amount to hundreds of nanoseconds; a timing error of one nanosecond can lead to a navigational error of 30 cm.
To account for the Sagnac effect, a hypothetical non-rotating reference frame is introduced. Time in this so-called Earth-Centered Inertial (ECI) Frame is adopted as the basis for GPS time; this is discussed in Section 2. Of course the earth’s mass encompasses the origin of the ECI frame and has signiﬁcant gravitational effects. To an extremely good approximation in the GPS, however, gravitational effects can be simply added to other effects arising from special relativity. In this article gravitational effects will not be considered. Even time dilation, which is an effect of second order in the small parameter v/c, where v is the velocity of some clock, will be neglected. I shall conﬁne this discussion to effects which are of ﬁrst order (linear) in velocities. The Sagnac effect is such an effect.
A description of the GPS system, of the signal structure, and the navigation message, needed to understand how navigation calculations are performed, is given in Section 3. In comparing synchronization processes in the ECI frame with those in the ECEF frame, taking into account relativity principles, it becomes evident that the Sagnac effect is a manifestation of the relativity of simultaneity. Observers in the rotating ECEF frame using Einstein synchronization will not agree that clocks in the ECI frame are synchronized, due to the relative motion. In fact observers in the rotating frame cannot even globally synchronize their own clocks, due to the rotation. This is discussed in Section 4. Section 5 discusses Sagnac corrections that are necessary when comparing remote clocks on earth by observations of GPS satellites in common-view. Section 6 introduces the GPS navigation equations and discusses synchronization processes from the point of view of the rotating ECEF frame. Section 7 develops implications of the fact that GPS navigation messages provide satellite ephemerides in the ECEF frame.

2. Local Inertial Frames
Einstein’s Principle of Equivalence allows one to discuss frames of reference which are freely falling in the gravitational ﬁelds of external bodies. Sufﬁciently near the origin of such a freely falling frame, the laws of physics are the same as they are in an inertial frame; in particular electromagnetic waves propagate with uniform speed c in all directions when measured with standard rods and atomic clocks. Such freely falling frames are called locally inertial

The Sagnac effect in the GPS

13

frames. For the GPS, it is very useful to introduce such a frame that is nonrotating, with its origin ﬁxed at earth’s center, and which falls freely along with the earth in the gravitational ﬁelds of the other solar system bodies. This is called an Earth-Centered Inertial (ECI) frame.
Clocks in the GPS are synchronized in the ECI frame, in which self - consistency can be achieved. Thus imagine the underlying ECI frame, unattached to the spin of the earth, but with its origin at the center of the earth. In this non-rotating frame a ﬁctitious set of standard clocks is introduced, available anywhere, all of them synchronized by the Einstein synchronization procedure and running at agreed rates such that synchronization is maintained. These clocks read the coordinate time t. Next one introduces the rotating earth with a set of standard clocks distributed around upon it, possibly roving around. One applies to each of the standard clocks a set of corrections based on the known positions and motions of the clocks. This generates a “coordinate clock time" in the earth-ﬁxed, rotating system. This time is such that at each instant the coordinate clock agrees with a ﬁctitious atomic clock at rest in the local inertial frame, whose position coincides with the earth-based standard clock at that instant. Thus coordinate time is equivalent to time which would be measured by standard clocks at rest in the local inertial frame. [4]
In the ECEF frame used in the GPS, the unit of time is the SI second as realized by the clock ensemble of the U. S. Naval Observatory, and the unit of length is the SI meter. In summary, the reference frame for navigation is the rotating WGS-84 frame, but clocks are synchronized in the underlying hypothetical ECI frame with a unit of time deﬁned by clocks (essentially on the geoid) and a unit of length determined by the deﬁned value of the speed of light, c = 299792458 m/s.

