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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ational Marine Electronics Association (NMEA), references to this protocol have been compiled from public records, allowing open source tools like gpsd to read the protocol without violating intellectual property laws. Other proprietary protocols exist as well, such as the SiRF and MTK protocols. Receivers can interface with other devices using methods including a serial connection, USB, or Blueto | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | oth.
Applications
While originally a military project, GPS is considered a dual-use technology, meaning it has significant civilian applications as well.
GPS has become a widely deployed and useful tool for commerce, scientific uses, tracking, and surveillance. GPS's accurate time facilitates everyday activities such as banking, mobile phone operations, and even the control of power grids by a | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | llowing well synchronized hand-off switching.
Civilian
Many civilian applications use one or more of GPS's three basic components: absolute location, relative movement, and time transfer.
Amateur radio: clock synchronization required for several digital modes such as FT8, FT4 and JS8; also used with APRS for position reporting; is often critical during emergency and disaster communications s | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | upport.
Atmosphere: studying the troposphere delays (recovery of the water vapor content) and ionosphere delays (recovery of the number of free electrons). Recovery of Earth surface displacements due to the atmospheric pressure loading.
Astronomy: both positional and clock synchronization data is used in astrometry and celestial mechanics and precise orbit determination. GPS is also used in bo | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | th amateur astronomy with small telescopes as well as by professional observatories for finding extrasolar planets.
Automated vehicle: applying location and routes for cars and trucks to function without a human driver.
Cartography: both civilian and military cartographers use GPS extensively.
Cellular telephony: clock synchronization enables time transfer, which is critical for synchronizing i | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ts spreading codes with other base stations to facilitate inter-cell handoff and support hybrid GPS/cellular position detection for mobile emergency calls and other applications. The first handsets with integrated GPS launched in the late 1990s. The U.S. Federal Communications Commission (FCC) mandated the feature in either the handset or in the towers (for use in triangulation) in 2002 so emergen | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | cy services could locate 911 callers. Third-party software developers later gained access to GPS APIs from Nextel upon launch, followed by Sprint in 2006, and Verizon soon thereafter.
Clock synchronization: the accuracy of GPS time signals (±10 ns) is second only to the atomic clocks they are based on, and is used in applications such as GPS disciplined oscillators.
Disaster relief/emergency ser | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | vices: many emergency services depend upon GPS for location and timing capabilities.
GPS-equipped radiosondes and dropsondes: measure and calculate the atmospheric pressure, wind speed and direction up to from the Earth's surface.
Radio occultation for weather and atmospheric science applications.
Fleet tracking: used to identify, locate and maintain contact reports with one or more fleet vehi | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | cles in real-time.
Geodesy: determination of Earth orientation parameters including the daily and sub-daily polar motion, and length-of-day variabilities, Earth's center-of-mass - geocenter motion, and low-degree gravity field parameters.
Geofencing: vehicle tracking systems, person tracking systems, and pet tracking systems use GPS to locate devices that are attached to or carried by a person, | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | vehicle, or pet. The application can provide continuous tracking and send notifications if the target leaves a designated (or "fenced-in") area.
Geotagging: applies location coordinates to digital objects such as photographs (in Exif data) and other documents for purposes such as creating map overlays with devices like Nikon GP-1
GPS aircraft tracking
GPS for mining: the use of RTK GPS has sign | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ificantly improved several mining operations such as drilling, shoveling, vehicle tracking, and surveying. RTK GPS provides centimeter-level positioning accuracy.
GPS data mining: It is possible to aggregate GPS data from multiple users to understand movement patterns, common trajectories and interesting locations.
GPS tours: location determines what content to display; for instance, information | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | about an approaching point of interest.
Navigation: navigators value digitally precise velocity and orientation measurements, as well as precise positions in real-time with a support of orbit and clock corrections.
Orbit determination of low-orbiting satellites with GPS receiver installed on board, such as GOCE, GRACE, Jason-1, Jason-2, TerraSAR-X, TanDEM-X, CHAMP, Sentinel-3, and some cubesats | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | , e.g., CubETH.
Phasor measurements: GPS enables highly accurate timestamping of power system measurements, making it possible to compute phasors.
Recreation: for example, Geocaching, Geodashing, GPS drawing, waymarking, and other kinds of location based mobile games such as Pokémon Go.
Reference frames: realization and densification of the terrestrial reference frames in the framework of Globa | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | l Geodetic Observing System. Co-location in space between Satellite laser ranging and microwave observations for deriving global geodetic parameters.
Robotics: self-navigating, autonomous robots using GPS sensors, which calculate latitude, longitude, time, speed, and heading.
Sport: used in football and rugby for the control and analysis of the training load.
Surveying: surveyors use absolute l | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ocations to make maps and determine property boundaries.
Tectonics: GPS enables direct fault motion measurement of earthquakes. Between earthquakes GPS can be used to measure crustal motion and deformation to estimate seismic strain buildup for creating seismic hazard maps.
Telematics: GPS technology integrated with computers and mobile communications technology in automotive navigation systems. | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System |
Restrictions on civilian use
The U.S. government controls the export of some civilian receivers. All GPS receivers capable of functioning above above sea level and , or designed or modified for use with unmanned missiles and aircraft, are classified as munitions (weapons)—which means they require State Department export licenses.
This rule applies even to otherwise purely civilian units that | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | only receive the L1 frequency and the C/A (Coarse/Acquisition) code.
Disabling operation above these limits exempts the receiver from classification as a munition. Vendor interpretations differ. The rule refers to operation at both the target altitude and speed, but some receivers stop operating even when stationary. This has caused problems with some amateur radio balloon launches that regularly | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | reach .
These limits only apply to units or components exported from the United States. A growing trade in various components exists, including GPS units from other countries. These are expressly sold as ITAR-free.
Military
As of 2009, military GPS applications include:
Navigation: Soldiers use GPS to find objectives, even in the dark or in unfamiliar territory, and to coordinate troop and s | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | upply movement. In the United States armed forces, commanders use the Commander's Digital Assistant and lower ranks use the Soldier Digital Assistant.
Target tracking: Various military weapons systems use GPS to track potential ground and air targets before flagging them as hostile. These weapon systems pass target coordinates to precision-guided munitions to allow them to engage targets accurate | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ly. Military aircraft, particularly in air-to-ground roles, use GPS to find targets.
Missile and projectile guidance: GPS allows accurate targeting of various military weapons including ICBMs, cruise missiles, precision-guided munitions and artillery shells. Embedded GPS receivers able to withstand accelerations of 12,000 g or about have been developed for use in howitzer shells.
Search and re | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | scue.
Reconnaissance: Patrol movement can be managed more closely.
GPS satellites carry a set of nuclear detonation detectors consisting of an optical sensor called a bhangmeter, an X-ray sensor, a dosimeter, and an electromagnetic pulse (EMP) sensor (W-sensor), that form a major portion of the United States Nuclear Detonation Detection System. General William Shelton has stated that future sate | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | llites may drop this feature to save money.
GPS type navigation was first used in war in the 1991 Persian Gulf War, before GPS was fully developed in 1995, to assist Coalition Forces to navigate and perform maneuvers in the war. The war also demonstrated the vulnerability of GPS to being jammed, when Iraqi forces installed jamming devices on likely targets that emitted radio noise, disrupting rec | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | eption of the weak GPS signal.
GPS's vulnerability to jamming is a threat that continues to grow as jamming equipment and experience grows. GPS signals have been reported to have been jammed many times over the years for military purposes. Russia seems to have several objectives for this behavior, such as intimidating neighbors while undermining confidence in their reliance on American systems, p | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | romoting their GLONASS alternative, disrupting Western military exercises, and protecting assets from drones. China uses jamming to discourage US surveillance aircraft near the contested Spratly Islands. North Korea has mounted several major jamming operations near its border with South Korea and offshore, disrupting flights, shipping and fishing operations. Iranian Armed Forces disrupted the civi | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | lian airliner plane Flight PS752's GPS when it shot down the aircraft.
Timekeeping
Leap seconds
While most clocks derive their time from Coordinated Universal Time (UTC), the atomic clocks on the satellites are set to "GPS time". The difference is that GPS time is not corrected to match the rotation of the Earth, so it does not contain leap seconds or other corrections that are periodically add | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ed to UTC. GPS time was set to match UTC in 1980, but has since diverged. The lack of corrections means that GPS time remains at a constant offset with International Atomic Time (TAI) (TAI - GPS = 19 seconds). Periodic corrections are performed to the on-board clocks to keep them synchronized with ground clocks.
The GPS navigation message includes the difference between GPS time and UTC. GPS tim | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | e is 18 seconds ahead of UTC because of the leap second added to UTC on December 31, 2016. Receivers subtract this offset from GPS time to calculate UTC and specific time zone values. New GPS units may not show the correct UTC time until after receiving the UTC offset message. The GPS-UTC offset field can accommodate 255 leap seconds (eight bits).
Accuracy
GPS time is theoretically accurate to a | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | bout 14 nanoseconds, due to the clock drift relative to International Atomic Time that the atomic clocks in GPS transmitters experience Most receivers lose some accuracy in their interpretation of the signals and are only accurate to about 100 nanoseconds.
Format
As opposed to the year, month, and day format of the Gregorian calendar, the GPS date is expressed as a week number and a seconds-int | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | o-week number. The week number is transmitted as a ten-bit field in the C/A and P(Y) navigation messages, and so it becomes zero again every 1,024 weeks (19.6 years). GPS week zero started at 00:00:00 UTC (00:00:19 TAI) on January 6, 1980, and the week number became zero again for the first time at 23:59:47 UTC on August 21, 1999 (00:00:19 TAI on August 22, 1999). It happened the second time at 23 | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | :59:42 UTC on April 6, 2019. To determine the current Gregorian date, a GPS receiver must be provided with the approximate date (to within 3,584 days) to correctly translate the GPS date signal. To address this concern in the future the modernized GPS civil navigation (CNAV) message will use a 13-bit field that only repeats every 8,192 weeks (157 years), thus lasting until 2137 (157 years after GP | [
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] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | S week zero).
