Algorithm for carrier-adjusted DGPS positioning and some

Journal of Geodesy (1997) 71: 411±422
Algorithm for carrier-adjusted DGPS positioning
and some numerical results
X. X. Jin
Delft Geodetic Computing Centre, Faculty of Geodetic Engineering, Delft University of Technology,
Thijsseweg 11, 2629 JA Delft, The Netherlands fax: +31 15 2783711; email: [email protected]
Received: 9 April 1996 / Accepted: 6 February 1997
Abstract. This paper derives a DGPS positioning algorithm, referred to as the algorithm for carrier-adjusted
DGPS positioning. This algorithm can be applied by a
DGPS user when code and carrier observations are
available and when the dynamic behaviours of both
mobile positions and receiver-clock biases can and
cannot be modelled. Since the algorithm directly uses
code and carrier observations, the stochastic model of
observations has a simple structure and can be easily
speci®ed. When the dynamic behaviour of mobile
positions can be modelled, the algorithm can provide
recursive solutions of the positions, on the other hand,
when the behaviour cannot be modelled, it can provide
their instantaneous solutions. Furthermore, the algorithm can integrate with a real-time quality-control
procedure so that the quality of the position estimates
can be guaranteed with a certain probability. Since in
the use of the algorithm there always exist redundant
observations unless the position parameters are inestimable, the quality control can even be performed when
only four satellites are tracked. Using the algorithm and
real GPS data collected at a 100-km baseline, this
contribution investigates how DGPS positioning accuracies vary with the type of observables used at reference
and mobile stations, and how important it is to choose
an elevation-dependent standard deviation for code
observations in DGPS data reduction. It was found
that using carrier observations along with code observations is more important at the reference station than
at the mobile station. Choosing an elevation-dependent
standard deviation for code observations can result in
better positioning accuracy than choosing a constant
standard deviation for code observations. For the 100km baseline, half-metre single-epoch positioning accuracy was achieved when dual-frequency data was used at
both reference and mobile stations. The positioning
accuracy became better than 0:75 m when the types of
observable used at the mobile station were replaced by
L1 code and carrier.
Key words. DGPS positioning algorithm dynamic
model, mobile station
1 Introduction
The performance of DGPS positioning is a function of
three elements: (1) generation of dierential corrections
at a known DGPS reference station, (2) transmission of
the corrections from the reference station to mobile
stations, and (3) computation of the mobile position. In
Jin (1995a) we discussed how to generate dierential
corrections. Here we focus on how to compute mobile
positions. As we know, when only code observations are
available, the positioning algorithm is quite simple,
particularly in the case of single-epoch positioning. But
when both code and carrier observations are available,
how to apply these two types of observables has been an
interesting topic for quite some time. Much research has
been conducted for the integration of code and carrier
observations. For example, Hatch (1982), Ashjaee
(1990) and Goad (1990) investigated how to generate
the so-called carrier-smoothed code observations by
using code and carrier observations. Kleusberg (1986),
Cannon (1987), Schwarz et al. (1989), Hwang and
Brown (1990) and Teunissen (1991) investigated how to
use code and carrier observations in GPS kinematic
relative positioning.
This contribution will ®rst derive in Sect. 2 an alternative dierential positioning algorithm based on code
and carrier observations, which is referred to as the algorithm for carrier-adjusted DGPS positioning, analogous to Teunissen (1991). Then, Sect. 3 will show the
eect of the types of observable used at reference and
mobile stations on the positioning accuracy and investigate the dierences in accuracy by choosing an elevation-dependent or constant standard deviation for
code observations. Finally, Sect. 4 will give some concluding remarks.
412
2 Carrier-adjusted DGPS positioning models
2.1 Positioning models
Assume that at a mobile station L1 code and carrier
observations are available. In equations they can be
expressed in metric units as
P i tk qi tk c dT tk ÿ dti tk T i tk
1
I i tk Ei tk ei tk
k1 /i tk qi tk c dT tk ÿ dti tk T i tk
ÿ I i tk Ei tk ÿ k1 N i gi tk
2
where
tk :
P:
i:
q:
GPS time at epoch k
L1 code observation (m)
satellite number
distance between the mobile station and the satellite
position computed from ephemeris data (m)
speed of light (m/s)
receiver clock bias (s)
satellite clock bias (including SA clock error) (s)
eect of ephemeris error (including SA orbit error)
(m)
ionospheric delay (m)
tropospheric delay (m)
code observation noise (m)
L1 wave length (m)
L1 carrier observation (cycles)
L1 carrier ambiguity (cycles), which is a real value.
L1 carrier observation noise (m)
c:
dT :
dt:
E:
I:
T:
e:
k1 :
/:
N:
g:
By using the broadcast navigation data,0 the approximate value for dti tk , denoted by dti tk , can be
computed. De®ne
i
i0
i
dt tk dt tk ÿ dt tk
rim tk c dT tk ÿ dti tk Ei tk I i tk T i tk 3
Then it follows from Eqs. (1)±(3) that
0
i0
k1 / tk ÿ c dt tk
i
q tk rim tk
ÿ k1 N i gi tk
i
ÿ 2I tk
4
By means of a radio link, the estimates of a dierential
correction and its rate of change which are generated at
a reference station can be transmitted to their users.