3. The GPS
The Global Positioning System can be described in terms of three principal “segments:" a Space Segment, a Control Segment, and a User Segment. The Space Segment consists essentially of 24 satellites carrying atomic clocks. (Spare satellites and spare clocks in satellites exist.) There are four satellites in each of six orbital planes inclined at 55◦ with respect to earth’s equatorial plane, distributed so that from any point on the earth, four or more satellites are almost always above the local horizon. Tied to the clocks are navigation and timing signals that will be discussed below.
The Control Segment is comprised of a number of ground-based monitoring stations which continually gather information from the satellites. These data are sent to a Master Control Station in Colorado Springs, CO, which analyzes the constellation and projects the satellite ephemerides and clock behav-

14

N. Ashby

ior forward for the next few hours. This information is then uploaded into the satellites for retransmission to users.
The User Segment consists of all users who, by receiving signals transmitted from the satellites, are able to determine their position, velocity, and the time on their local clocks.
The timing signals transmitted from each satellite are right circularly polarized. A carrier signal of frequency 1.542 MHz is modulated with a series of phase reversals; these phase reversals carry information bits from the transmitter to the receiver. Such phase reversals are conceptually important because the phase of an electromagnetic wave is a relativistic scalar. The phase reversals correspond to physical points in spacetime at which - for all observers - the electric and magnetic ﬁelds vanish.
The navigation message contained in these bit streams include values of parameters from which the receiver can compute the satellite’s position in the rotating ECEF frame, as a function of time of transmission. Also the GPS time on the satellite clock is indicated by a particular phase reversal in the sequence. A receiver distinguishes the signal from a particular satellite by comparing the bit streams, that are unique to each satellite, with bit streams generated by electronic circuitry within the receiver.
Additional information contained in the messages includes an almanac for the entire satellite constellation, information about satellite vehicle health, and information from which Universal Coordinated Time as maintained by the U. S. Naval Observatory–UTC(USNO)–can be determined.
The GPS is a navigation and timing system that is operated by the United States Department of Defense (DoD), and therefore has a number of aspects to it which are classiﬁed. Several organizations monitor GPS signals independently and provide services from which satellite ephemerides and clock behavior can be obtained. Accuracies in the neighborhood of 5-10 cm are not unusual. Carrier phase measurements of the transmitted signals are commonly done to better than a millimeter.
For purposes of the remainder of this article, I shall think of a signal from a GPS satellite as containing within itself information about the position and time of a transmission "event". The position is speciﬁed in the rotating ECEF frame. GPS time is time in an underlying local inertial frame. The signal propagates with speed c in a straight line in the ECI frame to the receiver, where it is decoded and its arrival time tR is compared to the time of transmission tT . The receiver can then form the so-called pseudoranges

ρ = c(tR − tT ).

(1.1)

A receiver continually forms such pseudoranges for each satellite being observed. A signal can be imagined abstractly as propagating with speed c from transmitter to receiver in a straight line in the ECI frame, with position and

The Sagnac effect in the GPS

15

u’

v

ux’

dx’

Figure 1.1. Synchronization by transmission of a signal

time of the transmission event “known" by the receiver. Possible clock biases in the receiver prevent the GPS time of the reception event from being known a priori.

4. Relativity of Simultaneity

To establish the connection between the Sagnac effect and the relativity of simultaneity, consider an observer moving with velocity v in the x direction relative to an inertial frame such as the ECI frame. To be speciﬁc, one can imagine measurements of unprimed quantities such as v and signal velocity u to be performed in the ECI frame, while primed quantities such as u are measured in the rest frame of the moving observer. Referring to Figure 1.1, let a signal be travelling with speed components (ux, uy) (measured in the moving observer’s frame). The vertical lines represent planes at x and x +dx . The signal travels a distance dx in the x direction and the moving observer desires to use this signal to transfer time from clocks in the plane at x to clocks in the plane at x + dx . Here I am neglecting higher-order terms in the velocity so dx = dx , there being no appreciable Lorentz contraction. Let the components of signal speed in the ECI frame be (ux, uy). The well-known Lorentz transformations for speed include the expression

ux

=

ux + v

1

+

uxv c2

.