Communication
The navigational signals transmitted by GPS satellites encode a variety of information including satellite positions, the state of the internal clocks, and the health of the network. These signals are transmitted on two separate carrier frequencies that are common to all satellites in the network. Two different encodings are used: a public encoding that enables lower | [
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] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | resolution navigation, and an encrypted encoding used by the U.S. military.
Message format
{|class="wikitable" style="float:right; margin:0 0 0.5em 1em;" border="1"
|+
! Subframes !! Description
|-
| 1 || Satellite clock,GPS time relationship
|-
| 2–3 || Ephemeris(precise satellite orbit)
|-
| 4–5 || Almanac component(satellite network synopsis,error correction)
|}
Each GPS satellite continuo | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | usly broadcasts a navigation message on L1 (C/A and P/Y) and L2 (P/Y) frequencies at a rate of 50 bits per second (see bitrate). Each complete message takes 750 seconds ( minutes) to complete. The message structure has a basic format of a 1500-bit-long frame made up of five subframes, each subframe being 300 bits (6 seconds) long. Subframes 4 and 5 are subcommutated 25 times each, so that a comple | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | te data message requires the transmission of 25 full frames. Each subframe consists of ten words, each 30 bits long. Thus, with 300 bits in a subframe times 5 subframes in a frame times 25 frames in a message, each message is 37,500 bits long. At a transmission rate of 50-bit/s, this gives 750 seconds to transmit an entire almanac message (GPS). Each 30-second frame begins precisely on the minute | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | or half-minute as indicated by the atomic clock on each satellite.
The first subframe of each frame encodes the week number and the time within the week, as well as the data about the health of the satellite. The second and the third subframes contain the ephemeris – the precise orbit for the satellite. The fourth and fifth subframes contain the almanac, which contains coarse orbit and status inf | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ormation for up to 32 satellites in the constellation as well as data related to error correction. Thus, to obtain an accurate satellite location from this transmitted message, the receiver must demodulate the message from each satellite it includes in its solution for 18 to 30 seconds. To collect all transmitted almanacs, the receiver must demodulate the message for 732 to 750 seconds or minutes | [
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] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | .
All satellites broadcast at the same frequencies, encoding signals using unique code-division multiple access (CDMA) so receivers can distinguish individual satellites from each other. The system uses two distinct CDMA encoding types: the coarse/acquisition (C/A) code, which is accessible by the general public, and the precise (P(Y)) code, which is encrypted so that only the U.S. military and o | [
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0.01922591,
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] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ther NATO nations who have been given access to the encryption code can access it.
The ephemeris is updated every 2 hours and is sufficiently stable for 4 hours, with provisions for updates every 6 hours or longer in non-nominal conditions. The almanac is updated typically every 24 hours. Additionally, data for a few weeks following is uploaded in case of transmission updates that delay data uplo | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ad.
Satellite frequencies
{|class="wikitable" style="float:right; width:30em; margin:0 0 0.5em 1em;" border="1"
|+
! Band !! Frequency !! Description
|-
| L1 || 1575.42 MHz || Coarse-acquisition (C/A) and encrypted precision (P(Y)) codes, plus the L1 civilian (L1C) and military (M) codes on Block III and newer satellites.
|-
| L2 || 1227.60 MHz || P(Y) code, plus the L2C and military codes on t | [
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-0.025217345,
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-0.045871552,
0.0027502251,
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0.019987082,
0.07290348,
0.040934823,
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0.104124956,
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0.049340606,
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | he Block IIR-M and newer satellites.
|-
| L3 || 1381.05 MHz || Used for nuclear detonation (NUDET) detection.
|-
| L4 || 1379.913 MHz || Being studied for additional ionospheric correction.
|-
| L5 || 1176.45 MHz || Used as a civilian safety-of-life (SoL) signal on Block IIF and newer satellites.
|}
All satellites broadcast at the same two frequencies, 1.57542 GHz (L1 signal) and 1.2276 GHz (L2 s | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ignal). The satellite network uses a CDMA spread-spectrum technique where the low-bitrate message data is encoded with a high-rate pseudo-random (PRN) sequence that is different for each satellite. The receiver must be aware of the PRN codes for each satellite to reconstruct the actual message data. The C/A code, for civilian use, transmits data at 1.023 million chips per second, whereas the P cod | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | e, for U.S. military use, transmits at 10.23 million chips per second. The actual internal reference of the satellites is 10.22999999543 MHz to compensate for relativistic effects that make observers on the Earth perceive a different time reference with respect to the transmitters in orbit. The L1 carrier is modulated by both the C/A and P codes, while the L2 carrier is only modulated by the P cod | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | e. The P code can be encrypted as a so-called P(Y) code that is only available to military equipment with a proper decryption key. Both the C/A and P(Y) codes impart the precise time-of-day to the user.
The L3 signal at a frequency of 1.38105 GHz is used to transmit data from the satellites to ground stations. This data is used by the United States Nuclear Detonation (NUDET) Detection System (USN | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | DS) to detect, locate, and report nuclear detonations (NUDETs) in the Earth's atmosphere and near space. One usage is the enforcement of nuclear test ban treaties.
The L4 band at 1.379913 GHz is being studied for additional ionospheric correction.
The L5 frequency band at 1.17645 GHz was added in the process of GPS modernization. This frequency falls into an internationally protected range for a | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | eronautical navigation, promising little or no interference under all circumstances. The first Block IIF satellite that provides this signal was launched in May 2010. On February 5th 2016, the 12th and final Block IIF satellite was launched. The L5 consists of two carrier components that are in phase quadrature with each other. Each carrier component is bi-phase shift key (BPSK) modulated by a sep | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | arate bit train. "L5, the third civil GPS signal, will eventually support safety-of-life applications for aviation and provide improved availability and accuracy."
In 2011, a conditional waiver was granted to LightSquared to operate a terrestrial broadband service near the L1 band. Although LightSquared had applied for a license to operate in the 1525 to 1559 band as early as 2003 and it was put | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | out for public comment, the FCC asked LightSquared to form a study group with the GPS community to test GPS receivers and identify issue that might arise due to the larger signal power from the LightSquared terrestrial network. The GPS community had not objected to the LightSquared (formerly MSV and SkyTerra) applications until November 2010, when LightSquared applied for a modification to its Anc | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | illary Terrestrial Component (ATC) authorization. This filing (SAT-MOD-20101118-00239) amounted to a request to run several orders of magnitude more power in the same frequency band for terrestrial base stations, essentially repurposing what was supposed to be a "quiet neighborhood" for signals from space as the equivalent of a cellular network. Testing in the first half of 2011 has demonstrated t | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | hat the impact of the lower 10 MHz of spectrum is minimal to GPS devices (less than 1% of the total GPS devices are affected). The upper 10 MHz intended for use by LightSquared may have some impact on GPS devices. There is some concern that this may seriously degrade the GPS signal for many consumer uses. Aviation Week magazine reports that the latest testing (June 2011) confirms "significant jamm | [
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] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ing" of GPS by LightSquared's system.
Demodulation and decoding
Because all of the satellite signals are modulated onto the same L1 carrier frequency, the signals must be separated after demodulation. This is done by assigning each satellite a unique binary sequence known as a Gold code. The signals are decoded after demodulation using addition of the Gold codes corresponding to the satellites | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | monitored by the receiver.
If the almanac information has previously been acquired, the receiver picks the satellites to listen for by their PRNs, unique numbers in the range 1 through 32. If the almanac information is not in memory, the receiver enters a search mode until a lock is obtained on one of the satellites. To obtain a lock, it is necessary that there be an unobstructed line of sight fr | [
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] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | om the receiver to the satellite. The receiver can then acquire the almanac and determine the satellites it should listen for. As it detects each satellite's signal, it identifies it by its distinct C/A code pattern. There can be a delay of up to 30 seconds before the first estimate of position because of the need to read the ephemeris data.
Processing of the navigation message enables the determ | [
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0.006759883,
-0.021645011,
0.03823015,
0.00072200235,
-0.023304863,
0.008018158,
0.03568683,
-0.020681227,
-0.0149119,
-0.04666327,
-0.038149834,
0.067250796,
0.06318148,
0.037560854,
-0.0537578,
-0.034160834,
0.033464767,
0.040800244,
-0.013104803,
-0.025513537,
0.033170275,
0.008707533,
0.008854778,
0.09429032,
-0.082403645,
-0.040478982,
0.012716611,
-0.043209706,
0.04449475,
-0.029154506,
-0.031483654,
0.028404893,
0.03236712,
-0.0013477935,
0.00030369277,
-0.0065490548,
-0.01683947,
0.017254435,
0.059005078,
0.0010775655,
-0.090435185,
0.011431565,
0.00046014055,
0.050438095,
0.0042600646,
-0.026584411,
0.024763927,
0.020065473,
-0.2816395,
0.019503266,
0.038015977,
-0.025700942,
0.010233525,
-0.040666383,
-0.0005923264,
-0.030974988,
-0.097181685,
0.019396178,
-0.037132505,
0.016665453,
-0.014122131,
-0.026289921,
0.025500152,
0.08979266,
0.076406755,
-0.046261694,
-0.06864293,
-0.10574866,
0.08331388,
0.013252048,
0.16984038,
0.048671156,
-0.02531275,
0.03499076,
-0.0061842888,
-0.009537458,
0.026022203,
0.00811186,
0.0676256,
-0.025647396,
0.039381336,
0.007629967,
-0.011224083,
0.04457507,
-0.007127995,
0.05196409,
-0.011685897,
-0.021283591,
-0.01151188,
0.016812699,
0.012074088,
-0.052740473,
0.10955026,
-0.042781357,
-0.051830232,
-0.07394375,
-0.06248541,
-0.01991823,
-0.017214276,
-0.043879002,
-0.021390678,
-0.028886786,
-0.0053041657,
0.02710646,
-0.045565628,
0.023987545,
0.0060203117,
0.012596138,
0.042593956,
-0.101358086,
0.07496107,
-0.009617774,
0.012482358
] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ination of the time of transmission and the satellite position at this time. For more information see Demodulation and Decoding, Advanced.
