GPS clock synchronization in ECI frame

[cianfa72]

TL;DR Summary

About physical process involved to synchronize GPS clocks in Earth Centered Inertial (ECI) frame

Hi,
starting from this old thread GPS clock synchronization I've a doubt about the physical process employed to synchronize clocks bolted on GPS system satellites.

We said that clock synchronization is frame dependent. In other words we must select a coordinate chart (aka reference frame) that implicitly defines the notion of 'the same time $t$' in that specific frame.

On the other hand each physical clock follows its own timelike worldline through spacetime, so we can ask: at a given coordinate time $t$ in the chosen coordinate chart (frame) which is the value of its own proper time read (or shown) from each physical clock ? By definition physical clocks are synchronized -- w.r.t. the chosen coordinate chart -- if in the above scenario they read (show) the same value.

So far, so good...coming back to the topic of thread: in the referenced thread we said that bolted clocks on GPS satellites

are and remain synchronized in the ECI frame so there is never any difference in timing in the ECI frame between any pair at any time in the ECI frame. They don't drift apart in the ECI frame, ever.

My question is: which is the process involved to synchronize (and keep synchronized) them in the ECI frame ?

 

[jbriggs444]

My question is: which is the process involved to synchronize (and keep synchronized) them in the ECI frame ?

https://space.stackexchange.com/que...onized-with-each-other-in-the-Earth's-frame-o

The clocks are designed to not tick according to their own proper time but to instead tick at a rate adjusted to align with ECI coordinate time. That gets them close. But they are still corrected periodically using signals from the ground. Since the satellites orbit at a fairly constant altitude at a constant ECI-relative speed, a fixed rate adjustment is all that is needed for this part.

According to the commentary on the above reference, the result of the periodic corrections from the ground is a sort of annotation: "my clock is currently offset by <such and such>".

 

[cianfa72]

As explained in the referenced link that update can occur every 8 hours, but in practice is done every 24 hours.

According to the commentary on the above reference, the result of the periodic corrections from the ground is a sort of annotation: "my clock is currently offset by <such and such>".

Does 'my clock' above (bold is mine) actually refer to the physical clock on the ground station ?

Not sure to grasp it: we send (actually encode) in the periodic update messages from ground station to GPS satellites the offset between the local ground clock and the time encoded in the messages sent from the GPS satellites ?

 

[jbriggs444]

Does 'my clock' above (bold is mine) actually refer to the physical clock on the ground station ?

No. It refers to the physical clock on the satellite. The satellite clock is <such and such offset> from the ECI standard.

 

[cianfa72]

No. It refers to the physical clock on the satellite. The satellite clock is <such and such offset> from the ECI standard.

ok, hence -- in order to check they are actually in synch -- we must subtract from the measured offset half of the round trip time as measured by the clock on ground (the clock that read/measure ECI standard).

In other words the synchronization processes employed for GPS system is logically the same as the Einstein synchronization procedure, I believe.

 

[jbriggs444]

ok, hence -- in order to check they are actually in synch -- we must subtract from the measured offset half of the round trip time as measured by the clock on ground (the clock that read/measure ECI standard).

In other words the synchronization processes employed for GPS system is logically the same as the Einstein synchronization procedure, I believe.

It is different. You get a coded time stamp from the satellite. You add the calculated travel time (and processing time, if necessary) to the time stamp to figure out what time it is "now", "over there" according to the satellite clock. And you calculate the offset from the time it is "now", "right here" according to the ground clock.

The calculated offset is the adjustment that needs to be applied to clock readings returned by the satellite clock.

Yes, if we pretend that the space-time region is suitably flat, this technique, like almost all other synchronization techniques, delivers roughly the same results as Einstein synchronization.

However, Einstein synchronization would have the problem that the ground station is moving in the ECI frame: the inbound and outbound travel times are not equal. Dividing round trip time by two would be the wrong thing to do.

 

[cianfa72]

The calculated offset is the adjustment that needs to be applied to clock readings returned by the satellite clock.

I took it as clocks on GPS satellites are never really 'adjusted': the aim of the synchronization process is just calculate the offset to apply to the clock readings (time) returned by GPS satellite clock.

