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I am currently working on a GPS project, so I am doing some research about the subject. I understand that the GPS receiver knows its location by calculating its distances from 3 satellites, and by eliminating one of the two points of the intersection of the 3 spheres.

  • Why then do we need the fourth satellite?
  • How accurate will the position b without the fourth satellite?
  • What do "clock accuracy" and "clock bias" mean?
  • Does time synchronization between the receiver and the satellite mean that the receiver clock will get the time very precisely?

I've found that some receivers use this method described below to compute the time of flight between the satellite and the device. Does it mean we don't need the fourth satellite to correct the clock bias?

Coarse/Acquisition (C/A) Code

A pseudorandom noise code (PRN) modulated onto a L1 signal which helps the GPS receiver to compute the distance from each satellite. Specifically, the difference between the pseudorandom number code generated by the GPS rover software and the pseudorandom number code coming in from the satellite is used to quickly compute the distance to a satellite and therefore calculate your position.

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8 Answers 8

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Why then do we need the fourth satellite ?

Because there is a fourth unknown: time. GPS works one-way, so the receiver's clock needs to be aligned with the senders' (the GPS satellites) clocks in order to compute the time-of-flight. Think about it this way: if the GPS satellite tells you "I sent this message at time X", and you receive it at time Y, you can compute time-of-flight only if your clocks are aligned. Typically, the clocks in GPS receivers is "bad", at least compared with the atomic clocks on the satellites, so the difference between the receiver clock and the GPS clock needs to be determined.

How much will be the position accuracy without the fourth satellite ?

It will be "useless", in te sense of (much) reduced accuracy. With 3 satellites you could do some attempt with the assumption that you are on a sphere, or using knowledge of where you were earlier, or using other inputs (e.g. altimeter), but it will not be as accurate.

What does it mean by "clock accuracy" and "clock bias" ?

See above.

Does time synchronization btw the receiver and the satellite mean that the receiver clock will get the time very precisely ?

Not "get", "compute". Basically the receiver determines a best guess at the correct time, "best" in the least-square-error sense.

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    $\begingroup$ @YahyaAouledAmer yes and maybe. Just an atomic clock doesn't help - it needs to be in sync with those on the GPS satellites also. You can probably do it with some effort, but not sure what the added value would be. $\endgroup$
    – Ludo
    Feb 14, 2022 at 12:25
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    $\begingroup$ @Ludo It's impossible to keep them in sync because time flows at different rates on the ground and in orbit. $\endgroup$
    – J...
    Feb 14, 2022 at 21:51
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    $\begingroup$ @J... but known different rates, surely? So one could account for those differences, and remain synchronised? $\endgroup$
    – Tim
    Feb 14, 2022 at 22:24
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    $\begingroup$ Real GPS receivers that I have seen will give a position with three sat fixes, by assuming you're near where you were before, and near the surface of the Earth. Current phones will have an altimeter and use "augments" to supplement the GPS. $\endgroup$
    – JDługosz
    Feb 14, 2022 at 23:07
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    $\begingroup$ the satellites clocks run slower on orbit than on the ground due to special relativity, and faster on orbit than on the ground due to general relativity. Both these effects (overall, the GR is about twice the opposite of SR, so they sum to about the negative of the SR term) are taken into account in designing and building the satellite clocks. the tricky part left over is the additional clock rate skewing from the Doppler shift, which depends on the projection of the satellite orbital velocity onto the line of sight from the receiver, which i describe in my answer. $\endgroup$
    – Ryan C
    Feb 15, 2022 at 5:21
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Your GPS cannot directly determine the distance from any satellite, it has to go indirectly. It gets a signal from the first satellite, say "it was exactly 10:30:25.123456789 seconds according to my extremely precise clock when this signal was sent", and it gets a signal from the second satellite, say "it was exactly 10:30:25.123556789 seconds according to my extremely precise clock when this signal was sent". The clocks are 0.0001 seconds apart. So the signal from the first satellite travelled 0.0001 seconds longer. At 299,792,458 meter/sec, that is 29,979.2458 meters difference. So you are 29,979 meters closer to the second satellite than to the first. And your GPS also knows the exact location of the satellites.