Navigation equations
Problem description
The receiver uses messages received from satellites to determine the satellite positions and time sent. The x, y, and z components of satellite position and the time sent (s) are designated as [xi, yi, zi, si] where t | [
-0.026965622,
0.023773555,
0.035713755,
-0.027940605,
-0.009322439,
-0.061383855,
0.05232853,
0.028822096,
0.0051720836,
-0.034992535,
0.028341282,
-0.057270225,
0.013075455,
0.014624743,
0.021703387,
-0.02378691,
-0.010284066,
0.009362507,
0.07019877,
0.0074993544,
0.07773151,
-0.0009858345,
-0.025322843,
0.0014883514,
0.107915916,
0.04738684,
-0.022144131,
-0.00020785166,
-0.0041236435,
-0.17512295,
-0.027539928,
-0.015492879,
-0.004768067,
0.009269015,
-0.064802974,
-0.024935521,
0.016093895,
-0.03576718,
-0.052782632,
0.026791995,
0.09626954,
0.028848808,
-0.008233931,
-0.013509523,
-0.043059517,
-0.050805956,
-0.040014364,
-0.014704878,
-0.033817213,
0.006587812,
-0.0057998123,
0.046345077,
0.073137075,
0.10748852,
0.03346996,
0.03202752,
0.06389477,
0.033550095,
-0.00800688,
0.010063693,
0.022865351,
-0.024267724,
-0.14264134,
0.047707383,
0.04551701,
0.012821693,
0.02434786,
-0.02398725,
-0.009636303,
0.07709043,
-0.056949683,
-0.018818505,
-0.08478344,
0.020661624,
0.02799403,
0.00008180507,
-0.003579389,
0.024027318,
0.0070051854,
-0.046425212,
0.050538838,
0.018017149,
-0.00027066626,
0.04196433,
0.0051954566,
-0.06421531,
0.045089617,
0.036621958,
0.011619659,
0.04367389,
-0.051981278,
-0.016841827,
-0.01414393,
0.03819796,
-0.052702498,
-0.051794294,
-0.01322237,
0.019085623,
0.006804846,
0.36199912,
-0.033282977,
0.060261954,
-0.0019132369,
0.0004950041,
-0.0046745753,
-0.04241843,
-0.043620467,
-0.03405762,
-0.016841827,
0.033683654,
-0.06020853,
0.032695316,
-0.015052132,
-0.003796423,
-0.044769075,
-0.009589558,
0.022424607,
0.041750636,
0.0034692027,
-0.027406368,
-0.01450454,
0.008167151,
0.0344583,
0.010651354,
0.04727999,
-0.0032738722,
0.06384134,
0.088523105,
-0.04468894,
0.021609895,
0.06416189,
-0.015840132,
-0.005245541,
0.016267523,
-0.00800688,
0.084462896,
-0.004784762,
-0.01945959,
-0.051206633,
-0.011980269,
-0.022437962,
-0.07377815,
0.06442901,
-0.0017529657,
-0.027059114,
0.10994601,
-0.0087281,
-0.0048348466,
0.04108284,
0.00082681555,
-0.034565147,
0.034485012,
0.051260058,
-0.05657572,
-0.044314973,
0.063734494,
0.00255933,
-0.013295828,
-0.081791714,
-0.009602914,
0.020728404,
-0.010831659,
-0.037129484,
0.101237945,
0.033202842,
-0.034858976,
-0.006110338,
0.020995522,
0.039987653,
-0.027780334,
-0.020741759,
0.036889076,
0.008895049,
-0.051260058,
0.05272921,
0.007666304,
0.004026813,
0.013088811,
-0.03376379,
0.00007335327,
-0.017309286,
-0.07554114,
0.043780737,
0.037850704,
0.0057397108,
0.00070076896,
-0.060422227,
0.014918573,
-0.023306098,
-0.027539928,
-0.0030735333,
-0.061971515,
-0.009315761,
0.027780334,
-0.042178027,
0.010885083,
-0.009669693,
-0.004153694,
-0.0053356937,
-0.0020501353,
0.06838236,
0.0013890166,
-0.009115422,
0.039132874,
-0.0051954566,
-0.017709963,
0.10219958,
-0.053049754,
-0.09108744,
0.030825486,
0.0021436268,
0.011305794,
-0.015546302,
-0.04338006,
0.046879314,
-0.035232943,
0.049336806,
-0.0051787617,
-0.052381955,
0.0281543,
-0.008474337,
-0.038571924,
-0.00026294487,
-0.3301586,
-0.035874028,
0.019833555,
0.037396602,
-0.019032199,
-0.049977887,
-0.037022635,
0.013903523,
0.04514304,
0.029035792,
0.096910626,
-0.020407861,
-0.025443045,
-0.079120524,
0.0043139653,
0.0073724734,
-0.035874028,
0.033870637,
-0.04094928,
-0.073137075,
-0.04971077,
0.031519994,
-0.016214099,
-0.018791793,
-0.027539928,
0.040468466,
0.0729768,
0.014477828,
0.056896262,
-0.0033156096,
0.063574225,
-0.008307388,
0.05115321,
0.032214504,
0.012200641,
0.0065377275,
-0.020648267,
0.03336311,
0.05491958,
-0.0055393716,
-0.04682589,
-0.000899021,
0.045890976,
0.094292864,
0.029810434,
-0.023639996,
-0.03480555,
-0.050538838,
-0.029676875,
-0.006127033,
0.057483923,
0.00038189613,
-0.04006779,
0.04583755,
0.048909415,
-0.035713755,
-0.025616674,
-0.048909415,
-0.045890976,
0.014290845,
-0.02024759,
0.047867652,
-0.038839042,
-0.0035459993,
-0.020648267,
0.03301586,
-0.026311181,
0.02283864,
0.02241125,
0.0024875419,
-0.015345964,
0.109518625,
-0.048054636,
-0.046184804,
0.02333281,
-0.059246905,
0.052836057,
-0.014584675,
0.04193762,
0.039373282,
0.035954162,
-0.042124603,
0.06298657,
0.03622128,
0.024094097,
-0.014477828,
0.053904533,
-0.020087318,
-0.022037284,
0.0006369109,
-0.057216804,
0.043460194,
-0.0067647784,
-0.05705653,
0.009489388,
-0.03550006,
-0.25664756,
0.037556875,
0.043620467,
-0.025282774,
-0.03357681,
-0.043887585,
-0.009749829,
-0.021422911,
-0.07260283,
0.011158879,
-0.09252988,
0.014183997,
-0.029035792,
-0.028047452,
0.015492879,
0.06838236,
0.073030226,
-0.05876609,
-0.06287972,
-0.082913615,
0.022170844,
0.0054692533,
0.16176702,
0.029543316,
0.030531656,
0.025322843,
-0.03336311,
0.00800688,
0.011205625,
-0.00019960855,
0.04423484,
-0.019833555,
0.07586168,
0.022224266,
-0.059407175,
0.024748538,
0.021596538,
0.015118913,
-0.014050438,
-0.011653049,
-0.055240124,
0.04725328,
-0.0059901346,
-0.009749829,
0.11603632,
0.028474841,
-0.06971795,
-0.099902354,
0.014998709,
-0.025416335,
-0.016601421,
-0.014531251,
-0.003602762,
-0.017055523,
-0.021663317,
-0.0017087243,
-0.006597829,
-0.026992334,
0.024601623,
0.034965824,
0.014384336,
-0.055667516,
0.073885,
-0.06320026,
0.017576404
] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | he subscript i denotes the satellite and has the value 1, 2, ..., n, where n ≥ 4. When the time of message reception indicated by the on-board receiver clock is t̃i, the true reception time is , where b is the receiver's clock bias from the much more accurate GPS clocks employed by the satellites. The receiver clock bias is the same for all received satellite signals (assuming the satellite clocks | [
-0.015263549,
-0.0006412764,
-0.017537698,
-0.054659825,
0.039516672,
-0.07341486,
0.040399577,
0.04513516,
0.02825295,
0.0056385505,
0.027691102,
-0.06934815,
0.01914298,
-0.020293431,
-0.033978455,
-0.0123406,
0.007946142,
-0.06742181,
0.07871229,
0.0291091,
0.08716677,
-0.0056084516,
-0.0079327645,
-0.006768936,
0.08984224,
0.033817925,
-0.023718031,
0.0014790326,
0.0042372737,
-0.15549824,
0.011658355,
-0.007029794,
0.015705002,
-0.033416606,
-0.023142805,
-0.03309555,
-0.006240531,
-0.009678508,
-0.06833147,
0.081013195,
0.065281436,
0.003892807,
0.006327484,
0.01208643,
-0.0582182,
-0.027584083,
-0.037135508,
0.04342286,
-0.043048292,
0.030152533,
-0.031731058,
0.05982348,
0.04864002,
0.046392627,
-0.01208643,
0.018153055,
0.023437107,
0.039596938,
0.044091724,
-0.0038894627,
-0.005267329,
0.037697356,
-0.1558193,
0.057201523,
0.02745031,
0.029831477,
-0.008802292,
-0.036065318,
-0.077535085,
0.021644542,
-0.0011170082,
-0.0054913997,
-0.015143153,
-0.023289956,
-0.0049830605,
-0.0021370307,
-0.019129602,
0.0029079001,
0.047115006,
-0.016815322,
0.021417128,
0.03585128,
-0.029751213,
0.0036954912,
0.031035438,
-0.018273452,
0.03408547,
0.025443709,
0.07164905,
0.03309555,
-0.00063667796,
-0.028359968,
-0.025202915,
0.031838078,
0.0060365265,
-0.0025166129,
-0.008240444,
-0.0068124123,
0.054526053,
0.30329114,
-0.042968027,
0.021778315,
-0.063569136,
0.06843849,
-0.002540023,
-0.0764649,
0.0038493306,
-0.08481236,
-0.03676094,
0.008113359,
-0.05358964,
0.00071067136,
0.042031612,
-0.03138325,
-0.041282482,
0.044653572,
0.057790127,
-0.040961426,
0.07106045,
-0.043289084,
-0.018219942,
-0.027931893,
0.06009103,
-0.028921816,
-0.009136726,
0.01359807,
0.0747526,
0.054472543,
-0.036092073,
0.0011663372,
0.019076092,
-0.06570951,
-0.016654793,
0.017698226,
0.033149056,
0.101721324,
-0.004123566,
0.02185858,
0.0096116215,
0.011457695,
-0.033443358,
-0.10632313,
0.051797073,
0.03555698,
-0.0029480322,
0.066993736,
-0.018928941,
0.030393325,
0.055061147,
0.0044546556,
-0.055328693,
0.03373766,
0.016079567,
-0.043342594,
-0.055114657,
0.06458581,
-0.040961426,
0.03932939,
-0.027477063,
-0.022353541,
0.007966208,
0.00059570983,
-0.0062539084,
0.05902084,
0.026420254,
-0.056024317,
0.004043302,
0.049068097,
0.030259551,
-0.06094718,
-0.03555698,
-0.04093467,
0.011370742,
-0.0565059,
0.036841206,
-0.0046319054,
-0.0051235226,
0.016828699,
0.040212296,
-0.053455867,
-0.009424339,
-0.06747532,
0.042727236,
0.048720285,
0.030393325,
-0.009778839,
