Error analysis for the Global Positioning System: Difference between revisions

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The position calculated by a GPS receiver requires the current time, the position of the satellite and the measured delay of the received signal. The position accuracy is primarily dependent on the satellite position and signal delay.

To measure the delay, the receiver compares the bit sequence received from the satellite with an internally generated version. By comparing the rising and trailing edges of the bit transitions, modern electronics can measure signal offset to within about one percent of a bit pulse width, <math>\frac{0.01 \times 300,000,000\ \mathrm{m/s}}{(1.023 \times 10^6 /\mathrm{s})}</math>, or approximately 10 nanoseconds for the C/A code. Since GPS signals propagate at the [[speed of light]], this represents an error of about 3 meters.

This component of position accuracy can be improved by a factor of 10 using the higher-chiprate P(Y) signal. Assuming the same one percent of bit pulse width accuracy, the high-frequency P(Y) signal results in an accuracy of <math>\frac {(0.01 \times 300,000,000\ \mathrm{m/s})} {(10.23 \times 10^6 / \mathrm{s})}</math> or about 30 centimeters.

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'''Ionospheric delay''' of a microwave signal depends on its frequency. It arises from ionized atmosphere (see [[Total electron content]]). This phenomenon is known as [[dispersion (optics)|dispersion]] and can be calculated from measurements of delays for two or more frequency bands, allowing delays at other frequencies to be estimated.<ref>The same principle, and the math behind it, can be found in descriptions of [[Dispersion measure|pulsar timing by astronomers]].</ref> Some military and expensive survey-grade civilian receivers calculate atmospheric dispersion from the different delays in the L1 and L2 frequencies, and apply a more precise correction. This can be done in civilian receivers without decrypting the P(Y) signal carried on L2, by tracking the [[carrier wave]] instead of the [[modulation|modulated]] code. To facilitate this on lower cost receivers, a new civilian code signal on L2, called L2C, was added to the Block IIR-M satellites, which was first launched in 2005. It allows a direct comparison of the L1 and L2 signals using the coded signal instead of the carrier wave.

The effects of the ionosphere generally change slowly, and can be averaged over time. Those for any particular geographical area can be easily calculated by comparing the GPS-measured position to a known surveyed ___location. This correction is also valid for other receivers in the same general ___location. Several systems send this information over radio or other links to allow L1-only receivers to make ionospheric corrections. The ionospheric data are transmitted via satellite in [[Satellite Based Augmentation System]]s (SBAS) such as [[Wide Area Augmentation System]] (WAAS) (available in North America and Hawaii), [[EGNOS]] (Europe and Asia), [[Multi-functional Satellite Augmentation System]] (MSAS) (Japan), and [[GPS Aided Geo Augmented Navigation]] (GAGAN) (India) which transmits it on the GPS frequency using a special pseudo-random[[pseudorandom noise]] sequence (PRN), so only one receiver and antenna are required.

[[Humidity]] also causes a variable delay, resulting in errors similar to ionospheric delay, but occurring in the [[troposphere]]. This effect is more localized than ionospheric effects, changes more quickly and is not frequency dependent. These traits make precise measurement and compensation of humidity errors more difficult than ionospheric effects.<ref>[https://web.archive.org/web/20140522193825/http://www.navipedia.net/index.php/Earth_Sciences#Troposphere_Monitoring Navipedia: Troposphere Monitoring]</ref>

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{{anchor|GPS_SA}}

== Selective availabilityAvailability ==

GPS formerly included a (currently disabled) feature called ''Selective Availability'' (''SA'') that addsadded intentional, time varying errors of up to 100 meters (328&nbsp;ft) to the publicly available navigation signals. This was intended to deny an enemy the use of civilian GPS receivers for precision weapon guidance.

SA errors are actually [[Pseudorandomness|pseudorandom]], generated by a cryptographic algorithm from a classified ''seed'' [[key (cryptography)|key]] available only to authorized users (the U.S. military, its allies and a few other users, mostly government) with a special military GPS receiver. Mere possession of the receiver is insufficient; it still needs the tightly controlled daily key.

