Error analysis for the Global Positioning System: Difference between revisions

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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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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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== Dilution of precision {{anchor|gdop}} ==

{{main articlesee|Dilution of precision (navigation)}}

{{merge to|Dilution of precision (navigation)#Dilution of precision|discuss=Talk:Dilution of precision (navigation)#Duplication|date=April 2021|section=TRUE}}

=== Computation of geometric dilution of precision ===

{{Duplication|section=yes|dupe=Dilution of precision (navigation)#Computation|discuss=Talk:Dilution of precision (navigation)#Duplication|date=June 2021}}

The concept of geometric dilution of precision was introduced in the section, ''error sources and analysis''. Computations were provided to show how PDOP was used and how it affected the receiver position error standard deviation.

When all visible GPS satellites are close together in the sky (i.e., small angular separation), the DOP values are high; when far apart, the DOP values are low. Conceptually, satellites that are close together cannot provide as much information as satellites that are widely separated. Low DOP values represent a better GPS positional accuracy due to the wider angular separation between the satellites used to calculate GPS receiver position. HDOP, VDOP, PDOP and TDOP are respectively Horizontal, Vertical, Position (3-D) and Time Dilution of Precision.

Figure 3.1 Dilution of Precision of Navstar GPS data from the U.S. Coast Guard provide a graphical indication of how geometry affect accuracy.<ref>{{Cite web |last=<!--Staff writer(s); no by-line.--> |date=September 1996 |title=NAVSTAR GPS User Equipment Introduction |url=http://www.navcen.uscg.gov/pubs/gps/gpsuser/gpsuser.pdf |access-date=July 5, 2014 |website=US Coast guard navigation center |publisher=US Coast Guard}}</ref>

We now take on the task of how to compute the dilution of precision terms. As a first step in computing DOP, consider the unit vector from the receiver to satellite i with components <math>\frac{(x_i- x)}{R_i}</math>, <math>\frac {(y_i-y)}{R_i}</math>, and <math>\frac {(z_i-z)}{R_i}</math> where the distance from receiver to the satellite, <math>\ R_i </math>, is given by:

:<math>R_i\,=\,\sqrt{(x_i- x)^2 + (y_i-y)^2 + (z_i-z)^2}</math>

where <math>\ x, y,</math> and <math>\ z</math> denote the position of the receiver and <math>\ x_i, y_i,</math> and <math>\ z_i</math> denote the position of satellite ''i''. These ''x'', ''y'', and ''z'' components may be components in a North, East, Down coordinate system, a South, East, Up coordinate system, or any other convenient system. Formulate the matrix ''A'' as:

:<math>A =

\begin{bmatrix}

\frac {(x_1- x)} {R_1} & \frac {(y_1-y)} {R_1} & \frac {(z_1-z)} {R_1} & 1 \\

\frac {(x_2- x)} {R_2} & \frac {(y_2-y)} {R_2} & \frac {(z_2-z)} {R_2} & 1 \\

\frac {(x_3- x)} {R_3} & \frac {(y_3-y)} {R_3} & \frac {(z_3-z)} {R_3} & 1 \\

\frac {(x_4- x)} {R_4} & \frac {(y_4-y)} {R_4} & \frac {(z_4-z)} {R_4} & 1

\end{bmatrix}</math>

The first three elements of each row of ''A'' are the components of a unit vector from the receiver to the indicated satellite. The elements in the fourth column are c where c denotes the speed of light. Formulate the matrix, ''Q'', as

:<math> Q = \left (A^T A \right )^{-1}

</math>

This computation is in accordance with Chapter 11 of The global positioning system by Parkinson and Spilker where the weighting matrix, ''P'', has been set to the identity matrix. The elements of the ''Q'' matrix are designated as:<ref>Parkinson (1996)</ref>

:<math>Q =

\begin{bmatrix}

d_x^2 & d_{xy}^2 & d_{xz}^2 & d_{xt}^2 \\

d_{xy}^2 & d_{y}^2 & d_{yz}^2 & d_{yt}^2 \\

d_{xz}^2 & d_{yz}^2 & d_{z}^2 & d_{zt}^2 \\

d_{xt}^2 & d_{yt}^2 & d_{zt}^2 & d_{t}^2

\end{bmatrix}

</math>

The Greek letter <math>\ \sigma</math> is used quite often where we have used ''d''. However the elements of the ''Q'' matrix do not represent variances and covariances as they are defined in probability and statistics. Instead they are strictly geometric terms. Therefore, d as in dilution of precision is used. PDOP, TDOP and GDOP are given by

:<math>\begin{align}

PDOP &= \sqrt{d_x^2 + d_y^2 + d_z^2} \\

TDOP &= \sqrt{d_{t}^2} = |d_{t}| \\

GDOP &= \sqrt{PDOP^2 + TDOP^2}

\end{align}</math>

in agreement with [https://web.archive.org/web/20141122153439/http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap1/149.htm "Section 1.4.9 of PRINCIPLES OF SATELLITE POSITIONING"].

