Civil Engineering Reference
In-Depth Information
(13.4)
where f
1
(
α
) and f
2
(
β
) are one-dimensional PDFs of
α
and
β
, respectively (Fig. 13.28,
above). f
1
(
α
) and f
2
(
β
) can be approximated by frequency distributions of the measured
α
and
β
values (Fig. 13.28, lower). The mean angles
can then be calculated as
the arithmetic means:
(13.5)
(13.6)
where
β
j
are the strike and dip angles of m mapped discontinuities of the cor-
responding set. The standard deviations
α
j
and
σ
α
and
σ
β
can also be estimated from the fre-
quency distributions of
α
and
β
:
(13.7)
(13.8)
Since
are assumed to be statistically independent, it follows that the standard
deviation of the two-dimensional orientation vector {
α
and
β
α
,
β
} is given by
(13.9)
The measured orientation data of a set often include horizontal or vertical orientations.
The statistical evaluation then requires a transformation of
in order to distin-
guish discontinuities which have the same strike direction but dip in opposite directions.
Thus, the range of strike angles is reduced to 0 ≤
α
and
β
≤ 180° in such a case.
If the horizontal orientation (
β
= 0) appears, the range of dip values is increased to
-90° ≤
≤ 90° and
α
and
β
must be transformed as follows (Wittke 1990):
(13.10)
(13.11)
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