However, Einstein synchronization would have the problem that the ground station is moving in the ECI frame: the inbound and outbound travel times are not equal. Dividing round trip time by two would be the wrong thing to do.

Do you mean the inbound and outbound travel time respectively from (to) the GPS satellite to (from) the ground station ?

 

[PeterDonis]

clocks bolted on GPS system satellites

They aren't "bolted on". You seem to be imagining analog clocks with faces somehow bolted on to the surfaces of the satellites.

The "clocks" are quartz crystals (a much more sophisticated version of the crystals inside quartz wristwatches) coupled to electronic circuitry. All of this is inside the satellites.

at a given coordinate time $t$ in the chosen coordinate chart (frame) which is the value of its own proper time read (or shown) from each physical clock ?

I'm not sure what you mean. The coordinate time $t$ and the proper time of the clock are two different things. And with the GPS satellites, there is a third time, the "adjusted" time of the clock--which is the physical proper time of the clock (the time the quartz crystals are keeping), adjusted by further circuitry so that its "flow rate" matches that of coordinate time $t$ in the ECI frame.

which is the process involved to synchronize (and keep synchronized) them in the ECI frame ?

As above, there is further circuitry in each satellite that produces an adjusted "clock time" by taking as input the physical proper time of the quartz crystal clock and applying a rate correction. The satellites also receive inputs from ground stations that allow them to correct errors that build up over time, since the rate correction applied on board each satellite won't be exactly correct (both because of unavoidable imprecision in the circuitry and because the satellites' orbital parameters change over time and the rate correction circuitry can't detect that on its own, it has to be told about it by the ground stations).

 

[cianfa72]

The "clocks" are quartz crystals (a much more sophisticated version of the crystals inside quartz wristwatches) coupled to electronic circuitry. All of this is inside the satellites.

Yes, of course.

I'm not sure what you mean. The coordinate time $t$ and the proper time of the clock are two different things.

About ECI frame coordinate time $t$ my understanding is as follows: as explained here we take 3 Earth-centered orthogonal axes fixed w.r.t. the stars. In imagination we can consider a 'latticework' of rods aligned with those 3 axes having standard clocks Einstein synchronized bolted on them. To do that in imagination we are neglecting the gravitation field (i.e. geodesic deviation). The time rate of those clocks is -- by definition -- the coordinate time $t$ of the ECI frame.

And with the GPS satellites, there is a third time, the "adjusted" time of the clock--which is the physical proper time of the clock (the time the quartz crystals are keeping), adjusted by further circuitry so that its "flow rate" matches that of coordinate time $t$ in the ECI frame.

Maybe I've not a clear understanding about this: the circuitry onboard the GPS satellite just 'adjusts' the 'flow rate' of the local clock against the 'flow rate' of the ECI coordinate time $t$ or really synchronize it ?

To me the adjustment of clock rate and the term synchronization are really two different things: in fact we could have clocks that beat the same 'flow time' however there is a fixed 'offset' between them.

 

[jbriggs444]

Maybe I've not a clear understanding about this: the circuitry onboard the GPS satellite just 'adjusts' the 'flow rate' of the local clock against the 'flow rate' of the ECI coordinate time $t$ or really synchronize it ?

Try Googling for "gps and relativity". That eliminates the chaff of folks advertising receivers and gets to system design.

To achieve this level of precision, the clock ticks from the GPS satellites must be known to an accuracy of 20-30 nanoseconds. However, because the satellites are constantly moving relative to observers on the Earth, effects predicted by the Special and General theories of Relativity must be taken into account to achieve the desired 20-30 nanosecond accuracy.

Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts that we should see their clocks ticking more slowly (see the Special Relativity lecture). Special Relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect of their relative motion [2].

Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth's mass is less than it is at the Earth's surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away (see the Black Holes lecture). As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day.

The combination of these two relativitic effects means that the clocks on-board each satellite should tick faster than identical clocks on the ground by about 38 microseconds per day (45-7=38)! This sounds small, but the high-precision required of the GPS system requires nanosecond accuracy, and 38 microseconds is 38,000 nanoseconds. If these effects were not properly taken into account, a navigational fix based on the GPS constellation would be false after only 2 minutes, and errors in global positions would continue to accumulate at a rate of about 10 kilometers each day! The whole system would be utterly worthless for navigation in a very short time.

