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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