PSEUDORANGE AND CARRIER PHASE OBSERVABLES - GPS Satellite Surveying - page 175

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5.2 DIFFERENCING

For receivers at stations k and m , observing the same satellite p at the nominal
times t k and t m , one can write two pseudorange equations (5.7) and two carrier phase
equations (5.10). Let t k be approximately equal to t m . Some manufacturers time-shift
the observations to make these nominal times equal, say, to the full second. Even if
the nominal times are the same, the respective signals leave the satellite p at slightly
different times. This is so because the distances between receivers and satellite differ.
Because the satellite clocks are highly stable, we assume that the satellite clock errors
are the same for these near-simultaneous transmissions. The same assumption is made
in regard to the internal satellite hardware delays. Under such a condition of near-
simultaneity, the single-difference phase observable (5.1) becomes

f
c ρ

[17

ϕ km, 1 (t)

p

km (t p )

N km, 1 ( 1 ) = + − f(d t k − d t m )

(5.17)

f
c T km (t)

I km, 1 ,ϕ (t) d km, 1 ,ϕ (t) ε km, 1 ,ϕ + + + d km, 1 ,ϕ (t) + +

Lin
—
6.8
——
Lon
PgE

To simplify the notation, we have used the symbol t to denote the time of observations
in those terms where the distinction between t k and t p

is not necessary. Computing

p
km requires two different emission times, one with respect to the observation from
receiver k and one with respect to receiver m . We introduce no additional notation
to label these two different emission times because it is clear from the context which
times must be used for ephemeris interpolation. Following the subscript convention
for differencing, we have

ρ

[17

p p

km (t p ) k (t p )

p

m (t p )

ρ = ρ − ρ (5.18)

N km ( 1 ) N k ( 1 )

N m ( 1 )

= − (5.19)

I km, 1 ,ϕ (t) I k, 1 ,ϕ (t) I m, 1 ,ϕ (t)

= − (5.20)

T km (t) T k (t)

T m (t)

= − (5.21)

d km, 1 ,ϕ (t) = d k, 1 ,ϕ (t) − d m, 1 ,ϕ (t) (5.22)

d km, 1 ,ϕ (t) d k, 1 ,ϕ (t) d m, 1 ,ϕ (t) = − (5.23)

ε km, 1 ,ϕ (t) ε k, 1 ,ϕ (t) ε m, 1 ,ϕ (t) = − (5.24)

W e notice that the satellite clock error and the satellite hardware delay have can-
celed in the single differences. However, the single-difference observations remain
sensitive to both receiver clock errors d t k and d t m , and to signal multipath at the
receiver.