The engineers who designed the GPS system included these relativistic effects when they designed and deployed the system. For example, to counteract the General Relativistic effect once on orbit, the onboard clocks were designed to "tick" at a slower frequency than ground reference clocks, so that once they were in their proper orbit stations their clocks would appear to tick at about the correct rate as compared to the reference atomic clocks at the GPS ground stations. Further, each GPS receiver has built into it a microcomputer that, in addition to performing the calculation of position using 3D trilateration, will also compute any additional special relativistic timing calculations required [3], using data provided by the satellites.

Reference 3 in the above is to: "GPS Interface Control Document ICD-GPS-200C (10 Oct 1993)"

To me the adjustment of clock rate and the term synchronization are really two different things: in fact we could have clocks that beat the same 'flow time' however there is a fixed 'offset' between them.

A little-used term for making rates match is "syntonization". If you want to synchronize clocks and not have to do it continuously, you want to syntonize them first.

 

[cianfa72]

A little-used term for making rates match is "syntonization". If you want to synchronize clocks and not have to do it continuously, you want to syntonize them first.

So the adjustment as described in the above note actually is intented to synthonize the rate of GPS clocks with the rate of ECI coordinate time, I believe.

 

[jbriggs444]

So the adjustment as described in the above note actually is intented to synthonize the rate of GPS clocks with the rate of ECI coordinate time, I believe.

Yes. That portion that I'd highlighted in bold is for rate syntonization.

 

[cianfa72]

Yes. That portion that I'd highlighted in bold is for rate syntonization.

ok, and what about time synchronization? To do that I believe some 'convention' is needed such as Einstein synchronization convention.

 

[jbriggs444]

ok, and what about time synchronization? To do that I believe some 'convention' is needed such as Einstein synchronization convention.

Asked and answered already.

The satellite sends a coded signal that includes its current time stamp. The ground station receives this, computes expected travel and processing time for the signal and deduces the current error in the satellite clock. The computed value is communicated and used to offset the readings reported by the satellite clock.

The ECI convention is embedded in the computation of the expected travel time for the signal.

Note that Einstein has a synchronization procedure, not a convention. The convention is the choice of a standard of rest to use for the procedure.

 

[cianfa72]

The ground station receives this, computes expected travel and processing time for the signal and deduces the current error in the satellite clock. The computed value is communicated and used to offset the readings reported by the satellite clock.

Sorry, not sure to grasp this point: the computed value you were talking about is not communicated back to the GPS satellite, it is employed on Earth receivers just to offset the readings reported (sent) by the satellite clock, right ?

The ECI convention is embedded in the computation of the expected travel time for the signal.

Can you elaborate please this point ? Thank you.

 

[cianfa72]

Note that Einstein has a synchronization procedure, not a convention. The convention is the choice of a standard of rest to use for the procedure.

ok, so which could be an example of 'convention for standard rest' to use for the Einstein synchronization procedure ?

 

[jbriggs444]

Sorry, not sure to grasp this point: the computed value you were talking about is not communicated back to the GPS satellite, it is employed on the receivers on the Earth just to offset the readings reported (sent) by the satellite clock, right ?

I believe that it is communicated back to the satellite and sent from there to the receivers as part of the satellite's normal signaling. But that's just window dressing. It does not matter how the data gets to the receiver as long as the receiver gets it.

Can you elaborate please this point ? Thank you.

This is about how the ECI convention is embedded in the computation of signal travel time for a clock reading being sent from satellite to ground station.

So you have this ground station. Its position is accurately known. It is slowly moving as the Earth rotates against the ECI frame. Its time is accurately known and kept synchronized with all of the other ground stations using the ECI synchronization convention.

You have this satellite. Its trajectory is accurately known based on methodology that we need not go into. The satellite has an on-board clock. We want to compare that on-board clock to ECI standard time.

