The carrier phase \(\phi _i^k\) is equal to the difference between the phase \({\phi _i}\;\)of the receiver generated carrier signal at the signal reception time and the phase \({\phi _k}\) of the satellite generated carrier signal at the signal transmission time. Only the fractional carrier phase can be measured when a satellite signal is acquired, i.e. an integer number N of full cycles is unknown N is called the carrier phase ambiguity.

\(\begin{array}{l}\Phi _i^k\left( t \right) = \left| {\left| {{r^k}\left( {t - \tau _i^k} \right) + \delta {r^k}\left( {t - \tau _i^k} \right) - \left( {{r_i}\left( t \right) + \delta {r_i}\left( t \right)} \right)} \right|} \right| - I_i^k + T_i^k + c\left[ {d{t_i}\left( t \right) - \;d{t^k}\left( {t - \tau _i^k} \right)} \right] + \\c\left[ {{\delta _i}\left( t \right) + {\delta ^k}\left( {t - \tau _i^k} \right)} \right] + \delta m_i^k + \lambda \left[ {{\phi _0}\left( {{t_0}} \right) - {\phi ^k}\left( {{t_0}} \right)} \right] + \lambda N_i^k + \varepsilon _i^k\end{array}\)

\({\rm{\Phi }}_i^k\) carrier phase measurement.\(\;{\rm{\Phi }}_i^k = \lambda *\phi _i^k\;\)

\({r^k}\) position vector of the centre of mass

\(t\) time

\(\tau _i^k\) signal travel time from the transmitting antenna to the receiver antenna

\(\delta {r^k}\) eccentricity vector of the transmitting antenna

\(\delta {r_i}\) eccentricity vector of the receiving antenna

\(I_i^k\) ionospheric phase delay

\(T_i^k\) tropospheric delay

\(c\) speed of light

\(d{t_i}\) sum of true time,deviation of the actual oscillator frequency and effect of non zero

\(d{t^k}\) initial phase of the oscillator at the receiver

\({\delta ^k}\) sum of true time,deviation of the actual oscillator frequency and effect of non zero

\({\delta _i}\) initial phase of the oscillator at the transmitter

\(\delta {m^k}\) signal delay occuring between the signal generation and transmission

\(\delta {m_i}\) signal delay between the receiving antenna and the signal correlator in the receiver

\(\varepsilon _i^k\) multipath error at the satellite

\({\phi _i}\;\left( {{t_0}} \right)\) multipath error at the receiver

\({\phi ^k}\left( {{t_0}} \right)\) carrier phase measurement error

\(N_i^k\) non-zero initial phases at the receiver

\(\lambda \) \(\frac{c}{{{f_0}}}\), \(\lambda \;\)represents the nominal wave length of the carrier signal

Source: Teunissenn and Kleusberg (1996)

\(\begin{array}{l}\Phi _i^k\left( t \right) = \left| {\left| {{r^k}\left( {t - \tau _i^k} \right) + \delta {r^k}\left( {t - \tau _i^k} \right) - \left( {{r_i}\left( t \right) + \delta {r_i}\left( t \right)} \right)} \right|} \right| - I_i^k + T_i^k + c\left[ {d{t_i}\left( t \right) - \;d{t^k}\left( {t - \tau _i^k} \right)} \right] + \\c\left[ {{\delta _i}\left( t \right) + {\delta ^k}\left( {t - \tau _i^k} \right)} \right] + \delta m_i^k + \lambda \left[ {{\phi _0}\left( {{t_0}} \right) - {\phi ^k}\left( {{t_0}} \right)} \right] + \lambda N_i^k + \varepsilon _i^k\end{array}\)

\({\rm{\Phi }}_i^k\) carrier phase measurement.\(\;{\rm{\Phi }}_i^k = \lambda *\phi _i^k\;\)

\({r^k}\) position vector of the centre of mass

\(t\) time

\(\tau _i^k\) signal travel time from the transmitting antenna to the receiver antenna

\(\delta {r^k}\) eccentricity vector of the transmitting antenna

\(\delta {r_i}\) eccentricity vector of the receiving antenna

\(I_i^k\) ionospheric phase delay

\(T_i^k\) tropospheric delay

\(c\) speed of light

\(d{t_i}\) sum of true time,deviation of the actual oscillator frequency and effect of non zero

\(d{t^k}\) initial phase of the oscillator at the receiver

\({\delta ^k}\) sum of true time,deviation of the actual oscillator frequency and effect of non zero

\({\delta _i}\) initial phase of the oscillator at the transmitter

\(\delta {m^k}\) signal delay occuring between the signal generation and transmission

\(\delta {m_i}\) signal delay between the receiving antenna and the signal correlator in the receiver

\(\varepsilon _i^k\) multipath error at the satellite

\({\phi _i}\;\left( {{t_0}} \right)\) multipath error at the receiver

\({\phi ^k}\left( {{t_0}} \right)\) carrier phase measurement error

\(N_i^k\) non-zero initial phases at the receiver

\(\lambda \) \(\frac{c}{{{f_0}}}\), \(\lambda \;\)represents the nominal wave length of the carrier signal

Source: Teunissenn and Kleusberg (1996)