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Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level Basic
Year of Publication 2011

The linear observation model $\displaystyle{ {\mathbf Y}={\mathbf G}\;{\mathbf X} }$ can be solved using Kalman Filter, considering the carrier phase bias $\displaystyle{ B_{C}^i }$ as "constants" along continuous phase arcs, and white-noise at the instants when cycle-slips occurs.

The following stochastic model can be used for the filter:

• Carrier phase biases ($\displaystyle{ B_C }$) are taken as "constants" along continuous phase arcs, and "white-noise" when a cycle-slip happens (with $\displaystyle{ \sigma=10^4\,m }$, for instance), see Kalman Filter.

• Wet tropospheric delay ($\displaystyle{ \Delta T_{z,wet} }$) is taken as a random-walk process (a process noise with $\displaystyle{ d\sigma^2/dt= 1\,cm^2/h }$, initialised with $\displaystyle{ \sigma^2_0=0.25\, m^2 }$, can be used for most of the applications), see Kalman Filter.

• Receiver clock ($\displaystyle{ c\, \delta t }$) is taken as a white noise process (with $\displaystyle{ \sigma= 3\, 10^5\; m }$, i.e, $\displaystyle{ 1 }$ millisecond, for instance.), see Kalman Filter.

• Receiver coordinates ($\displaystyle{ dx,dy,dz }$)
For static positioning: the coordinates are taken as constants, see Kalman Filter.
For kinematic positioning: the coordinates are taken as white noise or a random walk process as in Kalman Filter.

This solving procedure is called to float ambiguities. Floating in the sense that they are estimated by the filter "as real numbers". The bias estimations $\displaystyle{ B^i }$ will converge into a solution after a transition time that depends on the observation geometry, model quality and data noise. In general, one must expect errors at the decimetre level in pure kinematic positioning (i.e., coordinates $\displaystyle{ (x,y,z) }$ modelled as white-noise) and at the centimetre level in static PPP.