The satellite sends a signal that includes a pattern that can be used as a "MARK!" reference and a coded transmission of the satellite's current clock reading.

We want to know how long it takes for that signal to get from the satellite to the ground station.

At the ground station with a known (ECI) location, we know the (ECI) time of arrival. And we can compute where (ECI location) the satellite would have been on its trajectory at the (ECI) time of sending to get a signal to arrive when (ECI time) it was observed to arrive.

So we can calculate the distance (ECI-relative) the light signal would have travelled. And we know the speed (invariant) of light. So we can calculate the (ECI-relative) time of flight.

[Plus some GR details, very likely]

But all of the things we know and compute -- position of the ground station, trajectory of the satellite, ECI time when the satellite sent the signal, travel time of the signal are based on an assumed ECI reference frame. It utterly pervades all of the calculations and the result.

We do not measure or compute any of these things using the frame of reference of a particle at CERN. Doing so would be ludicrous. So we don't do that. We use the ECI frame.

 

[PeterDonis]

About ECI frame coordinate time $t$ my understanding is as follows

This is basically correct as a definition of coordinate time in the ECI frame, yes. (There are some minor complications due to the fact that the Earth is in orbit about the Sun, but we can ignore them here.)

Maybe I've not a clear understanding about this: the circuitry onboard the GPS satellite just 'adjusts' the 'flow rate' of the local clock against the 'flow rate' of the ECI coordinate time $t$

It doesn't adjust the "flow rate" of the quartz crystals themselves; that can't be done. The extra circuitry just adjusts the output rate of the overall "clock" system.

or really synchronize it ?

You can't synchronize clocks to an abstraction, which coordinate time in any frame is. You can only try to synchronize clocks to each other. The GPS satellites try to synchronize their clocks to all the other clocks in the GPS system with the rate adjustments and corrections from ground stations already described.

we could have clocks that beat the same 'flow time' however there is a fixed 'offset' between them.

We could define a simultaneity convention that way, yes, but that is not how the GPS system's simultaneity convention is defined. In the GPS system, events at any single component (a satellite, ground station, or client receiver such as your smartphone) that that component labels with a given "GPS time" are treated as being simultaneous (happening at the same time) as events at all other components that are labeled with the same "GPS time".

 

[cianfa72]

It doesn't adjust the "flow rate" of the quartz crystals themselves; that can't be done. The extra circuitry just adjusts the output rate of the overall "clock" system.

so let me say in case of a digital clock that basically amount to implement an appropriate digital divider circuitry.

We could define a simultaneity convention that way, yes, but that is not how the GPS system's simultaneity convention is defined.

Sorry, which simultaneity convention were you talking about ?

In the GPS system, events at any single component (a satellite, ground station, or client receiver such as your smartphone) that that component labels with a given "GPS time" are treated as being simultaneous (happening at the same time) as events at all other components that are labeled with the same "GPS time".

So GPS system just 'synthonize' clocks are part of.

 

[jbriggs444]

ok, so let me say in case of a digital clock that basically amount to implement an appropriate digital divider circuitry.

Sorry, which simultaneity convention you were talking about ?

So GPS system just 'synthonize' clocks are part of.

The satellite clocks are designed so that their outputs keep reasonably accurate pace with the passage of ECI time on their own. That's syntonization. (there is no synth in syntonize).

The once-every-24-hour corrections are applied so that any drift away from accurate ECI time is compensated for.

I don't see that it matters much whether you count that as synchronizing once and correcting to maintain perfect syntonization thereafter or as approximately syntonizing once and periodically synchronizing thereafter. However, I prefer the latter interpretation.

In any case, you end up with a bunch of satellite clocks all over the place, all generating outputs that accurately reflect ECI time.

 

[PeterDonis]

which simultaneity convention were you talking about ?

The GPS system uses the simultaneity convention of the ECI frame.

 

[cianfa72]

The GPS system uses the simultaneity convention of the ECI frame.

Sorry, maybe the following point is unclear to me: using the ECI frame in GPS system are we implicitly neglecting the spacetime curvature in the spacetime region around the Earth (up to the GPS satellites orbits) ?

