Talk:Global Positioning System

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Section 6.2.1, Spheres, should be modified or removed, four satellites are required

Administrator Yamla at https://en.wikipedia.org/wiki/User_talk:Yamla has stated, "For the record, I believe RHB100 is correct here. I'm not an engineer, but I believe three satellites narrow your position down to two points and you need a fourth to get a unique position." We need a competent editor to remove or modify the article section 6.2.1, Spheres, in accordance with the paragraph I wrote above in the talk page section, Improvement for geometric interpretation of GPS navigation equations. RHB100 (talk) 02:43, 3 December 2018 (UTC)

Please explain how the source has been misinterpreted or misapplied, and provide another reliable source. Thanks. Strebe (talk) 03:29, 3 December 2018 (UTC)
Strebe, the source has been misinterpreted. The source, a paper called "The Mathematics of GPS" at https://web.archive.org/web/20050316051910/http://www.siam.org/siamnews/general/gps.htm states in the first paragraph "A handheld GPS receiver gives your position on the earth within 80 meters, and usually better, by measuring the range to four or more satellites". The second paragraph states that three spheres will intersect at two points. This provides a direct contradiction to the language in the section Spheres that there is a unique intersection of the surfaces of three spheres. The third paragraph of "The Mathematics of GPS" states "In reality we need a minimum of four satellites." There is nothing anywhere in the reference, "The Mathematics Of GPS" which justifies the statements made in the section Spheres.
Strebe, you also request that I provide another reliable source. Actually there is nothing wrong with the source, "The Mathematics of GPS". The problem is with its interpretation as explained above. Thus another source may not be needed. RHB100 (talk) 19:24, 3 December 2018 (UTC)
@RHB100: It is not clear to me that the caveat mentioned in the source still applies. It is quite old, and it states that a fourth satellite is needed because The receiver clock is not top quality, and the military intentionally dithers the satellite clock and coordinates. The dithering is not true anymore, and the clock quality may or may not have improved on the receiving end. Meanwhile the preceding paragraph states that three satellites are needed in principle. Possibly that is true in practice now. We need a modern source. Strebe (talk) 20:52, 3 December 2018 (UTC)
Strebe, I have said that there is nothing anywhere in the reference, "The Mathematics Of GPS" which justifies the statements made in the section Spheres. Do you agree or disagree with this statement. If you disagree, what statement in "The Mathematics Of GPS" justifies the statements made in the section Spheres? RHB100 (talk) 22:13, 3 December 2018 (UTC)
@RHB100: By using the time delays of the signals to calculate its distance to each satellite, the receiver knows where it is. In principle, three distance measurements should be enough. They specify spheres around three satellites, and the receiver lies at the point of intersection. This excerpt from the source expresses the identical concept as the article text. Strebe (talk) 22:47, 3 December 2018 (UTC)
We say in the Problem description section that we must have signals from at least 4 satellites. The surfaces of three spheres typically intersect at two points not one. RHB100 (talk) 00:50, 4 December 2018 (UTC)
One of the two intersections can be trivially discarded because it does not coincide with Earth’s surface, which is the fourth (implicit) sphere. As an idealized surface, Earth’s surface does not permit any accuracy for elevation, though. Given that the referenced source has slipped off into archive.org and is also not clear on why it says what it says, I recommend a new source. I also endorse clearer text for the article. Strebe (talk) 02:10, 4 December 2018 (UTC)
Yes, this is true you can choose the near earth solution. But another reason you need 4 or more satellite signals is so that you can get by with a cheap receiver clock. With 4 signals you can solve for both receiver position and time with a low cost receiver clock. RHB100 (talk) 04:51, 4 December 2018 (UTC)

It is proposed that the text below between the horizontal lines replace the current section called Spheres in the article.

Spheres

The solution of the equations in the problem description section result in corrected distances from satellites to receiver, ${\displaystyle {\bigl (}{\tilde {t}}_{i}-b-s_{i})c{\bigr )},\;i=1,2,\dots ,n}$ . These distances represent the radii of spheres, each centered on one of the transmitting satellites. The solution for the position of the receiver is then at or near the intersection of the surfaces of four or more of these spheres. [1][2][3] This near intersection of four or more sphere surfaces only exists when the clock bias, b, is at least approximately correct.

