*3.3. Doppler Shift Computation*

The Doppler shift of a GNSS signal is predominantly influenced by the relative velocity between the transmitter satellite and the receiver, along with a common offset that is proportional to the error in the receiver clock's frequency. However, as demonstrated in [24,25], various ionospheric effects, such as changes in the redistribution and density of electrons in the ionosphere, lead to frequency variations in the electromagnetic waves emitted by a stable transmitter. These variations are manifested as the Doppler shift and can be quantified as the time derivative of the phase path of the signal. When considering only the ionospheric delay term in the carrier phase observation model [26], the residual phase path expressed in units of cycles can be given by:

$$
\phi = \frac{\Delta\_{p\_{\rm{ions}}}}{\lambda} \tag{5}
$$

where *λ* is the wavelength of the GPS L1 frequency (0.1905 m). As the Doppler shift (*f <sup>d</sup>*) of a given signal corresponds to the rate of change of its carrier phase over time, it can be computed using the following equation:

$$f\_d = \frac{d\phi}{dt} \tag{6}$$