These corrections read as
r^ir t rir t li1 t
d r^ir t drir t

li2 t
dt
dt
5
6
2
Usually only the rate of change of dierential corrections is used to account for the unavoidable latency
when the corrections are applied at a mobile station
(RTCM SC-104 1994). For reasons of simplicity, let us
assume that the second-order derivative of the dierential correction, rir t, is a zero-mean white noise with
spectral density qr m2 =s3 . Then we arrive at
^i
^ i tk r
^ i tDC d rr tDC tk ÿ tDC
r
r
r
dt
i
drr tDC
i
i
i
l2 tDC
rr tDC l1 tDC
dt
tk ÿ tDC
drir tDC
tk ÿ tDC li1 tDC
dt
li2 tDC tk ÿ tDC
rir tDC
rir tk ÿ wi tk li1 tDC li2 tDC tk ÿ tDC 7
with [see (A.11) of Jin (1996b)]
Z
i
w tk
q
d 2 rir t

3
r
t

t
ÿ
t
dt
N
0;

ÿ
t

k
k
DC
3
dt2
tDC
tk
8
r2r^ i tk r2wi tk r2li tDC
r
1
tk ÿ tDC 2 r2li tDC tk ÿ tDC rli2 li1 tDC
2
q
r tk ÿ tDC 3 r2li tDC
1
3
2 2
tk ÿ tDC rli tDC tk ÿ tDC rli2 li1 tDC
2
P i tk ÿ c dti tk qi tk rim tk ei tk
i
with
(
)
li1 t
E
0
li2 t
2
3
(
)
r2li t SYM:
li1 t li1 t
5
4 1
E
li2 t li2 t
rli2 li1 t r2li t
9
In Appendix A it is shown that the dynamic noise wi tk
is not correlated with the estimate errors li1 tDC and
li2 tDC .
If we assume that the mobile station is close to the
reference station where the dierential corrections are
generated, it follows that
rim tk rir tk c dT tk
10
where dT is the dierence between receiver-clock biases
in the mobile and reference stations.
Substituting Eq. (10) into Eq. (4) and combining it
with Eq. (7) gives
413
2
3
2
0 0
7
6
6 P i tk ÿ c dti0 tk 7 6
5 41 1
4
i
i0
1 1
k1 / tk ÿ c dt tk
3
2 i
q tk
3
7 2 i
6
l tk
6 c dT tk 7
7
6
7
i
7 6 i
6
6 rr tk 7 4 e tk 5
7
6 i
gi tk
4 I tk 5
k1 N i
^ i t k
r
r
1
1
0
0
1
ÿ2
3
0
7
0 5
ÿ1
1
i
I tk I i tk k1 N i
2
11
li tk ÿwi tk li1 tDC tk ÿ tDC li2 tDC
r2li tk r2r^ i tk
12
r
Pre-multiplying Eq. (11) by the full-rank transformation
2
3
1 0 0
matrix 4 ÿ1 1 0 5 gives
ÿ1 0 1
3
2
^ i t k
r
r
7
6
6 P i tk ÿ c dti0 tk ÿ r
^ i tk 7
r
5
4
i
i
i0
^
k1 / tk ÿ c dt tk ÿ rr tk
3
2 i
q tk
2
36
7
0 0 1 0
0 6 c dT tk 7
7
6
6
7
i
7
41 1 0 0
0 56
6 rr tk 7
7
6 i
1 1 0 ÿ2 ÿ1 4 I tk 5
k1 N i
2
3
li t k
6 i
7
4 e tk ÿ li tk 5
13
gi tk ÿ li tk
^ i tk is a free variate
As can be seen from this equation, r
r
(Teunissen 1994). Therefore Eq. (13) can be re-written as
# 0
^ i tk
1
P i tk ÿ c dti tk ÿ r
r

i
i
i0
^
1
k1 / tk ÿ c dt tk ÿ rr tk
2 i
3
q tk
#
6
7 " i
6 c dT tk 7
e tk ÿ li tk
7
6
6 I i tk 7 gi t ÿ li t
k
k
4
5
i
k1 N
Then Eq. (14) becomes
"
#
0
^ i tk
P i tk ÿ c dti tk ÿ r
r
0
^ i t k
k1 /i tk ÿ c dti tk ÿ r
r
3
2 i
q

k
i
1 1 0 6
e tk ÿ li tk
7

4 c dT tk 5 i
1 1 ÿ2
g tk ÿ li tk
i
I tk
16
In order to estimate the mobile position, linearize the
satellite-receiver range qi as follows
with
"
15
1
0
0
1
ÿ2
ÿ1
0
qi tk qi tk
j1
Dxj tk
17
0
qi : approximate value of qi computed from the ephemeris data and the approximate position of the
mobile station
Dxj : xj ÿ x0j
xj : coordinate of the mobile station in Cartesian or
geocentric coordinate system j 1; 2; 3
x0j : approximate value of xj
Inserting Eq. (17) into Eq. (16) yields
3
2 0
0
0
oqi tk oqi tk oqi tk
1 0 7
6 ox1
ox2
ox3
7
y i tk 6
0
0
5
4 i0
oq tk oqi tk oqi tk
1 ÿ2
ox1
ox2
ox3
3
2
D x1 tk
7
6
6 D x2 tk 7 7
6
ei tk ÿ li tk
7
6
6 D x3 tk 7 i
7
6
g tk ÿ li tk
6 c dT tk 7
5
4
i
I t k
i
y tk
Obviously, in the preceding system of equations the
columns related to qi tk and c dT tk are dependent, as
are those related to I i tk and k1 N i . It will be clear later,
however, that the dependency between the columns related to the ®rst two parameters does not aect the determination of c dT tk and the position parameters
contained in qi tk when four or more satellites are
tracked. To overcome the dependency between the columns related to the last two parameters, which makes it
impossible to estimate I i tk and k1 N i separately, let us
de®ne
oxj
where
where
14
0
3
X
oqi tk
"
0
^ i tk ÿ qi0 tk
P i tk ÿ c dti tk ÿ r
r
0
^ i tk ÿ qi0 tk
k1 /i tk ÿ c dti tk ÿ r
r
18
#
19
In most situations, the dynamic behaviour of mobile
positions or receiver-clock biases is probably hard to
model. But there may be cases when the behaviour can
be well modelled. For example, when the user is static or
moves at sea, or a clock of good quality is available. It is
preferred to build up such function and stochastic
models that can be used in practice when the dynamic
behaviour of unknown parameters can or cannot be
modelled, so that in the former and latter cases one can
obtain recursive and instantaneous solutions, respectively. It will be shown later that a random dynamic
behaviour can be treated as a special random process.
