Precise Orbit Determination of Global Navigation Satellite System of ...

Chapter 7 Orbit Determination Using Carrier Phase Observation
order to use carrier phase observation for GEO satellite orbit determination, other independent methods such as
CCD camera and SLR could be used in data preprocessing. Using these techniques, the precise distance between
satellite and tracking station is obtained and the initial ambiguities of carrier phase observations can be
computed. Because the tracking stations for orbit determination are located in the open area, the locked signals in
the tracking stations will not be easily interrupted. Therefore initial ambiguities will not be easily changed.
7.3 Ambiguity Resolution Approaches
There are many ambiguity resolution approaches for GPS. Some typical methods may also be suitable for
GNSS-2/Galileo orbit determination when using carrier phase measurements.
Following Teunissen (1994), the ambiguity approach can be described as follows.
The carrier phase measurement can be represented as
ϖ ϖ
Φ= r − R + c( dt− dT) + λN + ε (7-15)
where
ϖ
r unknown receiver antenna position vector at signal reception time
ϖ
R given satellite antenna position vector at signal transmission time
c speed of light in vacuum
cdt receiver clock range offset
cdT satellite clock range offset
λ carrier wavelength of the signal
N carrier phase ambiguity
The term ε represents carrier phase measurement noise and biases such as satellite ephemeris errors, tropospheric
and ionospheric delays, and ranging errors caused by multipath effects. In the processing of carrier phase data it
is common to differentiate the carrier phase measurements between satellites and between receivers to eliminate
some common errors. This gives the single and double differenced carrier phase observation equations
∇Φ = ∇ r− R + c∇ dt+ λ∇ N+
∇ε
(7-16)
∆∇Φ = ∆∇ r− R + λ∆∇N+ ∆∇ε
(7-17)
∇ and ∆∇ stand for the single and double difference operator. In the following discussion a two-receiver
situation is assumed. The term ∇dt then denotes the single differenced clock error between the reference
receiver and the remote receiver. Because of double differencing this error vanishes in Eq.(7-17). In both
equations the ambiguity-terms ∇N and ∆∇N are known to be integers.
Linearization of the observation equations with respect to the unknown parameters and a collection of these
linearized equations into a linear system of equations gives:
ϖ ϖ ϖ ϖ
y = Aa+ Bb +ε (7-18)
ϖ ϖ ϖ
with E{} ε = 0,
D{} ε = D{} y = Qy where
ϖ
y vector of observed minus computed differenced carrier phase observations
ϖ
a vector of unknown ambiguities
ϖ
b vector of unknown coordinates of baseline or satellite position, velocity and related
dynamic parameters
A,B design matrices for unknown ambiguities and parameters
ϖ
ε vector of unmodeled errors (in the following ϖ Qy
ε is sufficiently small to be neglected)
variance-covariance matrix of the observations ϖ y
E{.} expectation operator
D{.} dispersion operator
Because ϖ a should be integers, it can be shown, that Eq. (7-18) may be solved in three steps. Firstly, estimating a
double difference float solution (real-valued ambiguities ϖ ϖ
a ∃ ∃b
with ), i.e.
ϖ ϖ ϖ 2
min y− Aa−Bb with
ab ,
Qy
ϖ ϖ
ab R n
, ∈ (7-19)
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