GNSS observables#

We understand by GNSS observables as

  • pseudorange

  • carrier phase

  • Doppler

  • Signal-to-Noise ratio

These are delivered usually in RINEX format

Observable model#

The GNSS observables can be modelled usign the following expressions for the code and phase

\[ P = \rho + c \cdot (dt_r - dt^s) + \frac{40.3}{f^2}\cdot STEC_r^s+T + b_P + \varepsilon_P \]
\[ L = \rho + c \cdot (dt_r - dt^s) - \frac{40.3}{f^2}\cdot STEC_r^s+ T + b_L + \lambda \cdot N + \varepsilon_P \]

where:

  • \(P\) and \(L\) represent the pseudorange and carrier-phase respectively

  • \(\rho\) is the geometric (Euclidean) distance between the receiver and satellite

  • \(dt_r\) and \(dt^s\) is the clock bias for the receiver and satellite. \(c\) is the speed of light

  • \(f\) is the frequency of the signal (and \(\lambda\) is the corresponding wavelength)

  • \(b_P\) and \(b_L\) are the code and phase hardware biases respectively

  • \(N\) is the integer phase ambiguity

Observable combination#

Ionospheric (geometry-free) combination#

Between frequencies \(a\) and \(b\). Assuming that \(f_b > f_a\), the code combination can be expressed as:

\[ PI_{a,b} = P_a - P_b = 40.3 \cdot \left ( \frac{1}{f_a^2} - \frac{1}{f_b^2} \right) \cdot STEC_r^s + b_{P,a} - b_{P,b} + \sqrt{2} \cdot \varepsilon_P \]

On the other hand, the ionospheric combination of phases can be expressed as:

(1)#\[LI_{a,b} = P_b - P_a = 40.3 \cdot \left ( \frac{1}{f_a^2} - \frac{1}{f_b^2} \right) \cdot STEC_r^s + b_{L,b} - b_{L,a} + {\lambda}_b \cdot N_b - {\lambda}_a \cdot N_a + \sqrt{2} \cdot \varepsilon_L\]

In GNSS textbooks, the term \(40.3 \cdot \left ( \frac{1}{f_a^2} - \frac{1}{f_b^2} \right)\) is usually referred to the alpha constant \(\alpha_{LI}\) for the given frequencies

For example, the \(\alpha_{LI}\) for GPS L1 and L2 frequencies can be computed as follows:

40.3 * (1 / (10.23e6*120)**2 - 1 / (10.23e6*154)**2)

1.0504595284873213e-17

On the other hand, the term \(b_{P,a} - b_{P,b}\) is known as the Differential Code Bias (DCB) between the codes \(P_a\) and \(P_b\) (i.e. \(DCB(P_a, P_b)\)).

This combination is usually employed when building models for ionospheric estimation due to the fact that the Slant Total Electron Content (STEC) is exposed.

Code minus carrier combination#

This combination (also called code/carrier divergence) is defined as follows

(2)#\[CMC_{a} = P_a - L_a = 2 \cdot \frac{40.3}{f_a^2} \cdot STEC_r^s + b_{P,a} - b_{L,a} - {\lambda}_a \cdot N_a + \varepsilon_P - \varepsilon_L\]

The combination has twice the ionospheric combination, plus several constant terms (biases, ambiguity) and the code and phase noises. Note however that the code noise is much higher than the phase one. Therefore, if the ionospheric contribution is removed (using a rolling window to detrend it or removing the STEC computed from e.g. IONEX maps) and the bias is removed, the CMC can serve as noise estimator of the observables.

In addition, for high-rate applications (> 1Hz), large jumps in the CMC (larger than the code noise level, ~ 1 m) can indicate the presence of cycle slips.