Therefore, in the following derivation of the function
and stochastic models, mobile positions and receiverclock biases are regarded as random processes.
414
Let us assume that the accelerations of
i
Dxj ; j 1; 2; 3; c dT and I are white noise with spectral densities qx ; qc and qI, respectively, and that no cycle
slips have occurred up to time tk . Then after noticing
i
that the rate of change of I t equals that of I i t, we
have the following satellite i related dynamic model
3
3
2
2
Dx1 tk
Dx1 tkÿ1
7
7
6
6
6 D_x1 tk 7
6 D_x1 tkÿ1 7
7
7
6
6
6 Dx t 7
6 Dx t 7
2 k
2 kÿ1
7
7
6
6
7
7
6
6
6 D_x2 tk 7
6 D_x2 tkÿ1 7
7
7
6
6
6 Dx t 7 2
6 Dx t 7
3
3 k
3 kÿ1
7
7
6
6
I3
7
7
6
6
6 D_x3 tk 7 6
6
D_x3 tkÿ1 7
7
7 4
7
6
6
F
1
5
k6
7
6ÿ ÿ ÿ ÿ ÿ7
7
6
6ÿ ÿ ÿ ÿ ÿ7
1
7
7
6
6
6 c dT tk 7
6 c dT tkÿ1 7
7
7
6
6
6 c dT_ t 7
6 c dT_ t 7
7
6
6
k
kÿ1 7
7
7
6
6
6ÿ ÿ ÿ ÿ ÿ7
6ÿ ÿ ÿ ÿ ÿ7
7
7
6
6
i
7
7
6
6 i
4 I tk 5
4 I tkÿ1 5
I_i tk
I_i tkÿ1
2
3
d~tk
6
7
4 d tk 5
20
i
d tk
where is the Kronecker product and missing elements
in a matrix are zeroes here and afterwards, and
2
1
I3 4 0
0
0
1
0
3
0
05 ;
1
Fk
1 Dtk
0 1
21
with
82
39
< d~tk =
E 4 d tk 5 0
: i
;
d tk
22
82
32
3 9 2
qx I3
< d~tk
d~tk =
E 4 d tk 54 d tk 5
4
: i
;
d tk
d i tk
3
qc
qI
5 Sk
23
where
"
Sk
1
3
3 Dtk
1
2
2 Dtk
SYM:
Dtk
#
24
Assuming that code and carrier observation noises are
zero-mean time-independent random errors and that
there is no correlation between them, then the satellite i
related measurement model reads
2
3
60 0
7
1
0
0
6 oqi tk
7
oqi tk
oqi tk
6
7
0
0
0
1
0
6
B
C
7
0
0
ox
ox
ox
1
2
3
B
C
7
y i tk 6
0
0
6@ i0
A
7
i
i
ÿ2
0
oq tk
oq tk
6 oq tk
7
0
0
0 1 0
6
7
ox1
ox2
ox3
4|{z}
5
i
A tk
2
3
Dx1 tk
6 D_x t 7
6
7
1 k
6
7
6 Dx2 tk 7
6
7
6 D_x t 7
6
7
2 k
6
7
6 Dx3 tk 7 "
#
6
7
ei tk ÿ li tk
6
7
25
6 D_x3 tk 7 i
6
7
g tk ÿ li tk
6 c dT tk 7 |{z}
6
7
6
7
ei tk
6 c dT_ tk 7
6
7
6ÿ ÿ ÿ ÿ ÿ7
6
7
6
7
i
4 I tk 5
I_i tk
This discussion is based on the case of one satellite. If
m satellites are tracked, then the full measurement model
reads
3
2
0 0
3 6 A 1 t k j
2 1
7
y tk
ÿ2 0
7
6
7
6
6 . 7 6 .
7
..
6 . 76 .
7
4 . 5 6 .
j
.
7
7
6
0 0 5
4 m
y m tk
|{z}
A tk j
ÿ2 0
yk
|{z}
Ak
3
2
Dx1 tk
7
6
D_
x1 tk 7
6
7
6
6 Dx2 tk 7
7
6
7
6
6 D_x2 tk 7
7
6
6 Dx3 tk 7
7
6
7
6
6 D_x3 tk 7 2 1
3
7
6
e t k
6 c dT tk 7
7 6 . 7
6
7 6
7
6
26
6 c dT_ tk 7 4 .. 5
7
6
6ÿ ÿ ÿ ÿ ÿ7
em tk
7 |{z}
6
7
6
i
6 I t k 7
ek
7
6
6 I_i t 7
k
7
6
7
6
7
..
6
7
6
.
7
6
m
7
6
4 I tk 5
I_m tk
|{z}
xk
with
Efek g 0
27
415
2
6
6
6
6
E ek ek 6
6
6
6
4
SYM:
r2r^ 1 tk
r2g r2r^ 1 tk
rr
r
3
!
r2e r2^ 1 tk
r
..
.
r2e

r2r^ m tk
r
and the full dynamic model reads
3
2
3
2
~tk
I3
d
7
6
7
6
6 d tk 7
1
7
6
7
6
7
6
1
d 1 t k 7
xk 6
7 Fk xkÿ1 6
7
6
7
6
..
6 .. 7
5
4
.
4 . 5
1
d m tk
|{z}
|{z}
Uk;kÿ1
29
dk
with
E fd k g 0
30
2
E dk dk
6
6
6
6
6
4
r2r^ m tk
r
qx I3
3
qc
qI
..
.
