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

Chapter 2 Observations of Orbit Determination
S ( t2
) the geocenter vector of ground tracking station at t 2
From Figure 2-1,
1
1
2
1
2
T
1
2
1
2
S = || r ( t ) − S ( t ) || = {[ r ( t ) − S ( t )] [ r ( t ) − S ( t )]}
(2-2)
2
1
S
= [ r(
t ) − S ( t )] [ r(
t ) − S ( t )] = [ r ( t ) r(
t ) − r ( t ) S ( t ) − S ( t ) r(
t ) + S ( t ) S ( t )
= r
T
1
( t ) r ( t ) + S
1
1
2
T
T
( t ) S ( t ) − 2r
( t ) S ( t )
2
1
2
2
T
1
T
1
2
1
T
1
Eq.(2-1) can be written as
L = c t − t ) = || r(
t ) − S ( t ) ||
(2-4)
( 2 1 1 2
By linearizing Eq. (2-4), Eq.(2-1) becomes
0
0 0 ∂L
∂L
L = L + δ L = || r ( t1)
− S ( t2)
|| + δr
( t1)
+ δS
( t2)
(2-5)
∂r
( t1)
∂S
( t2)
where the partial derivatives are
T
T
∂L
[ r(
t1)
− S ( t2
)] [ r ( t1)
− S ( t 2 )]
=
=
(2-6)
∂r
( t1)
|| r(
t1)
− S ( t2
) || L
T
T
∂L
[ r ( t1)
− S ( t2)]
[ r ( t1)
− S ( t2)]
= −
= −
(2-7)
∂S
( t ) || r ( t ) − S ( t ) || L
2
1
2
∂r(
t1)
∂r
( t1)
∂p(
t1)
δ r( t1)
= δr(
t0)
+
δp(
t0)
(2-8)
∂r(
t0)
∂p(
t1)
∂p(
t0)
where
p dynamical model parameters of satellite orbit
Inserting Eq.(2-6) - Eq.(2-8) into Eq.(2-5), the final linear observation equation can be written as
T
T
[ r(
t1)
− S ( t 2 )] ∂r(
t1)
∂r
( t1)
∂p(
t1)
[ r ( t1)
− S ( t 2 )]
δL
= [ δr(
t0
) +
δp(
t0
)] −
δS
( t 2 )
(2-9)
0 r(
t ) p(
t ) p(
t )
0
L ∂
∂ ∂
L
where
t 0
initial epoch
0
1
0
2.1.1.2 Error Budgets
The accuracy of range observations is dependent on the chip rate or frequency. Discussion in detail about
influence on range measurements from various error sources such as tropospheric, ionospheric errors and
multipath effect are given in Chapter 4. In this section the major error sources of range measurement related to
the receiver are discussed and the error budget based on experiences of current satellite navigation systems is
presented.
Range measurements are implemented inside the receiver in two ways: 1) tone range; 2) pseudocode ranging like
GPS C/A code measurements. The major error sources related to receivers are the following two types.
i) Tone Range Error
The accuracy of ranging is set by the highest major carrier frequency and the signal-to-noise ratio:
c 1
σ r =
2 2πf
s
where
K
S / N
(2-10)
σ r distance error
c speed of light
fs major tone frequency
S/N signal-to-noise ratio
K receiver-related parameter
ii) Pseudonoise(PN)-Code Ranging Error with Delay Lock Loop (DLL)
For a Delay Lock Loop, the accuracy of range measurement can be expressed by
4
2
T
2
1
T
2
2
(2-3)