-0.039971504,
-0.0064779785,
0.02905559,
0.020908788,
-0.05912786,
-0.043957952,
-0.044733837,
0.041469764,
-0.018019281,
-0.029777966,
-0.0139124375,
-0.004300816,
0.045616742,
0.017444056,
0.089681715,
-0.00034697485,
-0.0081869345,
-0.03646664,
0.02830646,
0.05554273,
0.049014587,
-0.07459207,
0.00458174,
0.014822097,
0.017791867,
-0.016427377,
0.008708651,
-0.034379773,
-0.0037456562,
-0.035075396,
0.056024317,
0.04834572,
-0.012782052,
-0.00637096,
-0.0547936,
-0.020989053,
-0.043770667,
-0.28638217,
-0.017256774,
0.01455455,
0.018313583,
0.014795342,
-0.027129253,
-0.040346067,
0.004708825,
0.043770667,
0.013678334,
0.05998401,
-0.019009205,
-0.012922514,
-0.09342737,
-0.002407922,
0.028359968,
-0.035369698,
-0.021724807,
-0.022634465,
-0.100062534,
-0.04971021,
0.06073314,
-0.03983773,
0.007370916,
0.035342943,
0.02185858,
0.080799155,
-0.052064624,
0.05142251,
-0.06485336,
0.04674044,
-0.05484711,
0.06094718,
0.038259204,
-0.013236881,
0.011150016,
-0.004160354,
0.029269628,
0.033068795,
0.0016663154,
-0.065441966,
-0.0048894193,
0.026768064,
0.04508165,
-0.021015806,
-0.027717857,
0.016681548,
-0.029938495,
-0.056933977,
-0.02044058,
-0.010588167,
-0.03122272,
-0.012327222,
0.08021055,
0.077695616,
0.001901255,
-0.06977622,
-0.031169211,
-0.03579777,
0.007952831,
0.035209168,
-0.0018728281,
-0.06934815,
0.01304291,
0.021176334,
0.041309237,
0.0019029272,
-0.0024564147,
0.0143672675,
0.04007852,
0.042727236,
0.069080606,
-0.060144536,
-0.016173208,
0.02211275,
-0.05302779,
0.09203613,
0.0024564147,
-0.010360753,
0.024614312,
0.054365523,
-0.012909137,
0.052599717,
0.008180246,
-0.0055114655,
-0.001632872,
0.012106496,
0.005655272,
-0.0045650187,
0.06458581,
-0.06410423,
0.022353541,
0.012594769,
-0.03831271,
-0.028814798,
-0.01828683,
-0.30692977,
-0.0017791868,
0.026915215,
-0.016333736,
-0.032587208,
-0.034914866,
0.00027298142,
0.028574007,
-0.16020706,
-0.028012158,
-0.057683107,
0.112262666,
-0.020280054,
0.005635206,
-0.0027456998,
0.06089367,
0.0890396,
-0.078979835,
-0.039730713,
-0.06298053,
0.07197011,
-0.0074712466,
0.136984,
0.05399096,
-0.0016646432,
0.06314106,
-0.01299609,
0.031436756,
0.029216118,
0.043182064,
-0.009009641,
-0.009792215,
0.082404435,
0.03847324,
-0.003882774,
0.08647115,
-0.055221677,
-0.033496868,
0.026460387,
0.0074110483,
-0.01923662,
0.053803675,
-0.015022757,
-0.0066552283,
0.1455455,
0.00886249,
-0.06587004,
-0.094604574,
0.02991174,
-0.033871435,
-0.06255246,
-0.016534397,
-0.037403055,
-0.04690097,
-0.035075396,
0.048479494,
-0.015878906,
-0.031142456,
0.023985578,
0.04703474,
0.000023867586,
-0.044439536,
0.05655941,
-0.007477935,
-0.01903596
] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | are all perfectly synchronized). The message's transit time is , where si is the satellite time. Assuming the message traveled at the speed of light, c, the distance traveled is .
For n satellites, the equations to satisfy are:
where di is the geometric distance or range between receiver and satellite i (the values without subscripts are the x, y, and z components of receiver position):
Defini | [
-0.00018058228,
0.009870855,
-0.00079970655,
-0.038707137,
-0.031372588,
-0.03378175,
0.042775936,
0.06643922,
0.03439742,
-0.010379455,
0.017921457,
-0.039108664,
0.008050603,
0.01726563,
-0.00143462,
-0.029418495,
0.015793368,
-0.022458704,
0.014401411,
0.0046945116,
0.10032804,
-0.0026132672,
-0.036003526,
-0.0061734673,
0.13823213,
0.059747115,
-0.031934727,
-0.016676726,
-0.003881421,
-0.17121083,
-0.024372647,
-0.007649076,
-0.0076089236,
-0.013571589,
-0.07072217,
-0.023850663,
-0.0013175082,
-0.036940422,
-0.0114836525,
0.039911717,
0.053965136,
0.024533257,
0.030756915,
0.02043769,
-0.03677981,
-0.04116983,
-0.023864048,
0.0025547112,
-0.026581042,
-0.0008775023,
-0.014976932,
0.025189085,
0.022191022,
0.07366669,
0.020170005,
0.041758735,
0.038867746,
0.03477218,
0.061674442,
-0.022431936,
-0.008197829,
-0.011751337,
-0.14711924,
0.060603704,
0.026741652,
-0.0055745235,
0.039751105,
-0.028428063,
-0.06788471,
0.10241598,
-0.021347815,
0.028080074,
-0.03035539,
0.05830162,
0.005972704,
0.003955034,
-0.0025112124,
-0.0069597894,
-0.0037643092,
-0.040580925,
0.03592322,
0.020103084,
-0.036324747,
0.02221779,
-0.019795246,
-0.013698739,
0.009984621,
0.0011794835,
0.030034168,
0.017533315,
-0.048183158,
0.0009410773,
-0.04001879,
0.009743705,
-0.06285225,
-0.0722212,
-0.049601883,
-0.009529558,
0.050217558,
0.3295728,
-0.024519874,
0.06301286,
0.0010874671,
0.028909894,
-0.011041974,
-0.0061232764,
0.011456884,
-0.066225074,
-0.008880423,
0.0026969183,
-0.040795073,
0.0029060466,
0.016890874,
-0.06788471,
-0.016061053,
-0.00027960452,
0.059211746,
0.00073529506,
0.019032348,
-0.014307721,
-0.035227243,
0.07168583,
-0.007876608,
-0.028320989,
0.0188182,
-0.0019842093,
0.053777758,
0.05776625,
0.02279331,
0.032951925,
0.08897823,
0.007823071,
0.025603995,
0.03645859,
-0.014254184,
0.05254641,
0.03410297,
-0.03525401,
-0.03798439,
-0.0040821843,
-0.023368832,
-0.10964345,
0.08233966,
0.027076257,
-0.006417729,
0.09170861,
0.027103025,
0.029820021,
0.0012288379,
-0.03421004,
0.010693984,
0.010747521,
0.049949873,
-0.059640042,
-0.031051368,
0.043150693,
0.0065917238,
0.033166073,
-0.09652693,
-0.008358439,
0.024104962,
0.020999826,
0.018108837,
0.071418144,
0.03851976,
-0.030703379,
0.0027136486,
0.045372475,
0.028936664,
-0.06718874,
-0.0012254918,
0.032818083,
0.056749053,
-0.016542884,
0.027678547,
-0.015391842,
-0.028026536,
0.013136603,
0.0038078078,
-0.00048601415,
-0.014936779,
-0.06884838,
0.07056156,
0.05251964,
0.03388882,
-0.005029117,
-0.07377377,
0.051127683,
0.008820195,
0.0023238335,
0.006220312,
-0.041624896,
0.0076624607,
0.032255948,
-0.092136905,
-0.02603229,
-0.01687749,
-0.011376579,
0.0016002497,
-0.019527564,
0.0752728,
0.021802878,
-0.013959732,
-0.029124042,
-0.04116983,
-0.09111971,
0.08346394,
-0.03592322,
-0.06220981,
0.048129622,
-0.015525684,
-0.03303223,
-0.029364958,
-0.0677241,
-0.0053135315,
-0.021227358,
0.06949082,
0.037609633,
-0.019366952,
0.045800768,
-0.07645061,
0.007970298,
-0.03525401,
-0.29852143,
-0.04055416,
0.020397536,
0.065743245,
0.05185043,
-0.025349695,
-0.058408692,
0.032871623,
0.061942127,
0.005410567,
0.1257045,
-0.028561905,
-0.020250311,
-0.1123203,
0.03016801,
0.011791489,
-0.0589976,
0.043150693,
-0.02765178,
-0.07998404,
-0.012146171,
0.042990085,
-0.049655423,
-0.024198653,
0.026540888,
0.024948169,
0.081376,
-0.028535137,
0.06285225,
-0.058569305,
0.041972883,
-0.05616015,
-0.014762784,
0.030462462,
-0.0064645736,
-0.017145174,
-0.027397478,
0.037154566,
0.03798439,
0.0017767539,
-0.04015263,
0.025603995,
0.008284826,
0.03881421,
-0.0065314947,
-0.015217847,
0.039296042,
-0.021187205,
-0.015953979,
0.0048919288,
0.018858353,
-0.009402408,
-0.029953863,
0.058408692,
0.04293655,
-0.019206341,
-0.030060936,
-0.04625583,
-0.055571243,
0.027357327,
-0.06237042,
0.015391842,
-0.04408759,
0.013973116,
-0.012172939,
0.0022201058,
-0.0027788966,
-0.03980464,
-0.0103727635,
0.014173879,
0.032255948,
0.07355962,
-0.07152522,
0.007963605,
0.079288065,
-0.0300074,
0.04151782,
0.059425894,
0.036405053,
0.05648137,
0.05610661,
-0.045533083,
0.07168583,
0.028802821,
0.020986442,
-0.02295392,
0.07179291,
-0.047701325,
-0.064030066,
-0.021655653,
-0.0025279426,
0.00040780017,
0.0155123,
-0.010098387,
-0.034879252,
-0.040179398,
-0.26939738,
0.025510306,
0.0246671,
-0.036860116,
-0.050297864,
-0.023636516,
0.025711069,
0.031051368,
-0.104557455,
-0.053991906,
-0.03442419,
0.047835167,
-0.0054975646,
0.0236499,
0.007394776,
0.0272101,
0.09310057,
-0.075058654,
-0.08426699,
-0.08763981,
0.064458355,
-0.026540888,
0.13694724,
0.045533083,
-0.012534313,
0.017332552,
-0.030837221,
0.06847362,
0.018523747,
0.0076022316,
0.023516057,
0.014736015,
0.023275143,
-0.043953747,
-0.037208106,
0.03742225,
0.039215736,
0.021133669,
0.0320418,
-0.004456942,
-0.0137322,
0.029980632,
-0.009302027,
-0.028347759,
0.108626254,
0.016194895,
-0.097437054,
-0.09824011,
-0.007066863,
-0.015552453,
-0.012259937,
0.03324638,
-0.0064712656,
-0.028187146,
0.0075420025,
0.052011043,
-0.051770125,
-0.049896337,
0.00081560033,
0.013785737,
-0.010111771,
-0.007381392,
0.051609516,
-0.058355156,
-0.017131789
] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ng pseudoranges as , we see they are biased versions of the true range:
.