The horizontal dilution of precision, <math> HDOP = \sqrt{d_x^2 + d_y^2}</math>, and the vertical dilution of precision, <math>\ VDOP = \sqrt{d_{z}^2} = |d_z|</math>, are both dependent on the coordinate system used. To correspond to the local horizon plane and the local vertical, ''x'', ''y'', and ''z'' should denote positions in either a North, East, Down coordinate system or a South, East, Up coordinate system.

=== Derivation of equations for computing geometric dilution of precision ===

The equations for computing the geometric dilution of precision terms have been described in the previous section. This section describes the derivation of these equations. The method used here is similar to that used in [https://books.google.com/books?id=lvI1a5J_4ewC&pg=PA474 "Global Positioning System (preview) by Parkinson and Spiker"]

Consider the position error vector, <math>\mathbf{e}</math>, defined as the vector from the intersection of the four sphere surfaces corresponding to the pseudoranges to the true position of the receiver.<math>\mathbf{e} = e_x\hat{x} + e_y\hat{y} + e_z\hat{z} </math> where bold denotes a vector and <math>\hat{x}</math>, <math>\hat{y}</math>, and <math>\hat{z}</math> denote unit vectors along the x, y, and z axes respectively. Let <math>\ e_t</math> denote the time error, the true time minus the receiver indicated time. Assume that the mean value of the three components of <math>\mathbf {e}</math> and <math>\ e_t</math> are zero.

:<math>A

\begin{bmatrix}

e_x \\ e_y \\ e_z \\ e_t

\end{bmatrix} =

\begin{bmatrix}

\frac {(x_1- x)} {R_1} & \frac {(y_1-y)} {R_1} & \frac {(z_1-z)} {R_1} & 1 \\

\frac {(x_2- x)} {R_2} & \frac {(y_2-y)} {R_2} & \frac {(z_2-z)} {R_2} & 1 \\

\frac {(x_3- x)} {R_3} & \frac {(y_3-y)} {R_3} & \frac {(z_3-z)} {R_3} & 1 \\

\frac {(x_4- x)} {R_4} & \frac {(y_4-y)} {R_4} & \frac {(z_4-z)} {R_4} & 1

\end{bmatrix}

\begin{bmatrix}

e_x \\ e_y \\ e_z \\ e_t

\end{bmatrix} =

\begin{bmatrix}

e_1 \\ e_2 \\ e_3 \\ e_4

\end{bmatrix}

\ (1)</math>

where <math>e_1</math>, <math>e_2</math>, <math>e_3</math>, and <math>e_4</math> are the errors in pseudoranges 1 through 4 respectively. This equation comes from linearizing [[Global Positioning System#Multidimensional Newton-Raphson calculations|the Newton-Raphson equation]] relating pseudoranges to receiver position, satellite positions, and receiver clock errors. Multiplying both sides by <math>A^{-1}</math> there results

:<math>

\begin{bmatrix}

e_x \\ e_y \\ e_z \\ e_t

\end{bmatrix} =

A^{-1}

\begin{bmatrix}

e_1 \\ e_2 \\ e_3 \\ e_4

\end{bmatrix} \ (2)</math> .

Transposing both sides:

:<math>

\begin{bmatrix}

e_x & e_y & e_z & e_t

\end{bmatrix} =

\begin{bmatrix}

e_1 & e_2 & e_3 & e_4

\end{bmatrix}\left (A^{-1} \right )^T \ (3)</math> .

Post multiplying the matrices on both sides of equation (2) by the corresponding matrices in equation (3), there results

:<math>

\begin{bmatrix}

e_x \\ e_y \\ e_z \\ e_t

\end{bmatrix}

\begin{bmatrix}

e_x & e_y & e_z & e_t

\end{bmatrix} =

A^{-1}

\begin{bmatrix}

e_1 \\ e_2 \\ e_3 \\ e_4

\end{bmatrix}

\begin{bmatrix}

e_1 & e_2 & e_3 & e_4

\end{bmatrix}\left (A^{-1} \right )^T \ (4)

</math> .

