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last 
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(11) 
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12) 
her 
Therefore weihgted zero residuals can be computed by 
VizVigdgi. nor EVE dur 
Assuming observation i with a gross error vi , then 
o 
m : x 
Vis Vir E gjct uoo F1 m, ped 
j= 
Gross error Vi is reverl completely in its weighted zero 
residual. It must be noted that when ai: is very small 
, the compoents related no-gross error observations are 
enlarged evidently in &; . It is possible that Vi 
will have a big magnitude even the observation do not 
have any gross error. Therefore one using standardized 
residual (symbolized v, ) as statistical equantity to do 
rigorous statistical test for each iteration is quite 
reasonable. 
3.2 Let pi,k 0.7 
We take p >0.7 as the critical value of correlation of 
residuals and we symbolized MCCVi as the main component 
coefficient of standardized residual vi . In observation 
i and observation k of which main component coefficient 
of standardized residual is as follows : 
MCCY, — 6, / 
mac 
  
(13) 
When? 0.7, the denominator of equation(12) will be 
very small and some value of &, » à, ,(j=1,m ,jxi ,kxi) 
would be enlarged evidently. Because of residual i and 
residual k are strong correlation. Usually, there are 
several elements of i th and k th column of Qvv.P matrix 
satisified that 4; Yu +. But £u ., $a » is still 
equal 1.0. After the residuals have been standardized 
, one would find that the magnitude of MCCVi or MCCVk 
will be decreased, as compared with the magnitude 
computed by using o - 9  , and the capability and the 
relibility of gross errors localizing would be decreased 
as well. 
Frome the discussions above, we give some conclusions 
about weighted zero residual as fellows : 
3.2.1 Gross error can be revealed in its weighted 
zero residual completly. Generally speaking, the 
observation contained gross error its weighted zero 
residual is of rather large magnitude. 
3.2.2 Weighted zero residual is not suitable as 
statistical quantity for statistical test. It is 
necessary to be transformed to standardized residual 
in order to get rigerous statistical test. 
3.2.3 The maximum value of main component coe- 
fficient of standardized residual MCCVi is a 
of which magnitude is determined by design matrix. 
It is impossible to enlarge its value by the way of 
iteration weighting any small value to the observation. 
3.2.4 If any two observations are assigned zero 
weight, after iteration, the correlation coefficient 
of the two observational residuals must be zero. 
3.2.5 For any two observations in which residuals 
are strong correlation and weighted zero, the value of 
main component coefficient of the standardized residuals 
will be decreased evidently. 
The correlation of residuals is an important factor to 
make mistakes of localizing gross errors. It is 
difficult to overcome this mistakes. by the way of 
improving the iterational weight function. It is better 
from the statistical point view to investigate .gross 
errors localization by ponti the property of weighted 
zero residual. 
-4. CALCULATION EXAMPLES 
We take the calculation of photo relative orientation 
parameters with simulated data for example. 
design matrix A design matrix B 
g .0 1.0 .0-1.0 4 09 -1.0 1 0-10 
0. .0 4.0-1.0  .0 A .0-1.0-1.0 .0 
4-140 2.0 0-1.0 0-30-28 : 0-20 
-1.0 .0 2.0 -1.0 740 -1.0 ..0 -2.0 -1.0 .0 
0 1.0: 2.0: ..0 -1.0 4 1.0 -2.0 , .0 -1.0 
1.0 .8 2.0 -1.0 0 1.0 ..0 2.0 -1.0 .0 
-8 -2 20 -.8 -.2 -2 -.8 -2.0 r.2 -.8 
4D. ..8 1.04 .0 -1.0 -8. 7272.0 8 -.2 
-.04 -.16 1.04 -.2 -.8 52 7.8-2.0 -.2 -.8 
16.04 1.04 -.8 _-.2 18. +2 =2.0 3.8.-.2 
The vector of simulated observational errors of 
vertical parallax is 
E =(-.87 -.39 1.5.51 .44 ,08 -.87 -.75 2.11 -. 15) 
4.1 Calculating Qvv.P Matrix by Fast Recursive Algorithm 
  
First we take design matrix A and P - I, according to 
equation(2) to compute G, then using 
Pe{..9;.2 «7 .4 5 5 6 .3..8 0 ) 
According to equation(2) and equation(9) to compute G 
respectively. The discrepancy of the elements of matrix 
G computed in the two ways is very small and the average 
of the absolute value of the discrepany is equal to 
0.917*10'* , Design matrix B is computed in Same way 
with good results. However there are two cases in which 
the mastake will be made by using fast recursive 
algorithm. 
4.1.1 Example 1 First we take design matrix B 
and P= I computed matrix G with equation(2). the 
~ 
elements of 1 th and 2 th column of matrix G is 
qui = ( .37 -.37 -.13 .13 -.13 .13 -.08 .08 -.08 .08 ) 
{~-.37 .37 .13 -.13 .13-.13 .08 -.08 .08 -.08 ) 
Qai 
Because the correlation coefficient of residual between 
observation 1 and 2 is equal to 1.0, the donominator of 
S; will be equal to zero and the computed results will 
be wrong after by the fast recuvsive algorithm with 
weight matrix P=diag( 0 0 1 1 1 1 1 1 1 1) and 
equation(9). 
4.1.2 Example 2 First We taking matrix B and 
weight matrix -P=diag(1 1 0 1.1.1 1.1.1.1) and 
computed G with equation(2). Then use weight matrix 
Dsding( ;9 .2-.7 54 .5..8 .6 .9 .8. 0) according 
to fast recursive algorithm , because the denominator 
of Sj; equal to zero, the computed G is also wrong. 
The above results give the examples of limitation 
by fast recursive algorithm for calculating Qvv.P 
matrix. The first example can not be overcome by 
any way except changing the design matrix. However the 
second one can be treated in an approximate way, for 
instance, one takes p = 0.01 instead of p = 0, the 
computed results will be correct. 
4.2 Weighted Zero Residual 
  
4.2.1 The Main Component Coefficient of Standardized 
Residual We take matrix A and P - 1, and any two