Geology Reference
In-Depth Information
S
(
z
) can be represented by two functions
x
D
`
1
(
y
)and
x
D
`
2
(
y
) connecting (
x
1
,
y
1
)to(
x
2
,
y
2
).
For any function
f
(
x
,
y
) on the surface of the
lamina it results:
Finally, substitution in (
5.19
) provides:
—
nC1
Z
X
N
.
z
/ Š
nD1
—
n
.'
n
y
C
“
n
/
Z
Z
y
2
`
2
.y/
Z
C
z
2
/
q
'
n
C 1
y
2
C 2'
n
“
n
y C “
n
C
z
2
dy
I
.
y
2
f.x;y/dxdy
D
dy
f.x;y/dx
y
1
S.
z
/
`
1
.y/
—
NC1
—
1
(5.22)
Z
y
2
D
dy ŒF .`
2
.y/;y/
F.`
1
.y/;y/
The solution of these integrals gives:
y
1
I
X
N
D
F.x;y/dy
(5.17)
.
z
/
Š
Œarctan
n
.Ÿ
nC1
;—
nC1
;
z
/
n
D
1
B.
z
/
arctan
n
.Ÿ
n
;—
n
;
z
/
(5.23)
where
B
(
z
) is the boundary of
S
(
z
). Therefore,
the quantity (
z
)in(
5.16
) assumes the following
expression:
where:
I
n
.x;y;
z
/
D
x
.
z
/
D
C
z
2
/
p
x
2
C
z
2
dy
z
.
“
n
y'
n
z
2
/
x
Œ.
1C'
n
/
z
2
C“
n
.
'
n
z
2
C“
n
/
p
x
2
Cy
2
C
z
2
(5.24)
.y
2
C
y
2
B.
z
/
(5.18)
This integral can be calculated by approximat-
ing the perimeter
B
(
z
) of the lamina through a
polygon having vertices (Ÿ
1
,—
1
),(Ÿ
2
,—
2
), :::,(Ÿ
n
,—
n
),
as shown in Fig.
5.5
. This is equivalent to
approximate the functions `
1
(
y
)and`
2
(
y
)by
piecewise first-order polynomials. Therefore, the
line integral (
5.18
) will be converted into a sum
of simple integrals:
Substituting the solution (
5.23
)into(
5.15
)
provides the vertical component of gravity at
the origin. In general, integration over
z
can be
performed using standard numerical techniques
and should not constitute a problem. The basic
idea of converting a surface integral into a
line integral around the surface boundary also
represents the starting point of the method
proposed by Talwani et al. (
1959
) for calculating
the gravity anomalies of two-dimensional bodies.
A geological structure having a linear trend, for
example a long horizontal cylinder, generates
linear magnetic or gravity anomalies and can
be modelled by sources, respectively magnetic
or gravitational, that are invariant along the
direction parallel to the long side. In this case
the
y
axis is often chosen parallel to the invariant
direction (Fig.
5.6
), leaving calculations to be
performed only with respect to the
x
and
z
dimensions. We say that the corresponding
problem is two-dimensional. This class of
forward
-
modelling
problems can be solved by
approximating the cross-section of the body by
nC1
Z
X
N
x
.
z
/
Š
C
z
2
/
p
x
2
C
z
2
dy
I
.y
2
C
y
2
n
D
1
n
—
NC1
—
1
(5.19)
The variable
x
in this equation can be easily
expressed in terms of
y
, because the path is
composed by straight line segments:
x
D
'
n
y
C
“
n
(5.20)
where:
Ÿ
nC1
Ÿ
n
—
nC1
—
n
I
“
n
D
Ÿ
n
'
n
—
n
'
n
D
(5.21)
