zotero/storage/PI6XT8IJ/.zotero-ft-cache

GPS AND RELATIVITY: AN ENGINEERING OVERVIEW

Henry F. Fliegel and Raymond S. DiEsposti GPS Joint Program Offlce The Aerospace Corporation
El Segundo, California 09245, USA

Abstract
We give and explain in detail the fonnuh for the re&fivisficcorwtiona to be implemented in high-speed airwaJ?,or when using other safellifes in conneclion with GPS,or when using GPS from Molher satellite. We explain how to urc these formdas in various scencvios, give numerical examples, and itemize the pitfalls to be avoided by (for esamplc) receiver manufacturn.

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INTRODUCTION

The Operational Control System (OCS) of the Global Positioning System (GPS) does not

include the rigorous transformations between coordinate systems that Einstein's general theory
of relativity would seem to require - transformations to and from the individual space vehicles

(SVs), the Monitor Stations (MSs), and the users on the surface of the rotating earth, and the

geocentric Earth Centered Inertial System (ECI)in which the SV orbits are calculated. There

is a very good reason for the omission: the effects of relativity, where they are different from

the effects predicted by classical mechanics and electromagnetic theory, are too small to matter

- less than one centimeter, for users on or near the earth. However, a new class of users, who

employ satellites that obtain time and position in space from GPS, cannot be satisfied with

the approximations in the current OCS. Furthermore, because those approximations have not

been publicly analyzed and presented, there is much confusion in the GPS literature. Eminent

scientists have been divided amongst themselves, wondering whether the OCS software does

cnuotrrneenet dspteocibfeicaretiwonristt-en6,

expecially meters in

since User

RthaengDeepEarrrtomr e(nUtRofE)D-efaernesetoisbneowtigrhetqeunieridnguntdheart

the the

Accuracy Improvement Initiative (AII). In this paper, we compare the predictions of relativity

to those of intuitive, classical, Newtonian physics; we show how large or small the differences

I
t

are, and how and for what applications those difference are large enough to make it necessary to correct the formulas of classical physics.

1

RELATIVITY: THE FORMULAS
- If two observers determine what intuitively we call the same quantity the distance between
two points, or the time interval between two events - they will measure different lengths and times, if (1) they are moving with respect to each other, (2) one is higher or lower than another in a gravitational field, or (3) one is accelerating with respect to the other. Users of GPS

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1. REPORT DATE
DEC 1996

2. REPORT TYPE

3. DATES COVERED
00-00-1996 to 00-00-1996

4. TITLE AND SUBTITLE
GPS and Relativity: An Engineering Overview
6. AUTHOR(S)
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
The Aerospace Corporation,GPS Joint Program Office,El Segundo,CA,90245

5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 8. PERFORMING ORGANIZATION REPORT NUMBER

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Approved for public release; distribution unlimited

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13. SUPPLEMENTARY NOTES
See also ADA419480. 28th Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting, Reston, VA, 3-5 Dec 1996

14. ABSTRACT
see report

15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF:

a. REPORT
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b. ABSTRACT
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Standard Form 298 (Rev. 8-98)
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encounter all three effects, and should correct their measurements accordingly, by formulas which we now explain.

The Velocity Effect (Lorentz Contraction)

Case (1) is the province of special relativity. In classical physics, if a transmitter in a vacuum sends out a radio frequency F in its own frame of reference, and if a receiver is moving with

1

velocity v at an angle A to the beam coming from the transmitter to the receiver, then the

signal will be received at the Doppler shifted frequency f , where f < F if A is less than 90"