7
7
7
7 Sk
7
5
31
qI
For the sake of simplicity, the measurement noises
here are assumed not only to be uncorrelated among
observables and epochs but also to have a constant
standard deviation. It was shown in Euler and Goad
(1991), Jin (1995b) and Jin and Jong (1996), however,
that the standard deviation for code observations should
be modelled better as a function of satellite elevations.
On the basis of the preceding dynamic and measurement models, recursive solutions of the state vector
can be obtained by using the standard Kalman-®lter
algorithm. The initial values of the ®lter state vector and
their covariance matrix can be determined by solving the
®rst two epochs simultaneously by least squares.
2.2 Discussion
In the previous subsection, we derived the carrieradjusted DGPS positioning models based on the
assumptions that L1 code and carrier observations are
available and that the dynamic behaviours of both
receiver-clock biases and mobile positions can be
modelled. Although in many cases the assumption on
the dynamic behaviours may not be realistic, it may not
be a problem in practice. As we know, the spectral
density is a measure of the uncertainty of dynamic
noises. A large spectral density represents great dynamic
noises, i.e., a rather random dynamic behaviour,
whereas a small spectral density represents less dynamic
noises, i.e., a quite smooth dynamic behaviour. There-
SYM:
r2g
r2r^ m tk
7
7
7
7
7
7
!7
7
5
28
r
fore, when the dynamic behaviour of mobile positions or
receiver-clock biases cannot be well modelled, it can in
practice be treated as a random process with a very large
spectral density; whereas, when mobile positions are
static, their dynamic behaviour can be considered as a
random process with a spectral density of zero (Axelrad
and Brown 1996). In other words, the previously derived
positioning models can be applied in the case when the
dynamic behaviours of receiver-clock biases and mobile
positions can be well modelled or not. In the former
case, the models can suciently use the dynamic
information in the estimation of mobile positions,
whereas in the latter case, the models can provide
single-epoch solutions of mobile positions.
In the derivation of the positioning models, we assumed that the second-order time derivatives of both
mobile positions and receiver-clock biases are white
noise. In fact, the assumption can also be otherwise for
example, that the ®rst-order time derivatives of mobile
positions and the third-order time derivatives of receiver-clock biases are white noise. Naturally, in this
case the measurement and dynamic models need to be
appropriately adapted.
It is probably usual that DGPS mobile stations are
equipped with single-frequency receivers. But there may
be a time when dual-frequency data is available in a
mobile station. It is shown in Appendix B that in this
case the positioning models can be easily adapted.
Since the positioning models directly use code and
carrier observations as inputs, the stochastic observation
model has a simple structure and can be easily speci®ed.
In addition, as long as four or more satellites are
tracked, the positioning models always (except in the
initial two epochs) have redundancy, even though no
dynamic models are introduced for mobile positions and
receiver-clock biases. Let us look at a situation where
only four satellites are tracked and no dynamic models
are introduced for the position and clock bias states.
The states to be estimated are the three position components Dxj ; j 1; . . . ; 3, the clock bias c dT , the four
i
pairs of satellite related parameters I (combination of
L1 carrier ambiguity and ionospheric delay) and I_i (rate
of change of ionospheric delays), i 1; . . . ; 4. Therefore,
the total number of states is 12. The measurements
available in this case consist of two types: the four
pairs of L1 code and carrier observations P i and
/i ; i 1; . . . ; 4, and the four pairs of predicted obi
i
servations Î tk jtkÿ1 and Î_ tk jtkÿ1 ; i 1; . . . ; 4. Therefore, the total number of measurements is 16. It follows
416
then the number of redundancy is 4. Note that the
predicted observations result from the modelling of the
dynamic behaviour of ionospheric delays and the constant property of the carrier ambiguity. Concerning the
modelling of ionospheric delays, Jin (1996a) showed the
modelling accuracy could be within a few millimetres.
As can be easily seen, the number of redundancy is actually identical to the number of tracked satellites multiplied by the number of carrier observables in use, no
matter whether any dynamic models are introduced for
mobile positions and receiver clock biases or not.
Another important property of the carrier-adjusted
positioning models is that they can be easily integrated
with a real-time quality-control procedure (Teunissen
1990a,b), so that the quality of position estimates can be
assured with a certain probability. Since the positioning
models always have redundancy unless the position
parameters are inestimable, the quality-control procedure can even be performed when only four satellites are
tracked.
Note that the approach used in the algorithm to solve
the problem that the ionospheric delay I i tk and the L1
carrier ambiguity k1 N i are not separately estimable is to
combine them into a new state (see Eq. 15). In fact the
problem can also be solved by estimating the change of
I i tk with respect to I i t0 and a combination of k1 N i
and I i t0 . From the theoretical point of view, the latter
should be used since the solutions based on the former
are only sub-optimal (Hwang and Brown 1990). But the
former has the advantage that it reduces the order of the
®lter (one state per satellite) and consequently saves
computational time and memory. Since most DGPS
users may need real-time solutions by the use of only
personal computers, the compromise over the optimality
was assumed necessary in the derivation of the positioning models.
3 DGPS positioning experiments
3.1 Data description and choices of a priori parameters
The GPS data used here was collected at two known
stations: DE18 in Delft and KO25 in Kootwijk, The
Netherlands, on 7 June 1995. In the DE18 station a
Trimble 4000-SSE receiver was used, whereas in the
KO25 station a TurboRogue SNR-8000 receiver was
used. The data was collected from 08:00 to 09:00 UTC
and sampling interval was 1 s. During the data
collection, the cut-o angle was 10 ; there were seven
to eight satellites in view. Figure 1 pictures the
information on the number of visible satellites and
DOP values versus time. It was found that during data
collection, both AS and SA were active. Since under AS
condition P1 and P2 codes are encrypted into Y1 and Y2
codes, the types of observable available in the RINEX
data were C1 (i.e. C/A) code, L1 and L2 carriers, and C2
(i.e. C1+(Y2-Y1)) code. The distance between the two
stations is about 100 km.
