Geoscience Reference
In-Depth Information
mes A
\
ð
R
nB
Þ
X
¼
mes
R
nB
where
R
is a set consisting of a random point in the study area (
T
). Problems of this
type will be solved later in this topic (Chap.
14
).
The Minkowski operations discussed in this section have been implemented in
various Geographic Information Systems (GIS's) and are useful in practical appli-
cations. It should, however, be kept in mind that these operations are basically
linear. For example, they cannot be used for nonlinear dilatations or erosions as
might be desirable if the curve representing the contact between two rock types is
the intersection between a curved surface and the topographic surface and one
would wish to account for strike/dip of the contact. In such situations, more flexible
methods are required along the lines of those discussed in Sect.
1.4
.
1.5.3 Boundaries and Edge Effects
Geoscience projects generally are conducted in a study area with a shape that is
either rectangular or curved. Various statistical techniques applied to variables
observed at points within the study area are subject to edge effects in the vicinity
of the boundaries of the study area. Such edge effects arise when observations are
used for extrapolations into their immediate neighborhoods. In a simpler situation
this kind of problem occurs in 1D as well; for example, in time series analysis edge
effects can occur at the beginning and end of a series of observations. In 2-D and
3-D applications, edge effects generally present a more serious problem because
'relatively' many more data points occur near study area boundaries and a relatively
simple 1D method such as reflection of a series around its end points in order to
obtain extra observations with locations outside the range of observation cannot
be used.
In GIS applications, pixels used for representation of attributes are situated on
a regular grid. Curved lines on maps also can be represented in vector mode
meaning that they are approximated by sequences of densely spaced points on the
curves so that these are approximated precisely by strings of very short straight-
line segments. In 2-D, edge effects are easier to avoid if the study area is
rectangular in shape as it is frequently in remote sensing, geophysical and
regional geochemical applications because the data then are averages in which
abrupt changes such as those related to contacts between different rock types have
been masked out. If map data are averaged in such applications, it is often possible
to keep the unit areas for which values are averaged within the boundaries of the
study area. In geological map applications, however, the boundaries of study
areas often are curvilinear in shape. Examplesarethe“plays”oftenusedinthe
oil industry and delineations of permissive areas for occurrence of different types
of mineral deposits in economic geology.
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