The GNSS integer ambiguities: estimation and validation

of reception diminished with the signal travel time. This gives the following basic code
observation equation:
with:
p s r,j(t) = c [tr(t) − t s (t − τ s r )] + e s r,j(t) (2.1)
ps r,j : code observation at receiver r from satellite s on frequency j [m]
t : time of observation in GPS time [s]
c : velocity of light in vacuum [m/s]
tr : reception time at receiver r [s]
ts : transmission time from satellite s [s]
τ : signal travel time [s]
e : code measurement error
Both the receiver clock time and satellite clock time are not exactly equal to GPS time.
Therefore, the respective clock errors dtr and dt s need to be taken into account:
tr(t) = t + dtr(t) (2.2)
t s (t − τ s r ) = t − τ s r,j + dt s (t − τ s r ) (2.3)
Inserting equations (2.2) and (2.3) in (2.1) yields
p s r,j(t) = cτ s r,j + c [dtr(t) − dt s (t − τ s r )] + e s r,j(t) (2.4)
In order to determine the geometric distance between the satellite and the receiver, the
signal travel time τ s r,j needs to be corrected for instrumental delays at the satellite and
the receiver, and also for atmospheric effects and multipath:
with:
τ s r,j = δτ s r,j + dr,j + d s ,j
δτ s r,j = 1 s
ρr + da
c
s r,j + dm s
r,j
δτ : signal travel time from satellite antenna to receiver antenna [s]
dr : instrumental code delay in receiver [s]
ds : instrumental code delay in satellite [s]
ρ : geometric distance between satellite and receiver [m]
da : atmospheric code error [m]
dm : code multipath error [m]
(2.5)
(2.6)
Insertion of equations (2.5) and (2.6) into (2.4) gives the following code observation
equation:
p s r,j(t) =ρ s r(t, t − τ s r ) + da s r,j(t) + dm s r,j(t)
+ c dtr(t) − dt s (t − τ s r ) + dr,j(t) + d s ,j(t − τ s r ) + e s r,j(t)
(2.7)
8 GNSS observation model and quality control