(receiver moving away from the transmitter). Now, if the transmitter and receiver are replaced

by two units of hardware each of which both transmits and receives, each unit will measure

the other's frequency as "too low." We accept intuitively the fact that there is no real paradox

here. If the two units were colocated, then if one ran slower than the second, the second must

run faster than the first; but if they are not colocated, what an 0 b S e ~ emr easures depends not

only on the hardware, but on the state of motion between them; and each observer perceives

the motion the same way: "that other is receding from me at a rate v cos(A)." If our problem

is to measure f (which is what we observe), and to calculate F (which we must infer), then in

classical physics

where c is the speed of light, or radio, in free space. If the receiver is moving at right angles to the line of sight, -(A) = 0, and F = f; there is no Doppler shift. But the relativistic equivalent of the same formula is

where 7 is defined to be always equal or greater than one:

4

The 7 formula looks like the formula for going from sine to secant in ordinary trigonometry, and for good reason. The effect of special relativity (see [I]) is a coordinate rotation through an angle, the arctangent of ivlc. There is a related angle, a real number P , such that mn(P) = v/c, and 7 = sec(P); this provides a convenient way of calculating 7. Like the secant, the 7 factor (1) is very nearly equal to one for small angles (or v/c), because (2) it increases with the square of the angle V / C for small angles.

Notice that, even if cosA = 0, the Doppler shift is no longer zero, but 7; and so the frequency

as received is lower than the frequency transmitted. Since we naively expect, from intuition

and from classical physics, a Doppler shift of zero, the usual way

result is receiver's

to say frame

othf arteftehreenmceov-inagpptreaanrssmtoittbeer'srufnrneiqnugenslcoyw.staAngdaainrd,

to or

dcleoscckrii-e

the relativistic moving in the

as in classical physics, each

observer can consider himself at rest and the others to be moving; and it is no paradox that

each measures the others as "too slow." The relativistic formulas defy our intuitions of time

and space, but not the basic principles of logic; they can stand.

Now, the angle A should be corrected for aberration. From either a classical or a relativistic point of view, aberration is the change in the apparent direction from which a wavetrain appears to be coming because of the observer's velocity. In Eqs. (1) and (2) above, the cosine of the angle A should be calculated in the observer's frame of reference. If A' is the angle
that is calculated in (for example, the so-called "Earth Centered Inertial" or ECI system of
coordinates, then the relativistic equation to go from cosine of A' to the cosine of A is

and this equation is exact. The reader may wonder why the omnipresent 7 factor does not appear. It would, in the formulas for sin(A) and tm(A). However, we need ws(A), and the 7 factor cancels out. (For the derivations of all the above equations, see [I].)

Sa oGtPhSens,attoellciatelc-ulfaotre

the Doppler correction by which we correct the r e c e example, in crosslink ranging from one SV to another

~ e dfrequency from
- we first calculate

the relative speed of the transmitter to the receiver in the ECI frame, and the angle A' between

relative velocity and the line of sight between transmitter and receiver, also in the ECI frame.

We then apply Eq. (4) to obtain m ( A ) , and then Eqs. (2)-(3) to transform the observed

frequency f to the true transmitted frequency F, in the transmitter's frame of reference.

Gravitational Effect
We now turn to Case (2) above, where the transmitter is higher or lower than the receiver, in a gravitational field. If a photon (whether of light or of radio) falls into a potential well, it is shifted to a higher frequency; climbing up, it loses frequency. The quantum acts like a material particle, gaining kinetic energy as it falls through the gravity field. In quantum mechanics, it is shown that the energy of the photon is h f , where h is Planck's constant and f is the frequency. If it were a materiai particle, then from special relativity its total energy would be m 2, where
the kinetic energy is included in the increase of the mass m from the rest mass m. Since h f is equivalent to m 2, the mass m of the particle must correspond to h f/2 of the photon. Then,
since the change in potential in going from one point to another is the change in energy of the particle per unit mass, that correspondence requires that

In the earth's gravitational field, if a radio signal passes from a satellite at distance R, from earth's center to a receiver (whether on another satellite or on the ground) at distance 8,
a good approximation for the frequency shift correction, to transform from the frequency measured on the ground to the frequency transmitted by the satellite, is