Can you explain your dissatisfaction with the current text? Your new text includes notation that is duplicated in the Solution methods section but unhelpful to the Geometric interpretation section. It also deletes what appears to me to be helpful explanations. Strebe (talk) 20:00, 4 December 2018 (UTC)
Well first of all, I did not know that you had modified the article. I thought we were in a rational discussion and would compare proposed changes before changing the article as I have done. Please quote the text you have accused me of duplicating. I don't believe such duplication exists. I find my discussion of spheres and their radii to be quite helpful for geometric interpretation. You accusation that they are unhelpful sounds more like an expression of hostility than a rational explanation. I think my clear and concise explanation is much better than your long and wordy blah blah blah. RHB100 (talk) 20:57, 4 December 2018 (UTC)
The purpose of the Geometric interpretation section, as I infer from its title, description, and content, is to offer intuitive ways to think about the problem. The Solution methods section is to formulate solutions. What you have proposed uses mathematical notation unnecessarily, notation that the casual reader won’t understand and won’t be inclined to engage with, and references three different sources for its allegedly simple description (why?), two of which are already described in detail in Solution methods. What is the point in repeating what is already described as the problem in Problem statement by stating, The solution of the equations in the problem description section result in corrected distances from satellites to receiver? Your proposal eliminates observations from the source I cited, to wit: 1) Three satellites suffice for ground position without elevation, and why; 2) satellites beyond the minimum improve accuracy by canceling out time errors, and why.
My purpose for modifying what was there was to eliminate the reference that was only available on the Wayback Machine as well as to clarify text that was unclear while largely preserving its intent. You shouldn’t get distressed over the fact that I changed something without coming to an agreement. Nothing is indelible.
As for your hostile “long wordy blah blah blah”… well: hypocrite. You’re the hostile one. You have failed to demonstrate excess verbiage while alleging excess verbiage in unWP:CIVIL terms. Meanwhile mathematical notation is not a substitute for a description, especially not your ill-formed notation with mismatching parenthesis that you mis-copied from the Problem description (not Solution methods; sorry). Is that what you’re calling “much better”? Strebe (talk) 21:47, 4 December 2018 (UTC)
Alright, I understand your criticisms a little better now. At first I objected to your changes, but I know see the wisdom of your changes. My reason for using the slight bit of mathematical notation was to be consistent with the Problem description section. But it could be eliminated. What I would like to get rid of is this boring text, "The measured ranges, called pseudoranges, contain clock errors. In a simplified idealization in which the ranges are synchronized,". We don't need to do any simplified idealizing. We can replace this text with "The solution of the equations in the problem description section result in corrected distances from satellites to receiver. Also the case where we have only 3 satellites is the degraded mode. We shouldn't be talking about the degraded mode. We should instead talk about the case when we have 4 or more satellites. Also we can get by with only one reference. When we do this we get the following text for the article section Spheres which is shown between the horizontal lines below. This covers your criticisms of the notation and 3 references. It also provides a concise description for a realistic situation.

Spheres

The solution of the equations in the problem description section result in corrected distances from satellites to receiver. These distances represent the radii of spheres, each centered on one of the transmitting satellites. The solution for the position of the receiver is then at or near the intersection of the surfaces of four or more of these spheres.[4] This near intersection of four or more sphere surfaces only exists when the clock bias, b, is at least approximately correct.