Before it was turned off on May 2, 2000, typical SA errors were about 50&nbsp;m (164&nbsp;ft) horizontally and about 100&nbsp;m (328&nbsp;ft) vertically.<ref>{{Cite book |last=Grewal (2001)|first=Mohinder S. |title=Global positioning systems, pinertial navigation, and integration |last2=Weill |first2=Lawrence Randolph |last3=Andrews |first3=Angus P. |date=2001 |publisher=Wiley |isbn=978-0-471-20071-0 |___location=New York, NY |pages=103.}}</ref> Because SA affects every GPS receiver in a given area almost equally, a fixed station with an accurately known position can measure the SA error values and transmit them to the local GPS receivers so they may correct their position fixes. This is called [[Differential GPS]] or ''(DGPS''). [[Differential GPS|DGPS]] also corrects for several other important sources of GPS errors, particularly ionospheric delay, so it continues to be widely used even though SA has been turned off. The ineffectiveness of SA in the face of widely available DGPS was a common argument for turning off SA, and this was finally done by order of President [[Bill Clinton|Clinton]] in 2000.<ref>{{Cite web |title=President Clinton Orders the Cessation of GPS Selective Availability |url=https://clintonwhitehouse4.archives.gov/WH/EOP/OSTP/html/0053.html }}</ref>

DGPS services are widely available from both commercial and government sources. The latter include WAAS and the [[US Coast Guard|U.S. Coast Guard's]] network of [[Low frequency|LF]] marine navigation beacons. The accuracy of the corrections depends on the distance between the user and the DGPS receiver. As the distance increases, the errors at the two sites will not correlate as well, resulting in less precise differential corrections.

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== Relativity ==

The [[Theorytheory of Relativityrelativity]] introduces several effects that need to be taken into account when dealing with precise time measurements. First, accordingAccording to [[special relativity]], time passes differently for objects in relative motion. That is known as "kinetic" [[time dilation]]: in an inertial reference frame, the faster an object moves, the slower its time appears to pass

(as measured by the frame's clocks). [[General relativity]] takes into account also the effects that gravity has on the passage of time. In the context of GPS the most prominent correction introduced by general relativity is [[gravitational time dilation]]: the clocks located deeper in the gravitational potential well (i.e. closer to the attracting body) appear to tick slower.

[[File:Orbit times.svg|thumb|Satellite clocks are slowed by their orbital speed but sped up by their distance out of the Earth's gravitational well.]]

=== [[Special Relativityrelativity]] (SR) ===

SRSpecial relativity predicts that as the velocity of an object increases (in a given frame), it'sits time slows down (as measured in that frame). For instance, the frequency of the atomic clocks moving at GPS orbital speeds will tick more slowly than stationary clocks by a factor of <math>{v^{2}}/{2c^{2}}\approx 10 ^{-10}</math> where the orbital velocity is ''v'' = 4&nbsp;km/s and ''c'' =is the [[speed of light]], approximately <math>3\times 10^8</math>m/s. The result is an error of about -7.2 μs/day in the satellite. The SRspecial relativistic effect is due to the constant movement of GPS clocks relative to the Earth-centered, non-rotating approximately inertial [[special relativity#Reference frames, coordinates and the Lorentz transformation|reference frame]]. In short, the clocks on the satellites are slowed down by the velocity of the satellite. This [[time dilation]] effect has been measured and verified using the GPS.

=== [[General Relativityrelativity]] (GR) ===

SRSpecial relativity allows tothe comparecomparison of clocks only in a flat [[spacetime]], which neglects gravitational effects on the passage of time. According to GRgeneral relativity, the presence of gravitating bodies (like Earth) curves spacetime, which makes comparing clocks not as straightforward as in SRspecial relativity. However, one can often account for most of the discrepancy by the introduction of [[gravitational time dilation]], the slowing down of time near gravitating bodies. In case of the GPS, the receivers are closer to the center of Earth than the satellites, causing the locksclocks at the altitude of the satellite to be faster by a factor of 5×10⁻¹⁰, or about +45.8 μs/day. This gravitational frequency shift is measurable. During early development some{{who|date=January 2024}} believed that GPS would not be affected by GRgeneral relativistic effects, but the [[Hafele–Keating experiment]] showed that it would be.

=== Combined kinetic and gravitational time dilations ===

Combined, these sources of time dilation cause the clocks on the satellites countto extragain +38.6 microseconds per day, comparedrelative to the clocks on the ground. This is a difference of 4.465 parts in 10¹⁰.<ref>Rizos, Chris. [[University of New South Wales]]. [http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap3/312.htm GPS Satellite Signals] {{Webarchive|url=https://web.archive.org/web/20100612004027/http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap3/312.htm |date=2010-06-12}}. 1999.</ref> Without correction, errors of roughly 11.4&nbsp;km/day would accumulate in the position.<ref>{{Cite book |last=Faraoni |first=Valerio |url=https://books.google.com/books?id=NuS9BAAAQBAJ |title=Special Relativity |publisher=Springer Science & Business Media |year=2013 |isbn=978-3-319-01107-3 |edition=illustrated |page=54}} [https://books.google.com/books?id=NuS9BAAAQBAJ&pg=PA54 Extract of page 54]</ref> This initial pseudorange error is corrected in the process of solving the [[GPS#Navigation equations|navigation equations]]. In addition, the elliptical, rather than perfectly circular, satellite orbits cause the time dilation and gravitational frequency shift effects to vary with time. This eccentricity effect causes the clock rate difference between a GPS satellite and a receiver to increase or decrease depending on the altitude of the satellite.

