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Concept
Satellite navigation solution
Summary
Satellite navigation solution for the receiver's position (geopositioning) involves an algorithm. In essence, a GNSS receiver measures the transmitting time of GNSS signals emitted from four or more GNSS satellites (giving the pseudorange) and these measurements are used to obtain its position (i.e., spatial coordinates) and reception time.
A global-navigation-satellite-system (GNSS) receiver measures the apparent transmitting time, , or "phase", of GNSS signals emitted from four or more GNSS satellites ( ), simultaneously.
GNSS satellites broadcast the messages of satellites' ephemeris, , and intrinsic clock bias (i.e., clock advance), as the functions of (atomic) standard time, e.g., GPST.
The transmitting time of GNSS satellite signals, , is thus derived from the non-closed-form equations and , where is the relativistic clock bias, periodically risen from the satellite's orbital eccentricity and Earth's gravity field. The satellite's position and velocity are determined by as follows: and .
In the field of GNSS, "geometric range", , is defined as straight range, or 3-dimensional distance, from to in inertial frame (e.g., Earth-centered inertial (ECI) one), not in rotating frame.
The receiver's position, , and reception time, , satisfy the light-cone equation of in inertial frame, where is the speed of light. The signal time of flight from satellite to receiver is .
The above is extended to the satellite-navigation positioning equation, , where is atmospheric delay (= ionospheric delay + tropospheric delay) along signal path and is the measurement error.
The Gauss–Newton method can be used to solve the nonlinear least-squares problem for the solution: , where . Note that should be regarded as a function of and .
The posterior distribution of and is proportional to , whose mode is . Their inference is formalized as maximum a posteriori estimation.
The posterior distribution of is proportional to .
Image:Light cones.svg|Essentially, the solution, \scriptstyle (\hat{\boldsymbol{r_{\text{rec,, \hat{t}_{\text{rec), is the intersection of [[light cone]]s.
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