Why does GPS require an accurate clock?

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Why do GPS need a cesium clock accurate to 10ns? Is this strictly for time keeping or is there some synchronization between satellites?

Janus
Staff Emeritus
Gold Member
The satellites are not in geostationary orbits, which means they are over different points of the Earth at different times . To know where a particular one is when using it to get a position fix, you have to know when it is.

Svein
And - of course - the knowledge of where the satellites are and the differences in the received clock signals are used to calculate your position.

jbriggs444
Homework Helper
Why do GPS need a cesium clock accurate to 10ns? Is this strictly for time keeping or is there some synchronization between satellites?
Simplistically, the signal from a particular satellite gives you your current distance from that satellite. This is determined by subtracting the time when the signal was sent from the time it was received. The accuracy of those times determines the accuracy of the position fix.

Of course, that simple strategy would require your own GPS unit to carry a well-synchronized atomic clock. So the truth is a bit more complex. With four satellites in view, one can receive signals from all four and solve a set of four simultaneous equations to determine your 3 dimensional position plus current time.

Edit: As Grace Hopper was fond of pointing out, one nanosecond is approximately one foot at light speed. As a rule of thumb, 10 ns time error is a 10 foot error in position fix. [I got a salt packet full of picoseconds at one of her talks -- she'd given up on handing out nanoseconds by that time]

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jim mcnamara
Dale
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Why do GPS need a cesium clock accurate to 10ns? Is this strictly for time keeping or is there some synchronization between satellites?
If you have an accurate clock, and a satellite with a known position broadcasts a time signal, then you can compare the time on your accurate clock with the time signal from the satellite and you know the distance to the satellite. That tells you that your position is somewhere on a sphere of a known radius centered around that satellite.

Two such signals puts your position somewhere on the circle where the two spheres intersect. Three such signals narrows your position down to one of two points where all three spheres intersect. And four or more such signals only intersect in a single point. But you need an accurate clock for determining each radius.

There are some tricks you can do to use more satellites and a cheaper clock.

russ_watters
russ_watters
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Why do GPS need a cesium clock accurate to 10ns? Is this strictly for time keeping or is there some synchronization between satellites?
Because GPS *is* a collection of clocks. Comparing the timing of the signals from the satellites is how the system works!

russ_watters
Mentor
The satellites are not in geostationary orbits, which means they are over different points of the Earth at different times . To know where a particular one is when using it to get a position fix, you have to know when it is.
You'd still need clocks if they were stationary because GPS essentially uses ranging via signal delay.

jbriggs444
sophiecentaur
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A GPS system is based on the same method as was used in the old terrestrial hyperbolic navigation systems such as Decca Navigator, which had transmitters at fixed points on Earth. A ship could compare the times of arrival signals from three (or more) transmitters and those times would place the ship's position on a point where intersecting hyperbolae crossed.
The GPS satellites are on the move constantly so it's a much more complicated problem. Their positions also have to be known at all times and they themselves use ground based references to create a network of their relative positions and times in order that your iPhone (even) can do the hyperbolic geometry from the received signals and know where it is.
Because the satellites are travelling so fast, Special Relativity affects their local clocks and gives them a time drift of about 24μs every day. 24μs represents a long distance for light to travel and that would be a massive distance error. So it has to be corrected for too.

russ_watters, Tech2025 and Asymptotic
Simplistically, the signal from a particular satellite gives you your current distance from that satellite. This is determined by subtracting the time when the signal was sent from the time it was received. The accuracy of those times determines the accuracy of the position fix.

Of course, that simple strategy would require your own GPS unit to carry a well-synchronized atomic clock. So the truth is a bit more complex. With four satellites in view, one can receive signals from all four and solve a set of four simultaneous equations to determine your 3 dimensional position plus current time.

It is commonly said that GPS clocks are corrected for SR/GR, because otherwise they would drift away from Earth clocks by something like 38 microseconds per day. Fine, I believe that SR/GR is true and that GPS clocks actually do drift away, and I can even believe they are corrected for this. What I don't understand is WHY they need to be corrected for?

