then further processed on line (real time) for achieving 
decimetre accuracy positioning and lppm accuracy by 
postprocessing. Finally experimental results for aircraft 
positioning are given, which confirm the achievable 
decimetre-level positioning capability 
2. DETECTION AND REPAIR OF CARRIER PHASE 
CYCLE SLIPS 
We assume the use of dual frequency GPS receivers with 
C/A code and/or P code pseudorange measurements. Thus, 
there are four possible observation equations (Dong & Bock, 
1989): 
Ri=p+Ifi® +en (1) 
qQi7p -Uf^-4N* £pl 2) 
R;-p-lo? t£ 3) 
Q227p -I6? 4, N, + E42 (4) 
where R, is C/A code or P1 code pseudorange, and R; is P2 
code pseudorange; «; and q» are the L1 and L2 carrier 
phase measurements in units of cycles whose frequencies 
are f, and f; and wavelengths A, and A, , N, and N; denote 
integer cycle ambiguities, p is the geometric range of the 
receiver to a satellite; I is defined by Total Electron Content 
(TEC) and e denotes the noise in the various measurements. 
The linear combination of ¢; and q; creates a new 
observable (Han & Rizos, 1995) 
ei 7ipitjq (5) 
where i,j denote arbitrary integers. Its integer ambiguity and 
wavelength are 
Ni =iN;+jN, (6) 
Ai 7c f; *jf) () 
where c is the speed of the light. According to the equations 
1-4, the real-valued ambiguity N;; for the combined phase 
@;; can be written as: 
Ni = ¢jj-ou Ry + oz Ry (8a) 
where 
où = [ 9240(i+j) +289] / 2329, (8b) 
a = [ 9240(i-+j) + 289 j ] / 2329, (80) 
Formula (8) is designed for those cases where C/A code or 
Pl-code and P2 -code pseudorange measurements are 
available. Studies have shown its good suitability for 
widelane ambiguity estimation (i=1j=-1), but it is 
inaccurate for other phase combinations (Han & Rizos, 
1995). Therefore, the following ionosphere-biased formula 
is considered as an alternative: 
Ni 794- Raj vg (/fi^) (9) 
where 
VW (uj * BY X (103 
où; = (4620- i + 5929 j )/(4620 i +3600j) (10b) 
100 for R=R, 
ß= {1.647 for R=R, (10c) 
1.323 for R- (R; * Rzy2 
R stands for R; or R, or the average of R, and Ra, depending 
on the measurements available. 
Now, we wish to select some linear combinations of « and 
Q2 for cycle slip detection and repair. In principle, the 
combinations which have 10 and 100 times of the L1 or L2 
wavelengths would be considered as the most appropriate 
ones. The second factor is the noise of a combined phase 
ambiguity N;; to be computed by Eq (8) or (9). The smaller 
the noise of the combined ambiguity and the larger its 
wavelength, the better the capacity of identifying the 
combined cycle slips. The third factor is the effect of the 
time-varying ionosphere biases on the determination of the 
combined ambiguity N;; Table 1 gives the standard 
deviation ( STD or 16 error ) estimates of several typical 
combined ambiguities Nj; and their coefficient y;; , of the 
time-varying ionospheric biases, where «;, .& is the 
ionosphere-free phase observable. It is noted that the 
maximum wavelength for possible L1 and L2 combinations 
is 14.653 metres, achieved by ¢.; 5 . Considering all the 
three factors, the best possible choice for cycle slip detection 
is to use the combinations N, ;, defined by Eq (8) or Eq (9) 
and No, defined by Eq (9). 
Table 1 STDs for determining combined ambiguities 
  
  
  
  
  
  
  
  
Phase Mm) o(cy)- o(cy) Y-Eq (10) 
Eq(8) Eq(9) (R=R;) 
$i .190 8.085 1.579 10.510 
© 244 8.024 1.220 10.839 
Q71.50 .006 142.02 |47.679 | 158.948 
E ‚862 0.248 0.348 -0.329 
P12 341 7.971 0.862 -11.167 
P34 1.682 7.886 0.191 11.825 
Q9 14653 |15740 |0.116 23.979 
  
  
  
  
  
  
Note: Assumptions for STD computations: 
0517 O4 0.01 cycles, og; 7 og; 0.3 metres 
  
  
66 
The detectability of the cycle slips then depends on the 
temporal and spatial predictability of the ionosphere biases 
in Eq (9). Over a period of up to a few minutes, the regular 
variation of ionosphere biases can be represented as a linear 
function of time. This implies that a linear model of N; ;, 
estimated from the observations within a moving time 
window in the past, can be used to predict the N;; values at 
the next epoch. Using the predicted value N;; (k, k-1) and 
observation value Nj;(k) at the epoch k, the differences 
DN;;(k) are obtained to detect and repair the cycle slips at 
this epoch: 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B1. Vienna 1996 
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