Geoscience Reference
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inherently limited to two dimensions. These authors have presented the structural
field interpolation (SFI) algorithm to overcome these limitations by constructing
3-D structural form lines from the vector components of strike/dip measurements. It
is beyond the scope of this topic to explain SFI in detail but an example of its
application will be given in the next section to illustrate how geological structures
can be modeled in 3-D. A few introductory remarks on representation of lines and
planes in 3-D space are as follows.
Any point P
(
x
,
y
,
z
) in a 3-D Cartesian coordinate system with origin O
determines a vector OP. Suppose that
α
,
ʲ
, and
ʳ
are the angles of this vector
with
X
-,
Y
-, and
Z
- axes. The cosines of these angles
ʻ ¼
cos
α
,
ʼ ¼
cos
ʲ
and
ʽ ¼
¼
2
+
2
+
2
cos
ʳ
are the direction cosines of OP. They satisfy the relation
ʻ
ʼ
ʽ
¼
1.
The equation
|OP| represents a plane in 3-D. The line OP is the
normal of this plane. OP is a unit vector if its length |OP|
ʻ
x
+
ʼ
y
+
ʽ
z
¼
1. In structural geology,
a plane is characterized by its strike and dip. If the north direction points in the
negative
X
-direction, the strike
¼
ʴ
of the plane satisfies tan
ʴ ¼ ʼ
/
ʻ
and its dip angle
is equal to
.
Statistics of directional features will be discussed in more detail in Chap.
8
.
However, for a better understanding of the SFI example in the next section, it is
pointed out here that one method of estimating the mean direction of
n
unit vectors
in 3-D is to maximize
ʳ
cos
2
ʸ
i
represents the angle
between the
i
-th observed unit vector and the mean to be estimated. This method
was first applied by Scheidegger (
1964
) in connection with the analysis of fault-
plane solutions of earthquakes and by Loudon (
1964
) for orientation data in
structural geology. It can be shown that the resulting average unit vector has
direction cosines equal to those of the first (dominant) eigenvector of the following
matrix:
Σ
ʸ
i
(
i
¼
1, 2,
...
,
n
) where
2
4
3
5
2
i
Σʻ
Σʻ
i
ʼ
i
Σʻ
i
v
i
2
i
M
¼
Σʼ
i
Σʼ
Σʼ
i
v
i
v
i
Σ
v
i
ʻ
i
Σ
v
i
ʼ
i
Σ
A useful interpolation method in 2-D or 3-D is inverse distance weighting. It
means that the value of an attribute of a rock such as a chemical concentration value
or the strike and dip of a plane at an arbitrary point are estimated from the known
values in their surroundings by weighting every value according to the inverse of a
power of its distance to the arbitrary point. In SFI this method is called IDW
(inverse distance weighted) interpolation. Usually the weights are raised to a
power called IDW exponent before they are applied. This exponent is often set
equal to 2. The SFI algorithm can employ an anisotropic inverse distance weighting
scheme derived from eigen analysis of the poles to strike/dip measurements within
a neighborhood of user defined dimension and shape (ellipsoidal to spherical).
When the matrix
M
is used, all its nine elements are multiplied by the same weight
that is different for every strike and dip depending on the distance from the arbitrary
point and on direction of the connecting line if an anisotropic weighting scheme
is used.
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