So to the Doppler shift given by Eqs. (Z), (3), and (4) we add the frequency shift from Eq. (7). If R, is greater than Q, the correction is negative, and so by adding it to the received frequency f we obtain the frequency F of the transmitter in its own frame of reference.
Acceleration Effect
This is the correction to be made if an observer (= the receiver) is accelerating. In fact, the receiver, whether in an SV or on the ground, is certainly accelerated; it is either in free fall in orbit or subject to centripetal acceleration on the rotating earth. Acceleration effect can be absorbed in the gravity potential. The formal justification for doing so is the Principle of Equivalence in Einstein's general theory of relativity, which is best explained by an example. Suppose that a photon of light is radiated across a chamber through a distance r from wall A to wall B, along a line perpendicular to both walls. Suppose that the whole chamber is accelerating along this line, in the same direction as the photon is moving, by g meterstsecond squared. Define the initial velocity as zero at the instant that the photon is radiated; we are entitled to do this by the principles of special relativity. Now, by the time the photon arrives at the opposite wall, that wall is receding at a velocity v = -st, where t is r/c, and as measured by a receiver on the receding wall the photon will be Doppler-downshifted by an amount (in classical approximation)

But now consider a corresponding scenario in which the chamber is not accelerated, but in which a gravitational field produces a local acceleration due to gravity, g. As we have seen (Eqs. (5)-(7) above), the proportional frequency shift is the difference of gravitational potential
divided by c squared. By the Principle of Equivalence, the product gr in Eq. (8) corresponds to this potential difference, and we can allow for the observer's acceleration by modifying the
gravity term.

But we see that there is no need to do so, if we remember what this modification would be.

The -@/c of Eq. (8) is receiver velocity during

simply the correction to the the signal propagation time

-Dionppoluerr

shift due example,

to the change in the time it takes

the the

photon to traverse (r). But if we compute the Doppler effect by Eq. (Z), using the relative

velocity between transmitter and receiver at the time of reception of the signal, we include the

effect of Eq. (8) automatically. Only if we were to label signal events at the transmission time

would we need to include Eq. (8) explicitly. Among GPS users, hardly anyone does so. If, for

any reason, it is necessary so to do, rewrite Eq. (8) as

where Z is the user's from transmitter to

vector acceleration in an inertial frame (e.g, the ECI), and r' is the vector
receiver - at either transmission or reception time, in the "weak field

approximation." This is in the frequency domain; integrate, for timing receivers.

Summary of Corrections T o Be Made
Here is the summary. In general, to correct the frequency measured by a receiver for all relativistic effects, to obtain the frequency at the transmitter, we perform the following steps.

v (i) Obtain the velocity of the transmitting satellite, and the velocity u'of the receiver, in an
ECI coordinate frame, for the timcs of reception. Calculate the angle between the receiver's velocity vector and the line from transmitter to the receiver, in the ECI frame. That is the angle A' of Eq. (4).
(ii) Apply Eq. (4) to obtain the corresponding angle A in the receiver's frame of reference.
(iii) Calculate the magnitude of the velocity of the GPS satellite with respect to the receiver,
in the ECI frame. That is simply the magnitude of the vector difference C-3. Using Eq. (3),
calculate 7.
(iv) Calculate the Aflf frequency shifts using Eqs. (2) and (7), and add them together. This is
the total correction to be added to the received frequency to obtain the transmitted frequency.
To obtain the correction to add to the transmitted frequency in order to obtain the received frequency, perform the above steps and reverse the sign. To convert to the time domain, integrate, and assume any convenient zero of time. Without a timing receiver obsewing four satellites, one cannot synchronize to GPS time.
Since GPS receivers work in the time and not in the frequency domain, they handle the velocity, gravity, and acceleration shifts differently than d e d b e d above. First, each GPS space vehicle (SV) clock is offset from its nominal rate by about -4.45xlO-'O (= -38 microseconds per day) to allow for the relativistic offsets between the differences between the SV and the ground. Of this -38 microseconds per day, about -45 are due to the gravitational potential difference between the SV at its mean distance and the earth's surface, and +7 to the mean SV speed, which is about 3.87 kmlsec. To this mean correction, each receiver must add a term due to the eccentricity of the GPS orbit. It can be shown that this effect produces a variation in the SV clock, as seen from the earth, of
where E? is the vector of position of the SV from the ECI, and ? the velocity vector. This is
the equation given in ICD-GPS-200. It is appropriate for users on or near the earth's surface, but not users in space, who should apply the frequency correction equations given above, or their integrals to transform to the time domain.
STICKY WICKETS
We now turn to the misunderstandings that have arisen over the 15 years or so that GPS has been in operation. Most of the disagreements have semantics for a father. A few arose from the mathematics of relativity, in going from a rigorous treatment to the approximations that are found sufficient for the practical use of GPS.
T h e Newtonian World of the Operational Control System
The GPS Operational Control System (GPS) corrects the pseudoranges measured by its Monitor Stations for the sum of the gravitational and the velocity effects as discussed in the previous section, but computes the velocity effect only for the mean orbital speed of the satellite, which is about 3.87 kdsec. The question has been raised: why doesn't the operational s o h a r e calculate the velocity effect using the speed of each satellite relative to each Monitor Station, which varies

