International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
single GPS receiver assisted with precise orbit and clock 
products is called Precise Point Positioning (PPP). The word 
“precise” is used here to distinguish it from the conventional 
SPP method. The method developed at The University of 
Calgary is described in the following [Gao and Shen, 2002]. 
2.1 Observation Combination 
The observation equations for code and carrier phase 
measurements on the Z; frequency (/ — 1, 2) are shown in 
Equations 1 and 2. 
P(Liy=p+cldt —dT) +d, +dp,, +d, ib 
d ult / P(Li) tei PU 
GL = pt c(dt zm dT) =+ d + drop == don! Li 
*AN;Mj(b (09, Li) - 6,9, Li) — O) 
+d ulti dLi) Ts (e(Li)) 
where: 
P(Li) — Measured pseudorange on L; (m) 
®(Li) — Measured carrier phase on L; (m) 
p — True geometric range (m) 
e — Speed of light (m/s) 
dt — Satellite clock error (s) 
dT — Receiver clock error (s) 
d ib — Satellite orbital error (m) 
drop — Tropospheric delay (m) 
ion Li — lonospheric delay on L; (m) 
À ; — L; wavelength (m) 
N; — Integer ambiguity on L; (cycle) 
0, (fo, Li) — Initial phase of receiver oscillator 
Q.(fo, Li) — Initial phase of satellite oscillator 
duis periy — Multipath effect in measured pseudorange on 
L; (m) 
d uult / D. Multipath effect in measured carrier phase on 
L; (m) 
e(.) — Measurement noise (m) 
Note that the initial phase of the receiver and satellite 
oscillators, always less than half of the corresponding 
wavelength (Gabor, 2000), is commonly ignored in 
conventional carrier phase based double differenced systems. If 
it is combined with the integer phase components into a single 
term, Equation 2 can be rewritten as: 
Q(Li) 2 pt c(dt - dT) d, + drop -d 
ion / Li 
+ € (D(Li)) 
orb trop 
TAIN +d 
mult / (Li) 
where MN, is no longer an integer term if the initial phase value 
becomes significant. 
846 
In order to mitigate the ionospheric effect, which is the largest 
error source in GPS positioning after SA was turned off, the 
following ionosphere-free combinations can be formed: 
Py 11 =0.5[P(L1)+d(L1)] 
z p—edT * d, * 0.5A,N, (4) 
trop 
+0.5d +0.5¢ (P(L1)+®(L1)) 
mult / P( LY) 
Pir, 12 =0.5/ P(L2)+®(L2)] 
=p-cdl+d,,. $0.55, N, (5) 
trop 
+0.5d +0.5e (P(L2)+®(L2)) 
mult / P( L2) 
o,-[fp-o(L)- f2-o(L2) T f - £31 
rs NES 
trop 7 12 9 2 2 
zp-cdf td 
d ni / o(Li-L2) +E (S(LI + L2)) 
Note the application of corrections from precise orbit and clock 
products have been applied in the above equations to eliminate 
the satellite orbit and clock error terms. A combination of 
Equations 4, 5 and 6 yields a new observation model for PPP. 
Different from the traditional ionosphere-free model, the new 
model is capable of estimating the ambiguities associated with 
L, and L, frequencies separately. This makes it possible to 
exploit the integer properties of both L, and L; ambiguities, 
which is essential for real-time kinematic positioning. The 
unknowns to be estimated in precise point positioning therefore 
include the position coordinates, receiver clock offset, 
troposphere, and ambiguity terms. 
2.2 Error Mitigation 
In equations (4) to (5), the ionosphere-free code and carrier 
phase combinations are used to mitigate the effect of the 
ionospheric error. The troposphere cannot be mitigated in this 
manner due to its non-dispersive nature. However, it can be 
modelled or estimated along with other parameters. 
To facilitate high precision position determination, a number of 
unconventional error corrections have to be applied. These 
unconventional errors, related to un-differenced observations 
and precise satellite orbit/clock products, include satellite 
antenna phase centre, earth tide and ocean loading etc. The 
satellite antenna phase center correction is necessary for Block 
IIIA satellites because the phase centers and centers of mass 
of these satellites do not coincide. Earth tide and ocean loading 
models are necessary because errors associated with them can 
reach several decimetres. Similarly, a satellite phase windup 
correction is necessary since the error can reach half a cycle. 
Note that these corrections are commonly ignored in double 
differenced positioning approaches because they can be 
cancelled out by the differencing procedure that is implemented 
between satellites and receivers. In the case of un-differenced 
code and carrier phase observations, however, these errors do 
not cancel out and their sizes are relatively large, influencing 
the accuracy of the point positioning solution. 
In 
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