International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004 
  
relativistic corrections, polar motion and tidal earth. State space 
vector is estimated by Kalman filtering and include the receiver 
dependent parameter (receiver clock) and the satellite dependent 
parameters (carrier phase ambiguities, ionospheric and 
tropospheric delays). Due to the low redundancy of the system, 
some parameters as orbit errors, satellite clock estimation errors 
and multipath are neglected. 
3. EQUATIONS FOR BIASES CORRECTION 
Undifferenced equations are suitable for use in bias estimation, 
necessary to build the bias model and then broadcast the 
network parameters, to correct the observations in the rover 
station. A refined way to calculate the GPS signal biases is to 
estimate every single bias from undifferenced equations: all the 
error sources that is possible to model, are estimated in the 
reference station by Kalman filter, applying deterministic and 
stochastic models in the observation and in the parameter space. 
This procedure is named one way biases estimation. The 
observation equations in the reference station, identified by the 
subscript £, are: 
AT, 
Pl=pl—E +p. AT, +c- (AT, - At‘) +1/+Tr/ TM, I +5,] te, 
muli 1r «M, s +e 
> 
n 
OD 2 p/ — E! D] - AT, *c- (AT, 
> 
Dyk “b,k [n 
( ) 
Pfi=pi-Eitpi-AT, +c-(A7, —ar) 
(A7, ZA) LT +M, 1 +6 
(AT ^n] 
0, - pj - EL - pj -AT, *c- 
(1) 
pi Geometric range between the receiver and the 
satellite antenna reference point. 
Pi Topocentric range rate. 
Ej Ephemeris geometric radial error. 
AT, Receiver clock error. 
Ar! Satellite clock error. 
I lonospheric delay. 
Trj Tropospheric delay. 
5 posp y 
M! Multipath affecting the observation Q.. 
8, Hardware delay. 
e, Observation noise. 
N/ Integer ambiguity. 
nk e © y 
AsA, Wavelengths of the L1 and L2 signals. 
c Light speed in the vacuum. 
V 4. Rate between the two squared frequencies. 
^2 
f 
Subtracting the terms known from the ephemeris, the range p, 
the topocentric range rate and the satellite clock error, we obtain 
for every observation: 
B,= Pl-pl-pitcar= [Jeamemes +E] +c 08" +M 189,1] 
B,- Pi-pi-pi+e-ar= [cA +Tg ]+vi{ +[El+cda eM, [+8,]] 
Bu= ®{-pi-pi+e-a= [cAT + |=1+AN,  +[E[+c-0ar +M, 1+6,;{] 
B,2= Di-pi-pircar= [CAT ATK ]-VE AAN} [E80 ML i91] 
(2) 
In these equations, the ephemeris error can be negligible by using 
appropriate ephemeris, such as ultra rapid IGS available in real 
time, or can be estimated by modelling over large networks. In 
+, TAN 
A 
7 -AU)-VII TE AM Ito, AA No} 
ultra rapid IGS ephemeris the satellite clock error is available 
with ~ 5 ns = 1.5 m RMS, and the geometric error as ~10 cm 
RMS. The terms in these equations are ordered as geometric 
biases (receiver clock error and tropospheric delay), frequency 
dependant biases (ionospheric delay and phase ambiguities) and 
negligible biases (ephemeris errors, multipath and hardware 
delays). The geometric terms are separable by modelling the 
clock offset and drift, and the zenithal tropospheric delay; their 
estimation may be conditioned by model uncertainties. 
Moreover, the negligible terms are mainly absorbed by the 
residuals, however they affect the parameter estimation. 
Simplifying, we can write: 
ade 
AT, +Tr! |+vr; (3) 
[Ee 
an [ea0 r0 1-7 AN 
eel 
I | 
CAT, + Tr} [=v] + LN] 
By using a good oscillator, such as a rubidium clock or better, 
the clock error can be modelled linearly with 
c- AT, za, a (t, 74.) # 
where à, is the clock offset and «a, the clock drift. The zenital 
tropospheric delay can be estimated introducing a mapping 
function m: 
Tr, VI (5) 
The state vector and the design matrix will be: 
  
  
  
  
m [eat | Ii A MANS] (6) 
I 
Ll 6: (ei) te 0 & Pla 
1 
v 
D 6r Um (ei) nt 0 = Ra 
À = s 
=i I 
le dy m (el) Es fy el 
=i ] 
my l 
+ ór m (el) e in L2 
= l 
  
  
  
  
  
The total number of parameters to be estimated is: 
p=3+3-sat (8) 
that are 21 parameters with 6 tracked satellites. 
3.1 Weighting observations 
The Signal to Noise Ratio (SNR) is a function of antenna gain 
pattern and of atmospheric diffraction; it decrease with the 
     
   
   
    
   
   
   
     
    
    
    
   
      
   
   
     
    
   
  
  
  
   
   
    
     
     
   
     
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