With the third satellite, you also learn how much closer or further away you are to the third satellite compared to the first and the second. You can turn that into three rather complicated equations, and try to solve those equations, but there is not just one solution: There is a whole curve of solutions.

Now if three satellites is all you've got, your GPS can make a guess: It guesses that you are located on the surface of the earth. Your GPS has likely a map of roads in your area, but it also has a map of elevations. So it guesses first that you are at height zero and calculates where that curve intersects with the earth surface at height zero. That might be off a bit because you are in a hilly area, 1000 meters above sea level. But the GPS knows your approximate location, so it guesses you are about 1,000 meters above sea level, recalculates where you are, and that location might be 980 meters above sea level, and then the next calculation gives you your precise location. But only if you are on the surface. If you are at the top of a church tower, your location will be guessed wrong. If you are on an airplane, with a window seat so your GPS gets a signal, it will be quite imprecise, maybe kilometres off if you are 10,000 meters above ground.

With a fourth satellite, there are four ways to take three satellites and calculate the curve where you should be, so you get four curves. And then the GPS picks the point that it is closest to all four curves. That gives you your location quite precisely, and at the same time, if the curves don't meet exactly in one point but are maybe ten meters apart, then you also know the precision of your location.

(Some smartphones nowadays have a barometer. That could also be used to estimate your height above sea level, not very precise, because air pressure also depends on the weather, and help you get your location if you are high above ground. I don't think anyone does that. )

If you had a very precise atomic clock, you'd need only three satellites. But atomic clocks are big and expensive, so there isn't one in your mobile phone.

PS. If you think that it's kind of unfair that you need four satellites to get three coordinates, you are actually getting four. You also get the time with very high precision (maybe 100ns). Annoyingly no phone that I have seen uses this ability to set its clock. Actually, just a single satellite gives you the time with less than 100ms error: The fact alone that you can receive the satellite gives you your location with a ridiculous error of thousands and thousands of miles - but if you divide this error by the speed of light, then you get a better approximation for the time than your wristwatch will give you.

PPS. With four satellites your position is already overspecified. There are four curves, and with infinite precision they would intersect exactly at the point where the GPS is. But we don't have infinite precision, so we take the point that is closest to them. Five or six satellites would work exactly the same, except you have more curves.

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    $\begingroup$ I find this answer the easiest to understand, but have I got this right? 3 satellites give a curve in space that (hopefully) intersects the Earth in one (or two) places where the satellites are visible, so some combination of assumptions and barometric pressure can provide a likely position. The fourth satellite narrows the fix to a single point on that curve, thus producing an assumption-less fix? $\endgroup$
    – uhoh
    Feb 14, 2022 at 23:33
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    $\begingroup$ The problem with "three satellites plus surface of the Earth" is that it tends to produce large errors. If you're standing on a second-floor balcony, the equivalent "surface of the Earth" location might be hundreds of meters away. $\endgroup$
    – Mark
    Feb 15, 2022 at 0:49
  • $\begingroup$ @uhoh that's a bit too simple. each pair of satellites gives you a surface, with a thickness dependent on the uncertainty (take the pancakes from the bistatic radar post and stretch them into other shapes). due to measurement error, most likely there is NO point at which ALL the surfaces intersect, but you can solve for which point is the closest to all of them, averaged in a way that involves weighting by the estimated uncertainties. more detail coming in my answer the the OP. $\endgroup$
    – Ryan C
    Feb 15, 2022 at 5:32
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    $\begingroup$ @RyanC "most likely there is NO point at which ALL the surfaces intersect" I do not think that that applies to either three satellites plus the Earth surface + altitude solution or the four satellite without surface assumption solution that are being talked about here for the OP's question "Why does GPS need the fourth satellite?". I could be wrong, but I don't think those are over-specified scenarios. It's only when you have more information than that where the problem becomes over-specified and you have to start thinking about how to handle it. $\endgroup$
    – uhoh
    Feb 15, 2022 at 5:50
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    $\begingroup$ @RyanC I certainly understand error, but when a problem is underspecified or exactly specified as discussed here, it is not the same thing as when the problem is over-specified. A GPS unit is working with three or four satellites as discussed here, it does not have to reconcile with non-intersecting surfaces. Certainly the final number on the screen that the user sees is a best fit taking account the thicknesses of the surfaces but the non-intersection you mentioned only happens when there is more information available (e.g. 4 satellites plus Earth surface assumption, or 5+ satellites etc. $\endgroup$
    – uhoh
    Feb 15, 2022 at 6:26
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The receiver has no very precise time, the fourth satellite is needed to calculate a 3D position without knowing the precise time.