To compensate for the discrepancy, the frequency standard on board each satellite is given a rate offset prior to launch, making it run slightly slower than the desired frequency on Earth; specifically, at 10.22999999543&nbsp;MHz instead of 10.23&nbsp;MHz.<ref name="Nelson">[http://www.aticourses.com/global_positioning_system.htm The Global Positioning System by Robert A. Nelson Via Satellite] {{Webarchive|url=https://web.archive.org/web/20100718150217/http://www.aticourses.com/global_positioning_system.htm |date=2010-07-18 }}, November 1999</ref> Since the atomic clocks on board the GPS satellites are precisely tuned, it makes the system a practical engineering application of the scientific theory of relativity in a real-world environment.<ref>Pogge,{{cite web|last=Pogge|first=Richard W.; [|url=http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html "|title=Real-World Relativity: The GPS Navigation System"]. Retrieved 25 January |access-date=2008.-01-25}}</ref> Placing atomic clocks on artificial satellites to test Einstein's general theory was proposed by [[Friedwardt Winterberg]] in 1955.<ref>{{Cite web |date=1956-08-10 |title=Astronautica Acta II, 25 (1956). |url=http://bourabai.kz/winter/satelliten.htm |access-date=2009-10-23 |archive-date=2014-07-03 |archive-url=https://web.archive.org/web/20140703080406/http://bourabai.kz/winter/satelliten.htm |url-status=dead }}</ref>

{| class="wikitable" style="margin:1em auto;"

|+ SR and GR combined

|-

! Time dilation !! Value !! Notes

|-

| Kinetic || -7.2 μs/day || Clocks slowed in satellites due to Velocity

|-

| Gravitational || +45.8 μs/day || Clocks sped up in satellites due to higher altitude

|-

| Total (Combined) || +38.6 μs/day ||

|}

To compensate for the discrepancy, the frequency standard on board each satellite is given a rate offset prior to launch, making it run slightly slower than the desired frequency on Earth; specifically, at 10.22999999543&nbsp;MHz instead of 10.23&nbsp;MHz.<ref name="Nelson">[http://www.aticourses.com/global_positioning_system.htm The Global Positioning System by Robert A. Nelson Via Satellite] {{Webarchive|url=https://web.archive.org/web/20100718150217/http://www.aticourses.com/global_positioning_system.htm |date=2010-07-18 }}, November 1999</ref> Since the atomic clocks on board the GPS satellites are precisely tuned, it makes the system a practical engineering application of the scientific theory of relativity in a real-world environment.<ref>Pogge, Richard W.; [http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html "Real-World Relativity: The GPS Navigation System"]. Retrieved 25 January 2008.</ref> Placing atomic clocks on artificial satellites to test Einstein's general theory was proposed by [[Friedwardt Winterberg]] in 1955.<ref>{{Cite web |date=1956-08-10 |title=Astronautica Acta II, 25 (1956). |url=http://bourabai.kz/winter/satelliten.htm |access-date=2009-10-23 |archive-date=2014-07-03 |archive-url=https://web.archive.org/web/20140703080406/http://bourabai.kz/winter/satelliten.htm |url-status=dead }}</ref>

=== Calculations ===

To calculate the amount of daily time dilation experienced by GPS satellites relative to Earth we need to separately determine the amounts due to the satellite's velocity and altitude, and add them together.

==== Kinetic time dilation ====

The amount due to velocity will beis determined using the [[Lorentz transformation]]. The time measured by an object moving with velocity <math>v</math> changes by (the inverse of) the [[Lorentz factor]]:

: <math> \frac{1}{\gamma } = \sqrt{1-\frac{v^2}{c^2}} </math>

For small values of ''v/c'' this approximates to:

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: <math> -8.349\times 10^{-11}\times 60\times 60\times 24\times 10^9\approx -7214 \text{ ns} </math>

That is, the satellites' clocks loseare slower than Earth's clocks by 7214 nanoseconds a day due to their velocity.

