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unless all measured vectors are parallel to one another. The vector R points in the
same direction as the vector sum of all observed unit vectors, and the quantity 1
|
R
|
provides a measure of the scatter of the measurements around their mean vector.
8.2 Unit Vector Fields
Methods of unit vector field analysis include 2-dimensional polynomial trend-
surface fitting applied separately to the three direction cosines of the measurements
followed by combining the estimated values to construct the unit vector field. A
new mathematical derivation of the underlying theory was presented in Agterberg
(
2012
). Examples of application are to B-axes for Hercynian minor folds and
schistosity planes in quartzphyllites belonging to the crystalline basement of the
Italian Dolomites. During Alpine orogeny, these
s
-planes were not only refolded
but there were large-scale sliding movements along the Hercynian schistosity
resulting in various kinds of rotations of the B-axes. The unit vector fields help to
outline these deformation patterns. The unit vector field for the B-axes in
quartzphyllites in the San Stefano area east of the Dolomites shows existence of
an SW vergent Alpine anticlinal structure. Two examples for quartzphyllites in the
Pustertal adjacent to the Periadriatic Lineament can be explained as part of the
south vergent upper-crustal response to the north and north-eastward directed
subduction of the Adria microplate below the Eastern Alps from the late Miocene
onward. Because of this motion the crystalline basement north of the Dolomites
was subjected to significant N-S shortening with probably a sinistral component in
the vicinity of the TRANSALP profile near Bruneck. Box
8.1
contains a brief
review of the mathematics of polynomial unit vector field fitting. It will be followed
by applications to B-axes in quartzphyllites south of the Periadriatic Lineament.
Box 8.1: Mathematics of Polynomial Unit Vector Field Fitting
Let
U
oi
represent an observed unit vector with strength |
U
oi
|
¼
1 at the
i
-th
observation point (
i
1 the corresponding
estimated unit vector.
U
ei
has the same direction as a vector
R
i
for which the
quantity
¼
1, 2,
...
,
n
), and
U
ei
with |
U
ei
|
¼
R
i
|
2
will be minimized over the
n
observations. Suppose that
the three direction cosines of observed unit vectors
U
oi
, the estimated vectors
R
i
, and the estimated unit vectors
U
ei
are written as
Σ
|
U
oi
λ
hi
(
h
'
hi
,
λ
hi
and
¼
1, 2, 3),
3
2
3
2
|
R
i
|
2
; and
3
h
¼1
λ
2
respectively. Then
h
¼ 1
'
hi
¼
1;
h
¼ 1
λ
hi
¼
hi
¼
1. In the
∑
∑
∑
Cartesian coordinate system to be used, the
x
-axis (
h
¼
1) points northwards,
the
y
-axis (
h
3) upwards, respectively.
Suppose that trend surface analysis is applied to each direction cosine
'
h
yielding three polynomial equations of degree
p
in terms of the geographical
coordinates
x
and
y
with
¼
2) eastwards, and the
z
-axis (
h
¼
¼
∑
j
+
k p
b
hjk
x
i
y
i
where
j
λ
hi
¼
λ
h
(
x
i
,
y
i
)
¼
0, 1,
...
,
p
;
(continued)
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