Taking the expected value of both sides and taking the non-random matrices outside the expectation operator, E, there results:

:<math>E

\left (\begin{bmatrix}

e_x \\ e_y \\ e_z \\ e_t

\end{bmatrix}

\begin{bmatrix}

e_x & e_y & e_z & e_t

\end{bmatrix} \right ) =

A^{-1} E

\left (\begin{bmatrix}

e_1 \\ e_2 \\ e_3 \\ e_4

\end{bmatrix}

\begin{bmatrix}

e_1 & e_2 & e_3 & e_4

\end{bmatrix} \right )

\left (A^{-1} \right )^T \ (5)

</math>

Assuming the pseudorange errors are uncorrelated and have the same variance, the covariance matrix on the right side can be expressed as a scalar times the identity matrix. Thus

:<math>

\begin{bmatrix}

\sigma_x^2 & \sigma_{xy}^2 & \sigma_{xz}^2 & \sigma_{xt}^2 \\

\sigma_{xy}^2 & \sigma_{y}^2 & \sigma_{yz}^2 & \sigma_{yt}^2 \\

\sigma_{xz}^2 & \sigma_{yz}^2 & \sigma_{z}^2 & \sigma_{zt}^2 \\

\sigma_{xt}^2 & \sigma_{yt}^2 & \sigma_{zt}^2 & \sigma_{t}^2

\end{bmatrix} = \sigma_R^2 \ A^{-1} \left (A^{-1} \right )^T =

\sigma_R^2 \ \left (A^T A \right )^{-1} \ (6)</math>

since <math>\ A^{-1} \left (A^{-1} \right )^T \left (A^T A \right ) = I </math>

Note: <math>\left( A^{-1} \right)^T = \left( A^{T} \right)^{-1},</math> since <math>I = \left( A A^{-1} \right)^T = \left( A^{-1}\right)^T A^T</math>

Substituting for <math>\left( A^T A \right)^{-1} = Q</math> there follows

:<math>

\begin{bmatrix}

\sigma_x^2 & \sigma_{xy}^2 & \sigma_{xz}^2 & \sigma_{xt}^2 \\

\sigma_{xy}^2 & \sigma_{y}^2 & \sigma_{yz}^2 & \sigma_{yt}^2 \\

\sigma_{xz}^2 & \sigma_{yz}^2 & \sigma_{z}^2 & \sigma_{zt}^2 \\

\sigma_{xt}^2 & \sigma_{yt}^2 & \sigma_{zt}^2 & \sigma_{t}^2

\end{bmatrix} = \sigma_R^2

\begin{bmatrix}

d_x^2 & d_{xy}^2 & d_{xz}^2 & d_{xt}^2 \\

d_{xy}^2 & d_{y}^2 & d_{yz}^2 & d_{yt}^2 \\

d_{xz}^2 & d_{yz}^2 & d_{z}^2 & d_{zt}^2 \\

d_{xt}^2 & d_{yt}^2 & d_{zt}^2 & d_{t}^2

\end{bmatrix} \ (7)

</math>

From equation (7), it follows that the variances of indicated receiver position and time are

:<math>\sigma_{rc}^2 = \sigma_x^2 + \sigma_y^2 + \sigma_z^2 = \sigma_R^2\left(d_x^2 + d_y^2 + d_z^2\right) = PDOP^2 \sigma_R^2</math> and

:<math>\sigma_t^2 = \sigma_R^2 d_t^2 = TDOP^2 \sigma_R^2</math>

The remaining position and time error variance terms follow in a straightforward manner.

{{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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== 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&nbsp;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.

== Relativity ==

The [[theory of relativity]] introduces several effects that need to be taken into account when dealing with precise time measurements. According 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

[[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) due to the different effects.

(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) tick slower.

[[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.

=== [[Special Relativity (SR)relativity]] ===

SRSpecial relativity predicts that clocksas the velocity of an object increases (in a given frame), its time slowslows down (as velocitymeasured increases.in that Thatframe). isFor instance, the frequency of the atomic clocks moving at GPS orbital speeds will tick more slowly than stationary ground 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 theirthe constant movement andof heightGPS 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 Relativity (GR)relativity]] ===

GRSpecial hasrelativity allows the oppositecomparison effect.of clocks GRonly predictsin thata clocksflat speed[[spacetime]], upwhich asneglects theygravitational geteffects furtheron awaythe frompassage aof massivetime. objectAccording to general relativity, (the Earthpresence inof thisgravitating case).bodies (like TheEarth) effectcurves ofspacetime, gravitationalwhich frequencymakes shiftcomparing onclocks thenot GPSas duestraightforward toas [[generalin special relativity]]. isHowever, thatone acan clockoften closeraccount tofor amost massiveof objectthe willdiscrepancy beby slowerthe thanintroduction aof clock[[gravitational farthertime awaydilation]], the slowing down of time near gravitating bodies. AppliedIn tocase of the GPS, the receivers are much closer to the center of Earth than the satellites, causing the GPS clocks inat the satellitesaltitude 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. In fact, duringDuring early development, itsome{{who|date=January was2024}} believed that GPS would not be affected by GRgeneral relativistic effects, but the [[Hafele–Keating_experimentHafele–Keating experiment]] showed that it didwould be.