If I understand the working principle of GPS correctly, it does not matter whether the GPS sattelite clocks are synced with the ground clocks at all. As far as I understand, the receiver needs at least four sattelites such that it knows the position of each sattelite and the exact time at which the message was sent. The fourth sattelite is required to determine the "bias" or what you call "current time". The fact that your receiver clock is not accurate enough is built into the design, and the fourth sattelite gives the required information to determine the offset or bias between the accurate GPS time and the inaccurate receiver time.

Why then does it matter that GPS clocks are synced up with the ground? They have to be synced up with other GPS clocks, sure, but since they are all in the same orbital height (and therefore velocity), the effect of SR/GR would be the same for all of them and they would not drift away from each other (only from the ground). The "bias" or offset between Earth time and GPS time would grow quickly but in fact it does not matter because it is determined from the fourth sattelite.

With SR/GR corrections I suppose the bias / offset is small, but there is no reason for it to be small as far as I can tell? It can be seconds, minutes, years... Doesn't matter, right?

Basically summarized, GPS uses the time difference between the four (or more) sattelites to determine location. The local time of the receiver is not necessary to know and hence there would be no problem if SR/GR was not corrected for. The GPS clocks would be wildly different than receiver clocks, but position would still be accurate.

The only thing I can think of is that the GPS sattelite orbits are not exactly identical (due to different orientations of the orbits, the Earth is not exactly spherical, gravity of the Earth is not exactly uniform, etc). Are those effects taken into account and is this in fact what is being corrected for? Are these effects large enough that correction is necessary? Especially since I think the clocks and positions of GPS are already corrected at regular intervals (daily?) from ground stations.

Can anyone enlighten me? :)

jbriggs444
Homework Helper
While you could run the atomic clocks in GPS satellites at rates which have nothing to do with terrestrial clocks, it would be a needless complication. Why bother? You have to convert the raw frequency into a usable counter value anyway. Why not convert it into one that matches Earth clocks? One tick rate is as good as another.

As I recall, the GPS clocks are adjusted daily to match terrestrial timekeeping (to avoid long term drift). It presumably simplifies things a bit to have them tick at the same rate as the terrestrial master clock(s) they are synchronized against.

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russ_watters and sophiecentaur
sophiecentaur
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Can anyone enlighten me? :)
I imagine that it's easier to have the times running as near correctly as possible so that the drift is easier to compensate for. The satellites are constantly being updated, afaiia by referencing known positions and times on earth stations.
I see @jbriggs444 has the same idea. Both out of the trap at the same time.

jbriggs444
First of all, I don't see how having the times "as near correctly as possible" simplifies the daily drift corrections. Does it matter if you have to correct 2 nanoseconds or 2 milliseconds or even 2 years drift? It seems just as easy to me regardless of how much drift you are correcting for.

But alright, I can see that it would be preferrable to have the clocks stay roughly in sync with Earth just because it "would be nice". But I think you both agree that it is not strictly necessary? That means that in fact all of these articles (I'm not going to bother finding sources but look at any random article about GPS) claiming "GPS would not work without SR/GR corrections" are wrong? In fact GPS would work just fine without the corrections.

sophiecentaur
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First of all, I don't see how having the times "as near correctly as possible" simplifies the daily drift corrections. Does it matter if you have to correct 2 nanoseconds or 2 milliseconds or even 2 years drift? It seems just as easy to me regardless of how much drift you are correcting for.

But alright, I can see that it would be preferrable to have the clocks stay roughly in sync with Earth just because it "would be nice". But I think you both agree that it is not strictly necessary? That means that in fact all of these articles (I'm not going to bother finding sources but look at any random article about GPS) claiming "GPS would not work without SR/GR corrections" are wrong? In fact GPS would work just fine without the corrections.
We are talking about a control loop here. When you design a system that involves control loops you have to consider stability, loop delay and the acceptable error. Dragging a very stable oscillator back to an agreed frequency value (and the ones in all the other satellites too) can introduce problems so, even if they could get away with a lower stability 'on a good day', why make life difficult?

