Geology Reference
In-Depth Information
where,
'
n
log
r
n
C
1
r
n
C
.™
nC1
™
n
/
(5.47)
“
n
1
C
'
n
X
n
D
log
r
n
C
1
r
n
'
n
.™
nC1
™
n
/
(5.48)
“
n
1
C
'
n
Z
n
D
Therefore, substituting these expressions into
(
5.44
)and(
5.45
)wehave:
@X
n
@x
cos I sin '
C
sin I
X
N
@X
n
@
z
Fig. 5.8
Crustal field
F
and components of the IGRF
field
F
in a local frame of reference. Declination
D
0
is
the angle between the North direction and the horizontal
projection of the field, measured clockwise. The angle be-
tween projection
H
and the field vector
F
is the
inclination
I
0
, positive if
F
is directed downward
0
M
2
F
x
D
n
D
1
(5.49)
@Z
n
@x
cos I sin '
C
sin I
X
N
@Z
n
@
z
0
M
2
F
z
D
n
D
1
(5.50)
can be easily obtained by the following simple
transformation:
8
<
In Sect.
5.1
we have shown that a total field
anomaly can be calculated by projecting the
anomalous field vector
F
onto the IGRF field
axis (Eq.
5.3
). Let
D
0
and
I
0
be respectively the
declination and inclination of the reference field
(Fig.
5.8
). The components of the anomalous
field vector
F
that are calculated through
(
5.49
)and(
5.50
) are expressed in the local
(
x
,
y
,
z
) frame of a prism. In order to combine
the contributions of several blocks through the
superposition principle, we must represent the
anomalous vector components in a common
reference frame. Then, the expected total field
anomaly,
T
, associated with the crustal field
F
D
F
(
r
), can be calculated by projecting
the vector
F
onto the axis of the present-day
reference field
F
, which has declination,
D
0
,
and inclination,
I
0
,inthe(
X
,
Y
,
Z
) local reference
frame of Fig.
5.8
. Therefore, it is convenient
to express the components of the anomalous
magnetic field vectors generated by each block
in the standard (
X
,
Y
,
Z
) coordinate system. Let
“ (0
ı
“<360
ı
) be the local strike of the
y
axis, measured clockwise from the North. As it is
shown below, this quantity is determined by the
local trend of the flow line. Then, the components
of the vector
F
in the local frame of reference
F
X
D
F
x
sin “
F
Y
D
F
x
cos “
F
Z
D
F
z
(5.51)
:
It should be noted that an anomalous field vec-
tor
F
does not have
y
-component in the (
x
,
y
,
z
)
reference frame of a magnetized prism, whereas
it has a non-zero
Y
-component with respect to the
standard (
X
,
Y
,
Z
) geomagnetic coordinate system.
Therefore, the total field anomaly,
T
,atan
observation point
r
is given by:
T
D
F
F
D
F
X
F
X
C
F
Y
F
Y
C
F
Z
F
Z
D
cos I
0
.F
X
cos D
0
C
F
Y
sin D
0
/
C
F
Z
sin I
0
(5.52)
Evaluation of derivatives in formulae (
5.49
)
and (
5.50
)issimple:
.Ÿ
n
C
1
Ÿ
n
/
2
R
2
@Z
n
@
z
D
—
nC1
—
n
r
n
.™
nC1
™
n
/
Ÿ
nC1
Ÿ
n
log
r
n
C
1
P
(5.53)