After knowing the position, the receiver may calculate the GPS signal delay between a satellite and the receiver. Using the delay the the precise time for the receiver may be calculated from the time of the satellite.

The receiver clock frequency may be a little bit too small or too large, that is the clock accuracy. The time of the receiver may be a little bit too early or too late, that is the clock bias.

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  • $\begingroup$ If I use the time provided by the receiver clock, does it mean I will get the position but with very low accuracy, and if use the fourth satellite I will get good accuracy ? $\endgroup$ Feb 14, 2022 at 12:08
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    $\begingroup$ @YahyaAouledAmer Generally yes, but most standalone receivers have at best mediocre clocks. With something like a smartphone with a solid network link you can generally have an accurate enough estimate of the ‘exact’ time to not need the fourth satellite though, but if you’ve got such a device you can also triangulate off the cell network to get a generic location which you can use to narrow things down with just three satellites (or fewer possibly), or even use an internal altimeter/barometer to provide that extra bit of data. $\endgroup$ Feb 15, 2022 at 2:30
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    $\begingroup$ @YahyaAouledAmer: At the speed of light, even a microsecond difference is a 300 meter difference. That's why GPS requires 4 atomic clocks: they can achieve nanosecond accuracy. The smartphones bit in the comment above is mostly nonsense - smartphones use the mobile network to download the satellite orbits, so they can acquire the 4th signal faster. Even 5G doesn't have atomic clocks in the network. Knowing your 5G tower gets you the same ~300 meter error and therefore doesn't really help. $\endgroup$
    – MSalters
    Feb 15, 2022 at 12:43
  • $\begingroup$ what exactly does the satellite orbits information contain ? $\endgroup$ Feb 15, 2022 at 12:54
  • $\begingroup$ @MSalters yes, I doubt mobile networks can provide a time accuracy much better than NTP, which is ~ms. But tower triangulation can be surprisingly good in ideal circumstances - or (as I found the other day) over a mile out; then I picked up the satellites and saw the dot jump to where I knew I was, and the error circle vanish $\endgroup$
    – Chris H
    Feb 15, 2022 at 14:02
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GPS is much easier to understand in the one-dimensional case.

You are somewhere on the road between Springfield and Newtown. The road is straight and the two towns are six kilometers apart. Around noon, you hear the church bells of Springfield. Ten seconds later, you also hear the bells of Newtown.

  • Where exactly are you?
  • What exact time is it?

The speed of sound is 300m/s. You heard the bells of Springfield 10s earlier, so you must be 3km closer to Springfield than to Newtown. The distance is 6km in total, which means you are 1.5km from Springfield and 4.5km from Newtown. The bells rang at noon and sound needs 15s to cover the 4.5km, so it's now 12:00:15 when you hear the bells of Newtown.

In the example above, we needed two satellites to solve for two variables - position and time. If you hear only the bells of Springfield, you cannot determine your position at all. That is, unless you have an accurate clock to measure the time of arrival directly instead of calculating backwards from time difference and total distance. In the three-dimensional case, you need four satellites to solve for the three position variables and for time.

To answer your questions one by one:

Why then do we need the fourth satellite?

