Bernese GPS Software Version 5.0 - Bernese GNSS Software

2. Fundamentals
δi error of satellite clock at signal emission time t − τ
rk(t) position of receiver k at signal reception time t,
ri (t − τ) position of satellite i at signal emission time t − τ,
˙r i (t) velocity of satellite at signal reception time,
̺i k geometric distance between satellite i (at signal emission time t −τ) and receiver
k (at signal reception time t).
The geometric distance ̺ i k
may be written as
(c is the velocity of light) and at the same time as
Using the approximation
̺ i k
= c τ (2.24)
̺ i k = |rk(t) − r i (t − τ)| . (2.25)
r i (t − τ) = r i (t) − ˙r i (t) τ (2.26)
we obtain the following equation which may be solved for τ:
c 2 − ˙r i (t) · ˙r i
(t) τ 2 − 2 ˙r i
(t) rk(t) − r i
(t) τ−
− rk(t) · rk(t) − 2 rk(t) · r i (t) + r i (t) · r i
(t) = 0 . (2.27)
2.3.1 Code Pseudoranges
Using the known codes modulated onto the GPS carriers, receivers are able to measure the
quantity
P i k = c ((t + δk) − (t − τ + δ i )) , (2.28)
which is called pseudorange (because it is biased by satellite and receiver clock errors).
Introducing the geometric distance ̺i k the code pseudorange for frequency F may also be
written as
2.3.2 Phase Pseudoranges
P i Fk = ̺i k + c δk − c δ i . (2.29)
The GPS receiver measures the difference between two phases. The basic form of the observation
equation is (see Eqn. (2.4))
where
ψ i Fk (t) = φFk(t) − φ i F (t − τ) + ni Fk
, (2.30)
ψi Fk (t) is the phase measurement (in cycles) at epoch t and frequency F,
φFk(t) is the phase generated by the receiver oscillator at signal reception time t,
φi F (t − τ) is the phase of the carrier at emission time t − τ, and
is an unknown integer number of cycles (the so-called initial phase ambiguity).
n i Fk
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