Geoscience Reference
In-Depth Information
Box 8.1 (continued)
k
¼
0, 1,
...
,
p
;and
j
+
k
p
. Because
R
i
and
U
ei
point in the same direction, the
X
3
h
X
3
h
¼1
λ
hi
'
hi
1
λ
hi
'
hi
¼
angle
ʸ
i
between
U
oi
and
U
ei
satisfies: cos
ˑ
i
¼
Rjj
;
¼
h i
½
. The difference vector between
U
oi
and
R
i
, which can be
written as
d
i
¼
X
3
h
¼1
λ
2
hi
Rjj
¼
strength |
d
i
| wth|
d
i
|
2
1+|
R
i
|
2
U
oi
R
i
, has
¼
2|
R
i
|
3
h
¼ 1
2
3
h
¼ 1
2
3
h
¼1
3
'
hi
λ
hi
)
2
.The
cos
ˑ
i
¼
∑
'
hi
+
λ
hi
2
λ
hi
'
hi
¼
∑
h
¼ 1
(
∑
∑
∂
b
mjk
¼ 2
X
n
i
¼1
X
3
¼
Σ
|
d
i
|
2
is a minimum when
∂
T
sum of squares
T
ð
'
hi
λ
hi
Þ
h
¼1
∂
λ
hi
∂
b
mjk
¼
0 for all coefficients
b
mjk
(
m
¼
1, 2, 3;
j
¼
1,
...
,
p
;
k
¼
0, 1,
...
,
p
;
j
+
k
p
). Because the three polynomial functions
λ
h
(
x
i
,
y
i
) are linear in
b
mjk
,
their partial derivatives satisfy
∂
λ
hi
∂
h
and
∂
λ
hi
∂
x
i
y
i
if
m
b
mjk
¼
¼
b
mjk
¼
0if
m
6
¼
h
. This
allows us to split the
n
Gaussian normal equations into 3 groups (for
m
¼
1, 2
'
mi
∑
j
+
k p
b
hjk
x
i
y
i
]
x
i
y
i
¼
i
and 3, respectively) with
¼1
[
0where
s
¼
0,
∑
1,
p
. The simultaneous linear equations for each
group can be solved yielding estimates of the coefficients required to estimate
λ
h
(
x
i
,
y
i
). Finally, normalization gives estimates of
...
,
p
;
t
¼
0, 1,
...
,
p
;
s
+
t
λ
h
(
x
i
,
y
i
)(
h
1, 2, 3). For
representation of the results as continuous functions on a map, it is convenient
to use the azimuth
ʱ
(
x
i
,
y
i
) and dip
ˆ
(
x
i
,
y
i
) of the unit vectors with tan
ʱ
(
x
,
y
)
¼
¼
λ
2
(
x
,
y
)/
λ
1
(
x
,
y
) and sin
¼
λ
3
(
x
,
y
).
ˆ
(
x
,
y
)
8.2.1 San Stefano Quartzphyllites Example
The following example illustrates usefulness of the mean unit vector. Fig.
8.6
(after
Agterberg
1961
) shows locations and average values of B-axes and schistosity-
planes in the San Stefano area east of the Dolomites. Trace of axial plane of
neo-Alpine anticline in Fig.
8.6
approximately coincides with the line between E
and SE dipping average minor folds, and those dipping W to N. Plots on the
Schmidt net for measurements from subareas A and B (outlined on Fig
8.6
) are
shown in Fig.
8.7
. Mean values plotted in Figs.
8.6
and
8.7
were calculated by
means of a simple, approximate method of averaging azimuths, strikes and dips.
Note that the preferred orientation of the B-axes in subarea A is markedly different
from that in Subarea B. Fig.
8.8
(after Agterberg
1985
) shows original measure-
ments or average B-axes for (100 m
100 m) squares in comparison with unit
vector means based on larger samples, combining all measurements from within
(1 km
2 km) circular neighborhoods. A clear regional pattern
confirming the global pattern of Fig.
8.6
emerges in Fig.
8.8b
from the mean unit
vectors that are based on the larger samples. Structural interpretation of the change
in average dip of the B-axes is as follows. The ESE-striking schistosity-planes and
the B-axes are Hercynian in age because they occur in large boulders within the
1 km) or (2 km
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