: Note that this speed of {{val|3874|u=m/s}} is measured relative to Earth's center rather than its surface where the GPS receivers (and users) are. This is because Earth's equipotential makes net time dilation equal across its geodesic surface.<ref>{{Cite web |last=S. P. Drake |date=January 2006 |title=The equivalence principle as a stepping stone from special to general relativity |url=http://www.phys.unsw.edu.au/einsteinlight/jw/2006AJP.pdf |website=Am. J. Phys., Vol. 74, No. 1 |pages=22–25}}</ref> That is, the combination of Special and General effects make the net time dilation at the equator equal to that of the poles, which in turn are at rest relative to the center. Hence we use the center as a reference point to represent the entire surface.

==== Gravitational time dilation ====

The amount of dilation due to gravity will beis determined using the [[gravitational time dilation]] equation:

: <math> \frac{t_r}{t_\infty} =\sqrt{1-\frac{2G M}{r c^2}} </math>

where <math>t_r</math> is the time passed at a distance <math>r</math> from the center of the Earth and <math>t_\infty</math> is the time passed for a far away observer.

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: <math> 5.307\times 10^{-10}\times 60\times 60\times 24\times 10^9\approx 45850 \text{ ns} </math>

That is, the satellites' clocks gain 45850 nanoseconds a day due to gravitational time dilation.

==== Combined time dilation effects ====

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: 45850 – 7210 = 38640 ns

Hence the satellites' clocks gain approximately 38,640 nanoseconds a day or 38.6 μs per day due to relativityrelativistic effects in total.

In order to compensate for this gain, a GPS clock's frequency needs to be slowed by the fraction:

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: (1 – &nbsp;{{val|4.472|e=-10}}) × 10.23 = 10.22999999543

That is, we need to slow the clocks down from 10.23&nbsp;MHz to 10.22999999543&nbsp;MHz in order to negate both time dilation effects.

=== Sagnac distortion ===

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Since GPS signals at terrestrial receivers tend to be relatively weak, natural radio signals or scattering of the GPS signals can [[Desensitization (telecommunications)|desensitize]] the receiver, making acquiring and tracking the satellite signals difficult or impossible.

[[Space weather]] degrades GPS operation in two ways, direct interference by solar radio burst noise in the same frequency band<ref>Cerruti, A., P. M. Kintner, D. E. Gary, A. J. Mannucci, R. F. Meyer, P. H. Doherty, and A. J. Coster (2008), Effect of intense December 2006 solar radio bursts on GPS receivers, Space Weather, {{doi|10.1029/2007SW000375}}, October 19, 2008</ref> or by scattering of the GPS radio signal in ionospheric irregularities referred to as scintillation.<ref>{{Cite journal |last1=Aarons, Jules |last2=Basu, Santimay |year=1994 |title=Ionospheric amplitude and phase fluctuations at the GPS frequencies |journal=Proceedings of ION GPS |volume=2 |pages=1569–1578}}</ref> Both forms of degradation follow the 11 year [[solar cycle]] and are a maximum at sunspot maximum although they can occur at any time. Solar radio bursts are associated with [[solar flares]] and [[coronal mass ejection]]s (CMEs)<ref>S. Mancuso and J. C. Raymond, "Coronal transients and metric type II radio bursts. I. Effects of geometry, 2004, Astronomy and Astrophysics, v.413, p.363-371'</ref> and their impact can affect reception over the half of the Earth facing the sun. Scintillation occurs most frequently at tropical latitudes where it is a night time phenomenon. It occurs less frequently at high latitudes or mid-latitudes where magnetic storms can lead to scintillation.<ref>{{Cite journal |last1=Ledvina, B. M. |last2=J. J. Makela |last3=P. M. Kintner |name-list-style=amp |year=2002 |title=First observations of intense GPS L1 amplitude scintillations at midlatitude |journal=Geophysical Research Letters |volume=29 |issue=14 |page=1659 |bibcode=2002GeoRL..29.1659L |doi=10.1029/2002GL014770|s2cid=133701419 |doi-access=free }}</ref> In addition to producing scintillation, magnetic storms can produce strong ionospheric gradients that degrade the accuracy of SBAS systems.<ref>Tom Diehl, [http://www.faa.gov/about/office_org/headquarters_offices/ato/service_units/techops/navservices/gnss/library/satNav/media/SATNAV_0604.PDF Solar Flares Hit the Earth- WAAS Bends but Does Not Break], SatNav News, volume 23, June 2004.</ref>

== Artificial sources of interference ==