 

[PeterDonis]

using the ECI frame in GPS system are we implicitly neglecting the spacetime curvature in the spacetime region around the Earth (up to the GPS satellites orbits) ?

Not quite. We are ignoring the spacetime curvature due to the Sun (and, strictly speaking, the Moon), but we are only "sort of" ignoring the spacetime curvature due to the Earth.

The name "Earth Centered Inertial" for the ECI frame is a bit of a misnomer; the "inertial" part, as regards the gravity of the Earth itself, has to be taken in the old Newtonian sense, where we treat the Earth's gravity as a "force" in some respects (so that the GPS satellites are not actually moving "inertially" in this "inertial" frame, even though from a straight GR standpoint the satellites are "inertial" since they are in free-fall orbits).

Also, the "rate of time flow" of the ECI frame is adjusted to match the "natural" rate of clocks on the geoid, i.e., on the surface of constant gravitational potential due to the Earth that is exactly at "sea level". Note that this is a surface of constant gravitational potential on the rotating Earth, so the altitude (distance from Earth's center) of this surface is not constant, and the "rate of time flow" on it does not match the "natural" rate of time flow of inertial observers (in any sense--observers at rest on the rotating Earth are not "inertial" in either the Newtonian or the GR sense) at any point on the geoid except the North and South poles.

 

[cianfa72]

The name "Earth Centered Inertial" for the ECI frame is a bit of a misnomer; the "inertial" part, as regards the gravity of the Earth itself, has to be taken in the old Newtonian sense, where we treat the Earth's gravity as a "force" in some respects (so that the GPS satellites are not actually moving "inertially" in this "inertial" frame, even though from a straight GR standpoint the satellites are "inertial" since they are in free-fall orbits).

Also, the "rate of time flow" of the ECI frame is adjusted to match the "natural" rate of clocks on the geoid, i.e., on the surface of constant gravitational potential due to the Earth that is exactly at "sea level".

So GPS system basically uses a flat model of spacetime around the Earth in an Earth-centered inertial (Lorentz) coordinate chart (ECI frame) in which, however, the coordinate time $t$ is actually the "natural" rate of clocks on the geoid.

In other words coming back to the 'latticework' model of my post #9 clocks bolted on it run at the "natural" rate of clocks on the geoid.

 

[PeterDonis]

So GPS system basically uses a flat model of spacetime around the Earth in an Earth-centered inertial (Lorentz) coordinate chart (ECI frame) in which, however, the coordinate time is actually the "natural" rate of clocks on the geoid.

A "flat" model in which there is a Newtonian gravity force due to the Earth. But yes, it's an inertial (non-rotating) frame centered on the Earth's center of mass, with the "rate of time flow" of coordinate time set as you describe.

 

[cianfa72]

A "flat" model in which there is a Newtonian gravity force due to the Earth. But yes, it's an inertial (non-rotating) frame centered on the Earth's center of mass, with the "rate of time flow" of coordinate time set as you describe.

ok, so using this model of spacetime & coordinate chart the scenario is quite the same as in SR: there are clocks moving w.rt. that coordinate chart (ECI frame) and the aim is 'adjust' their rate in order to be in synch w.r.t. ECI frame.

Since the synchronization procedure described in post #17 by @jbriggs444, I believe the main point is dispose of an accurate ECI time reference at ground stations.

 

[jbriggs444]

ok, so using this model of spacetime & coordinate chart the scenario is quite the same as in SR: there are clocks moving w.rt. that coordinate chart (ECI frame) and the aim is 'adjust' their rate in order to be in synch w.r.t. ECI frame.

The point is to physically realize a set of satellite clocks that all [at least effectively] maintain accurate time according to some agreed-upon coordinate system.

To achieve this and to maintain this, it is helpful for the clock rates to closely match the coordinate system. But the important part is that each clock's current [adjusted] reading always closely matches the current time according to the coordinate system.

It is clock reading that is important to position determination, not clock rate.

 

[cianfa72]

It is clock reading that is important to position determination, not clock rate.

I take it as the important thing is that when a 'time indicatation' sent from the clocks onboard on GPS satellites arrives on receivers on the Earth (e.g my and your smartphone) it is in synch with the ECI coordinate time.

