GNSS positioning calculation

The global navigation satellite system (GNSS) positioning for receiver's position is derived through the calculation steps, or algorithm, given below. In essence, a GNSS receiver measures the transmitting time of GNSS signals emitted from four or more GNSS satellites and these measurements are used to obtain its position (i.e., spatial coordinates) and reception time.

Essentially, the solution, ${\displaystyle \scriptstyle ({\hat {\boldsymbol {r}}}_{\text{rec}},\,{\hat {t}}_{\text{rec}})}$, is the intersection of light cones.

The posterior distribution of the solution is derived from the product of the distribution of propagating spherical surfaces. (See animation.)

in which ${\displaystyle \scriptstyle E_{i}}$ is the orbital eccentric anomaly of satellite ${\displaystyle i}$, ${\displaystyle \scriptstyle M_{i}}$ is the mean anomaly, ${\displaystyle \scriptstyle e_{i}}$ is the eccentricity, and ${\displaystyle \scriptstyle \delta t_{{\text{clock}},i}(t_{i},\,E_{i})\;=\;\delta t_{{\text{clock,sv}},i}(t_{i})\,+\,\delta t_{{\text{orbit-relativ}},i}(E_{i})}$.

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