ver high
ervations
We have
receivers
| the raw
) satellite
ation.
lon is to
modelled
, without
one way
reference
cript m:
UN
Im
4 vi
1 term for
o master
2m
(2)
In the rover station, applying the correction terms estimated by
master station, we obtain the single differences equations. In
these equations, the differential ephemeris error AZ is quite
small to be negligible, for baselines up to 500 km. Residual
biases has to be studied for long baseline, but are mitigated in
MRS or NRTK.
G -p) -E| «e (AT,- AU) +I] 4T! «Mi
*M, 7
€ im
= +Ej} -c-(AT„-At') 1} -T.
m m m m
pl +AK" *c- (AT, - AT, ) «AU AT! AM,
F nm
i A -E| «e(AT,- A) -r +T/ +M,’ +AN}
= +E} -c-(aT,-A#) +1} -T} My AN.
z p) - AK! +c (AT, AT, ) «AP «AT! 4AM,T A (NI - NV)
(3)
2.2 One way biases estimation
We make use of the equations (2), modified subtracting the
satellite clock error, known from ephemeris:
se =(- El ve je ATOETS
Un m m m m
TOSAT.FTÀ- VI
m m m
SB.
m I+ ep
N
m m m m
[A
0, =(-E +8," +c AT, +I) - I, + 4 N,,
(rs)
CAT +7 -vIi + LN,
m m m 2m
7
oD, E + Ep
(4)
In these equations, all the term out of the round bracket are
modelled and estimated via Kalman filter; the terms in the
round bracket are included into the filter residuals or affect the
modelled terms, because of is not possible to model them. The
non modelled biases are the ephemeris error (E) and other (5)
that are multipath, hardware delay, relativistic effects, etc.
The state vector of the system, includes all the modelled
parameters:
TG RM ZT JAN, AN, | (5)
Where: a, receiver clock offset
a, receiver clock drift
a, receiver clock parabolicity
ZTD zenithal tropospheric delay
1 satellite ionospheric delay
ambiguities
AN}
the number of parameter is constellation dependant:
par = 4+3 sat (6)
that is i.e. 22 state parameters for 6 satellite constellation.
3. STATE SPACE MODEL
In this section we examine the kinematic model of the rover
state. Each kinematic model will be defined in terms of its state
vector x, transition matrix and covariance matrix of the
system noise C,,. No discussion of the state space approach and
79
Kalman filtering will be given here; we want only to remember
the prediction equations:
X, m FX, QE, (7)
;
Q E PO + C. (8)
where Q, is the variance-covariance matrix of the state vector.
3.1 Constant velocity model
In this case the state vector contains three position states, three
velocity states and two clock states, plus ambiguities. The state
vector is given for the single frequency form:
we x7 Sy ZamlN | (9)
The corresponding transition matrix is of the form:
DICAT 0
10
F9: qo
-
0 45091
where all submatrix are diagonal.
4. TEST AND RESULTS
4.1 The software ALARIS
Several functions described in this paper and others are
implemented in the self-made software ALARIS. Here, we want
only to give an overview to these capabilities:
e Reference station:
o Raw biases estimation
o One way biases estimation
e Rover station:
o Dynamic positioning (no kinematic)
o Constant velocity model kinematic
positioning
o Single or double frequency data processing
e Real time quality control (DIA procedure) as described in
(Teunissen 1998, Tiberius 1998).
e. On The Fly ambiguity fixing to the nearest integer
e Satellite tracking with broadcast or precise ephemeredes
The software ALARIS is developed at the Department of
Georesource and Territory (Politecnico di Torino), is written in
FORTRANODQ, and is intended to perform kinematic positioning
using a Multi Reference Station approach.
4.2 Test on corrections
Using the algorithm implemented in ALARIS, we have
estimated the bias terms in the reference station, that are the
corrections for the rover station. The algorithm uses the
equations (4) to estimate the atmospheric delays, that we have
compared with the RTCM corrections PRC (Pseudo Range
Correction) and RRC (Range Rate Correction).
The software ALARIS uses the IGS ultrarapid predicted
ephemeris, while the RTCM corrections are generated by the
reference stations using the broadcast ephemeris, that are not so
good as the IGS ultrarapid.