 

[jbriggs444]

I take it as the important thing is that when a 'time indicatation' sent from the clocks onboard on GPS satellites arrive on receivers on the Earth (e.g my and yor smartphone) it is in synch with the ECI coordinate time.

It will not be. That is rather the point.

The signals are in synch when sent. They are not in synch when received. The amount by which the received signals are all out of synch with each other is the basis for how position (and time) is calculated at the receiver.

 

[pervect]

Summary:: About physical process involved to synchronize GPS clocks in Earth Centered Inertial (ECI) frame

Hi,
starting from this old thread GPS clock synchronization I've a doubt about the physical process employed to synchronize clocks bolted on GPS system satellites.

GPS implicitly defines a coordinate system. If we approximate the Earth as a sphere, a useful conceptual model would be defining all clocks that received a light pulse from the center of the sphere at the same time as being synchronized.

The nonspherical nature of the actual Earth means we'd have to add corrections to this simple idea.

For a more formal treatment, I personally like Misner's "Precis of General Relativity", https://arxiv.org/abs/gr-qc/9508043, as I've probably mentioned a few times.

The point of view there is that a line element defines the coordinates - something I'll say quickly, but probably needs deep thought to be fully appreciated. Misner gives the line element for the ECI frame as:

dτ^2 = [1 + 2(V − Φ0)/c^2]dt^2 − [1 − 2V/c^2](dx^2 + dy^2 + dz^2)/c^2

This simple expression can then be regarded as the defintion of ECI coordinates.

There are standards for time and frequency transfer as well, the underpinnings of the transfer standards though are the standards that define the coordinates themselves. From what I recall, there is a two way satellite synchronization standard, and a GPS standard in actual use.

You'll often see the effects of the Earth's rotation lumped into terms called the "Sagnac effect", https://en.wikipedia.org/wiki/Sagnac_effect.

https://tf.nist.gov/general/pdf/836.pdf may be of some help, though a quick skim of it may give the unfortunate impression that the sagnac effect is unique to satellite time transfer. This is not true, in my opinion, the sagnac effect wold also affect syncrhonization carried out through any two-way communications medium, but it might not be obvious that this is true from reading this paper.

 

[cianfa72]

For a more formal treatment, I personally like Misner's "Precis of General Relativity", https://arxiv.org/abs/gr-qc/9508043, as I've probably mentioned a few times.

The point of view there is that a line element defines the coordinates - something I'll say quickly, but probably needs deep thought to be fully appreciated. Misner gives the line element for the ECI frame as:

dτ^2 = [1 + 2(V − Φ0)/c^2]dt^2 − [1 − 2V/c^2](dx^2 + dy^2 + dz^2)/c^2

This simple expression can then be regarded as the defintion of ECI coordinates.

I read that note that employs the following metric (with Newtonian gravitational potential $V$ variable with $r$):
$$d \tau^2 = [1 + 2(V − \Phi_0)/c^2 ]dt^2 − [ 1 - 2V/c^2 ] (dx^2 + dy^2 + dz^2)/ c^2$$ it actually entails a not Euclidean spatial geometry changing with the radius $r$, I believe.

Sorry, coming back to the GPS satellites clock synchronization, in post #29 has been said that GPS clock time is in synch when sent and not when received from receivers at ground.

Just to help me in understanding: consider for instance the time indication 12:00. To be in synch when signal is sent does mean the indication 12:00 at the event of sending the signal from the GPS satellite clock (that actually encode the time value 12:00) is the same as the indication 12:00 of an imaginary 'ECI coordinate clock' spatially co-located with that event ?

Your help is really appreciated !

 

[Ibix]

As I understand it, the system works by the satellites basically sending signals saying "Satellite #56, time 23:20:00, satellite #56, time 23:20:01..." and your GPS receiver looks up where satellite #56 is at 23:20:00. If your GPS receives the 23:20:00 signal from satellite #34 1ms later than the one from #56 then it knows it's on the plane one light millisecond closer to the location of #56 than #34. If it looks up #34's position at that time too then you've got a partial fix - with a third satellite you can reduce the plane to a line, which is enough for a location if you additionally assume you're on the surface of the Earth.

