a correlation between measurements occurs. 
The correlation, that is expressed as a covari- 
ance between measurements, appears in cases 
where several measurements are involved in a 
computation. In such cases, using only vari- 
ances in order to determine the SD is insuffi- 
cient, and covariances have to be included. In 
addition, the covariance relevancy increases, 
since in many computations the observations 
are close one to another. The following tables 
demonstrate the altimetric component's vari- 
ance-covariance matrices as extracted for some 
representative cases. 
Table 1. Variance-covariance (1 km difference) 
  
0 km 1 km 2 km 3 km 4 km 5 km 
  
  
Okm [4360 11.70 9.73 7.74 573 2.07 
Tkm || 11.70 - 10.50 931 8.07 6.82 4.29 
2 km 9.73 9.31 8.83 8.35 7.86 6.46 
3 km 7.74 8.07 8.35 8.63 8.89 8.63 
4 km 3.73 6.82 7.86 8.89 9.00 10.70 
5 km 2.07 4.29 6.46 8.63 + 10.70 13.50 
  
Table 1 demonstrates the covariance 
rates between measurements that were col- 
lected at a difference of 1 km between adjacent 
points. The covariances values indicates that 
the correlation between the measurements is by 
any means not negligible. For example, for 
medium distances the correlation increases to 
80%, and for a distance of 5 km, it decreases 
only to 50%. 
Table 2. Variance-covariance (50 m difference) 
  
0m 50m 
0m 13.6 13.6 
50m 13.6 13.6 
  
  
  
Table 2 depicts the variance-covariance matrix 
of two measurements collected at a distance of 
50 m between them. The covariance value, as 
for this case, has the variances own value. Al- 
though a bit surprising, this value is quiet pre- 
dictable, since those two observations were 
collected in an almost similar position, and 
therefore should propagate, quiet, if not al- 
most, the same. 
14 
The previous examples indicate that the 
covariances are too big to be ignored. The fol- 
lowing sub-chapters demonstrate the manner 
by which the covariance are to be included in 
the SD evaluation procedure. 
4.2 The SD of a calculated distance 
A calculated planimetric distance is a 
simple and commonly used function for the 
purpose of distance evaluation. The distance is 
defined by the well known equation: 
  
Ds Gl -x2) «(y1-y2Y 
It's SD is usually evaluated by using the a vari- 
ance's based error propagation, and by applying 
the valid assumption that: 
m?, - m?, - m2, - m?, 
the SD is expressed as follows: 
mb -2" my, 
On the other hand by using the matrix 
form of the error propagation, where the vari- 
ance-covariance matrix is formed by extracting 
the planimetric variances and the relevant co- 
variances for each point, and where the trans- 
formation matrix is formulated by deriving the 
distance function, the SD is expressed as fol- 
lows: 
2 dx dy‘ -dx -dy "y «|; D 
DD Do:D D "m 
  
where } isthe variance-covariance matrix 
of the two points. 
The SD that was evaluated by the vari- 
ance based error propagation, causes a result 
that is, for medium distances, twice larger 
(meaning twice less accurate) than the one 
evaluated b 
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