sinusoidally as the Monitor Station is carried around the rotating earth? In principle, it should, but the effect is negligible for two reasons: (1) the nature of the pseudorange measurement; and (2) the size of the effect.

The pseudorange measurement can be regarded from two points of view. On the one hand,

pseudorange is simply range, once the clock offsets are removed. As d e s c n i d in the previous

section, the ranges measured by a moving observer are foreshortened by the 7 factor. For an

SV speed of 3.87 kmhec, incurred neglecting the q

7 - 1 is 8.33~10-". factor is 30,000 km

The range multiplied

bisyty8p.3ic3a~ll1y0a-b"o-utth3a0t,0i0s,02k.5m.mTillhime eetrerrosr.

Close enough for government work.

On the other hand, the pseudorange measurement should be equivalent to accumulated Doppler, once the ionospheric delays are removed. If the MS accumulated Doppler measurements over a pass, then, since the OCS assumes Newtonian physics, it would use Eq. (1). In the more
nearly correct world of special relativity, we should use Eq. (2), which differs from Eq. (1) by the 7 factor. Not only the velocity term, but the observed frequency f , is multiplied by 7.
And 7 is not constant, because the rotation of the earth (up to 0.465 kmlsec at the equator) vectorially adds to the 3.87 kmlsec of the satellite when we compute the satellite speed relative to a Monitor Station. Corresponding to a possible v of 3.87 f 0.465 km/sec, 7 can vary from 8.33xlO-"f -1.88~10-". The constant part is absorbed in the mean relativistic offset described above, but the variable part would alias into the accumulated Doppler range. For example, over one hour = 3,600 seconds, the error could be 1.88~10-" times 3,600 seconds times c = 20.3 meters. Clearly, this would be a major source of error in the Operational Control System.

But this error would be incurred only if the station clocks were independent of the GPS satellite clocks, each MS keeping its own time. That's not the way the OCS works. Station time is estimated in the Kalman filter together with the SV clocks, and each MS clock is effectively being updated continuously by the satellites. The station clock is used only to bridge the gap in time between measurements; but since several satellites are always in view, these measurements
are vjrtually instantaneous, except for the different signal propagation times from different satellites. These times are about 0.1 sec, and, multiplying by 7,once more we derive an error of about 2.5 millimeters.

