Satellite navigation solution

1. A global-navigation-satellite-system (GNSS) receiver measures the apparent transmitting time, ${\displaystyle \displaystyle {\tilde {t}}_{i}}$, or "phase", of GNSS signals emitted from four or more GNSS satellites (${\displaystyle \displaystyle i\;=\;1,\,2,\,3,\,4,\,..,\,n}$ ), simultaneously.[1]
2. GNSS satellites broadcast the messages of satellites' ephemeris, ${\displaystyle \displaystyle {\boldsymbol {r}}_{i}(t)}$, and intrinsic clock bias (i.e., clock advance), ${\displaystyle \displaystyle \delta t_{{\text{clock,sv}},i}(t)}$ as the functions of (atomic) standard time, e.g., GPST.[2]
3. The transmitting time of GNSS satellite signals, ${\displaystyle \displaystyle t_{i}}$, is thus derived from the non-closed-form equations ${\displaystyle \displaystyle {\tilde {t}}_{i}\;=\;t_{i}\,+\,\delta t_{{\text{clock}},i}(t_{i})}$ and ${\displaystyle \displaystyle \delta t_{{\text{clock}},i}(t_{i})\;=\;\delta t_{{\text{clock,sv}},i}(t_{i})\,+\,\delta t_{{\text{orbit-relativ}},\,i}({\boldsymbol {r}}_{i},\,{\dot {\boldsymbol {r}}}_{i})}$, where ${\displaystyle \displaystyle \delta t_{{\text{orbit-relativ}},i}({\boldsymbol {r}}_{i},\,{\dot {\boldsymbol {r}}}_{i})}$ is the relativistic clock bias, periodically risen from the satellite's orbital eccentricity and Earth's gravity field.[2] The satellite's position and velocity are determined by ${\displaystyle \displaystyle t_{i}}$ as follows: ${\displaystyle \displaystyle {\boldsymbol {r}}_{i}\;=\;{\boldsymbol {r}}_{i}(t_{i})}$ and ${\displaystyle \displaystyle {\dot {\boldsymbol {r}}}_{i}\;=\;{\dot {\boldsymbol {r}}}_{i}(t_{i})}$.
4. In the field of GNSS, "geometric range", ${\displaystyle \displaystyle r({\boldsymbol {r}}_{A},\,{\boldsymbol {r}}_{B})}$, is defined as straight range, or 3-dimensional distance,[3] from ${\displaystyle \displaystyle {\boldsymbol {r}}_{A}}$ to ${\displaystyle \displaystyle {\boldsymbol {r}}_{B}}$ in inertial frame (e.g., Earth-centered inertial (ECI) one), not in rotating frame.[2]
5. The receiver's position, ${\displaystyle \displaystyle {\boldsymbol {r}}_{\text{rec}}}$, and reception time, ${\displaystyle \displaystyle t_{\text{rec}}}$, satisfy the light-cone equation of ${\displaystyle \displaystyle r({\boldsymbol {r}}_{i},\,{\boldsymbol {r}}_{\text{rec}})/c\,+\,(t_{i}-t_{\text{rec}})\;=\;0}$ in inertial frame, where ${\displaystyle \displaystyle c}$ is the speed of light. The signal time of flight from satellite to receiver is ${\displaystyle \displaystyle -(t_{i}\,-\,t_{\text{rec}})}$.
6. The above is extended to the satellite-navigation positioning equation, ${\displaystyle \displaystyle r({\boldsymbol {r}}_{i},\,{\boldsymbol {r}}_{\text{rec}})/c\,+\,(t_{i}\,-\,t_{\text{rec}})\,+\,\delta t_{{\text{atmos}},i}\,-\,\delta t_{{\text{meas-err}},i}\;=\;0}$, where ${\displaystyle \displaystyle \delta t_{{\text{atmos}},i}}$ is atmospheric delay (= ionospheric delay + tropospheric delay) along signal path and ${\displaystyle \displaystyle \delta t_{{\text{meas-err}},i}}$ is the measurement error.
7. The Gauss–Newton method can be used to solve the nonlinear least-squares problem for the solution: ${\displaystyle \displaystyle ({\hat {\boldsymbol {r}}}_{\text{rec}},\,{\hat {t}}_{\text{rec}})\;=\;\arg \min \phi ({\boldsymbol {r}}_{\text{rec}},\,t_{\text{rec}})}$, where ${\displaystyle \displaystyle \phi ({\boldsymbol {r}}_{\text{rec}},\,t_{\text{rec}})\;=\;\sum _{i=1}^{n}(\delta t_{{\text{meas-err}},i}/\sigma _{\delta t_{{\text{meas-err}},i}})^{2}}$. Note that ${\displaystyle \displaystyle \delta t_{{\text{meas-err}},i}}$ should be regarded as a function of ${\displaystyle \displaystyle {\boldsymbol {r}}_{\text{rec}}}$ and ${\displaystyle \displaystyle t_{\text{rec}}}$.
8. The posterior distribution of ${\displaystyle \displaystyle {\boldsymbol {r}}_{\text{rec}}}$ and ${\displaystyle \displaystyle t_{\text{rec}}}$ is proportional to ${\displaystyle \displaystyle \exp(-{\frac {1}{2}}\phi ({\boldsymbol {r}}_{\text{rec}},\,t_{\text{rec}}))}$, whose mode is ${\displaystyle \displaystyle ({\hat {\boldsymbol {r}}}_{\text{rec}},\,{\hat {t}}_{\text{rec}})}$. Their inference is formalized as maximum a posteriori estimation.
9. The posterior distribution of ${\displaystyle \displaystyle {\boldsymbol {r}}_{\text{rec}}}$ is proportional to ${\displaystyle \displaystyle \int _{-\infty }^{\infty }\exp(-{\frac {1}{2}}\phi ({\boldsymbol {r}}_{\text{rec}},\,t_{\text{rec}}))\,dt_{\text{rec}}}$.

