Geology Reference
In-Depth Information
where
A
refers to a gravitational or magnetic field and is
a function of (
x
,
y
,
z
).
In the case of a two-dimensional field there is no varia-
tion along one of the horizontal directions so that
A
is a
function of
x
and
z
only and equation (6.23) simplifies to
the Fourier transform (Section 2.3). If, then, instead
of the terms (
a
cos
kx
+
b
sin
kx
) in equation (6.25) or
(
A
0
sin
kx
) in equation (6.27), the full Fourier spectrum,
derived by Fourier transformation of the field into
the wavenumber domain, is substituted, the results of
equations (6.28) and (6.29) remain valid.
These latter equations show that the field measured
at the surface can be used to predict the field at any
level above or below the plane of observation.This is the
basis of the upward and downward field continuation
methods in which the potential field above or below
the original plane of measurement is calculated in order
to accentuate the effects of deep or shallow structures
respectively.
Upward continuation methods
are employed in gravity
interpretation to determine the form of regional gravity
variation over a survey area, since the regional field is as-
sumed to originate from relatively deep-seated struc-
tures. Figure 6.24(a) is a Bouguer anomaly map of the
Saguenay area in Quebec, Canada, and Fig. 6.24(b) rep-
resents the field continued upward to an elevation of
16 km. Comparison of the two figures clearly illustrates
how the high-wavenumber components of the observed
field have been effectively removed by the continuation
process. The upward continued field must result from
relatively deep structures and consequently represents
a valid regional field for the area. Upward continuation
is also useful in the interpretation of magnetic anomaly
fields (see Chapter 7) over areas containing many
near-surface magnetic sources such as dykes and other
intrusions. Upward continuation attenuates the high-
wavenumber anomalies associated with such features
and enhances, relatively, the anomalies of the deeper-
seated sources.
Downward continuation
of potential fields is of more re-
stricted application. The technique may be used in the
resolution of the separate anomalies caused by adjacent
structures whose effects overlap at the level of observa-
tion. On downward continuation, high-wavenumber
components are relatively enhanced and the anomalies
show extreme fluctuations if the field is continued to a
depth greater than that of its causative structure. The
level at which these fluctuations commence provides an
estimate of the limiting depth of the anomalous body.
The effectiveness of this method is diminished if the
potential field is contaminated with noise, as the noise is
accentuated on downward continuation.
The selective enhancement of the low- or high-
wavenumber components of potential fields may be
achieved in a different but analogous manner by the
2
2
∂
∂
A
x
∂
∂
A
z
+
=
0
(6.24)
2
2
Solution of this partial differential equation is easily per-
formed by separation of variables
kz
A
x z
,
)
=
a
cos
kx
+
b
sin
kx
)
e
(
(
(6.25)
k
where
a
and
b
are constants, the positive variable
k
is the
spatial frequency or wavenumber,
A
k
is the potential
field amplitude corresponding to that wavenumber and
z
is the level of observation. Equation (6.25) shows that a
potential field can be represented in terms of sine and co-
sine waves whose amplitude is controlled exponentially
by the level of observation.
Consider the simplest possible case where the two-
dimensional anomaly measured at the surface
A
(
x
, 0) is a
sine wave
(
)
=
Ax
,
0
A
sin
kx
(6.26)
0
where
A
0
is a constant and
k
the wavenumber of the sine
wave. Equation (6.25) enables the general form of the
equation to be stated for any value of
z
kz
Axz
,
)
=
(
A
sin
kx
e
(
)
(6.27)
0
The field at a height
h
above the surface can then be
determined by substitution in equation (6.27)
-
kh
Ax h
,
-
)
=
(
A
sin
kx
e
(6.28)
(
)
0
and the field at depth
d
below the surface
Axd
,
)
=
(
A
sin
kx
e
kd
(
)
(6.29)
0
The sign of
h
and
d
is important as the
z
-axis is normally
defined as positive downwards.
Equation (6.27) is an over-simplification in that a
potential field is never a function of a single sine wave.
Invariably such a field is composed of a range of
wavenumbers. However, the technique is still valid as
long as the field can be expressed in terms of all its com-
ponent wavenumbers, a task easily performed by use of