The DE18 station was chosen as a reference station
and the KO25 station as a mobile station. The Trimble
Fig. 1 Number of visible satellites and PDOP, VDOP and HDOP
values in the KO25 station
4000-SSE data collected at the reference station was
processed using the algorithm for carrier-adjusted differential corrections (Jin 1995a). The TurboRogue
SNR-8000 data collected at the mobile station was
processed using the algorithm derived in the previous
section. In order to see the positioning accuracy in the
worst case that a DGPS user is in random dynamic
conditions, we computed the single-epoch positioning
accuracy by assuming that the velocities of both mobile
positions and receiver-clock biases are random noises
with a very large spectral density. Since in practice it is
unavoidable to take some time to compute dierential
corrections and to transmit them from a reference station to a user, the latency of dierential corrections was
chosen to be 5 s. In addition, to investigate the importance of choosing an elevation-dependent standard
deviation for code observations, the processing was
carried out twice with the two sets of a priori parameters
given in Table 1.
Note that the only dierence between these two sets
of a priori parameters is that at both reference and
mobile stations, set 1 accounts for the information of
satellite elevation in the standard deviation of code observations, whereas set 2 does not. In addition, it is
worth mentioning that Ashkenazi et al. (1994) reported
similar work to account for the elevation information in
DGPS positioning, but no details can be found in the
reference. Furthermore, it should be pointed out that the
correlation between C1 and C2 code observations was
neglected in the data processing. This neglect may result
in a precision too optimal for state estimates but may
not have signi®cant eect on the estimates themselves
(Roberts and Cross 1993).
3.2 Numerical results and discussion
Based on the dierent type of observables used at the
reference and mobile stations, Table 2 shows the RMS
417
Table 1. Two sets of a priori
parameters used in DGPS positioning experiments
Set 1
reference
mobile
ÿEl=18
re 0:12 1:3 e
m
rg 0:003 m
rg~ 0:003 m
r~e 0:15 1:3 eÿEl=18 m
qI 10ÿ8 m2 =s3
qs_ 10ÿ2 m2 =s5
qb 10ÿ3 m2 =s3
ÿEl=14
re 0:3 1:0 e
m
rg 0:003 m
rg~ 0:003 m
r~e 0:5 1:0 eÿEl=14 m
qI 10ÿ8 m2 =s3
qc_ 104 m2 =s
qx_ 104 m2 =s
Set 2
reference
mobile
re 0:3 m
rg 0:003 m
r~g 0:003 m
r~e 0:5 m
qI 10ÿ8 m2 =s3
qs_ 10ÿ2 m2 =s5
qb 10ÿ3 m2 =s3
re 0:5 m
rg 0:003 m
rg~ 0:003 m
r~e 0:8 m
qI 10ÿ8 m2 =s3
qc_ 104 m2 =s
qx_ 104 m2 =s
El: satellite elevation (degrees)
Table 2. RMS errors of DGPS
positioning with an elevationdependent standard deviation
for code observations
Observables used at reference station
Observables used at mobile station
C1
C1
C1, L1
C1, L1, L2
C1, L1, L2, C2
north
east
height
total
north
east
height
total
north
east
height
total
north
east
height
total
errors of DGPS positioning experiments in north, east,
and height components by using set 1 of a priori
parameters. The formula for computing the RMS error
P
12
e
^ i ÿ wi 2 , where w
^ i and wi are estimated
w
is n1e ni1
and true quantities, respectively, and ne is the number of
epochs. As can be seen from the table, the improvement
of the positioning accuracy using carrier observations
along with L1 code observations is evident. For
example, when C1 code and L1 plus L2 carriers were
used at the reference station, the total RMS error was
0.912 m in the case of only C1 code used at the mobile
station, and it decreased to 0.738 m in the case of C1
code and L1 carrier used at the mobile station.
Moreover, the use of carrier observations is more
important at the reference station than at the mobile
station. For instance, the total RMS error was 1.506 m
when C1 code was used at the reference station and C1
code and L1 carrier at the mobile station, whereas it was
improved to 1.124 m when C1 code and L1 carrier were
used at the reference station and C1 code at the mobile
station. The reason for these phenomena is that when
only C1 code observations were used at the reference
station, code noises had great in¯uence on the estimates
of both dierential corrections and correction rates.
When such noisy estimates were applied at the mobile
station, the noises were propagated to the corrected code
and carrier observations. This then made the positioning
errors include code observation noises at reference and
mobile stations, whereas when carrier observations were
0.833
0.489
1.279
1.603
0.778
0.454
1.207
1.506
0.754
0.438
1.153
1.446
0.754
0.438
1.153
1.446
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
C1, L1
C1, L1, L2
C1, L1, L2, C2
0.588
0.329
0.900
1.124
0.506
0.262
0.793
0.976
0.481
0.228
0.687
0.869
0.481
0.228
0.687
0.869
0.441
0.259
0.755
0.912
0.335
0.180
0.632
0.738
0.237
0.107
0.425
0.498
0.237
0.107
0.425
0.498
0.443
0.258
0.753
0.911
0.337
0.179
0.629
0.736
0.238
0.106
0.422
0.496
0.238
0.106
0.422
0.496
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
used at the reference station, carrier observations played
a much greater role than code observations in the
generation of dierential corrections, since the former
are much more precise than the latter. In this case, the
estimates of dierential corrections and particular
correction rates were much less noisy. When these
estimates were applied at the mobile station, the
corrected observations were actually not aected by
code noises at the reference station. This then made the
positioning errors less or much less than those in the
previous case. In short, in the case of carrier observations used at the mobile station, the positioning errors
mainly resulted from code noises at both reference and
mobile stations, whereas, in the case of carrier
observations used at the reference station, the positioning errors mainly resulted from code noises at only the
mobile station. Therefore, using carrier observations is
more important in the generation of dierential
corrections than in the estimation of mobile positions.