Since the equations have four unknowns [x, y, z, b]—the three components of GPS receiver position and the clock bias—signals from at least four satellites are necessary to attempt solving these equations. They can be solved by algebraic or numerical methods. Existence and uniqueness of GPS solutions are discussed by Abell a | [
-0.06555465,
-0.026123181,
0.037951287,
-0.044378735,
0.029816963,
-0.065394625,
-0.04389868,
-0.026536563,
0.016615355,
-0.039178103,
0.045045484,
-0.067261524,
0.0006754989,
0.015508555,
0.025563112,
0.002171931,
-0.021669304,
0.01996243,
0.059847284,
-0.0027219981,
0.03691116,
0.020375814,
-0.037844606,
0.039284784,
0.032910675,
0.0687017,
0.013581653,
-0.015455215,
0.019602386,
-0.19223674,
-0.03307069,
-0.018415576,
-0.018282225,
-0.025109723,
-0.07398234,
0.014788467,
-0.0033954133,
0.02178932,
-0.02017579,
0.023856238,
0.062460937,
0.025589783,
0.013294952,
-0.024376301,
-0.05568678,
0.012841564,
-0.02781672,
0.06656811,
-0.05245972,
-0.039231442,
-0.062407598,
0.054273274,
-0.028563477,
0.058940507,
0.0039004749,
0.03717786,
0.021095902,
0.028883517,
-0.0066274735,
0.012194818,
0.02178932,
-0.034430858,
-0.12822895,
0.04565889,
0.051632952,
0.029923642,
0.020762527,
-0.065768,
-0.026949948,
0.06427449,
-0.014241734,
0.03733788,
-0.031283807,
0.0004263019,
-0.021055896,
0.014388419,
0.004523884,
0.03456421,
-0.014988491,
0.0101945745,
-0.011354716,
-0.014521768,
0.015001826,
0.050379466,
0.024189612,
-0.022429397,
0.050619494,
0.03739122,
0.01718876,
-0.0014168392,
-0.005667357,
-0.004400536,
-0.016321987,
-0.014508433,
0.01802886,
-0.0073342263,
-0.021922668,
0.031283807,
0.020589173,
0.35268295,
0.013334957,
0.011041344,
0.025349753,
0.065768,
-0.044805456,
-0.030563721,
-0.027016623,
-0.066034704,
-0.046832368,
0.019389028,
0.024256287,
-0.04008488,
0.05267308,
-0.07280886,
-0.07152871,
0.041738413,
-0.0077009373,
-0.016068622,
0.0059707267,
-0.024336295,
-0.07131535,
-0.007867624,
0.07259551,
0.022242708,
0.029550264,
0.0068875053,
-0.024923034,
0.06091408,
-0.019789075,
0.051339585,
0.030163672,
-0.04315192,
-0.0035204287,
0.03435085,
0.001516018,
0.050486147,
0.044512086,
0.035497654,
0.010187907,
0.015908603,
-0.07542252,
-0.038751386,
0.07211544,
-0.016575351,
-0.019309018,
0.08166327,
-0.048912622,
0.019255677,
0.06742154,
-0.01412172,
-0.023962917,
0.015641904,
0.0336841,
-0.01969573,
-0.046992388,
0.0216293,
-0.038884733,
0.0027470011,
-0.09606503,
0.020775862,
-0.015908603,
0.0037804602,
-0.035257626,
0.11233368,
0.05520672,
-0.04344529,
0.02760336,
0.065981366,
0.032270595,
-0.020695852,
-0.041605067,
0.02294946,
0.009707849,
-0.020642513,
0.04357864,
0.009927875,
-0.015628569,
0.018962309,
-0.0009242792,
0.01078798,
-0.055260062,
-0.04053827,
-0.0019985766,
0.019562382,
-0.0027069962,
-0.0017452125,
-0.017868843,
-0.013294952,
-0.014615113,
-0.021762649,
-0.019682396,
-0.038644705,
-0.044805456,
0.06699482,
-0.0038037964,
-0.07675601,
-0.037764598,
-0.028910186,
0.04893929,
-0.004360531,
0.05235304,
-0.015748585,
0.009481154,
-0.010334591,
0.057980392,
-0.0008909418,
0.09030433,
-0.069181755,
0.0010834653,
0.03403081,
0.008714395,
-0.014068379,
0.003643777,
-0.05245972,
-0.042698532,
-0.06939512,
0.0143617485,
0.016655361,
-0.00690084,
0.021322595,
-0.014001705,
-0.03360409,
-0.03707118,
-0.3518295,
-0.054486632,
0.028296778,
-0.0047372435,
0.038538024,
-0.019402362,
-0.03083042,
0.016655361,
-0.0007984305,
0.03725787,
0.08977093,
0.008347683,
0.0008409357,
-0.047072396,
-0.02294946,
0.014601777,
-0.07755611,
0.031790536,
-0.04675236,
-0.042538512,
-0.03456421,
0.030217012,
-0.06907508,
0.0051339585,
0.023829568,
0.016722037,
0.11286707,
-0.060380682,
0.02316282,
-0.04925933,
0.051872984,
-0.081876636,
0.017535469,
0.10427936,
0.038111307,
0.014441758,
0.024082933,
0.010707971,
0.07190209,
-0.0043271934,
-0.01818888,
0.0072142114,
0.020415818,
0.0736623,
-0.021775985,
0.010647963,
-0.048032515,
0.045258842,
0.026843268,
0.020535834,
0.0017552137,
-0.010821317,
-0.009241125,
-0.028776838,
0.048539244,
-0.017628813,
-0.060754064,
-0.023309505,
-0.04880594,
0.039684832,
-0.016455337,
-0.02673659,
-0.022109358,
-0.00539399,
0.05029946,
0.047659136,
0.05221969,
0.023976251,
0.01303492,
0.029363574,
0.010007885,
0.078409545,
-0.010821317,
0.0069608474,
0.00064049463,
-0.054619983,
0.09062437,
-0.034137487,
0.010914662,
-0.014015039,
-0.0048772604,
0.0101945745,
0.041125007,
-0.0014135054,
-0.020709189,
0.008821074,
-0.029896973,
-0.016575351,
-0.017815502,
0.021482615,
-0.024336295,
0.02440297,
0.03640443,
-0.019842416,
0.05867381,
-0.032350603,
-0.26200524,
-0.012908238,
0.03339073,
-0.016815381,
-0.014228399,
-0.05195299,
0.028376788,
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0.103212565,
0.02776338,
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0.069448456,
0.044405404,
0.032190587,
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0.045072153,
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0.072328806,
0.041871764,
0.18786287,
0.03357742,
0.0005517338,
0.018922303,
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0.032030568,
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-0.028696828,
-0.029950313,
-0.0044572093,
0.016988736,
-0.050726175,
0.06331438,
-0.023576204,
0.004157173
] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | nd Chaffee. When n is greater than four, this system is overdetermined and a fitting method must be used.
The amount of error in the results varies with the received satellites' locations in the sky, since certain configurations (when the received satellites are close together in the sky) cause larger errors. Receivers usually calculate a running estimate of the error in the calculated position. | [
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] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | This is done by multiplying the basic resolution of the receiver by quantities called the geometric dilution of position (GDOP) factors, calculated from the relative sky directions of the satellites used. The receiver location is expressed in a specific coordinate system, such as latitude and longitude using the WGS 84 geodetic datum or a country-specific system.