Nugatory
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That means that in fact all of these articles (I'm not going to bother finding sources but look at any random article about GPS) claiming "GPS would not work without SR/GR corrections" are wrong? In fact GPS would work just fine without the corrections.
No. The relativistic corrections are needed to account for the relative motions between satellites and receivers, and for gravitational effects on signals between satellites and receivers.

anorlunda
Staff Emeritus
In fact GPS would work just fine without the correction
One answer is that the role of GPS is to determine our location in 4D space (including time), not just 3D.

A better answer is that the receiver's answer is arrived at iteratively. The initial guess for 4D position is part of the calculation. If the ground clocks were significantly off, then the TOF (see below) would also be significantly off and the iterations would fail to converge.

You can test that by turning off the power to a GPS for weeks or months, transport it somewhere else in the world while off, then see how long it takes to find a fix when you turn it back on. I recall cases where it took 10-15 minutes of furious calculations by the receiver to re-synch. That includes the time for the receiver to figure out where in the sky to look for satellites.

I also recall cases from experience where my GPS failed to converge, giving me positions wrong by as much as 30km for up to 30 seconds before correcting. The faster the receiver's CPU, and the more satellites in contact, the less likely that is to happen.

https://en.wikipedia.org/wiki/Global_Positioning_System#More_detailed_description said:
Each GPS satellite continually broadcasts a signal (carrier wave with modulation) that includes:

• A pseudorandom code (sequence of ones and zeros) that is known to the receiver. By time-aligning a receiver-generated version and the receiver-measured version of the code, the time of arrival (TOA) of a defined point in the code sequence, called an epoch, can be found in the receiver clock time scale
• A message that includes the time of transmission (TOT) of the code epoch (in GPS time scale) and the satellite position at that time
Conceptually, the receiver measures the TOAs (according to its own clock) of four satellite signals. From the TOAs and the TOTs, the receiver forms four time of flight(TOF) values, which are (given the speed of light) approximately equivalent to receiver-satellite ranges. The receiver then computes its three-dimensional position and clock deviation from the four TOFs.

In practice the receiver position (in three dimensional Cartesian coordinates with origin at the Earth's center) and the offset of the receiver clock relative to the GPS time are computed simultaneously, using the navigation equations to process the TOFs.

...
It is sometimes incorrectly said that the user location is at the intersection of three spheres. While simpler to visualize, this is only the case if the receiver has a clock synchronized with the satellite clocks (i.e., the receiver measures true ranges to the satellites rather than range differences). There are significant performance benefits to the user carrying a clock synchronized with the satellites. Foremost is that only three satellites are needed to compute a position solution. If this were an essential part of the GPS concept so that all users needed to carry a synchronized clock, then a smaller number of satellites could be deployed. However, the cost and complexity of the user equipment would increase significantly.

DrClaude and sophiecentaur
russ_watters
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First of all, I don't see how having the times "as near correctly as possible" simplifies the daily drift corrections. Does it matter if you have to correct 2 nanoseconds or 2 milliseconds or even 2 years drift? It seems just as easy to me regardless of how much drift you are correcting for.
You missed the point/the daily correction issue is pulling you off in a different direction. The problem started is that every receiver would need an additional, real-time correction built into the position calculation. Otherwise by the end of the day they'd be way off. in short: 1-2ns isnt a big deal, but 38ns is. This is a continuous problem, separate from/on top of the daily calibrations.
But alright, I can see that it would be preferrable to have the clocks stay roughly in sync with Earth just because it "would be nice". But I think you both agree that it is not strictly necessary? That means that in fact all of these articles (I'm not going to bother finding sources but look at any random article about GPS) claiming "GPS would not work without SR/GR corrections" are wrong? In fact GPS would work just fine without the corrections.
It's a bit of a nitpick: the way they are designed, the correction at the satellite clock level is a [zero effort] feature, so it must work properly for the system to work properly. But the system could have been designed differently - to include an additional calculation in the every receiver's workload as a workaround - and function almost as well.

But we engineers don't really go for "almost as good" if there is no other benefit.

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nsaspook, sophiecentaur and jbriggs444
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I can see that it would be preferrable to have the clocks stay roughly in sync with Earth just because it "would be nice". But I think you both agree that it is not strictly necessary?
No, it is not just nice, it is necessary in order for GPS to meet its accuracy targets. However, it would be possible to move the correction to a different segment.