(1.2)

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N. Ashby

The terms in the denominator of this expression arise from the time-component of the ordinary Lorentz transformation. In particular the second term in the denominator arises from the relativity of simultaneity, a consequence of the constancy of the speed of light. We wish to compare the propagation time of this signal, measured by the moving observer, with the propagation time measured in the ECI frame. The analysis is performed in the ECI frame.
If the moving observer moves a distance vdt in time dt, then the total distance travelled by the signal in the x-direction is uxdt, which is comprised of two contributions: the distance dx, plus the distance vdt required to catch up to the plane at x + dx . Thus

uxdt = dx + vdt,

(1.3)

and therefore the time required is

dx

dt =

.

(1.4)

ux − v

But from the expression for the Lorentz transformation of speed, keeping only terms of linear order in v,

ux

−v

≈

ux

1

+

uxv c2

.

(1.5)

and therefore

dx vdx

dt = + ux

c2

.

(1.6)

The ﬁrst term in this result is just the time required, in the moving frame, for

the signal to travel from the x plane to the x + dx plane. If the moving

observer ignores the motion relative to the ECI frame, this would be the time

used to synchronize clocks in the x + dx plane to clocks in the x plane. The

second term is the additional time required to synchronize the clocks in the ECI

frame. Note that in this second term, the value of ux has cancelled out, so that the value of the signal speed is irrelevant. The signal could be a light signal

travelling in a ﬁber of index of refraction n, or it could even be an acoustic

signal. The signal speed could even be variable, the last term would not be

affected.

Consider for example an optical ﬁber loop of length L and index of refrac-

tion n which by means of a system of pulleys is made to move with speed v

around in a closed circuit, relative to an inertial frame. The circuit itself could

be of any shape, such as a ﬁgure 8 or an oval. In such a case it is not useful

to speak of rotation, although Eq. 1.6 applies to the rotational case as well.

Eq. 1.6 applies to each inﬁnitesimal segment of the moving loop, since one

can imagine a sequence of moving reference frames each of which is instan-

taneously at rest with respect to the moving ﬁber loop and in which Eq. 1.6 is

The Sagnac effect in the GPS

17

valid. If a signal travels around the loop in a direction parallel to the velocity, then from Eq. 1.6, the total time required for the signal to make one circuit is

dx vL Δt+ = dt = ux + c2 ,

(1.7)

and the time required for the signal to make one circuit in the direction opposite

to the velocity is

dx vL Δt− = ux − c2 ,

(1.8)

The difference is

2vL

Δt = Δt+ − Δt− = c2 ,

(1.9)

and for two counterpropagating monochromatic beams this can be converted

into an observable interference fringe shift. If the beams are recombined in the

ECI frame where they have angular frequency ω, then the phase difference will

be

Δφ = ωΔt .

(1.10)

The Sagnac effect in a moving ﬁber loop is independent of the ﬁber’s index of refraction or of the shape of the loop. This has been conﬁrmed in recent experiments.[5]
For example for electrons of energy E = ω, the phase difference will be

2EvL Δφ = c2 .

(1.11)

Interference experiments with electrons have been reported in reference [6],

which also has a comprehensive discussion of the many different points of

view of the Sagnac effect that can be taken.

In the GPS, a decision was made to synchronize GPS clocks in the ECI

reference frame. The above discussion demonstrates that observers on earth, in

the ECEF frame, must apply a “Sagnac" correction (the second term in Eq. 1.6)

to their synchronization processes in order to synchronize their clocks to GPS

time.

The correction can be generalized slightly by noting that the distance dx is

in the same direction as the relative velocity v. If dr is the vector increment

of path in the direction of signal propagation, then the Sagnac correction term

can be written

v · dr dtSagnac = c2 .

(1.12)

For applications in the GPS, it is useful to describe this correction term another way, in terms of accounting for motion of the receiver during propagation of signals from transmitters to receivers. Henceforth only signals propagating

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N. Ashby

with speed c will be considered. This assumption also applies to measurements made locally by the moving observer in the ECEF frame, since at each instant the measurements of distance and time intervals are the same as they would be in an inertial frame which instantaneously coincides with the observer in the ECEF frame and which moves with the instantaneous velocity v of the ECEF observer. In Eq. 1.3, the velocity v is present to account for the fact that the signal must catch up to the position at x + dx which is moving with velocity v, and to ﬁrst order in the small quantity v/c leads directly to the Sagnac correction term in Eqs. 1.3 and 1.12. The Sagnac correction can thus be interpreted as an effect which arises in the ECEF frame when one accounts for motion of the receiver during propagation of the electromagnetic signal with speed c.