However, "springy"

t-hitshahtapips,y

result it has

depends entirely on the Kalman filter. At present, the filter is very a short time constant, because it is adjusted to absorb ephemeris

errors which can change rapidly (for example, at the onset of eclipse). Station clocks, satellite

clocks, and ephemeris parameters are all allowed to move together, and so Monitor Station

time is controlled entirely by the satellite clocks. (If several satellites are in view, the Monitor

Station clock is mathematically redundant, and in practice it drops out of the solution.) But

if the filter were retuned to be very "stiff," with long time constant and reasonable weights

assigned to the Monitor Station clocks, then the appropriate time over which the relativistic

effects would act would no longer nearly equal the signal propagation time, but would be

several hours or days; and then the error incurred in the hypothetical accumulated

dduoepptoletrhsecenneagrleioct-edm7anfyacmtoertwerosu. ld

approximate

that

In principle, the critics of GPS in the relativity debate have not been completely wrong. The
neglected 7 factor could hurt us. The OCS software should be reformulated. Nevertheless, in practice, neglect of relativity does not now contribute measurably to the GPS ermr budget, as the OCS software is currently configured.

The ECI Coordinate System
It has been said (e.g., [?I, and [dl) that the Operational Control System (OCS)of GPS makes
all calculations with respect to an ECI ("Earth Centered Inertial") frame of reference. This is a perfectly defensible statement, depending on how one chooses to describe the GPS time scale.

In actual practice, GPS operates in a mixed coordinate system: the spatial coordinates are

ECI, but the time rate is appropriate to observers on the surface of the rotating earth,

that is, in the ECEE GPS time is steered as closely as possible to U.S. Naval Observatory

time (UTCIUSNO]), which in turn is steered closely to International Atomic Time (TAI =

Temps International Atomique). By an important theorem of general relativity, ideal frequency

standards on the rotating geoid run at the same rate, because the effect of the earth's rotation

(by which clocks on the equator would run more slowly than clocks at the poles) is cancelled

by the earth's gravitational potential (since clocks on the equator are farther from the earth's

center, and so at higher potential, than clocks at the poles). So a frequency standard anywhere

I

on the earth's surface runs at the same rate as one at either pole, and is offset by a constant rate from a (hypothetical) clock at the earth's center. By the definitions adopted by the
International Astronomical Union (IAU), the ideal time that GPS, UTC, and TAI time scales

realize as nearly as possible is called Terrestrial Time (TT),and the time interval unit of TT is

the Standard International (SI) second on the geoid, also called the coordinate second. Then,

in at least one sense, the time scale of GPS is defined in the ECEF coordinate frame. Also

by Recommendation T1 of the Sub-Group on Time of the IAU Working Group on Reference

Systems, "coordinate times in (a) non-rotating reference system having ... spatial origin at the

P

geocenter ...(is) designated as Geocentric Coordinate Time (TCG)"; this definition was adopted

by the IAU in its General Assembly in 1991. Therefore, if GPS were to operate in the ECI

coordinate system in the strict sense, its time scale would be TCG, which is offset from TT,

1

etc. by the potential difference between earth's center and its poles. This GPS does not do.

If a satellite user of GPS were naively to map all measurements "to an ECI frame," using

I

Geocentric Coordinate Time (TCG), an error would be incurred of roughly 60 microseconds

per day.

However, there is a sense in which the TAI time coordinate is related to the ECI. Although all ideal frequency standards run at the same rate on the geoid, clocks cannot be synchronized
unless some sort of signals pass between them. Because of the earth's rotation, when clocks on different continents are compared by travelling frequency standards, satellite signals, or other means, corrections must be made for the relativistic Sagnac effect. To calculate these corrections, one must know the rate of the earth's rotation, that is, the angular rate of the ECEF frame relative to the ECI. Implicitly, then, in the process of forming TAI, to which GPS time is steered, reference is made to the ECI.

Missing Relativity Terms?
Oversimplifications such as in 141, which disseminated the mistaken notion that GPS time is calculated "in the ECI," ignoring the earth's rotation, misled Steven Deines, in his paper entitled, "Uncompensated relativity effects for a ground-based GPS receiver."[sl Deines argued that
The current ...GPS relativity corrections were based on
an Earth centered inertial reference frame. The derivation assumed [that] the receiver obtains inertial GPS coordinate

time from the treated tacitly

assatbeleliintegs.stHatoiowneavreyr,inthtehereicneeirvteiarlhfarsambeee..n.

relativity effects for a ground-based receiver include

gravity and Earth rotation. Airborne GPS receivers have

larger effects, and spaceborne GPS receivers have the worst

uncompensated relativity effects.