• 
  Essentially, the solution, ${\displaystyle \scriptstyle ({\hat {\boldsymbol {r}}}_{\text{rec}},\,{\hat {t}}_{\text{rec}})}$, is the intersection of light cones.
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  The posterior distribution of the solution is derived from the product of the distribution of propagating spherical surfaces. (See animation.)

• For Global Positioning System (GPS),[2] the non-closed-form equations in step 3 result in

${\displaystyle \scriptstyle {\begin{cases}\scriptstyle \Delta t_{i}(t_{i},\,E_{i})\;\triangleq \;t_{i}\,+\,\delta t_{{\text{clock}},i}(t_{i},\,E_{i})\,-\,{\tilde {t}}_{i}\;=\;0,\\\scriptstyle \Delta M_{i}(t_{i},\,E_{i})\;\triangleq \;M_{i}(t_{i})\,-\,(E_{i}\,-\,e_{i}\sin E_{i})\;=\;0,\end{cases}}}$

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 ${\displaystyle \scriptstyle \delta t_{{\text{clock}},i}(t_{i},\,E_{i})\;=\;\delta t_{{\text{clock,sv}},i}(t_{i})\,+\,\delta t_{{\text{orbit-relativ}},i}(E_{i})}$.

• The above can be solved by using the bivariate Newton–Raphson method on ${\displaystyle \scriptstyle t_{i}}$ and ${\displaystyle \scriptstyle E_{i}}$. Two times of iteration will be necessary and sufficient in most cases. Its iterative update will be described by using the approximated inverse of Jacobian matrix as follows:

${\displaystyle \scriptstyle {\begin{pmatrix}t_{i}\\E_{i}\\\end{pmatrix}}\leftarrow {\begin{pmatrix}t_{i}\\E_{i}\\\end{pmatrix}}-{\begin{pmatrix}1&&0\\{\frac {{\dot {M}}_{i}(t_{i})}{1-e_{i}\cos E_{i}}}&&-{\frac {1}{1-e_{i}\cos E_{i}}}\\\end{pmatrix}}{\begin{pmatrix}\Delta t_{i}\\\Delta M_{i}\\\end{pmatrix}}}$

• Tropospheric delay should not be ignored, while the Global Positioning System (GPS) specification[2] doesn't provide its detailed description.