5. Time Transfer with the GPS

In the GPS navigation is accomplished by means of signals from four or
more satellites, whose arrival times are measured at the location of the receiver. I now consider one such signal in space, transmitted from satellite position rT at GPS time tT . Let the receiver position at GPS time tT be rR, and let the receiver have velocity v in the ECI frame. Let the signal (considered abstractly as a pulse) arrive at the receiver at time tR. During the time interval Δt = tR − tT , the displacement of the receiver is vΔt. Since the signal travels with speed c, the constancy of the speed of light c implies that

c2(Δt)2 = (rR + vΔt − rT )2 .

(1.13)

To simplify the equation, I deﬁne

R = rR − rT . Then to leading order in v,
c2(Δt)2 = (R + vΔt)2 ≈ R2 + 2v · RΔt .

(1.14) (1.15)

Taking the square root of both sides of Eq. (1.15) and again expanding to leading order in v gives

cΔt = R + v · RΔt . R

(1.16)

This equation can be solved approximately for Δt to give

R v·R

Δt = + c

c2

.

(1.17)

The second term in Eq. 1.17 is the Sagnac correction term, which arises when one accounts for motion of the receiver while the signal propagates from transmitter to receiver. This is illustrated in Figure 1.2.

The Sagnac effect in the GPS

19

ω

v Δt

c Δt R

tR = tT +

_R_ c

+

_R_._v_ c2

t*

T

(Sagnac correction)

Figure 1.2. Sagnac correction arising from motion of the ECEF observer.

Suppose that the receiver is ﬁxed to the surface of the earth, at a well-

surveyed location so that the receiver position rR is well known at all times.

The velocity of the receiver will be just that due to rotation of the earth with

angular velocity ωE, so

v = ωE × rR ,

(1.18)

We take rR to be the vector from earth’s center to the receiver position. Then the Sagnac correction term can be rewritten as

ΔtSagnac

=

ωE

× rR c2

·R

=

2ωE c2

·

1 2 rR × R

.

(1.19)

The quantity 2ωE/c2 has the value

2ωE c2

= 1.6227 × 10−21

s/m2

=

1.6227 × 10−6

ns/km2 .

The last factor in Eq. 1.19 can be interpreted as a vector area A:

(1.20)

1 A = 2 rR × R .

(1.21)

The only component of A which contributes to the Sagnac correction is along earth’s angular velocity vector ωE, because of the dot product that appears in the expression. This component is the projection of the area onto a plane normal to earth’s angular velocity vector. This leads to a simple description

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N. Ashby

of the Sagnac correction: ΔtSagnac is 2ωE/c2 time the area swept out by the electromagnetic pulse as it travels from the GPS transmitter to the receiver, projected onto earth’s equatorial plane. This is depicted in Figure 1.3, in which the receiver is on earth’s surface at the tip of the path vector R.
In the early 1980s clocks in remotely situated timing laboratories were being compared by using GPS satellites in "common view", that is when one GPS satellite is observed at the same time by more than one timing laboratory. In one experiment[7] signals from GPS satellites were utilized in simultaneous common view between three pairs of earth timing centers to accomplish a circumnavigation of the globe. The centers were the National Bureau of Standards (now the National Institute of Standards and Technology) in Boulder, Colorado; Physikalisch-Technische Bundesanstalt in Braunschweig, West Germany; and Tokyo Astronomical Observatory. A typical geometrical conﬁguration of ground stations and satellites, with the corresponding projected areas, is illustrated in Figure 1.4. The size of the Sagnac effect calculated varies from about 240 ns to 350 ns depending on the location of the satellites at a particular moment. Sufﬁcient data were collected to perform 90 independent circumnavigations. As Figure 1.4 shows, when a satellite is eastward of one timing center and westward of another, one of the Sagnac corrections is positive and the other is negative, so when computing the difference of times between the two terrestrial clocks, the Sagnac corrections actually add up in a positive sense.
The mean value of the residuals over 90 days of observation was 5 ns, less than 2 percent of the magnitude of the calculated total Sagnac correction. A signiﬁcant part of these residuals can be attributed to random noise processes in the clocks.
Sagnac corrections of the form of Eq. 1.19 are routinely used in comparisons between distant time standards laboratories on earth.