Because we cannot measure the time of arrival accurately, only the differences. We thus cannot use the three-spheres method. We have to work backwards from time differences and total distances, as in the example of Springfield and Newtown. This requires one extra satellite, but as a bonus it gives you time as a calculated variable.

How much will be the position accuracy without the fourth satellite?
What does it mean by "clock accuracy" and "clock bias"?

With the three-spheres method, positioning error is proportional to clock error of the receiver clock. The proportionality constant is the speed of light, which is 300m/μs. If the clock of your receiver device differs from global atomic time by merely 10μs, your position will be off by 3km. Good luck with that.

Does time synchronization between the receiver and the satellite mean that the receiver clock will get the time very precisely?

Yes, a GPS receiver is by far the most accurate clock you can buy. It receives its signals directly from atomic clocks aboard the satellites and corrects them for time of flight, using the known speed of light and receiver position. Traditional radio time standards like DCF77 or WWVB don't correct for time of flight, which makes them greatly inferior to GPS.

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A GPS satellite sends out a constant stream of "I transmitted this at time x" messages. There are two ways of turning this into a location fix:

The three spheres method you describe requires a fourth parameter: time. If you have an atomic clock synchronized to the clocks of the GPS satellites, you can compute your distance from each satellite, find the intersection of three spheres, and determine your location.

Most GPS receivers don't include a synchronized atomic clock, though. In this case, you can use the time information to compute the difference in distances between two satellites. This tells you that you are somewhere on the surface of a hyperboloid of revolution; three satellites gives you three pairs, three hyperboloids, and, unfortunately, many intersection points. You need the fourth satellite to boost the number of pairs to six, giving you six hyperboloids and hopefully a single intersection point.

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  • $\begingroup$ Why is the shape of all the places that have some delta D of the distances to points A and B a hyperboloid? (I get the of revolution part) $\endgroup$
    – Eugene
    Feb 15, 2022 at 0:31
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    $\begingroup$ @Eugene, because that's the definition of a hyperbola/hyperboloid: the set of all points such that the difference between the distances to the two foci is constant. $\endgroup$
    – Mark
    Feb 15, 2022 at 0:42
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    $\begingroup$ I've been thinking about this, and I think there's something missing. You don't have just the delta, you have accurate readings of A and B, which are synchronized with atomic clocks, then you can tell which "half" of the hyperbola you're on: if A<B, you are on the one that's closer to A and visa versa. Three of these half hyperboloids have only a single intersect. $\endgroup$
    – Eugene
    Feb 15, 2022 at 8:09
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    $\begingroup$ @YahyaAouledAmer It's easier to grok in 2D 1st. If you have an accurate local time, then you can measure the "time of flight" and therefore the exact distance to a satellite A, the shape of where you could be with relation to A is a circle. If you don't have an accurate local time, but you can measure the difference between A and B times, the shape of where you could be with relation to A and B is a hyperbola. If you rotate those shapes to get a 3d object, you get a sphere and a hyperboloid respectively. $\endgroup$
    – Eugene
    Feb 15, 2022 at 18:02
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    $\begingroup$ @YahyaAouledAmer There are no spheres with GPS. Anyone talking about spheres in the context of GPS is oversimplifying it in an attempt to explain to laypeople who have never heard of a hyperboloid but can imagine a sphere. $\endgroup$
    – TooTea
    Feb 16, 2022 at 9:45
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The "spheres" illustration in regard to GPS functioning is simple, but slightly misleading.

One could get meaningful spheres if the time when the clock pulses were transmitted by the satellites was exactly known in the frame of reference of the receiver .

This time is not known with any meaningful precision because of number of reasons, including, but not limited to, relativistic differences between the clock rates and the limited long-term precision of the receiver's clock.

In reality, one gets a rotational hyperboloid of possible positions from the time difference between two satellites and a complex curve of equally possible positions from 3 satellites. When you have 4 satellites, you get a point.

