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slips at 
DN,,1(k) = N1,1(k) - Nia (k k-1) (13) 
DN-79(k) = N.79(k) - N.7,9(k,k-1) (14) 
This principle can also be used to estimate cycle slips which 
may have occurred during data gaps, that is during periods 
where no navigation solution was possible (due to shielding 
a.0.). It is difficult to determine the maximum allowable 
length of these data gaps as the noise and ionosphere 
irregularity and scintillation as well the multipath effects 
accumulate with prediction time. In practice, the valid 
predictable time may be identified through data analysis of 
observation sequences. 
After determining the integer cycle slips of DN; ; and 
DN, denoted by CS, ; and CS, the cycle slips CS, and 
CS; of «; and ©, are finally determined by the following 
relations: 
CS, == (CS; 5 +9 CS,.1 )/2 (15) 
CS, = (CS +7 CS,.1) #2 (16) 
This procedure does not necessarily produce unique integer 
solutions but integer candidates for cycle slips of each 
combination because of the error accumulation with the 
growth of data gaps and existence of code biases and 
multipath. In order to determine unique integer candidates 
without on the fly ambiguity resolution, it is referred to the 
test procedure proposed by Han (1995), which consists of a 
normality and two Fisher tests and has been proven very 
efficient for different dynamic environments. 
3. DYNAMIC FILTERING AND SMOOTHING FOR CODE 
AND PHASE SOLUTIONS 
The main feature of this method is to use delta-position 
solutions of first-order phase differences between epochs to 
filter and smooth the position solutions of psuedorange 
measurements. The process includes three steps: first, the 
double differenced ionosphere-free phase and pseudorange 
measurements are used to create the delta position and 
positions in three dimensions in real time (on-line); second, 
a on-line Kalman filter is designed to improve the positions 
with delta-position solution from metre-level accuracy to 
decimetre in real time, and third, optimal smoothing is 
performed off-line for achieving better position accuracy. 
3.1 Phase delta-position and code position solutions 
Precise delta coordinates of the rover receiver can be 
computed from the double  differenced ionosphere-free 
phase measurements as follows: 
AV®7.60(k) = AV p(k) + 477,50 AV N77,60 + AVE, (17) 
where k denotes the time epoch. Differencing the equations 
for epochs k+1, and k, we have 
SAV®7760(k+1)= AV p(k+1)- AV p(k)+dAVe,(k+1) 
(18) 
where SAV®7760(k+1)= AVE7760(k+1)- AVE77,60(k). The 
equation for double differenced ionosphere-free code 
measurements is 
AVR (k+1) = AV p(k+1) + AVer(k+1) (19) 
where R, stands for 
measurements. Linearising the equations (18) and (19) 
leads to 
ionosphere-free pseudorange 
9AV Qr, ak 1)-[Q(AVp(k))/0X(k) J5X ct LE) 
+[OAVp(k+1)/2X(k+1)-O(AVp(k))/0X(k)] 5X(k+1) 
+[AVp‘(k+1) - AVp°(k)]+5AVe,(k+1) (20) 
AVR.(k+1)=[@(AVp(k+1))/0X(k+1)]5X(k+1)+AV p°(+1) 
+ AVer(k+1) @1) 
where 5&X(k+1,k)= &X(k+1) - &X(k) denotes the delta 
position vector between the epoch k+1 and k. The value 
p(k) is computed from approximate coordinate vector 
X°(k). A minimum of four visible satellites is required to 
form three equations for both types of code and phase 
measurements. Thus, the least square solutions of both 
6X(k*1,k) and 8X(k*-1) can be obtained by the joint use of 
those two equations. It is noticed that this is a quite loose 
combination as the coefficient of 8X(k+1) are and there is a 
significant difference between carrier phase noise and code 
ranging noise. The former is normally in the level of 
millimetre to centimetres and the latter is in the order of 
metres to tens of metres, depending on the user receivers 
and environments. 
3.2 Optimal Filtering Of Phase Delta-Position And Code 
Position Solutions 
The following filter algorithm is applied in order to achieve 
decimetre accuracy of the position solution. The filter 
equations consist of a set of state equation and vector 
Observations 
z (K)= D(kk-1)2(k-1) + w(k-1) (22) 
y(k)=A(k) z(k) +e(k) (23) 
where z(k) is the 6x1 state vector consisting of the 3x1 
position vector x and the 3x1 velocity vector v, y(k) is a 6x1 
67 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B1. Vienna 1996