RHB100 (talk) 01:39, 5 December 2018 (UTC)

References

1. ^ Langley, R. B., "The Mathematics of GPS," GPS World, Vol. 2, No. 7, July/August 1991, pp. 45-50
2. ^ Cite error: The named reference GPS_BASICS_Blewitt was invoked but never defined (see the help page).
3. ^ Bancroft method
4. ^ Langley, R. B., "The Mathematics of GPS," GPS World, Vol. 2, No. 7, July/August 1991, pp. 45-50
Thanks for the commentary, RHB100. I still prefer the extant text for its didactic value. Your proposed description asserts several things but does not explain why those things are true. It seems unlikely to me that most readers who do not already understand the content will reach an understanding through reading the text you propose. The reason that the text I cited mentions three satellites and then four and then more is to help the reader progress from a simple, idealized model of surface location to a model that also provides elevation, and then to one that includes “extra” satellites that, as it turns out, are valuable for improving accuracy. The “degraded” state is thus far from superfluous; its inclusion offers the most basic geometry to consider and build upon in order to eventually reach an understanding of the typical, accurate, but more complicated state. Two separate sources both chose this treatment as a way to teach how GPS functions; I have to suppose the educators who wrote them considered how a layperson could best approach the concepts. Strebe (talk) 09:22, 5 December 2018 (UTC)

Well Strebe, I strongly disagree with many of the statements you have made particularly these statement, "Meanwhile the preceding paragraph states that three satellites are needed in principle. Possibly that is true in practice now. We need a modern source." which you made on 3 December 2018. RHB100 (talk) 17:55, 5 December 2018 (UTC)

The article text does not claim that only three are needed in practice. Strebe (talk) 18:18, 5 December 2018 (UTC)

User:Strebe has been changing this section that makes it quite muddled. It is no help to invoke the geoid. The editor interprets the given source incorrectly. Yes, in an idealised version, assuming a position on the geoid and using only 3 satellites, a 2 dimensional position can be calculated. However this is not the correct 2-dimensional position on the surface. The error caused by assumption of the geoid affects all 3-D coordinates, ie. it is not limited to only the elevation. Therefore the situation as stated is misleading and should be reverted.

More essential even, is that when solving for the clock bias simultaneously, the equations to be solved are NOT spheres, but spherical cones (as explained in the relevant section).

Additionally the statement about clock errors cancelling on average is plain wrong, because the biggest clock error arises from the offset of the measuring device from the satellite clocks and it cannot cancel against itself. If the editor means small random fluctuations in the clock, they may cancel, but this is in no way certain and only a small component of the error.

Woodstone (talk) 05:23, 7 December 2018 (UTC)

1. Additionally the statement about clock errors cancelling on average is plain wrong. The verbiage does not state "clock error"; it states "random clock errors". What you describe as the largest error is systematic, not random, and so your criticism is false. That said, I agree the problem is not random clock error; the low resolution afforded by the oscillator frequency is, in practice, subdivided far more finely digitally. The statement should simply be about "random error", regardless of source.
2. The text needs an explanation for why more satellites are better. If you disagree that the reason is to cancel out random errors, then please provide another reference that does explain it. This business of just stating things without explanation does not help readers understand anything and does not serve the stated purpose of the section.
3. The editor interprets the given source incorrectly. Then it behooves you to reconcile what the sources (both the source as given now and the source you reverted to) mean with the article's current text, since both of them state what appears to be the same thing. Is it your position that the sources are incorrect? In the case of three intersecting spheres, you have two candidate points for the surface location. The calculations presume true ranges (as stated in the text, and therefore without clock bias). Using the idealized sphere (not the "geoid", which is exactly not an idealized sphere) is for disambiguating the two candidate points. That is all. Therefore, I am not convinced by your criticism. Strebe (talk) 17:45, 7 December 2018 (UTC)
1&2. This section is about geometric interpretation in terms of spheres. This is only applicable with 0 clock bias. Random clock errors (from whatever source apply equally to any calculation and interpretation method. This digression does not belong in this section.
3a. Three spheres intersect in 0, 1 or 2 points, in practical cases always 2. One of them is the location of the measuring device. The other one is usually not on Earth's surface, but can be. A better method to distinguish the two points is that in successive measurements the desired one will be almost stationary, while the other will be moving very rapidly.
3b. In reality, at least one more satellite is needed in order to obtain an elevation. Ranges from three satellites with 0 clock bias yield a fully spatial solution. Elevation (and lat-long) can be calculated by subtracting the geoid. No need for more satellites. The case for needing 4 satellites applies only to solve simultaneously for the clock bias. That results in equations describing spherical cones (not spheres) and does not belong in this section. One can also solve with 2 satellites and a surface sphere (or geoid), in that case not only the elevation will be lost, but also the lat-long will be somewhat off.
Woodstone (talk) 05:19, 8 December 2018 (UTC)
This section is about geometric interpretation in terms of spheres. This is only applicable with 0 clock bias. That is a good point. However, the section still claims that more satellites are better without explaining why, and gives a reference that does not actually support this claim or the use of RMS calculation to improve accuracy.
I agree with the remainder of your comments here. Strebe (talk) 23:19, 10 December 2018 (UTC)
It currenty says "all spheres may not have an intersection point, because of inaccuracies in the data". That includes all kinds of errors, random clock variations, atmospheric diffraction. Not mentioned is the spread of the satellites. If all satellites used are close together, the equations are ill conditioned and the solution inaccurate. The more satellites the more spatial spread. I think this last point should (perhaps already is) be discussed in a more general section than geometric interpretation. −Woodstone (talk) 14:28, 11 December 2018 (UTC)
I’m confused how these comments relate to mine. I will make some edits. Strebe (talk) 17:35, 11 December 2018 (UTC)

