PROCESSING PSEUDORANGES AND CARRIER PHASES - GPS Satellite Surveying - page 254

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x 1 y 1 z 1 P k

−

x 2 y 2 z 2 P k

−

=

B (7.57)

x 3 y 3 z 3 P k

−

x 4 y 4 z 4 P k

−

The solution of (7.52) is

x k
cd t k

MB − 1 (

= Λ τ + α ) (7.58)

We note, however, that Λ is also a function of the unknowns x k and d t k . We substitute

(7.58) into (7.53), giving
B − 1

[25

τ Λ 2 B − 1 α − 1 Λ + B − 1 α = , B − 1

2

, B − 1 , B − 1 τ + τ α 0 (7.59)

Lin
—
-1.
——
Nor
*PgE

This is a quadratic equation of . Substituting its roots into (7.58) gives two solutions

for the station coordinates x k . Converting the solution to geodetic coordinates and
inspecting the respective ellipsoidal heights readily identifies the valid solution.

Λ

7.5 PRECISE POINT POSITIONING

Precise point positioning (PPP) refers to centimeter position accuracy of a single
static receiver using a long observation series, and to subdecimeter accuracy of a
roving receiver using the ionospheric-free pseudorange and carrier phase functions.
The receiver clock error d t k and the zenith tropospheric delay T k (no superscript
here) are estimated for each epoch, in addition to a constant R k . When using PPP,
one must avoid any simplifying assumptions, i.e., all known corrections must be ap-
plied to the observations and the corrections must be consistent. The satellite posi-
tions x k at transmission times are typically computed from the postprocessed precise
ephemeris available from the IGS or its associated processing centers. A crucial el-
ement in achieving centimeter position accuracy with PPP is accurate satellite clock
corrections d

[25

t p , which are part of the precise ephemeris. The ionospheric effects are
eliminated by using the ionospheric-free functions (6.91) and (6.94). The L1 and
L2 integer ambiguities are combined into a rational constant R k , which also serves
to absorb unmodeled receiver and satellite hardware delays that might change with
time, as well as the initial phase windup angles. The PPP model is

¯

P k, IF = ρ

p
k

T k, 0 +

dT k m(ϑ p ) − cd t k + (7.60)

f 1
c ρ

f 1 f 1

c sin ϑ p dT k

ϕ k, IF =

p
k

R k

c T k, 0 +

− f 1 d t k + + (7.61)

where the approximate tropospheric slant total delay (STD)