6. GPS Navigation Equations and the ECEF Frame
The navigation problem in GPS is to determine the position of the receiver in the ECEF reference frame. A by-product of this process is the accurate determination of GPS time at the receiver. In general neither the position nor the time is known, so the assumptions used in previous sections regarding the Sagnac effect are of little use. The principles of position determination and time transfer in the GPS can be very simply stated. Let there be four synchronized atomic clocks which transmit sharply deﬁned pulses from the positions rj at times tj, with j = 1, 2, 3, 4 an index labelling the different transmission events. Suppose that these four signals are received at position r at one and the same instant t. This is called "time-tagging at the receiver", meaning that observations of the various signals are made simultaneously at the receiver at

The Sagnac effect in the GPS
R

21
ω
Receiver

Figure 1.3. Sagnac correction arising from motion of the ECEF observer.

time t. Then from the principle of the constancy of the speed of light,

c2(t − tj)2 = |r − rj|2 , j = 1, 2, 3, 4.

(1.22)

These four equations can be solved for the unknown space-time coordinates of the reception event, (t, r). The solution will provide the position of the receiver at the time of the simultaneous reception events, t. No knowledge of the receiver velocity is needed. The Sagnac effect becomes irrelevant. At most one can say that because the solution gives the ﬁnal position and time of the reception event, the Sagnac effect has been automatically accounted for.
However there are complications from the fact that the navigation equations, Eqs. 1.22, are valid in the ECI frame, whereas users almost always want to know their position in the ECEF frame. For discussions of relativity, the particular choice of ECEF frame is immaterial. Also, the fact the the earth truly rotates about an axis slightly different from the WGS-84 axis, with a variable rotation rate, has little consequence for relativity and I shall not go into this here.
It should be emphasized strongly that the transmitted navigation messages provide the user only with a function from which the satellite position can be calculated in the ECEF as a function of the transmission time. Usually the satellite transmission times tj are unequal, so the coordinate system in which the satellite positions are speciﬁed changes orientation from one measurement

22

N. Ashby

ω
P N T

Figure 1.4. Common-view signals from three satellites provide an Around-the-World Sagnac experiment.

to the next. Therefore to implement Eqs. (1.22), the receiver must generally perform a different rotation for each measurement made, into some common inertial frame, so that Eqs. (1.22) apply. After solving the propagation delay equations, a ﬁnal rotation must then be performed into the ECEF to determine the receiver’s position. This can become exceedingly complicated and confusing. I shall discuss this in a later section.
The purpose of the present discussion is to examine ﬁrst-order relativistic effects from the point of view of the ECEF frame. Consider the simplest instance of a transformation from an inertial frame, in which the space-time is Minkowskian, to a rotating frame of reference. Thus ignoring gravitational potentials, the metric in an inertial frame in cylindrical coordinates is

−ds2 = −(c dt)2 + dr2 + r2dφ2 + dz2 ,

(1.23)

and the transformation to a coordinate system {t , r , φ , z } rotating at the uniform angular rate ωE is

t = t , r = r , φ = φ + ωEt , z = z .

(1.24)

This results in the following well-known metric (Langevin metric) in the rotating frame:

−ds2 = −

1

−

ωE2 r c2

2

(cdt )2 + 2ωEr 2dφ dt + (dσ )2 ,

(1.25)

The Sagnac effect in the GPS

23

where the abbreviated expression (dσ )2 = (dr )2 + (r dφ )2 + (dz )2 for the

square of the coordinate distance has been used.

The time transformation t = t in Eqs. (1.24) is a result of the convention to

determine time t in the rotating frame in terms of time in the underlying ECI

frame.

