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# Combination of GNSS Measurements

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Fundamentals
Title Combination of GNSS Measurements
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level Intermediate
Year of Publication 2011

Starting from the basic observables (described in GNSS Basic Observables) the following combinations can be defined (where $R_{_{Pi}}$ and $\Phi_{_{Li}}$, $i=1,2$, indicate measurements in the frequencies $f_1$ and $f_2$):

• Ionosphere-free combination: It removes the first order (up to 99.9%) ionospheric effect, which depends on the inverse square of the frequency ($\alpha_i \propto \frac{1}{f_i^2}$, see article Ionospheric Delay).

$\Phi_{_{LC}}=\frac{f_1^2\;\Phi_{_{L1}}-f_2^2\;\Phi_{_{L2}}}{f_1^2-f_2^2}~~~~~;~~~~~ R_{_{PC}}=\frac{f_1^2\;R_{_{P1}}-f_2^2\;R_{_{P2}}}{f_1^2-f_2^2} \qquad \mbox{(1)}$

Satellite clocks are defined relative to $R_{_{PC}}$ combination (see article Combining pairs of signals and clock definition).

• Geometry-free combination: it cancels the geometric part of the measurement, leaving all the frequency-dependent effects (i.e., ionospheric refraction, instrumental delays, wind-up) besides multipath and measurement noise. It can be used to estimate the ionospheric electron content, to detect cycle-slips in the carrier phase, or also to estimate antenna rotations as well. Note the change of terms order in $\Phi_{_{LI}}$ and $R_{_{PI}}$.

$\Phi_{_{LI}}=\Phi_{_{L1}}-\Phi_{_{L2}}~~~~~;~~~~~ R_{_{PI}}=R_{_{P2}}-R_{_{P1}} \qquad \mbox{(2)}$

• Wide-laning combinations: These combinations are used to create a signal with a significantly wide wavelength. This longer wavelength is useful for cycle-slips detection and ambiguity fixing. Other feature of this combination is the change of the sign in the ionospheric term, which is exploited by the Melbourne-Wübbena combination to remove the ionospheric refraction.

$\Phi_{_{LW}}=\frac{f_1\;\Phi_{_{L1}}-f_2\;\Phi_{_{L2}}}{f_1-f_2}~~~;~~~ R_{_{PW}}=\frac{f_1\;R_{_{P1}}-f_2\;R_{_{P2}}}{f_1-f_2} \qquad \mbox{(3)}$

• Narrow-laning combinations: These combinations create signals with a narrow wavelength. The signal in this combination has a lower noise than each separated component. It is used to reduce the code noise in the Melbourne-Wübbena combination to estimate the wide-lane ambiguity.

$\Phi_{_{LN}}=\frac{f_1\;\Phi_{_{L1}}+f_2\;\Phi_{_{L2}}}{f_1+f_2}~~~;~~~ R_{_{PN}}=\frac{f_1\;R_{_{P1}}+f_2\;R_{_{P2}}}{f_1+f_2} \qquad \mbox{(4)}$