Geometric interpretation
The GP | [
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] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | S equations can be solved by numerical and analytical methods. Geometrical interpretations can enhance the understanding of these solution methods.
Spheres
The measured ranges, called pseudoranges, contain clock errors. In a simplified idealization in which the ranges are synchronized, these true ranges represent the radii of spheres, each centered on one of the transmitting satellites. The sol | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ution for the position of the receiver is then at the intersection of the surfaces of these spheres; see trilateration (more generally, true-range multilateration). Signals from at minimum three satellites are required, and their three spheres would typically intersect at two points. One of the points is the location of the receiver, and the other moves rapidly in successive measurements and would | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | not usually be on Earth's surface.
In practice, there are many sources of inaccuracy besides clock bias, including random errors as well as the potential for precision loss from subtracting numbers close to each other if the centers of the spheres are relatively close together. This means that the position calculated from three satellites alone is unlikely to be accurate enough. Data from more s | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | atellites can help because of the tendency for random errors to cancel out and also by giving a larger spread between the sphere centers. But at the same time, more spheres will not generally intersect at one point. Therefore, a near intersection gets computed, typically via least squares. The more signals available, the better the approximation is likely to be.
Hyperboloids
If the pseudorange | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | between the receiver and satellite i and the pseudorange between the receiver and satellite j are subtracted, , the common receiver clock bias (b) cancels out, resulting in a difference of distances . The locus of points having a constant difference in distance to two points (here, two satellites) is a hyperbola on a plane and a hyperboloid of revolution (more specifically, a two-sheeted hyperbolo | [
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0.00013213349,
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0.025827996,
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] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | id) in 3D space (see Multilateration). Thus, from four pseudorange measurements, the receiver can be placed at the intersection of the surfaces of three hyperboloids each with foci at a pair of satellites. With additional satellites, the multiple intersections are not necessarily unique, and a best-fitting solution is sought instead.
Inscribed sphere
The receiver position can be interpreted as | [
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0.0057310243
] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | the center of an inscribed sphere (insphere) of radius bc, given by the receiver clock bias b (scaled by the speed of light c). The insphere location is such that it touches other spheres. The circumscribing spheres are centered at the GPS satellites, whose radii equal the measured pseudoranges pi. This configuration is distinct from the one described above, in which the spheres' radii were the un | [
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] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | biased or geometric ranges di.
Hypercones
The clock in the receiver is usually not of the same quality as the ones in the satellites and will not be accurately synchronized to them. This produces pseudoranges with large differences compared to the true distances to the satellites. Therefore, in practice, the time difference between the receiver clock and the satellite time is defined as an unkno | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | wn clock bias b. The equations are then solved simultaneously for the receiver position and the clock bias. The solution space [x, y, z, b] can be seen as a four-dimensional spacetime, and signals from at minimum four satellites are needed. In that case each of the equations describes a hypercone (or spherical cone), with the cusp located at the satellite, and the base a sphere around the satelli | [
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] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | te. The receiver is at the intersection of four or more of such hypercones.
Solution methods
Least squares
When more than four satellites are available, the calculation can use the four best, or more than four simultaneously (up to all visible satellites), depending on the number of receiver channels, processing capability, and geometric dilution of precision (GDOP).
Using more than four invol | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ves an over-determined system of equations with no unique solution; such a system can be solved by a least-squares or weighted least squares method.
Iterative
Both the equations for four satellites, or the least squares equations for more than four, are non-linear and need special solution methods. A common approach is by iteration on a linearized form of the equations, such as the Gauss–Newton | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | algorithm.
The GPS was initially developed assuming use of a numerical least-squares solution method—i.e., before closed-form solutions were found.
Closed-form
One closed-form solution to the above set of equations was developed by S. Bancroft. Its properties are well known; in particular, proponents claim it is superior in low-GDOP situations, compared to iterative least squares methods.
Banc | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | roft's method is algebraic, as opposed to numerical, and can be used for four or more satellites. When four satellites are used, the key steps are inversion of a 4x4 matrix and solution of a single-variable quadratic equation. Bancroft's method provides one or two solutions for the unknown quantities. When there are two (usually the case), only one is a near-Earth sensible solution.
When a receiv | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | er uses more than four satellites for a solution, Bancroft uses the generalized inverse (i.e., the pseudoinverse) to find a solution. A case has been made that iterative methods, such as the Gauss–Newton algorithm approach for solving over-determined non-linear least squares (NLLS) problems, generally provide more accurate solutions.
Leick et al. (2015) states that "Bancroft's (1985) solution is | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | a very early, if not the first, closed-form solution."
Other closed-form solutions were published afterwards, although their adoption in practice is unclear.
Error sources and analysis
GPS error analysis examines error sources in GPS results and the expected size of those errors. GPS makes corrections for receiver clock errors and other effects, but some residual errors remain uncorrected. Erro | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | r sources include signal arrival time measurements, numerical calculations, atmospheric effects (ionospheric/tropospheric delays), ephemeris and clock data, multipath signals, and natural and artificial interference. Magnitude of residual errors from these sources depends on geometric dilution of precision. Artificial errors may result from jamming devices and threaten ships and aircraft or from i | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ntentional signal degradation through selective availability, which limited accuracy to ≈ , but has been switched off since May 1, 2000.
Accuracy enhancement and surveying
Augmentation
Integrating external information into the calculation process can materially improve accuracy. Such augmentation systems are generally named or described based on how the information arrives. Some systems transmi | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | t additional error information (such as clock drift, ephemera, or ionospheric delay), others characterize prior errors, while a third group provides additional navigational or vehicle information.
Examples of augmentation systems include the Wide Area Augmentation System (WAAS), European Geostationary Navigation Overlay Service (EGNOS), Differential GPS (DGPS), inertial navigation systems (INS) a | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | nd Assisted GPS. The standard accuracy of about can be augmented to with DGPS, and to about with WAAS.
Precise monitoring
Accuracy can be improved through precise monitoring and measurement of existing GPS signals in additional or alternative ways.
The largest remaining error is usually the unpredictable delay through the ionosphere. The spacecraft broadcast ionospheric model parameters, but | [
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0.048622847,
-0.044880573,
-0.08115391,
-0.010191017,
-0.004012922,
-0.0458696,
0.01920589,
-0.070835926,
0.04418558,
0.010458322,
0.034268547,
-0.016800143,
-0.016506106,
-0.0013473862,
0.06538289,
-0.02316201,
0.017561963,
0.0069365744,
-0.0022487063,
-0.01395334,
-0.0039293887,
-0.021050299,
-0.10815175,
-0.027559184,
0.005406251,
-0.08548426,
0.007818682,
-0.005072119,
0.024445076,
-0.038171206,
0.06682634,
0.055920284,
0.00016226272,
-0.06078524,
-0.025754873,
0.08361312,
-0.032103375,
-0.026596883,
0.009743281,
-0.013338538,
0.0047680596,
-0.039962154,
0.0261291,
0.03429528,
-0.050975133,
0.031060884,
0.037743516,
0.0034198381,
0.008139448,
-0.31606185,
-0.09184612,
0.012155712,
0.03290529,
-0.005038706,
-0.04576268,
0.06677288,
0.07334859,
-0.027358705,
-0.0034281914,
0.07040823,
-0.025113339,
0.015503711,
-0.031648956,
0.020662704,
0.04918419,
-0.051162247,
0.0051222392,
-0.04736651,
-0.000091938426,
-0.056027204,
0.050868213,
-0.06586404,
-0.06420675,
-0.000752214,
0.033145867,
0.06030409,
-0.066986725,
0.098742604,
-0.10339372,
0.03942754,
-0.008079304,
0.027532453,
0.009796741,
0.04274213,
0.031809337,
0.033252787,
0.05538567,
-0.00064111524,
-0.065436356,
-0.017642153,
0.005703628,
0.08238351,
0.040977914,
0.0133251725,
-0.0063618678,
-0.05827257,
-0.032156836,
-0.0101175085,
-0.044613265,
0.008694107,
-0.026543424,
-0.012356191,
-0.05228493,
0.104142174,
0.01468843,
-0.07495242,
0.00380576,
0.0008148637,
0.009910347,
-0.01334522,
0.01717437,
0.038572166,
-0.001110988,
0.00946261,
-0.020849818,
0.011540909,
0.010291257,
0.058593336,
0.014528046,
-0.008012478,
0.17663538,
-0.087729625,
-0.01207552,
0.028788788,
-0.044479612,
0.008500311,
-0.03595257,
-0.009609628,
-0.009683137,
0.047072474,
0.009669771,
0.04592306,
0.009549485,
-0.0010082425,
0.0017942872,
0.042528283,
0.02022165,
-0.05423626,
-0.034509122,
-0.04124522,
-0.025821699,
0.023937196,
-0.041084833,
0.04782093,
-0.017896095,
-0.24356864,
-0.008079304,
-0.004854934,
-0.018751472,
-0.05725681,
-0.009168574,
-0.018069843,
0.025888525,
-0.043223277,
0.007110323,
-0.032210298,
0.057042964,
-0.022360094,
-0.0076516164,
0.062389072,
0.079282776,
0.020569148,
-0.028548213,
0.056454893,
-0.062923685,
0.09494687,
0.05875372,
0.1453072,
0.02047559,
0.004246814,
0.04581614,
-0.0039561195,
0.010478371,
-0.008513676,
-0.049718797,
0.0030071854,
0.0049150772,
0.010538514,
0.09884953,
0.030766848,
0.088264234,
0.006923209,
0.0407908,
-0.02559449,
0.00205491,
0.01545025,
-0.008045891,
0.040470034,
-0.029056093,
0.122746624,
-0.020542417,
-0.03442893,
-0.01819013,
-0.008907951,
0.011540909,
-0.022507112,
-0.058486413,
-0.01740158,
-0.042982705,
-0.023950562,
-0.0019546705,
0.0075380113,
-0.04442615,
-0.017842634,
0.0091953045,
0.07276052,
-0.032691445,
0.044880573,
-0.07062208,
0.047419973
] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | some errors remain. This is one reason GPS spacecraft transmit on at least two frequencies, L1 and L2. Ionospheric delay is a well-defined function of frequency and the total electron content (TEC) along the path, so measuring the arrival time difference between the frequencies determines TEC and thus the precise ionospheric delay at each frequency.