That's what @PeroK meant by the signals being in sync when sent (all the satellites send at the same ECI time) but not when received (the differing flight times of the radio pulses give you the difference in distances from the satellites).

So all we need on the ground is that all of the satellites report the time when they sent the signal in some agreed time standard. We choose the ECI and adjust the satellite clocks to tick that time standard.

 

[pervect]

Neil Ashby says that the clocks in the actual GPS implementation use the ECEF (earth centered, Earth fixed) frame. https://link.springer.com/article/10.12942/lrr-2003-1

Almost all users of GPS are at fixed locations on the rotating earth, or else are moving very slowly over earth’s surface. This led to an early design decision to broadcast the satellite ephemerides in a model earth-centered, earth-fixed, reference frame (ECEF frame), in which the model Earth rotates about a fixed axis with a defined rotation rate

I haven't been able to find written confirmation of my impression that clocks in the ECEF frame co-located with clocks in the ECI frame would show the same time. If ECEF and ECI share the same time coordinate for co-located clocks as I think they do, the only difference between ECEF and ECI s in how the spatial coordinates of the clocks are reported. But it would be good to have this in writing.

From a theoretical point of view, it's not necessary to use the ECEF frame, that's just the frame that GPS historically uses. For instance, see Minser "Precis of General Relatiavity"

Each clock maintains its own proper time (but may convert this via software into
other information when it transmits). We simplify to assume it transmits
its own proper time without random or systematic errors.

Conceptually using the proper times is "simpler" with idealized clocks in that it doesn't require adoption of any coordinate system for the clocks at all. The clocks report their proper time, and the software crunches the numbers to turn the proper times into whatever coordinates one wants to use. This follows the philosophy that coordinates are a convention that one adopts for reasons of convenience.

Furthermore, one can see that the synchronization of the clocks isn't critical to the question of what coordinates one wants. The fundamental issue is what coordinates one wants to use on the rotating Earth.

Practically, though, it probably is simpler to adopt a specific reference frame as the actual implmentation does, in order to simplify dealing with non-ideal effects that exist in the real world.

 

[PeterDonis]

Neil Ashby says that the clocks in the actual GPS implementation use the ECEF

The key point of the quote you give is not about the times broadcast by the satellites (as noted below, these are in fact the same as ECI coordinate times), but the spatial coordinates. Those are given in the ECEF frame, since, as Ashby notes, that's the most natural frame for GPS receivers.

I haven't been able to find written confirmation of my impression that clocks in the ECEF frame co-located with clocks in the ECI frame would show the same time.

It's later on in the Ashby article:

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.

The only complication is that one has to include a correction for gravitational redshift, as Ashby notes in the next paragraph.

 

[pervect]

It's later on in the Ashby article

Alright, thanks. I was pretty sure it had to be that way, but I wanted to find it in writing - it prevents mistakes, and is also more convincing.

This also gives us enough information to follow the outline of Misner's suggestions in "Precis of General Relativity". That would be to find the metric in the ECEF frame via the tensor transformation laws. I thought I had a source that had already done this, but a quick search didn't find one. With the information of the metric in ECI coordinates, and the coordinate transformations, we can compute the ECEF metric, though it'd be nice if we could find it also in writing somewhere to compare.

Then we can use the ECEF metric we compute to find the null paths that light would take in terms of ECEF coordinates, and use this information to confirm that the two-way time transmission standards proposed in https://tf.nist.gov/general/pdf/836.pdf work and are understood correctly. I'd have to go through this procedure myself to confirm my understanding.

What I would expect from this whole procedure is that when we compute the null paths above, that the coordinate speed of light in the ECEF frame, there is a linear relationship between position and coordinate time, but that the associated "speed" of this linear relationship is not the same east-west and west-east. It'd be easiest to do this on the equator, but if one was ambitious, one could do it at other lattitudes, and for paths that are not "due east" and "due west". One can then appreciate why Einstein's midpoint method for clock synchronization needs adjustment if done in these non-inertial coordinates, and exactly what this adjustment entails.