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.