The GNSS integer ambiguities: estimation and validation

2.2.2 Phase observations
The phase observation is a very precise but ambiguous measure of the geometric distance
between a satellite and the receiver. The phase measurement equals the difference
between the phase of the receiver-generated carrier signal at reception time, and the
phase of the carrier signal generated in the satellite at transmission time. An integer
number of full cycles is unknown since only the fractional phase is measured. This integer
number is the so-called carrier phase ambiguity. The basic carrier phase observation
equation is given by:
with:
ϕ s r,j(t) = ϕr,j(t) − ϕ s ,j(t − τ s r ) + N s r,j + ε s r,j(t) (2.8)
ϕ : carrier phase observation [cycles]
N : integer carrier phase ambiguity
ε : phase measurement error
The phases on the right hand side are equal to:
with:
ϕr,j(t) = fjtr(t) + ϕr,j(t0) = fj(t + dtr(t)) + ϕr,j(t0) (2.9)
ϕ s ,j(t) = fjt s (t − τ s r ) + ϕ s ,j(t0) = fj(t − τ s r,j + dt s (t − τ s r )) + ϕ s ,j(t0) (2.10)
f : nominal carrier frequency [s −1 ]
ϕr(t0) : initial phase in receiver at zero time [cycles]
ϕ s (t0) : initial phase in satellite at zero time [cycles]
The carrier phase observation equation becomes:
ϕ s s
r,j(t) = fj τr,j + dtr(t) − dt s (t − τ s r ) + ϕr,j(t0) − ϕ s ,j(t0) + N s r,j + ε s r,j(t)
(2.11)
This equation must be transformed to obtain units of meters and is therefore multiplied
with the nominal wavelength of the carrier signal:
φj = λjϕj, with λj = c
fj
The carrier signal travel time is expanded similarly as in equations (2.5) and (2.6). This
results in the following observation equation:
φ s r,j(t) =ρ s r(t, t − τ s r ) + δa s r,j(t) + δm s r,j(t)
+ c dtr(t) − dt s (t − τ s r ) + δr,j(t) + δ s ,j(t − τ s r )
+ φr,j(t0) + φ s ,j(t0) + λjN s r,j + ε s r,j(t)
(2.12)
Note that the atmospheric delays, the multipath error, and the instrumental delays are
different for code and phase measurements. Hence the different notation used in (2.12)
(δ instead of d). Also note that the phase measurement error is multiplied with the
wavelength, but the same notation ε is still used.
GNSS observation equations 9