Differential GPS - Understanding GPS: Principles and Applications - page 405

Pseudorange (Code) Double Difference
As in the case of the carrier-phase measurement, the receiver makes a pseudorange
measurement for each epoch for all satellites being actively tracked. The
pseudorange suffers from similar propagation and timing effects as is the case for
the carrier phase. The only basic difference is that where the ionosphere advances
the carrier phase, the pseudorange information experiences a group delay. In con-
sidering the propagation of electromagnetic waves through a plasma, of which the
ionosphere is an example, the propagation velocity ( v g ) of the modulation on a car-
rier is retarded, while the phase velocity ( v p ) of the carrier itself is advanced [28] (see
Section 7.2.4.1). The following relationship holds:

2

vv = c (8.17)

gp

where c is the speed of light. Thus, when the code DD is formed, the effects of the
ionospheric delay are additive. Formulation of the code DD begins with the
pseudorange equation as follows:

() () ()

p p p

Pt tt tt Q = − + +++ ττγ + δ

k p k iono tropo

(8.18)

k k

() () ()

p p p

Pt t t tt Q = − + ++ + ττ γ + δ

m m mp m iono tropo

where:

P is the receiver-measured pseudorange as a function of time in seconds
k, m refer to receiver/receiver antennas phase centers
p is the satellite-signal source
t p or t p is signal-reception time as measured by the receiver clocks
t p is signal-transmission time as determined from the SV clock
Q is noise (timing jitter) due to all sources (e.g., receiver, multipath)
τ is the associated satellite or receiver clock bias

iono represents group delay (seconds) of the modulation due to the ionosphere
tropo represents the delay (seconds) of the modulation due to the troposphere

Note the absence of the integer carrier-cycle ambiguity N —the pseudorange
measurement is unambiguous. In other words, code DD observables formed from
the pseudoranges measured by the receivers contain no carrier-cycle ambiguities.
Unfortunately, pseudorange cannot be measured as precisely as the carrier phase, so
it is noisier. Also of note is the change in the sign for the ionospheric effects from
that in (8.9) due to the group delay. The unambiguous nature of the code DD will
serve as the basis for code/carrier smoothing to be described in the next section.
Pseudorange SDs are now formed:

p p p

SD =+ + t Q τ

(8.19)

km km km km

q q p

SD =+ + t Q τ

km km km km