Military receivers can decode the P(Y) code tr | [
-0.03873728,
0.059056282,
0.01722698,
-0.029956473,
-0.020158377,
-0.011832673,
0.010668145,
0.039567173,
0.02915335,
-0.047437772,
0.08047289,
-0.06323252,
0.029822618,
0.019930825,
-0.0009537081,
-0.05405015,
0.0028360267,
-0.03903176,
0.022018945,
-0.016878959,
0.08786161,
0.031455636,
0.019609576,
0.017454531,
0.06971104,
0.049392037,
0.007904065,
0.01797656,
-0.028510852,
-0.19746107,
0.017160052,
-0.039138842,
-0.019315097,
0.0018890112,
-0.039888423,
-0.051801406,
0.004203008,
-0.05257776,
-0.05081089,
0.047464546,
0.029421058,
-0.029046267,
0.006271049,
-0.030866679,
-0.030893449,
-0.027654188,
-0.031321783,
0.023665346,
0.014710529,
0.013686547,
-0.03351698,
0.08866473,
0.01050752,
0.031910736,
0.015165632,
-0.003326266,
-0.033249274,
0.08898598,
0.027627418,
0.01742776,
-0.006431673,
-0.037264887,
-0.1771153,
0.08566641,
0.014710529,
-0.00046556012,
0.021978788,
-0.004922472,
-0.04765194,
-0.01341884,
0.005933068,
0.05016839,
-0.04275289,
0.050329015,
0.017334063,
0.021041812,
-0.028189603,
-0.0027155583,
0.016370315,
0.015848286,
0.046741735,
0.0043502473,
-0.047062982,
-0.027346324,
0.011618507,
-0.04058446,
0.041119877,
0.038603425,
0.0015811475,
0.006010034,
-0.0008144165,
-0.03517677,
-0.001206357,
0.048080273,
-0.05509421,
-0.012321239,
0.007663128,
0.037238117,
0.008673724,
0.35722893,
0.022366963,
0.028698247,
-0.037666447,
0.03458781,
0.013827094,
-0.011839366,
0.025887318,
-0.01869937,
-0.066766255,
0.011150018,
0.016477399,
-0.03841603,
0.03445396,
-0.056753997,
-0.09899824,
0.028510852,
0.036381453,
0.046581108,
0.029528141,
0.020493012,
-0.010079188,
0.00035701308,
0.039379776,
-0.022072487,
0.036327913,
0.013987718,
0.03868374,
0.054987125,
0.02006468,
0.042110395,
0.028243145,
-0.062697105,
-0.02365196,
0.02203233,
0.0002559953,
0.015326256,
-0.009952027,
-0.01707974,
-0.04518903,
-0.0038750665,
-0.031830426,
0.03383823,
0.023076389,
-0.04374341,
-0.057128787,
0.03485552,
0.01904739,
-0.0006801444,
-0.029956473,
0.0141215725,
-0.025713308,
0.013104283,
0.08207913,
-0.06799771,
-0.002170104,
0.068158336,
0.043609556,
0.006378132,
-0.036916867,
-0.0152057875,
0.016236462,
-0.0011168424,
-0.050998285,
0.100069076,
0.0013678181,
-0.013398762,
0.008553255,
-0.02061348,
0.036515307,
-0.043984346,
-0.017213594,
-0.028778559,
0.046366945,
-0.1074578,
0.04551028,
0.008104845,
-0.043850493,
0.046982672,
-0.039379776,
-0.023678731,
0.005089789,
-0.033757918,
0.043127682,
0.021911861,
0.06328606,
0.003724481,
-0.05803899,
0.011417726,
0.042297788,
-0.012977122,
0.014871153,
-0.052095886,
0.030759595,
-0.01670495,
-0.044974864,
-0.100069076,
-0.07950914,
-0.015781358,
0.013512538,
0.0056620142,
0.049204644,
-0.0062074685,
0.0035270466,
0.0497936,
0.0101260375,
0.010942545,
0.039513633,
-0.018819839,
-0.053220257,
0.07811706,
-0.019997753,
-0.0423781,
-0.0041762376,
-0.03852311,
0.015941983,
-0.023344096,
0.03236584,
0.017267136,
0.018016716,
0.0215103,
-0.025954245,
0.0101260375,
-0.0051801405,
-0.30625743,
-0.03809478,
-0.0061974293,
-0.0026670361,
0.008667031,
-0.03268709,
0.012876732,
0.043877263,
-0.00029991608,
-0.039192382,
0.09830221,
0.025084196,
-0.03865697,
-0.071210206,
-0.03019741,
0.027654188,
-0.033222504,
0.0373452,
0.037157804,
-0.03849634,
-0.022688214,
0.033918545,
-0.06157273,
0.009778017,
0.008499714,
0.0094969245,
0.06714105,
-0.060608983,
0.052765153,
-0.062750645,
0.041173417,
-0.029447827,
0.075172275,
0.032499693,
-0.00618739,
0.019970981,
0.027921895,
0.06002003,
0.015473495,
-0.012849961,
-0.06494585,
-0.023263784,
0.041173417,
0.031535946,
-0.016584482,
-0.010119344,
-0.009322914,
-0.022340193,
0.024120448,
-0.040049046,
0.070460625,
0.0017484648,
-0.031321783,
-0.028564394,
0.10231782,
-0.0032911294,
-0.03814832,
0.006726152,
-0.023732273,
-0.021229208,
0.0012749571,
0.058253158,
0.0094835395,
-0.0012582254,
0.047759023,
0.064196266,
0.030465117,
0.009255988,
0.06847958,
0.010487443,
0.0041461205,
0.16158827,
-0.08582703,
0.0013611255,
0.0053474577,
-0.059484612,
0.0049927454,
-0.012843269,
-0.008874505,
-0.01307082,
0.055254836,
0.060930233,
0.056753997,
-0.05220297,
-0.05857441,
-0.037853844,
0.018980464,
0.025927475,
-0.009791403,
0.022259882,
-0.05375567,
0.013432225,
0.05659337,
-0.029528141,
0.04160175,
0.004296706,
-0.27777332,
0.0052738385,
0.044626843,
-0.040316753,
-0.046126008,
-0.032794174,
0.012742878,
-0.004882316,
-0.11982589,
0.0030351342,
-0.042913515,
0.04050415,
-0.017856091,
0.012214156,
0.030063555,
0.0819185,
0.060662527,
-0.071317285,
-0.026155025,
-0.053809214,
0.06885438,
-0.006836581,
0.16362284,
0.018404892,
0.0007667311,
0.036783013,
-0.00085750065,
0.0169325,
0.02843054,
-0.04409143,
-0.0053508044,
0.023103159,
0.023638574,
-0.002431119,
-0.058467325,
0.036086977,
-0.035257082,
0.028216373,
-0.01366647,
-0.035765726,
-0.024334615,
0.019489108,
0.007924143,
-0.033918545,
0.10595864,
-0.03351698,
-0.060876694,
-0.086201824,
-0.01182598,
-0.028965954,
-0.05836024,
-0.044921324,
-0.07854539,
-0.017802551,
-0.013559386,
0.05509421,
-0.00515337,
-0.01675849,
0.040477376,
0.033570524,
0.04551028,
-0.057450034,
0.053835984,
-0.0515337,
0.012823191
] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ansmitted on both L1 and L2. Without decryption keys, it is still possible to use a codeless technique to compare the P(Y) codes on L1 and L2 to gain much of the same error information. This technique is slow, so it is currently available only on specialized surveying equipment. In the future, additional civilian codes are expected to be transmitted on the L2 and L5 frequencies. All users will the | [
-0.08121497,
-0.012160958,
-0.047739077,
-0.062002257,
-0.037254576,
-0.017363293,
-0.0005912501,
0.04558363,
0.006236815,
-0.02919162,
0.022193082,
-0.012679861,
0.014476064,
-0.005787764,
0.0038318993,
-0.0595541,
-0.018986529,
-0.016165825,
0.019904587,
0.015434038,
0.06450363,
0.017004052,
0.007218074,
0.0034626797,
0.0031150812,
0.018613983,
-0.033369455,
-0.018920003,
-0.025519384,
-0.21969555,
-0.025985066,
-0.0042077713,
0.07323185,
0.0067956336,
-0.031533338,
-0.03241148,
0.004517117,
0.017868891,
-0.011928117,
-0.0109036155,
0.01587311,
0.035471678,
0.04026155,
-0.033183184,
0.033582337,
-0.027009567,
-0.031985715,
-0.0047965264,
0.01945221,
-0.014023687,
0.021767316,
0.091539815,
0.00428095,
0.01643193,
-0.023443772,
-0.017882196,
-0.0058276798,
0.070411146,
-0.00031454337,
0.023457076,
0.020609763,
-0.020623067,
-0.14635725,
0.034939468,
0.009320296,
0.056626953,
0.05255556,
-0.079352245,
-0.020303743,
0.003775352,
0.015926331,
0.0019475494,
-0.033236403,
0.04848417,
0.030202815,
0.009832547,
-0.080257,
-0.057318825,
-0.023603434,
-0.014635726,
-0.036695756,
-0.03752068,
-0.011535613,
-0.037706953,
-0.017722534,
-0.055509318,
0.0016199087,
0.02714262,
0.027382113,
-0.0085086785,
0.019492125,
-0.03948985,
-0.019186106,
0.05375303,
0.0030651868,
-0.025971761,
-0.018134994,
0.053247433,
0.008122828,
0.39936903,
-0.058117136,
0.0042776237,
-0.03619016,
0.006699171,
0.058702566,
-0.033848442,
-0.06413109,
-0.068122655,
-0.008169396,
-0.0063632145,
0.04143241,
0.007936555,
0.0523959,
-0.011701928,
-0.08494043,
0.005544944,
0.040554266,
0.10878336,
0.0334759,
0.033928275,
-0.009287033,
-0.0447321,
0.015726753,
-0.06929351,
0.017602786,
0.013351774,
0.003377859,
0.07578645,
0.043055646,
0.09755377,
-0.025665741,
-0.06093784,
-0.019718314,
0.0016772874,
-0.02394937,
0.0019176127,
0.041166306,
0.046887543,
0.046195675,
-0.05114521,
0.017336683,
-0.039995447,
0.031107571,
0.024880733,
-0.010777216,
0.043295138,
-0.041725125,
0.014183349,
-0.03105435,
-0.015074798,
-0.003875141,
0.027488554,
0.019864671,
-0.052076574,
0.012067821,
0.033742,
0.06578094,
-0.011435824,
-0.017775754,
-0.016604897,
0.041831564,
-0.032810636,
-0.030655194,
0.06993216,
-0.020609763,
-0.035285406,
-0.020889172,
-0.04092681,
0.010065388,
-0.026730157,
0.00042784552,
-0.024601325,
-0.01055768,
-0.051597588,
-0.015367513,
0.0027741352,
-0.008635079,
-0.022485796,
0.037094913,
0.07786206,
-0.0460094,
-0.07722341,
0.05891545,
0.0063332776,
0.019611873,
0.019292548,
-0.051118597,
0.006030584,
0.012127695,
0.014023687,
0.017416514,
-0.025971761,
0.011422519,
0.05114521,
0.0024597999,
0.024987176,
-0.07701053,
-0.031320453,
-0.04340158,
-0.048138235,
0.053566758,
0.06838875,
0.023190973,
-0.034540314,
0.027009567,
0.036589313,
0.14784744,
-0.06913385,
-0.00761723,
0.08062954,
-0.00777024,
-0.04656822,
-0.01945221,
-0.053619977,
-0.011003405,
0.00920055,
0.039276965,
0.0039882353,
-0.026543885,
0.007317863,
-0.050107405,
-0.018534152,
-0.005488397,
-0.32443413,
-0.063279554,
0.0010477849,
-0.017988637,
0.016085993,
-0.08584519,
0.009127371,
0.035285406,
-0.08169396,
0.02916501,
0.067696884,
0.0069852327,
-0.08291804,
0.03701508,
-0.018973222,
0.029484335,
0.04670127,
0.018361183,
-0.022459187,
-0.017349988,
0.002097233,
0.039276965,
-0.038478654,
-0.05055978,
0.030149596,
0.006300015,
0.08552586,
-0.057904255,
0.07073047,
-0.02794093,
-0.044412777,
0.019026443,
0.024854124,
-0.029697219,
0.011249552,
-0.017735839,
-0.016165825,
0.056680176,
0.04534414,
-0.014941745,
-0.027435333,
0.02208664,
0.017349988,
-0.04265649,
-0.039809175,
-0.047579415,
-0.0324647,
-0.032837246,
-0.043162085,
0.0046534957,
-0.000769623,
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | n be able to perform dual-frequency measurements and directly compute ionospheric delay errors.