• The GLONASS ephemerides don't provide clock biases ${\displaystyle \scriptstyle \delta t_{{\text{clock,sv}},i}(t)}$, but ${\displaystyle \scriptstyle \delta t_{{\text{clock}},i}(t)}$.

• Time to first fix

• In the field of GNSS, ${\displaystyle \scriptstyle {\tilde {r}}_{i}\;=\;-c({\tilde {t}}_{i}\,-\,{\tilde {t}}_{\text{rec}})}$ is called pseudorange, where ${\displaystyle \scriptstyle {\tilde {t}}_{\text{rec}}}$ is a provisional reception time of the receiver. ${\displaystyle \scriptstyle \delta t_{\text{clock,rec}}\;=\;{\tilde {t}}_{\text{rec}}\,-\,t_{\text{rec}}}$ is called receiver's clock bias (i.e., clock advance).[1]
• Standard GNSS receivers output ${\displaystyle \scriptstyle {\tilde {r}}_{i}}$ and ${\displaystyle \scriptstyle {\tilde {t}}_{\text{rec}}}$ per an observation epoch.
• The temporal variation in the relativistic clock bias of satellite is linear if its orbit is circular (and thus its velocity is uniform in inertial frame).
• The signal time of flight from satellite to receiver is expressed as ${\displaystyle \scriptstyle -(t_{i}-t_{\text{rec}})\;=\;{\tilde {r}}_{i}/c\,+\,\delta t_{{\text{clock}},i}\,-\,\delta t_{\text{clock,rec}}}$, whose right side is round-off-error resistive during calculation.
• The geometric range is calculated as ${\displaystyle \scriptstyle r({\boldsymbol {r}}_{i},\,{\boldsymbol {r}}_{\text{rec}})\;=\;|\Omega _{\text{E}}(t_{i}\,-\,t_{\text{rec}}){\boldsymbol {r}}_{i,{\text{ECEF}}}\,-\,{\boldsymbol {r}}_{\text{rec,ECEF}}|}$, where the Earth-centred, Earth-fixed (ECEF) rotating frame (e.g., WGS84 or ITRF) is used in the right side and ${\displaystyle \scriptstyle \Omega _{\text{E}}}$ is the Earth rotating matrix with the argument of the signal transit time.[2] The matrix can be factorized as ${\displaystyle \scriptstyle \Omega _{\text{E}}(t_{i}\,-\,t_{\text{rec}})\;=\;\Omega _{\text{E}}(\delta t_{\text{clock,rec}})\Omega _{\text{E}}(-{\tilde {r}}_{i}/c\,-\,\delta t_{{\text{clock}},i})}$.
• The line-of-sight unit vector of satellite observed at ${\displaystyle \scriptstyle {\boldsymbol {r}}_{\text{rec,ECEF}}}$ is described as: ${\displaystyle \scriptstyle {\boldsymbol {e}}_{i,{\text{rec,ECEF}}}\;=\;-{\frac {\partial r({\boldsymbol {r}}_{i},\,{\boldsymbol {r}}_{\text{rec}})}{\partial {\boldsymbol {r}}_{\text{rec,ECEF}}}}}$.
• The satellite-navigation positioning equation may be expressed by using the variables ${\displaystyle \scriptstyle {\boldsymbol {r}}_{\text{rec,ECEF}}}$ and ${\displaystyle \scriptstyle \delta t_{\text{clock,rec}}}$.
• The nonlinearity of the vertical dependency of tropospheric delay degrades the convergence efficiency in the Gauss–Newton iterations in step 7.
• The above notation is different from that in the Wikipedia articles, 'Position calculation introduction' and 'Position calculation advanced', of Global Positioning System (GPS).

• Dilution of precision (navigation)
• Global Positioning System § Navigation equations
• Least squares adjustment
• Precise Point Positioning
• Real Time Kinematic

• PVT (Position, Velocity, Time): Calculation procedure in the open-source GNSS-SDR and the underlying RTKLIB