Additionally, one can also see from Table 2 that the
RMS error based on the use of C1, L1, L2 and C2 at the
mobile or reference station is either identical or very
close to its corresponding value based on the use of C1,
L1 and L2 at the same station. This can be explained by
the theory of least-squares adjustment. Since carrier
observations are much more precise than code observations, the weight of the former is much greater than
that of the latter when they are processed together.
When very precise L1 and L2 carrier observations are
used, the estimated parameters (e.g. dierential correc-
418
tion or mobile position) obtain much greater contributions from the carrier observations than from code observations (Jin 1995a, 1996b). Therefore, when L1 and
L2 carrier observations have already been used along
with L1 code observations, adding L2 code observations
at the reference or mobile station cannot signi®cantly
improve the positioning accuracy. Note that this does
not mean that the uselessness of L2 code observations in
DGPS positioning, since the use of the L2 code observable along with other type of observables can improve the positioning reliability.
Furthermore, Table 2 shows that the use of the different types of receiver at reference and mobile stations
may not be a problem in DGPS positioning applications, since the positioning accuracies in all cases concerned were better than 2 m, which were based on the
use of a Trimble 4000-SSE receiver at the reference
station and a TurboRogue-8000 SNR receiver at the
mobile station.
Table 3 gives the RMS errors of DGPS positioning
experiments by using set 2 of a priori parameters. After
comparisons of this table with Table 2, it can be seen
that when single-frequency data (i.e. L1 code alone or
along with carrier) was used at the reference or mobile
station, using an elevation-dependent standard deviation for code observations can clearly improve the positioning accuracy, particularly in the height component.
Table 4 shows the dierence between the total RMS
errors by using sets 1 and 2 of a priori parameters. As
can be seen from the table, when only C1 code was used
at the reference station, it was particularly important to
choose an elevation-dependent standard deviation for
code observations, no matter what type of observables
were used at the mobile station. When using more types
of carrier observable at the reference station, the improvement of the accuracy by choosing an elevationdependent standard deviation for code observations
became smaller. When the type of observables used at
both reference and mobile stations included L1 and L2
carriers, using an elevation-dependent standard deviation for code observations could not improve the positioning accuracy any more. Again, this is because code
observations could not play an important role in the
estimation of the position parameters and therefore
precisely choosing the standard deviation for code observations became less important.
Note that the improvement of the positioning accuracy using an elevation-dependent standard deviation
for code observations may vary with many factors, for
example, the number of satellites in low elevation, the
quality of receivers, the observation environment, and
the type of carrier observables in use. In our DGPS
positioning experiments the greatest improvement found
was about 0:3 m. In some cases it may be greater.
4 Concluding remarks
We have derived the algorithm for carrier-adjusted
DGPS positioning. This algorithm can be applied by a
DGPS user when code as well as carrier observations are
available. Since the algorithm directly uses code and
carrier observations in the measurement model of a
Kalman ®lter, the stochastic model of observations has a
Table 3. RMS errors of DGPS
positioning with a constant
standard deviation for code observations
Observables used at reference station
Observables used at mobile station
C1
C1, L1
C1, L1, L2
C1, L1, L2, C2
0.931
0.574
1.536
1.886
0.880
0.537
1.465
1.791
0.853
0.515
1.391
1.711
0.853
0.515
1.391
1.711
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
C1, L1, L2
C1, L1, L2, C2
0.611
0.358
1.038
1.257
0.520
0.284
0.910
1.077
0.490
0.239
0.773
0.946
0.490
0.237
0.773
0.945
0.445
0.271
0.836
0.985
0.332
0.168
0.695
0.788
0.220
0.081
0.429
0.489
0.220
0.081
0.428
0.488
0.446
0.271
0.835
0.985
0.333
0.168
0.692
0.786
0.220
0.081
0.427
0.487
0.220
0.081
0.427
0.487
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
Observables used at reference station
Observables
used at mobile
station
Table 4. Improvement of DGPS
positioning accuracy using an
elevation-dependent standard
deviation for code observations
C1
north
east
height
total
north
east
height
total
north
east
height
total
north
east
height
total
C1, L1
C1
C1
C1, L1
C1, L1, L2
C1, L1, L2, C2
0.283
0.285
0.265
0.265
m
m
m
m
C1, L1
C1, L1, L2
C1, L1, L2, C2
0.133
0.101
0.077
0.076
0.073
0.050
).009
).009
0.074
0.050
).009
).009
m
m
m
m
m
m
m
m
m
m
m
m
419
simple structure and can be easily speci®ed. The
algorithm always has, except in the initialization of the
®lter, redundancy as long as four or more satellites are
tracked. It can, therefore, integrate well with a real-time
quality-control procedure so as to guarantee the quality
of position estimates with a certain probability. The
algorithm can take into account the dynamic behaviour
of mobile positions and that of receiver clock biases.
When the dynamic behaviour of mobile positions can be
modelled, the algorithm can provide recursive solutions
for the positions. When the behaviour cannot be
modelled, it can provide instantaneous solutions.
By using the algorithm and real GPS data collected at
a 100-km baseline, some DGPS positioning experiments
have been conducted. In general, using carrier observations along with code observations at the reference
or mobile station can indeed improve the DGPS positioning accuracy. But using carrier observations is more
important at the reference station than at the mobile
station. The more types of carrier observable used in
generating dierential corrections, the better the improvement of the positioning accuracy.
It has been shown that using an elevation-dependent
standard deviation for code observations can improve
the DGPS positioning accuracy. The improvement is
related to the number of carrier observables used at
reference and mobile stations. When only code observations were available at the reference station, the
improvement was the most signi®cant. When C1 code
and L1 as well as L2 carrier observations were available
in both reference and mobile stations, choosing an elevation-dependent standard deviation for code observations became less important.