Deines used equations from Robert Nelson's work (see [6]) in an attempt to show that corrections depending on the earth's rotation should be added to those incorporated in existing GPS r e ~ i v e r saccording to ICD-GPS-200. A complete critique of Deines' derivations has been made by Neil Ashby..Pl Deines overlooked cancellations among several of his terms, by which the Nelson expressions reduce to very nearly the formulation of ICD-GPS-200.
Nelson shows ([6], p. 21) that the transformation between a time increment dt' expressed in a rotating andlor linearly accelerated reference frame and a time increment dt in a nonaccelerating frame is given by

dt

=

7

[

l

+

I+.,?
- c2

+

-B( - +dpG'
$dl?

x

$

]

dt'

where t? is the acceleration, the position vector of a GPS receiver, v' its velocity, and G the earth's angular rotation rate. If the ground station is motionless with respect to the earth's
surface, then the acceleration vector @ is determined by 3 and by 8:

and also the velocity ii:

Then one may rewrite Eq. (ll), letting the origin from which p'is measured to be the GPS
receiver, of which the position in ECI coordinates is the vector 8. Eq.(11) now becomes

As Nelson says, terms cancel in Eq. (14) to give a very simple result. The cancellation is as follows. As is shown in vector algebra,
Therefore, we can write
and Eq. (14) becomes

Eq. (17) "is just what one would expect by a Lorentz transformation from the center of rotation to the instantaneous rest frame of the accelerated origin" ([6], p. 23). Except for the leading 7 factor, it is the same as the formula derived in classical physics for the signal travel time from the GPS satellite to the ground station. As we have shown, introducing the y factor makes a change of only 2 or 3 millimeters to the classical result. In short, there are no "missing relativity terms." They cancel out.
8 REFERENCES
[I] L. Landau, and E. Lifshitz 1951, The Classical Theory of Fields, trans. M. Hamermesh (Addison-Wesley, Reading, Massachusetts, USA).
[2] P.K. Seidelmann (ed.) 1992, Explanatory Supplement to the Astronomical Almanac (University Science Books, Mill Valley, California, USA).
[s] N. Ashby, and J J . Spilker, Jr. 1996, "Introduction to relativistic effects on the Global Positioning System," in Global Positioning System: Theory and Applications, Vol. I, ed. B. Parkinson and J.J. Spilker, Jr. (American Institute of Aeronautics and Astronautics, Inc., Washington, D.C.. USA)
[4] P.S. Jorgensen 1988-89, "Relativity and intersatellite tracking, * Journal of the Institute of Navigation, 35, 429ff.
[5] S.D. Deines 1992, "Uncompensated relativity effects for a ground-based GPS re-
ceiver, " 0-7803-0468, IEEE.
161 R.A. Nelson 1991, "An analysis of g e n e d ~elativityin the Global Positioning
System time transfer algorithm, " report prepared for Sachflreeman Associates Inc.,
Landover, Maryland, USA.
[7] N. Ashby 1996, "Response to Steve Deines' Paper Entitled 'Missing Relativity
Terms in GPS',"University of Colorado, Boulder, Colorado, USA (private communica-
tion).
[8] "NAVSTAR GPS Space Segrnent/Navigation User Interfaces, " ICD-GPS-200C, GPS JPO, 15 Dec 1994.
[s]E. Kaplan (ed.) 1996, Understanding GPS: Principles and Applications (Artech
House, Norwood, Massachusetts, USA).
[lo] MCS software listing obtained from John Berg of the Aerospace Corp., 8 July 1996.
(111 "MAGR Computer Program Development Specification, " XP-RCVR-MAGR, Part I, 23 Jan 1991.

