Error analysis for the Global Positioning System: Difference between revisions

Revision as of 03:00, 6 July 2021

Revision as of 20:34, 30 September 2021

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[[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 errorserror 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==

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[[File:Accuracy of Navigation Systems.svg|thumb]]

[[File:Gps error diagram.svg|thumb|left| 300px|Geometric Error Diagram Showing Typical Relation of Indicated Receiver Position, Intersection of Sphere Surfaces, and True Receiver Position in Terms of Pseudorange Errors, PDOP, and Numerical Errors]]

User equivalent range errors (UERE) are shown in the table. There is also a [[numerical error]] with an estimated value, <math>\ \sigma_{num} </math>, of about {{convert|1|m|sp=us}}. The standard deviations, <math>\ \sigma_R</math>, for the coarse/acquisition (C/A) and precise codes are also shown in the table. These standard deviations are computed by taking the square root of the sum of the squares of the individual components (i.e., [[Residual sum of squares|RSS]] for root sum squares). To get the standard deviation of receiver position estimate, these range errors must be multiplied by the appropriate [[Dilution of precision (GPS)|dilution of precision]] terms and then RSS'ed with the numerical error. Electronics errors are one of several accuracy-degrading effects outlined in the table above. When taken together, autonomous civilian GPS horizontal position fixes are typically accurate to about 15 meters (50&nbsp;ft). These effects also reduce the more precise P(Y) code's accuracy. However, the advancement of technology means that in the present, civilian GPS fixes under a clear view of the sky are on average accurate to about 5 meters (16&nbsp;ft) horizontally.

The term user equivalent range error (UERE) refers to the error of a component in the distance from receiver to a satellite. These UERE errors are given as ± errors thereby implying that they are unbiased or zero mean errors. These UERE errors are therefore used in computing standard deviations. The standard deviation of the error in receiver position,

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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>

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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}

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== Relativity ==

[[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 the 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 and general relativity ===