Now consider a process in which observers in the rotating frame attempt to

use Einstein synchronization (that is, the principle of the constancy of the speed

of light) to establish a network of synchronized clocks. Light travels along a null worldline so I may set ds2 = 0 in Eq. (1.25). Also, it is sufﬁcient for

this discussion to keep only terms of ﬁrst order in the small parameter ωEr /c.

Then

(cdt )2 − 2ωEr 2dφ (cdt ) − (dσ )2 = 0 , c

(1.26)

and solving for (cdt ),

cdt = dσ + ωEr 2dφ . c

(1.27)

The quantity r 2dφ /2 is just the inﬁnitesimal area dAz in the rotating coordinate system swept out by a vector from the rotation axis to the light pulse, and projected onto a plane parallel to the equatorial plane. Thus the total time required for light to traverse some path is

dt =
path

dσ path c

+

2ωE c2

dAz .
path

[ light ]

(1.28)

Observers ﬁxed on the earth, who were unaware of earth rotation, would use just dσ /c for synchronizing their clock network. Observers at rest in the underlying inertial frame would say that this leads to signiﬁcant path-dependent inconsistencies, which are proportional to the projected area encompassed by the path. Consider for example a synchronization process which follows earth’s equator in the eastwards direction. For earth, 2ωE/c2 = 1.6227 × 10−21 s/m2 and the equatorial radius is a1 = 6, 378, 137 m, so the area is πa21 = 1.27802 × 1014 m2 . Thus the last term in Eq. (1.28) is

2ωE c2

dAz = 207.4 ns.
path

(1.29)

From the underlying inertial frame, this can be regarded as the additional travel time required by light to catch up to the moving reference point. Simpleminded use of Einstein synchronization in the rotating frame gives only dσ /c, and thus leads to a signiﬁcant error. Traversing the equator once eastward, the last clock in the synchronization path would lag the ﬁrst clock by 207.4 ns.

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N. Ashby

Traversing the equator once westward, the last clock in the synchronization path would lead the ﬁrst clock by 207.4 ns.
In an inertial frame a portable clock can be used to disseminate time. The clock must be moved so slowly that changes in the moving clock’s rate due to time dilation, relative to a reference clock at rest on earth’s surface, are extremely small. On the other hand, observers in a rotating frame who attempt this ﬁnd that the proper time elapsed on the portable clock is affected by earth’s rotation rate. Factoring (dt )2 out of the right side of Eq. (1.25), the proper time increment dτ on the moving clock is given by

(dτ )2 = (ds/c)2 = dt 2 1 −

ωE r c

2

−

2ωEr 2dφ c2dt

−

dσ cdt

2

.

(1.30) For a slowly moving clock (dσ /cdt )2 << 1 so the last term in brackets in

Eq. (1.30) can be neglected. Also, keeping only ﬁrst order terms in the small

quantity ωEr /c,

dτ

=

dt

−

ωEr 2dφ c2

(1.31)

which leads to

dt =
path

path

dτ

+

2ωe c2

dAz .
path

[ portable clock ]

(1.32)

This should be compared with Eq. (1.28). Path-dependent discrepancies in the rotating frame are thus inescapable whether one uses light or portable clocks to disseminate time, while synchronization in the underlying inertial frame using either process is self-consistent.
Eqs. 1.28 and 1.32 can be reinterpreted as a means of realizing coordinate time t = t in the rotating frame, if after performing a synchronization process appropriate corrections of the form +2ωE path dAz/c2 are applied. It is remarkable how many different ways this can be viewed. The different ways discussed so far in this article include the fact that from the inertial frame it appears that the reference clock from which the synchronization process starts is moving, requiring light to traverse a different path than it appears to traverse in the rotating frame. The Sagnac effect can also be regarded as arising from the relativity of simultaneity in a Lorentz transformation to a sequence of local inertial frames co-moving with points on the rotating earth, or as the difference between proper times of a slowly moving portable clock and a Master reference clock ﬁxed on earth’s surface.
This was recognized in the early 1980s by the Consultative Committee for the Deﬁnition of the Second and the International Radio Consultative Committee who formally adopted procedures incorporating such corrections for the

The Sagnac effect in the GPS

25

comparison of time standards located far apart on earth’s surface. For the GPS it means that synchronization of the entire system of ground-based and orbiting atomic clocks is performed in the local inertial frame, or ECI coordinate system.

