Pseudoranges
Each
receiver measures the range between itself and the transmitting
satellite. The measured range from signal delay isn't exact and contains
some errors. It is called a pseudorange. With different
corrections this pseudoranges come closer to the exact range and
can be used to calculate an accurate receiver position.
\( R^j = \rho^j + c(\delta t_{rec}  \delta t^j_{sat}) + T^j + \hat{\alpha}I^j + TGD^j + M^j + \epsilon^j \)
\( R^j \)  pseudorange between the GNSS receiver and the satellite at the time of transmission, calculated from signal delay

\(\rho^j\)  distance between the GNSS receiver and the satellite at the time of transmission

\(c\)  speed of light (299 792 458 m/s)

\( \delta t_{rec} \)  receiver clock error

\( \delta t^j_{sat} \)  satellite clock error

\( T^j \)  signal tropospheric delay


\( \hat{\alpha}I^j \)  signal ionospheric delay

\( TGD^j \)  instrumental delay

\( M^j \)  multipath errors

\(\epsilon^j \)  receiver measurement noise

Navigation equations
Each pseudorange is a sum of the true distance to the satellite which is unknown and different errors. True distance can be written as the distance between the unknown receiver coordinates and satellite coordinates.
\(\rho^j = \sqrt{(x_{rec}  x_{sat}^j)^2 + (y_{rec}  y_{sat}^j)^2 + (z_{rec}  z_{sat}^j)^2} \)
\(x_{rec},\space\space y_{rec}, \space \space z_{rec}\)  unknown receiver coordinates

\(x_{sat}^j, \space \space y_{sat}^j, \space \space z_{sat}^j\)  known satellite coordinates

All of the errors except the receiver clock error are dependent on the circumstances at the time of transmission primarily satellite possition with respect to the receiver. They are normally corrected with some models.
\( D^j =c\delta t^j_{sat} + T^j + \hat{\alpha}I^j + TGD^j + M^j + \epsilon^j \)
Pseudorange can than be written with the previous terms.
\( R^j = \sqrt{(x_{rec}  x_{sat}^j)^2 + (y_{rec}  y_{sat}^j)^2 + (z_{rec}  z_{sat}^j)^2} + c\delta t_{rec} + D^j \)
The \(D^j\) term is solved with error models and can be considered as a known term. We can transfer it to the left side.
\( R^j  D^j = \sqrt{(x_{rec}  x_{sat}^j)^2 + (y_{rec}  y_{sat}^j)^2 + (z_{rec}  z_{sat}^j)^2} + c\delta t_{rec}\)
There are now four variables in range equations that are unknown, the receiver coordinates \(x_{rec}, \space y_{rec}, \space z_{rec} \) and the receiver clock error \(\delta t_{rec}\). Therefore at least four pseudorange measurements from different satellites are needed to get a solution to this variables (a system of four equations with four variables).
Normally more than four satellites are used to obtain the solution. The solution is a best fit to the overdetermined (more equations than the unknown variables) navigation equation system.