Questions and Answers
STEVEN HUTSELL (USNO): You and I have been talking about this intermittently over the phone over the past couple months or so. Obviously, the question of trying to make the best utilization of the ephemeris and solar process noise values in the MCS Kalman filter is something to consider to try to best compensate for this. I have to plead ignorance on my
behalf because I haven't done any studies really or any simulations. I'm wondering if you or
anyone else at Aerospace has pursued this with different cboices for simulation.
HENRY FLIEGEL: No, I told our sponsor that for the limited amount of money that he paid us, he gets about this much and no more. It would really take quite a bit of effort to simulate in detail what that Kalman filter is doing. And frankly, I don't think I have the expertise myself, but of course there are other people at Aerospace Corporation that do. But we would have to be turned on by the Air Force or by somebody that has a charge number.
GERNOT WINKLER (INNOVATIVE SOLUTIONS I N T I ) : I have two comments or questions to ask you for your reaction to that. Number one, you said that the Kalman filter has no place to put these residuals; or better, that they will not be able to show up in the ephemeris part because it does not agree with the observations.
But there's one parameter which is very sensitive, and that's the frequency offset in the clocks. You have any error in the frequency offset in the clocks, it will, of course, accumulate since the clock offset will be larger proportional to the time offset since the last upload. But that's the most sensitive part, and I'm not sure if your conclusion is correct in that regard.
Number two, for most time users who use GPS, they're exactly those which you have excluded from your consideration, because they do observe the satellites in 30-minute passes; and an apparent frequency offset of 10 to the minus 10th in 30 minutes produces an offset of more than 10 nanoseconds. And that's really the explanation for the bowing effect, isn't it?
HENRY FLIEGEL: I wonder if it is. Let me deal with your last point first. Of course, we are concerned that some users may in fact be affected by this. That's why we're putting out the report. And I don't know whether perhaps the bowing effect is at least part due to this, other parts, ionosphere and what have you.
To get back to your first point, remember, though, that we're continually reporting the measured offset of each clock in the system from GPS time. So we're absorbing and reporting the very thing that you're taking about.
GERNOT WINKLER: Excuse me, yes, of course, during the time when your Monitor Station sees the clock. But you're uploading predicted information. It is here that the residual error in the frequency offset will become important.
HENRY FLIEGEL: It may. But remember, the effect repeats - a t least, in my calculations -
day after day after day. So essentially, if you take a mean value for a given day, probably the predictions will not be too badly impacted.
My other point was I said only it would be very difficult to alias clock error into ephemeris to any large extent. But I'm not saying that the clock error has just disappeared.
CARROLL ALLEY (UNIVERSITY OF MARYLAND): I'm happy to hear Henry agree at the end that one should look very carefully at the details of the relativistic physics. And it has been my contention that if one does that, software changes alone in the current system might bring these UREs down below the 1-meter level, and get actual user performance down a t the one to 2-meter level with the existing system. And it has been very difficult to get into the

system and to get the adequate support to do these studies. But I'm very happy that there seems to be an agreement that one should do this. It is not an expensive matter compared to the 10 or 12 billion that's being invested in the GPS.

I would like to make another point. When one looks at differential GPS,the correction that

needs to be made primarily is the difference between the radial distance in the ephemeris and

the time reading on the satellite. And I believe this comes in because of a mix-up, or aliasing

if you will, bemeen these two quantities in the iteration procedure that the Kalman filter is

-folIlomweinagn.,

And that it took a

if one perhaps does the explicit recognition of the long time to get general relativity down properly,

special relativistic effects but I think that is more

or less correct now. But it's the absence of any explicit acknowledgment of special relativistic

effects to the

due to the speed of light being the same whenever measured by an relativity of simultaneity and the associated Lorentz transformation

opbhsyesricvser-,

leading there's

nothing of that at all modeled in the current system, and I think it should be. Thank you.