Editing Global Positioning System (section)

=== Problem statement ===
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 [''x<sub>i</sub>, y<sub>i</sub>, z<sub>i</sub>, s<sub>i</sub>''] where the subscript ''i'' denotes the satellite and has the value 1, 2, ..., ''n'', where ''n''&nbsp;≥&nbsp;4. When the time of message reception indicated by the on-board receiver clock is <math>\tilde{t}_i</math>, the true reception time is <math>t_i = \tilde{t}_i - b</math>, 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 are all perfectly synchronized). The message's transit time is <math>\tilde{t}_i - b - s_i</math>, where ''s<sub>i</sub>'' is the satellite time. Assuming the message traveled at [[Speed of light|the speed of light]], ''c'', the distance traveled is <math>\left(\tilde{t}_i - b - s_i\right) c</math>.

For n satellites, the equations to satisfy are:
:<math>d_i = \left( \tilde{t}_i - b - s_i \right)c, \; i=1,2,\dots,n</math>
where ''d<sub>i</sub>'' 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):
:<math>d_i = \sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}</math>
Defining ''pseudoranges'' as <math> p_i = \left ( \tilde{t}_i - s_i \right )c</math>, we see they are biased versions of the true range:
:<math>p_i = d_i + bc, \;i=1,2,...,n</math> .<ref name=GPS_BASICS_Blewitt>section 4 beginning on page 15 [http://www.nbmg.unr.edu/staff/pdfs/Blewitt%20Basics%20of%20gps.pdf Geoffrey Blewitt: Basics of the GPS Technique] {{Webarchive|url=https://web.archive.org/web/20130922064413/http://www.nbmg.unr.edu/staff/pdfs/Blewitt%20Basics%20of%20gps.pdf |date=September 22, 2013 }}</ref><ref name=Bancroft>{{cite web|url=http://www.macalester.edu/~halverson/math36/GPS.pdf|archive-url=https://web.archive.org/web/20110719232148/http://www.macalester.edu/~halverson/math36/GPS.pdf|archive-date=July 19, 2011|title=Global Positioning Systems|access-date=October 15, 2010}}</ref>
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 and Chaffee.<ref name="Abel1" /> When ''n'' is greater than four, this system is [[Overdetermined system|overdetermined]] and a [[Mean|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. This is done by multiplying the basic resolution of the receiver by quantities called the [[Dilution of precision (navigation)|geometric dilution of position]] (GDOP) factors, calculated from the relative sky directions of the satellites used.<ref>{{cite web|url=http://www.colorado.edu/geography/gcraft/notes/gps/gps.html#Gdop|title=Geometric Dilution of Precision (GDOP) and Visibility|first=Peter H.|last=Dana|publisher=University of Colorado at Boulder|access-date=July 7, 2008|archive-url=https://web.archive.org/web/20050823013233/http://www.colorado.edu/geography/gcraft/notes/gps/gps.html#Gdop|archive-date=August 23, 2005}}</ref> The receiver location is expressed in a specific coordinate system, such as latitude and longitude using the [[WGS 84]] [[datum (geodesy)|geodetic datum]] or a country-specific system.<ref>{{cite web |author=Dana |first=Peter H. |title=Receiver Position, Velocity, and Time |url=http://www.colorado.edu/geography/gcraft/notes/gps/gps.html#PosVelTime |archive-url=https://web.archive.org/web/20050823013233/http://www.colorado.edu/geography/gcraft/notes/gps/gps.html#PosVelTime |archive-date=August 23, 2005 |access-date=July 7, 2008 |publisher=University of Colorado at Boulder}}</ref>