Suppose that the satellites were not corrected for that 38 ns/day circular orbit effect. All that would mean is that the satellites would be synchronized in a reference frame whose time coordinate was referenced to orbital time. The inaccurate receiver clocks would be systematically biased a little and would have larger delay times and need more frequent corrections. However, all of that would be in your “nice but not necessary” category.

Where it would become problematic is in a different segment of the operation. The GPS satellites don’t know where they are. The ground control segment of the system carefully measures their orbital path with ground based radar and updates the ephemeris on a regular basis. If the ground control segment were to generate the ephemeris based on geoid time and the satellites were to broadcast it based on orbital time then the receiver calculated ranges would be wrong, even with the receivers happily using orbital time.

So in the end, the system could have been designed to use orbital time instead of geoid time. But it would still need the 38 ns/day relativistic correction. The receivers would not require modification (nice but not necessary), but the ground control segment would (necessary). They would need to adjust their clocks by 38 ns/day in the opposite direction in order to produce an ephemeris that would be correct for the satellites to rebroadcast.

In addition, there are also relativistic effects that need to be corrected in the fact that the ground control clocks are not at rest on the geoid and the fact that the satellite clocks are not in perfectly circular orbits.

sophiecentaur
mfb
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The satellites are not in perfectly circular orbits around a perfectly circular Earth. A constant drift in the time wouldn't matter as long as the satellite positions are known, but the drift is not constant, it has periodic contributions over the course of an orbit. That alone, if uncorrected, would lead to errors of a couple of meters if I remember correctly.

sophiecentaur
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The final user is the crux of this matter. there are billions (?) of low cost receivers and only a few satellites, Any design has to be 'in favour' of all those users. That's not just a matter of "being nice". How many people would buy GPS receivers (or how many phone manufacturers would bundle them) if they had to cost £500+? The absolutely have to be cheap (and rugged) so they have to be presented with the simplest signals and be presented with the narrowest band system possible. It was mentioned earlier that, at switch on in a new location after a long time, a GPS receiver has to do a lot of self alignment before it knows where it is. With a less helpful system, the receiver would be having to do this constantly.

It will be interesting to hear the actual projected cost of the mooted UK go-it-alone, Galilileo. if the EU choose to use their new system (at high res) exclusively for EU members. (Despite the fact that brexiteers claim to have known all the facets and costs of brexit, I would doubt that more than a very few voters had an inkling of this problem.)

Have a look at https://en.wikipedia.org/wiki/GPS_signals and you'll see that rapid signal acquisition depends on having the receiver's clock within approx. 1.023 microsecond of the clock used for the received signal. A 32ns/day drift, over a month, would exceed that. Equivalently, so would an initial position displacement of the receiver by more than 1000 feet (radial from the received satellite) at power-up. Some clever algorithms in newer receivers can expedite re-acquisition outside these limits, but the basic reacquisition behaviour could require retrying all 1023 chips in the C/A code to find which is first in a cycle. That can be painfully slow. Of course, if there are supplemental sources of timing (such as a cellular signal) the time uncertainty at the receiver can be limited, expediting the reacquire.

anorlunda
If you have an accurate clock, and a satellite with a known position broadcasts a time signal, then you can compare the time on your accurate clock with the time signal from the satellite and you know the distance to the satellite. That tells you that your position is somewhere on a sphere of a known radius centered around that satellite.

Two such signals puts your position somewhere on the circle where the two spheres intersect. Three such signals narrows your position down to one of two points where all three spheres intersect. And four or more such signals only intersect in a single point. But you need an accurate clock for determining each radius.

There are some tricks you can do to use more satellites and a cheaper clock.
Remember that there is a continuous error. You have to know the altitude of the satellite and unfortunately their orbits are continuously degrading and so that accuracy is limited.

It presumably simplifies things a bit to have them tick at the same rate as the terrestrial master clock(s) they are synchronized against.
My understanding is the terrestrial clocks tick at one second per second, but the satellite clocks do not tick at one second per second with respect to their rest frame.

jbriggs444