In the case of 3 satellites, the curve in question may happen to be near-vertical near the Earth surface and, if lucky, to cross the surface in only one place near the satellites you hear. In this case the receiver may assume you are sufficiently near the surface and give you a position estimate based on this assumption (with a gross estimated inaccuracy of hundreds of meters or even kilometers).

A barometric altimeter, if properly calibrated, may help in this case by providing a better estimate for your altitude than the basic assumption that you are near the surface.

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  • $\begingroup$ Right. Thank you. $\endgroup$
    – fraxinus
    Feb 16, 2022 at 10:17
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I remember working on software that tracked the GPS satellite orbits and could therefore predict the accuracy of a fix at a given location in the future based on the position of the satellites at that time. In short: You need at least to receive signals from at least 3 satellites to get a location fix. With 4 satellites, you can tell if one satellite was giving out incorrect information such as it's location, as the accuracy of your fix would be compromised. Some more expensive GPS receivers can display the precision of your location based on data received. With 5+ satellites you can determine which data is inaccurate and safely ignore it. Obviously being to receive data from more satellites greatly improves the accuracy of your location.

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    $\begingroup$ You've got to be remembering wrong. Three satellites alone can't give you a location fix. You need three satellites plus time, or four satellites. And virtually every GPS receiver will give you an accuracy estimate -- it's not just a feature of the expensive ones. $\endgroup$
    – Mark
    Feb 15, 2022 at 4:09
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    $\begingroup$ This type of software is well-known in the aviation community, where it's called RAIM Prediction. Your numbers are incorrect for pure GPS: 5 satellites are required to detect a fault and 6 are required to determine which satellite to exclude. However if the receiver can use barometric or other augmentation, then you are correct with 4 and 5. $\endgroup$
    – Steve V.
    Feb 15, 2022 at 6:29
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All other answer contain a lot of fluff and unrelated tangentials, unnecessarily complicating the subject. Meanwhile, the actual "meat" needed to address your question is quite simple.

GPS works based on time dilation, which means that accurate time measurements are absolutely essential. The problem is, the clock installed in most GPS devices is worth less than 1$, while the clock installed on the fourth satellite is worth millions of dollars. The clock on the fourth satellite is used to periodically synchronize the cheap clock in GPS device to overcome this problem.

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    $\begingroup$ If we can synchronize the receiver clock to the satellite clock, does it mean we can can get our position to a centimeter accuracy? Then why in my smartphone I get an accuracy of 5-10 meters ? $\endgroup$ Feb 15, 2022 at 13:50
  • $\begingroup$ and without the fourth satellite, does it mean that we will get inaccuracy of kilometers ?? $\endgroup$ Feb 15, 2022 at 13:56
  • $\begingroup$ @YahyaAouledAmer Even with perfect synchronization, you still cannot get a centimeter accuracy because the signal gets distorted a bit by reflecting off buildings, terrain, cars, etc. and also, the signal could get distorted before it gets to Earth by solar flares, etc. Without fourth satellite, the cheap clock in your device would quickly drift away and the inaccuracy will keep decreasing and decreasing. And yes, you would get inaccuracy of kilometers, then as the cheap clock drifts away more it will be tens of kilometers, hundreds of kilometers, etc. and the GPS reading would get useless. $\endgroup$
    – user46610
    Feb 15, 2022 at 20:57
  • $\begingroup$ You can't "synchronize to the fourth satellite" because you don't know how far away it is. To overcome this, GPS does a combined "position and time" calculation that requires at least four satellites, and uses all of them equally. $\endgroup$
    – Mark
    Feb 15, 2022 at 23:00
  • $\begingroup$ user46610, time dilation doesn't matter (except we have to take it into account obviously). We start with no knowledge. When we receive the first signal, that gives us the time within 100ms. Knowing the time within 100ms, we can figure out where the satellites are within a few kilometres. With that, we get a quite precise time, maybe within microseconds. That gives us the satellite locations within metres, and that finally gives us the exact time and the exact location of the satellites. There is no need for a clock in the GPS receiver at all. $\endgroup$
    – gnasher729
    Feb 16, 2022 at 0:01

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