A second form of precise monitoring is called Carrier-Phase Enhancement (CPGPS). This corrects the error that arises because the pulse transition of the PRN is not instantaneous, and thus the correlation (satellite–receiver sequence matching) operation is imperfect. CPGPS uses the L1 carrier wave, whic | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | h has a period of , which is about one-thousandth of the C/A Gold code bit period of , to act as an additional clock signal and resolve the uncertainty. The phase difference error in the normal GPS amounts to of ambiguity. CPGPS working to within 1% of perfect transition reduces this error to of ambiguity. By eliminating this error source, CPGPS coupled with DGPS normally realizes between of ab | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | solute accuracy.
Relative Kinematic Positioning (RKP) is a third alternative for a precise GPS-based positioning system. In this approach, determination of range signal can be resolved to a precision of less than . This is done by resolving the number of cycles that the signal is transmitted and received by the receiver by using a combination of differential GPS (DGPS) correction data, transmitti | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | ng GPS signal phase information and ambiguity resolution techniques via statistical tests—possibly with processing in real-time (real-time kinematic positioning, RTK).
Carrier phase tracking (surveying)
Another method that is used in surveying applications is carrier phase tracking. The period of the carrier frequency multiplied by the speed of light gives the wavelength, which is about for th | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | e L1 carrier. Accuracy within 1% of wavelength in detecting the leading edge reduces this component of pseudorange error to as little as . This compares to for the C/A code and for the P code.
accuracy requires measuring the total phase—the number of waves multiplied by the wavelength plus the fractional wavelength, which requires specially equipped receivers. This method has many surveying ap | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | plications. It is accurate enough for real-time tracking of the very slow motions of tectonic plates, typically per year.
Triple differencing followed by numerical root finding, and the least squares technique can estimate the position of one receiver given the position of another. First, compute the difference between satellites, then between receivers, and finally between epochs. Other orders | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | of taking differences are equally valid. Detailed discussion of the errors is omitted.
The satellite carrier total phase can be measured with ambiguity as to the number of cycles. Let denote the phase of the carrier of satellite j measured by receiver i at time . This notation shows the meaning of the subscripts i, j, and k. The receiver (r), satellite (s), and time (t) come in alphabetical orde | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | r as arguments of and to balance readability and conciseness, let be a concise abbreviation. Also we define three functions, :, which return differences between receivers, satellites, and time points, respectively. Each function has variables with three subscripts as its arguments. These three functions are defined below. If is a function of the three integer arguments, i, j, and k then it is a | [
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] |
11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | valid argument for the functions, :, with the values defined as
,
, and
.
Also if are valid arguments for the three functions and a and b are constants then
is a valid argument with values defined as
,
, and
.
Receiver clock errors can be approximately eliminated by differencing the phases measured from satellite 1 with that from satellite 2 at the same epoch. This difference is designate | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | d as
Double differencing computes the difference of receiver 1's satellite difference from that of receiver 2. This approximately eliminates satellite clock errors. This double difference is:
Triple differencing subtracts the receiver difference from time 1 from that of time 2. This eliminates the ambiguity associated with the integral number of wavelengths in carrier phase provided this ambi | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | guity does not change with time. Thus the triple difference result eliminates practically all clock bias errors and the integer ambiguity. Atmospheric delay and satellite ephemeris errors have been significantly reduced. This triple difference is:
Triple difference results can be used to estimate unknown variables. For example, if the position of receiver 1 is known but the position of receiver 2 | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | unknown, it may be possible to estimate the position of receiver 2 using numerical root finding and least squares. Triple difference results for three independent time pairs may be sufficient to solve for receiver 2's three position components. This may require a numerical procedure. An approximation of receiver 2's position is required to use such a numerical method. This initial value can proba | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | bly be provided from the navigation message and the intersection of sphere surfaces. Such a reasonable estimate can be key to successful multidimensional root finding. Iterating from three time pairs and a fairly good initial value produces one observed triple difference result for receiver 2's position. Processing additional time pairs can improve accuracy, overdetermining the answer with multipl | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | e solutions. Least squares can estimate an overdetermined system. Least squares determines the position of receiver 2 that best fits the observed triple difference results for receiver 2 positions under the criterion of minimizing the sum of the squares.
Regulatory spectrum issues concerning GPS receivers
In the United States, GPS receivers are regulated under the Federal Communications Commissi | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | on's (FCC) Part 15 rules. As indicated in the manuals of GPS-enabled devices sold in the United States, as a Part 15 device, it "must accept any interference received, including interference that may cause undesired operation." With respect to GPS devices in particular, the FCC states that GPS receiver manufacturers, "must use receivers that reasonably discriminate against reception of signals out | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | side their allocated spectrum." For the last 30 years, GPS receivers have operated next to the Mobile Satellite Service band, and have discriminated against reception of mobile satellite services, such as Inmarsat, without any issue.
The spectrum allocated for GPS L1 use by the FCC is 1559 to 1610 MHz, while the spectrum allocated for satellite-to-ground use owned by Lightsquared is the Mobile Sa | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | tellite Service band. Since 1996, the FCC has authorized licensed use of the spectrum neighboring the GPS band of 1525 to 1559 MHz to the Virginia company LightSquared. On March 1, 2001, the FCC received an application from LightSquared's predecessor, Motient Services, to use their allocated frequencies for an integrated satellite-terrestrial service. In 2002, the U.S. GPS Industry Council came to | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | an out-of-band-emissions (OOBE) agreement with LightSquared to prevent transmissions from LightSquared's ground-based stations from emitting transmissions into the neighboring GPS band of 1559 to 1610 MHz. In 2004, the FCC adopted the OOBE agreement in its authorization for LightSquared to deploy a ground-based network ancillary to their satellite system – known as the Ancillary Tower Components | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | (ATCs) – "We will authorize MSS ATC subject to conditions that ensure that the added terrestrial component remains ancillary to the principal MSS offering. We do not intend, nor will we permit, the terrestrial component to become a stand-alone service." This authorization was reviewed and approved by the U.S. Interdepartment Radio Advisory Committee, which includes the U.S. Department of Agricultu | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | re, U.S. Space Force, U.S. Army, U.S. Coast Guard, Federal Aviation Administration, National Aeronautics and Space Administration (NASA), U.S. Department of the Interior, and U.S. Department of Transportation.
In January 2011, the FCC conditionally authorized LightSquared's wholesale customers—such as Best Buy, Sharp, and C Spire—to only purchase an integrated satellite-ground-based service from | [
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11866 | https://en.wikipedia.org/wiki/Global%20Positioning%20System | Global Positioning System | LightSquared and re-sell that integrated service on devices that are equipped to only use the ground-based signal using LightSquared's allocated frequencies of 1525 to 1559 MHz. In December 2010, GPS receiver manufacturers expressed concerns to the FCC that LightSquared's signal would interfere with GPS receiver devices although the FCC's policy considerations leading up to the January 2011 order | [
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