Combination of GNSS Measurements

Fundamentals
Title	Combination of GNSS Measurements
Author(s)	J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.
Level	Medium
Year of Publication	2011

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

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

[math]\displaystyle{ \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)} }[/math]

Satellite clocks are defined relative to [math]\displaystyle{ R_{_{PC}} }[/math] 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 [math]\displaystyle{ \Phi_{_{LI}} }[/math] and [math]\displaystyle{ R_{_{PI}} }[/math].

[math]\displaystyle{ \Phi_{_{LI}}=\Phi_{_{L1}}-\Phi_{_{L2}}~~~~~;~~~~~ R_{_{PI}}=R_{_{P2}}-R_{_{P1}} \qquad \mbox{(2)} }[/math]

• 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.

[math]\displaystyle{ \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)} }[/math]

• 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.

[math]\displaystyle{ \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)} }[/math]

For more information, please go to the article:

• Combining pairs of signals and clock definition