=== Combined SRkinetic and GRgravitational time dilations ===

WhenCombined, combiningthese SRsources andof GR,time dilation cause the discrepancyclocks ison aboutthe satellites to gain +38.6 microseconds per day relative 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.

<CENTER>

{| class="wikitable"

|+ SR and GR combined

|-

! Theory !! Value !! Notes

|-

| SR (Special Relativity) || -7.2 μs/day || Clocks slowed in Satellites due to Velocity

|-

| GR (General Relativity) || +45.8 μs/day || Clocks sped up in Satellites due to lower Gravity

|-

| 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,{{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> The conclusion is that the GPS satellites must compensate for GR, the physics of [[black holes]] and extreme gravity.

=== 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 [[specialthe relativity]]satellite's (velocity) and [[general relativity]] (gravity)altitude, and add them together.

==== SpecialKinetic Relativitytime (SR)dilation ====

The amount due to velocity will beis determined using the [[Lorentz transformation]]. ThisThe willtime bemeasured 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'', by using [[binomial expansion]] this approximates to:

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

The GPS satellites move at {{val|3874|u=m/s}} relative to Earth's center.<ref name="Nelson" /> We thus determine:

: <math> \frac{1}{\gamma } \approx 1-\frac{3874^2}{2 \left(2.998\times 10^8\right)^2} \approx 1-8.349\times 10^{-11} </math>

This difference below 1 of {{val|8.349|e=-11}} represents the fraction by which the satellites' clocks movetick slower than Earth'sthe stationary clocks. It is then multiplied by the number of nanoseconds in a day:

: <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 SRtheir effectsvelocity.

: 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.

==== GeneralGravitational Relativitytime (GR)dilation ====

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

: <math> \frac{1t_r}{t_\gamma 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.

For small values of ''M<math>GM/r'', by using [[binomial expansion]](rc^2)</math> this approximates to:

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

Determine the difference <math>\Delta t</math> between the satellite's time <math>t_{r_{\text{GPS}}}</math> and Earth time <math>t_{r_{\text{Earth}}}</math>:

We are again only interested in the fraction below 1, and in the difference between Earth and the satellites. To determine this difference we take:

: <math> \Delta t \leftapprox (1-\frac{1G M}{r_{\gammatext{GPS}} c^2}\right) \approx -(1-\frac{G M_M}{r_{\text{earthEarth}} c^2})= \frac{G M}{R_r_{\text{earthEarth}} c^2}-\frac{G M_{\text{earth}}M}{R_r_{\text{gpsGPS}} c^2} </math>

Earth has a radius of 6,357&nbsp;km (at the poles) making ''R<submath>earthr_{\text{Earth}}</submath>'' = 6,357,000 m and the satellites have an altitude of 20,184&nbsp;km<ref name="Nelson" /> making their orbit radius ''R<submath>gpsr_{\text{GPS}}</submath>'' = 26,541,000 m. Substituting these in the above equation, with Earth mass ''M_earth'' = {{val|5.974|e=24}}, ''G'' = {{val|6.674|e=-11}}, and ''c'' = {{val|2.998|e=8}} (all in [[International System of Units|SI]] units), gives:

: <math> \Delta \left(\frac{1}{\gamma }\right)t \approx 5.307\times 10^{-10} </math>

This represents the fraction by which the clocks at satellites' clocksaltitude movetick faster than on the surface of the Earth's. It is then multiplied by the number of nanoseconds in a day:

: <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 GRgravitational effects.time dilation.

==== Combined SRtime anddilation GReffects ====

These effects are added together to give (rounded to 10 ns):

: 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:

Line 353 ⟶ 184:

: (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 thetime SR and GRdilation effects of relativity.

=== Sagnac distortion ===

GPS observation processing must also compensate for the [[Sagnac effect]]. The GPS time scale is defined in an [[inertial]] system but observations are processed in an [[ECEF|Earth-centered, Earth-fixed]] (co-rotating) system, a system in which [[Relativity of simultaneity|simultaneity]] is not uniquely defined. A coordinate transformation is thus applied to convert from the inertial system to the ECEF system. The resulting signal run time correction has opposite algebraic signs for satellites in the Eastern and Western celestial hemispheres. Ignoring this effect will produce an east–west error on the order of hundreds of nanoseconds, or tens of meters in position.<ref>Ashby, Neil [http://www.ipgp.fr/~tarantola/Files/Professional/GPS/Neil_Ashby_Relativity_GPS.pdf Relativity and GPS]. [[Physics Today]], May 2002.</ref>

== Natural sources of interference ==

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 ==

Line 381 ⟶ 212:

== 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).