Error analysis for the Global Positioning System: Difference between revisions

Revision as of 10:06, 22 October 2022

Revision as of 18:59, 19 November 2022

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{{Short description|Detail of the global positioning system}}

[[File:GPS Satellite NASA art-iif.jpg|right|thumb|Artist's conception of GPS Block II-F satellite in orbit]]

The '''error analysis for the [[Global Positioning System]]''' is important for understanding how GPS works, and for knowing what magnitude of error should be expected. The GPS makes corrections for receiver clock errors and other effects but there are still residual errors which are not corrected. GPS receiver position is computed based on data received from the satellites. Errors depend on geometric dilution of precision and the sources listed in the table below.

== Overview ==

{{Disputed section|date=June 2016}}

{| class="wikitable" style="margin:.5em; float: right"

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== Anti-spoofing ==

{{seeSee also|GPS signals#Precision code}}

Another restriction on GPS, antispoofing, remains on. This encrypts the ''P-code'' so that it cannot be mimicked by a transmitter sending false information. Few civilian receivers have ever used the P-code, and the accuracy attainable with the public C/A code was much better than originally expected (especially with [[Differential GPS|DGPS]]), so much so that the antispoof policy has relatively little effect on most civilian users. Turning off antispoof would primarily benefit surveyors and some scientists who need extremely precise positions for experiments such as tracking tectonic plate motion.

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[[Special Relativity]] (SR) and [[General Relativity]] (GR) are two separate and distinct theories under the title of the [[Theory of Relativity]]. SR and GR make different (opposite) predictions when it comes to the clocks on-board GPS satellites. Note the opposite signs (plus and minus) below due to the different effects.

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

A number of sources of error exist due to [[Theory of relativity|relativistic]] effects<ref>Webb (2004), p. 32.</ref> that would render the system useless if uncorrected. Three relativistic effects are time dilation, gravitational frequency shift, and eccentricity effects. Examples include the relativistic time ''slowing'' due to the speed of the satellite of about 1 part in 10¹⁰, the gravitational time dilation that makes a satellite run about 5 parts in 10¹⁰ ''faster'' than an Earth-based clock, and the [[Sagnac effect]] due to rotation relative to receivers on Earth. These topics are examined below, one at a time.

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When combining SR and GR, the discrepancy is about +38 microseconds per day. 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.

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<CENTER>

{| class="wikitable"

|+ SR and GR combined

|-

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| Total (Combined) || +38.6 μs/day || GR is larger effect than SR

|}

</CENTER>

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}}</ref> The conclusion is that the GPS satellites must compensate for GR, the physics of [[black holes]] and extreme gravity.

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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 GR effects.

==== Combined SR and GR ====

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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, {{doiDOI|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 }}</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 ==

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== References ==

* {{Cite book |last1=Grewal, Mohinder S. |url=https://books.google.com/books?id=ZM7muB8Y35wC |title=Global positioning systems, inertial navigation, and integration |last2=Weill, Lawrence Randolph |last3=Andrews, Angus P. |publisher=John Wiley and Sons |year=2001 |isbn=978-0-47135471-03235032-3}}

* {{Cite book |last1=Parkinson |url=https://books.google.com/books?id=lvI1a5J_4ewC |title=The global positioning system |last2=Spilker |publisher=American Institute of Aeronautics & Astronomy |year=1996 |isbn=978-1-56347-106-3}}

* {{Cite book |last=Webb, Stephen |url=https://books.google.com/books?id=LzQcsSCdeLgC |title=Out of this world: colliding universes, branes, strings, and other wild ideas of modern physics |publisher=Springer |year=2004 |isbn=0-387-02930-3 |access-date=2013-08-16}}

== External links ==

{{commonsCommons|Global Positioning System}}

* [https://www.gps.gov/ GPS.gov]—General public education website created by the U.S. Government

* [http://www.gps.gov/technical/ps/2008-SPS-performance-standard.pdf GPS SPS Performance Standard]—The official Standard Positioning Service specification (2008 version).

* [http://www.gps.gov/technical/ps/2001-SPS-performance-standard.pdf GPS SPS Performance Standard]—The official Standard Positioning Service specification (2001 version).