7. Sagnac-like effects due to rotation of the ECEF frame

By design, the ephemerides (positions) of the GPS satellites are broadcast in such a way that the receiver can compute their positions at the instant of transmission in the rotating WGS-84 reference frame. For time-tagging at the receiver, the propagation delays from different satellites can vary from about 67 ms to 86 ms. During this approximately 19 ms transmission time variation, the ECEF reference frame can rotate more than a microradian and the positions of the satellites due to this rotation alone can vary by over 30 meters while the satellites move in inertial space by as much as 60 meters. If this is not carefully accounted for, unacceptable navigation errors can occur.
It would lead to serious error to assert Eqs. 1.22 were valid in the ECEF frame. What the receiver must do is rotate the positions of each of the satellites, that have been computed in the rotating frame, into some chosen ECI frame. Then Eqs. 1.22 are valid and can be solved in the ECI frame. The resulting position found in the ECI frame is ﬁnally rotated into the WGS-84 frame and used for navigation.
To illustrate that these rotations give rise to Sagnac-like effects, suppose the chosen ECI frame instantaneously coincides with the WGS-84 frame at the instant of arrival of the earliest of the four signals. I denote the GPS time of arrival of this particular signal by t1, and the position of this particular satellite at this time as r1. Let the time intervals between the arrival of this signal and the other three signals be denoted by

Δti = ti − t1 , i = 1, 2, 3, 4

(1.33)

where for simplicity I have taken Δt1 = 0. During the time interval Δti the ECEF frame has rotated the amount ωEΔti. An active rotation of the satellite position ri(ECEF ) by the amount +ωEΔti is necessary in order to express the position of satellite i in the inertial frame in which the position r1 is expressed. This rotation operation can be expressed as

ri(ECI) = ri(ECEF ) + ωE × ri(ECEF )Δti .

(1.34)

The navigation equations then become c2(t − ti)2 = |r − ri(ECEF ) − ωE × ri(ECEF )Δti|2

(1.35)

and if I put Δt = t − t1 (no subscript on t) and Ri = r − ri(ECEF ) I obtain

c2(Δt − Δti)2 = |Ri − ωE × ri(ECEF )Δti|2

(1.36)

26

N. Ashby

Eqs. 1.36 have within them the four unknowns (Δt, r). The position solution for r will be in the ECI frame chosen for computation. After ﬁnding this position, the result must then be rotated into the ECEF frame for navigation. Since the ECEF frame rotates an amount ωEΔt during the time interval Δt, the ﬁnal solution for the position in the ECEF frame will be

r(ECEF ) = r − ωE × rΔt .

(1.37)

The size of the correction term in this last equation can easily be estimated, since Δt ≈ .015 s and r ≈ 6.4 × 106 m. A typical value will be about 9 meters. Eq. 1.36 can be solved approximately for Δt by expanding the square
on the right side, keeping only linear terms in ωE, and then taking a square root, similar to the approximations made in deriving Eq. 1.17. The result is

Δt

=

Δti

+

Ri c

+

ωE

×

ri(ECEF ) cRi

·

Ri

Δti

.

(1.38)

The last term in the above equation is a Sagnac-like correction. I can estimate its magnitude by substituting in an approximate expression for Δti:

Δti

≈

Ri c

−

R1 c

(1.39)

So the correction term becomes, after interchanging dot and cross products,

ωE

·

ri (E C E F c2

)

×

Ri

(1

−

R1/Ri)

.

(1.40)

Is this really a Sagnac correction? It is linear in the rotational velocity, the

coefﬁcient can be interpreted in terms of an area, and it is relativistic (there is a factor 1/c2).