For a 100-km baseline with a PDOP value of about 2,
half-metre single-epoch positioning accuracy was
achieved when C1 code and L1 as well as L2 carrier
observations were used at both reference and mobile
stations. When the observables used at the mobile station were replaced by C1 code and L1 carrier, the positioning accuracy was better than 0:75 m.
Acknowledgements. The author appreciates the help of Prof. P. J.
G. Teunissen for his critical reading of the manuscript and giving
many valuable remarks. Thanks also go to Mr. C de Jong for his
improvements to the manuscript.
Appendix A
correlation between wi tk and li1 tDC can only result
from the correlation between the dynamic noises in
them. As we know, a dierential correction is the sum of
an ionosphere-free dierential correction and an ionospheric delay. Thus, a dynamic noise of a dierential
correction consists of a dynamic noise of an ionospherefree dierential correction and that of an ionospheric
delay. In the generation of dierential corrections (Jin
1995a), we assume that the dynamic noises of both
ionosphere-free dierential corrections and ionospheric
delays are white noise, i.e. the noises are uncorrelated, if
they are not related to the same time-interval. Since the
dynamic noises contained in wi tk are related to the
time-interval from tDC to tk and those in li1 tDC are
related to the time-interval from the initial time of the
®lter to tDC , there is no correlation between wi tk and
li1 tDC . In the same way we can prove the zero
correlation between wi tk and li2 tDC .
Appendix B
Carrier-adjusted DGPS positioning models based on dualfrequency data
Section 2 derived the carrier-adjusted DGPS positioning
models based on L1 code and carrier observations. This
appendix shows how to adapt the positioning models
when L2 carrier and code observations are also available.
B1 The case of L1 code and carrier plus L2 carrier observations
Let us de®ne the combination of the ionospheric delay
and the L2 carrier ambiguity, denoted by N~ , as
1
I~i tk I i tk k2 N~ i
2
and denote
3
2
0
0
P i tk ÿ c dti tk ÿ rîr tk ÿ qi tk
0
7
6
y i tk 4 k1 /i tk ÿ c dti tk ÿ rîr tk ÿ qi0 tk 5
0
0
k2 /~i tk ÿ c dti tk ÿ rîr tk ÿ qi tk
3
ei tk ÿ li tk
ei tk 4 gi tk ÿ li tk 5
g~i tk ÿ li tk
33
2
Zero correlations between dynamic noises and estimate
errors of dierential corrections
According to Eq. (8), the dynamic noise wi tk of
dierential corrections is a function of dynamic noises of
dierential corrections from time tDC to tk . It follows
from Eq. (5) that li tDC is the error of the estimated
dierential correction at time tDC and is, therefore, a
function of measurement noises and dynamic noises of
dierential corrections from the initial time of the ®lter
to the time tDC . Since measurement noises are assumed
to be uncorrelated with the dynamic noises, any
32
2
A i t k
0
oqi tk
6 ox0 1
6 oqi tk
6 ox
4 01
oqi tk
ox1
0
0
0
0
oqi tk
ox2
0
oqi tk
ox2
0
oqi tk
ox2
34
0
0
0
0
oqi tk
ox3
0
oqi tk
ox3
0
oqi tk
ox3
0
0
0
1
0
3
7
7
1 0 7 35
5
1 0
where k2 , /~ and g~ are the L2 carrier wavelength (m),
observation (cycles) and observation noise (m), respectively.
Then it can be easily veri®ed that the measurement
model reads
420
3
Dx1 tk
6 D_x1 tk 7
7
6
6 Dx2 tk 7
7
6
6 D_x2 tk 7
7
6
0
1
3 6 Dx3 tk 7
2
7
6
0 0
0
7
6
A
7 6 D_x3 tk 7
6 A1 tk j @ ÿ2 0
0
7 6 c dT tk 7 2 1
6
3
7
76
6
0 0 ÿ1 r
e t k
7 6 c dT_ tk 7
6
7 6
76
6
7
..
6 ...
7 6 ÿ ÿ ÿ ÿ ÿ 7 4 ... 5
j
. 0
7
76
6
1
1
7 6 I tk 7
6
0 0
0
em tk
7 |{z}
76
6
1
7
5
6
4 Am tk j
_
@ ÿ2 0
A
0
6 I 1 tk 7
ek
6 I~ tk 7
0 0 ÿ1 r
7
6
|{z} 6
7
..
7
6
Ak
.
7
6
6 I m tk 7
7
6
4 I_m t 5
k
I~m tk
|{z
}
2
3
y 1 tk
6 . 7
4 .. 5
2
m
y tk
|{z}
yk
36
xk
with
E f ek g 0
37
20
6B
6B
6B
6@
6
6
6
6
E e k ek 6
6
6
6
6
6
6
6
4
2
r2e rr
^ 1 tk
SYM:
r
2
rr
^ 1 t k
2
r2g rr
^ 1 tk
2
rr
^ 1 t k
2
rr
^ 1 tk
r
r
r
r
2
r2g~ rr
^ 1 tk
1
3
C
C
C
A
7
7
7
7
7
7
7
7
7
7
17
7
SYM:
7
C7
C7
C7
A5
2
r2g~ rr
^m
r
Uk;kÿ1
1 Dtk
F~k 4 0 1
0 Dtk
.
0
B
B
B
@
The symbol r represents the squared ratio of L1 and L2
frequencies r 1:647. The associated dynamic model
reads
3
3
2
2
I3 Fk
d~tk
7
6
6 d tk 7
Fk
7
7
6
6
7
7
6
6 1
~
Fk
xk 6
7 xkÿ1 6 d tk 7 39
7
6
6 . 7
..