I think the best thing to do at this time is to completely remove the section on Geometric interpretation. In my opinion it does nothing to enhance understanding of the solution method. It is clear to me that the most intelligent geometric interpretation of the 4 or more navigation equations is as a set of spheres. The near intersection of the surfaces of these spheres provides a solution. In my opinion making the decision to geometrically interpret the equations as spherical cones does not in any way enhance understanding of the solution methods. Geometrically interpreting the equations as representing a set of four or more spheres provides a much better understanding of the problem. RHB100 (talk) 22:10, 26 December 2018 (UTC)

The problem is that talking about four spheres is incorrect if the clock bias is one of the unknowns to be solved for. Four spheres do not usually have an intersection. Only if the clock bias is already known and is fixed in the equations, the resulting four spheres have a common intersection. Let's leave it as it is now. The hypothetical situation with perfect clock gives good insight using three spheres. −Woodstone (talk) 10:44, 27 December 2018 (UTC)
Well it is certainly not incorrect to talk about four or more of the navigation equations as equations of spheres. You can do so as is done in Langley, R. B., "The Mathematics of GPS," GPS World, Vol. 2, No. 7, July/August 1991, pp. 45-50. The navigation equations are equations of spheres. RHB100 (talk) 18:37, 28 December 2018 (UTC)
That article does not do as you say. Only for a fixed value of the clock bias the remaining equations represent spheres. When the clock bias is a variable, the equations are not quadratic in all coordinates an thus do not represent spheres. −Woodstone (talk) 14:00, 29 December 2018 (UTC)
For different values of clock bias, you have different radii of the spheres. For different values of clock bias you have different spheres. When you use the multi dimensional Newton Raphson method you get a sequence of values for the 4 unknowns and a sequence of spheres which converge to the solution. Talkinjg about spherical cones in no way aids in developing or understanding the solution methods. RHB100 (talk) 06:47, 30 December 2018 (UTC)

Jamming

User:Beland kindly added mention in the article about the Russian military jamming GPS in the are of recent NATO naval exercises. I reverted this because there have been many jamming incidents over the years, with no clear reason why some would be noteworthy and others not. I think it would be good to elaborate that there have, in fact, been many, and the significance of the problem. See, for example, [1]. Thanks. Strebe (talk) 02:32, 24 November 2018 (UTC)

Done as requested. -- Beland (talk) 02:34, 3 March 2019 (UTC)
Not. I’ll fix it. Strebe (talk) 02:50, 3 March 2019 (UTC)

Data rate

An earlier, now undone, edit changed the data rate from 50 bps to 50 kbps. As far as I can see, the original figure of 50 bps for the GPS data rate was already correct: take a look at https://gssc.esa.int/navipedia/index.php/GPS_Signal_Plan and https://ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/12-540-principles-of-the-global-positioning-system-spring-2012/lecture-notes/MIT12_540S12_lec7.pdf -- The Anome (talk) 14:13, 14 February 2019 (UTC)