Geoscience Reference
In-Depth Information
elementary methods of differentiation and integration. For example, average height
satisfies the equation Ave (
h
)
¼
R
h
·d
A
0
where
A
0
represents relative area that can
be expressed as a percentage. Average height on continents can be estimated
rapidly as follows: Firstly the area under the curve on the left of Fig.
1.5
is
determined graphically as 4.26 times as large as the unit of area (e.g., by counting
squares on graph paper). This value must be divided by 5 to account for the vertical
exaggeration. The result is 0.852 km for average height on continents, which, after
rounding off, duplicates the value of 850 m reported by Heiskanen and Vening
Meinesz (
1958
). This approximate equality may be fortuitous because the error of
the value estimated graphically from Fig.
1.5
probably exceeds 2 m.
A problem analogous to the one solved in the previous example consists of
calculating the average value over a given area for a variable of which the contour
map is given. For example, in the contour map of Fig.
7.28
, which will be discussed
later, the contours are for percent copper. A problem that can be solved in that
application is to determine the average percentage copper or copper bounded by the
0.5 contour on the map. By constructing a hypsometric curve with height replaced
by percentage copper, it was estimated that the average grade for the larger area is
approximately 0.99 % copper.
1.3.3 Graphical Curve-Fitting
The following example illustrates graphical integration in an application to an
isoclinal fold in cross-Section. A number of attitudes of strata suggesting existence
of an anticline were observed along a line across the topographic surface The
observation points are labelled 0, 1, 2,
4 in Fig.
1.6
). The objective is
to reconstruct a complete pattern for this fold in vertical cross-section.
It will be assumed that the isoclines (lines of equal dip) in the profile are parallel
to a line (
Y
-axis) through the origin (O) that is set at the first observation point
(Fig.
1.6
). The projections of points 0. 1, 2,
...
,
n
(
n
¼
...
,
n
on the corresponding
X
-axis
through O are called
x
i
(
i
¼
1, 2,
...
,
n
). Let
α
i
be the angles of dip with respect to
the
X
-axis. Then the curve for a function
f
(
x
)
¼
tan
α
can be constructed by using the
(
n
+ 1) known values of
α
i
.
The function
f
(
x
) plots as a smooth curve passing through the points
P
i
. Suppose
integration of
f
(
x
) gives the function y
α
i
. The points
P
i
in Fig.
1.6
have ordinates equal to tan
¼
R
f
(
x
)dx
F
(
x
)+
C
where
C
is an arbitrary
constant. In Fig.
1.6
use is made of the method of graphical integration. The curve
f
(
x
) is replaced by a staircase function
f*
(
x
) with the property:
Z
x
iþ
1
¼
Z
x
iþ
1
f
x
fx
ðÞ
dx
¼
ðÞ
dx
x
i
x
i
where
x
i
and
x
i+1
are the abscissae of two consecutive points
P
i
and
P
i+1
(
i
¼
1,
2,
,
n
-1). The function
f*
(
x
) is readily integrated yielding
F
*(
x
)+
C
. The constant
C
is specified by letting
F
*(
x
) pass through O. Note that
F
*(
x
) consists of a succession
...
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