5
4
4 .. 5
.
F~k
d m tk
|{z}
|{z}
where
2
..
dk
r
2
rr
^ m tk
2
r2g rr
^ m t k
2
rr
^ m tk
r
2
rr
^ m t k
r
r
40
r
r
with
Efdk g 0
41
2
6
6
6
E d k dk 6
6
4
qx I3 Sk
where
2
3
0
05
1
2
r2e rr
^ m tk
38
S~k
1
Dt3
63 k
6 1 Dt2
42 k
1
3
3 Dtk
3
qc Sk
SYM:
Dtk
1
2
2 Dtk
1
3
3 Dtk
qI S~k
..
.
qI S~k
7
7
7
7
7
5
42
3
7
7
5
43
421
2
Note that the variance of dierential corrections is
usually much greater than the variances of L1 and L2
2
2
2
carrier observations, i.e., rr
^ i tk rg and rg~. This
r
makes then the error covariance matrix E ek ek become
numerically singular (i.e. non-invertible). Therefore,
when solving for the state vector, the so-called B-form
recursive formulas should be used, instead of the A-form
ones (Teunissen 1990c).
A i t k
44
3
ei tk ÿ li tk
6 gi tk ÿ li tk 7
7
ei t k 6
4 g~i tk ÿ li tk 5
~ei tk ÿ li tk
45
0
0
0
0
0
oqi tk
ox2
0
oqi tk
ox2
0
oqi tk
ox2
0
oqi tk
ox2
0
0
0
0
0
oqi tk
ox3
0
oqi tk
ox3
0
oqi tk
ox3
0
oqi tk
ox3
0
1
0
1
0
0
0
3
7
7
07
7 46
7
1 07
5
1 0
where P~ and ~e are L2 code observation (m) and
observation noise (m).
Then the measurement model in this case reads
B2 The case of L1 and L2 code and carrier observations
Let us de®ne
3
0
0
P i tk ÿ c dti tk ÿ rîr tk ÿ qi tk
0
0
6
7
6 k /i t ÿ c dti tk ÿ rîr tk ÿ qi tk 7
y i tk 6 1 i k
7
0
0
i
4 k2 /~ tk ÿ c dt tk ÿ rîr tk ÿ qi tk 5
0
0
P~i tk ÿ c dti tk ÿ rîr tk ÿ qi tk
0
oqi tk
6 ox0 1
6 oqi tk
6 ox
6 01
6 oqi tk
6 ox
4 01
oqi tk
ox1
2
2
2
2
3
y 1 tk
6 . 7
6 . 7
4 . 5
y m tk
|{z}
yk
6
6 1
6 A tk
6
6
6
6
6
6
6 ...
6
6
6
6
6
6 Am t
k
6
4
0
0
0
0
0
0
0
3
1
7
7
7
7
0 ÿ1 r
7
7
0 0
0
rÿ1
7
7
7
..
7
j
.
0
17
7
0 0
0
0
7
B ÿ2 0
C7
0
0
B
C7
j
B
C7
7
@ 0 0 ÿ1 r
0 A5
0 0
0
rÿ1
|{z}
j
B ÿ2
B
B
@ 0
0 C
C
C
0 A
Ak
2
Dx1 tk
3
7
6
6 D_x1 tk 7
7
6
6 Dx2 tk 7
7
6
7
6
6 D_x2 tk 7
7
6
6 Dx3 tk 7
7
6
7
6
6 D_x3 tk 7
7
6
6 c dT tk 7
7
6
7
6
6 c dT_ tk 7 2 1
3
7
6
e tk
6ÿ ÿ ÿ ÿ ÿ7
7 6 . 7
6
7 6
7
1
6
6 I tk 7 4 .. 5
7
6
6 I_1 t 7
e m tk
k
7 |{z}
6
7
6 ~1
6 I tk 7
ek
7
6
6 I 1 t 7
k
7
6
7
6
..
7
6
7
6
.
7
6
m
7
6
6 I tk 7
7
6
6 I_m tk 7
7
6
7
6 ~m
4 I tk 5
m
I tk
|{z}
xk
47
422
with
20
6B
6B
6B
6B
6@
6
6
6
6
E ek e k 6
6
6
6
6
6
6
6
6
4
2
r2e rr
^ 1 t k
r
2
rr
^ 1 t k
r
2
rr
^ 1 t k
r
2
rr
^ 1 t k
r
SYM:
2
r2g rr
^ 1 t k
r
2
rr
^ 1 t k
r
2
rr
^ 1 t k
2
r2g~ rr
^ 1 tk
r
r
2
rr
^ t k
2
r~2e rr
^ 1 tk
r
1
3
C
C
C
C
A
7
7
7
7
7
7
7
7
7
7
17
7
7
C7
C7
C7
C7
A7
5
..
. 0
B
B
B
B
@
2
r2e rr
^ m t k
r
2
rr
^ m t k
r
2
rr
^ m t k
r
2
rr
^ m t k
r
SYM:
2
r2g rr
^ m t k
r
2
rr
^ m t k
r
2
rr
^ m t k
r
2
r2g~ rr
^ m t k
r
2
rr
^ m t k
r
2
r~2e rr
^ m t k
r
(48)
The form of the related dynamic model is identical to
Eq. (38), except that F~k of Eq. (39) and S~k of Eq. (42)
need to be replaced by the following, respectively:
2
1 Dtk
60 1
F~k 6
4 0 Dtk
0 Dtk
3
0 0
0 07
7 ;
1 05
0 1
21
S~k
3
3 Dtk
61 2
6 2 Dtk
6
6 1 Dt3
43 k
1
3
3 Dtk
SYM:
Dtk
1
2
2 Dtk
1
2
2 Dtk
1
3
3 Dtk
1
3
3 Dtk
1
3
3 Dtk
3
7
7
7
7
5
(49)
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