In the case of time-tagging at the transmitters, signals are chosen for pro-

cessing which leave the transmitters at some chosen time tT . Then the broad-

cast ephemerides will all be calculated by the receiver in one and the same

ECEF frame. It would then be natural to choose for application of the navi-

gation equations (Eqs. 1.22) an inertial frame which coincides with this ECEF

frame at the instant tT of GPS time. But then the signals do not arrive simul-

taneously at the receiver, and the receiver motion during the interval between

arrival of the ﬁrst and last signals must be accounted for.

To illustrate the size of the Sagnac-like effects that occur in this situation,

let r denote the receiver position at transmission time tT , and let ri denote the transmitter position at time tT . Imagine these positions to be expressed in an

inertial frame which coincides instantaneously with the ECEF frame at time

tT . Let ti denote the arrival time at the receiver, of the signal from the ith satellite. The receiver position at time ti will be modiﬁed by earth rotation and

will be

r + ωE × r(ti − tT ) .

(1.41)

REFERENCES

27

The navigation equations in this inertial reference frame will be

c2(ti − tT )2 = |r + ωE × r(ti − tT ) − ri|2

(1.42)

Because of the similarity of this equation to Eq. 1.13 it is clear that Sagnac-

like corrections will enter solution of the equations. The times ti are however

known only to within an added constant, because of a possible error or system-

atic bias in the receiver’s clock. If the arrival times actually measured in the

receiver are ti, then

ti = ti + b .

(1.43)

where b is the receiver clock bias then the navigation equations become

c2(ti + b − tT )2 = |r + ωE × r(ti + b − tT ) − ri|2

(1.44)

and the unknowns are (b, r). Obviously there are many other ways of formulating the problem of accounting for receiver motion. A technical note[8] discusses these issues in more detail, with numerical examples.

8. Summary
In the GPS, the Sagnac effect arises because the primary reference frame of interest for navigation is the rotating Earth-Centered, Earth-Fixed frame, whereas the speed of light is constant in a locally inertial frame, the EarthCentered Inertial frame. Additional Sagnac-like effects arise because the satellite ephemerides are broadcast in a form allowing the receiver to compute satellite positions in the ECEF frame. In the case of time-tagging of observations at the receiver, it is necessary to rotate the satellite positions into a common ECI reference frame in order apply the principle of the constancy of c. In the rotating frame of reference the effect appears to arise from a Coriolis-like term in the fundamental scalar invariant. Whether synchronization procedures are performed by using electromagnetic signals or slowly moving portable clocks, to leading order the same Sagnac effect arises. The effect is of signiﬁcant magnitude and must be taken into account for accurate navigation. It is also necessary to apply Sagnac corrections when comparing remote clocks on earth’s surface.

References

[1] World Geodetic System 1984, (National Imagery and Mapping Agency, 1994) Report No. 8350.2, Third Edition, Amendment 1. NIMA Stock No. DMATR83502WGS84, MSN 7643-01-402-0347.
[2] Malys, S., and Slater, “Maintenance and Enhancement of the World Geodetic System 1984", in Proceedings of the 7th International Technical Meeting of the Satellite Division of the Institute of Navigation, Sept.

28

N. Ashby

20-23, Salt Lake City, UT (ION GPS-94), 17-24. The Institute of Navigation (1994).
[3] Landau, L., and E. Lifshitz, The Classical Theory of Fields, 4th ed., Pergamon, New York p. 254 (1997).
[4] Ashby, N., and Allan, D.W., "Practical Implications of Relativity for a Global Coordinate Time Scale", Radio Science 14, 649-669 (1979).
[5] Wang, R., Y. Zheng, A. Yao, and D. Langley, "Modiﬁed Sagnac experiment for measuring travel-time difference between counter-propagating light beams in a uniformly moving ﬁber", Phys. Letts. A 312, pp 7-10, (2003).
[6] Hasselbach, F., and M. Nicklaus, Phys. Rev. A48, pp 143-151, (1993).
[7] Allan, D. W., M. Weiss, and N. Ashby, "Around-the-World Relativistic Sagnac Experiment", Science 228, pp 69-70 (5 April 1985).
[8] Ashby, N., and M. Weiss, "Global Positioning Receivers and Relativity", NIST Technical Note 1385, U. S. Government Printing Ofﬁce, Washington, D.C., March (1999).