anbul 2004
s available
as ~10 cm
geometric
frequency
uities) and
| hardware
delling the
delay; their
certainties.
ed by the
estimation.
x or better,
(4)
The zenital
i mapping
(5)
^
Ny
(6)
"i qn
P2
Li
1,2
(8)
enna gain
with the
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004
satellite elevation, and the observation standard deviation can be
modelled as an elevation dependant function. Many authors have
proposed an exponential function as:
6=G +60, exp(-e/e,) (9)
where e is the satellite elevations in units coherent with ej. The
function shape is given by the parameter Op, O; and ep, that
depend on the observation type, the receiver and the antenna
type. The approximate zenital standard deviation is oy (if e; «
90°) and the horizontal standard deviation is Op +0,. By using
the same values of Op, 0; and e, estimating the o for every
observation, we obtain
o (e) 0.08 - 4.5: exp(—e/10) (10)
where the numerical parameters depend mainly on the antenna
type. Observation weights are given by
Spon (11)
o(e).o
obs
Ww
obs T7
where w,,, is an a priori variance. Observation weighting leads to
other advantages, mainly to reduce multipath effects and
mapping function errors, more evident for low elevations.
3.2 Pseudo Range Correction estimation
The Pseudo Range Correction (PRC) value can be computed
with the estimated biases, and then broadcasted with the RTCM
signal to the rover user. The PRC parameter must include the
atmospheric delays and corrections to the observations derived
from other sources of error; the PRC is the sum of:
= (el ) Tr, tropospheric delay correction
[! ionospheric delay correction
Pig P, radial orbit error correction, or radial difference
between broadcast and predicted ranges,
referred to the satellite antenna reference point
c(Arj — At) satellite clock error correction
pi - p! radial datum difference, between user and
kdatl kdat2
satellite datum; includes reference antenna phase
centre variations
The application of the network corrections on the rover side,
leads to cancel satellite dependant errors, but residual
atmospheric biases are not negligible for long baselines. Residual
bias also can be estimated and recovered by parameterisation in
the state space domain.
4. STATE SPACE ESTIMATION
State model will be defined in terms of state vector x, transition
matrix F and covariance matrix of the system noise Cg. In the
Kalman filter formulation, the prediction is given by:
12
Xe TE: (12)
x EE
T :
Quis = FQ, ia T * Coa (15)
where Q, is the variance-covariance matrix of the state vector; €;
Is the system noise and C,, its a priori variance matrix. Then,
introducing the observation vector b, and the design matrix Ay,
the correction is given by using the gain matrix
x5
k[k 5E b A Sp) (14)
Q, "(FO RT +Cy ]- K, A, (20, 77 xc. (15)
where K, is the Kalman gain matrix defined by
Kı= (F, 0.7 tC, ) A; 4, (5, Q, FI *C, JA} tC, |
(16)
The recursive solution is locally optimal. Several strategies are
been proposed in the state space estimation:
KF standard Kalman filter
KFA augmented Kalman filter
AKF adaptive Kalman filter to estimate system parameters
2SKF two stage Kalman filter to estimate constant biases
4.1 Adaptive Kalman filter
State estimation given by Kalman filtering depends on system
and observation noise variance, usually defined a priori and as
constant. Adaptive algorithms are designed to refine this a priori
knowledge, analising the stochastic properties of the residuals,
following the dinamic variations in the system stochastic model
(Mohamed and Schwarz, 1999). Adaptive filtering leads to faster
convergence and produce self-calibrating covariance matrices.
Introducing the observation noise e, and its a priori variance
matrix C,,, the vector of predicted residuals and its covariance
matrix are given by
jm d Xu (17)
V, - Var(v,) 9 AQu Ai + Cu (48)
In adaptive filtering, the a priori knowledge of the noise variance
matrix can be refined by using the estimated covariance matrix of
the predicted residuals vector, given by
A 1 v = E
“= k- 1 > (v, M Xv, mu ) 19)
CSI a
I<
y--Xv (20)
D
^ ^. k-1l hj 5)
Po =/ ; ob E v,) (21)
c ru tq Y.
VL = ^E feet es E (22)
Finally, the noise variance matrix can be estimated with
>
€ = Ouid: (23)
substituting the a priori C.