Geoscience Reference
In-Depth Information
These gradients emphasise changes in the measured
parameter, but only in the direction in which they are
calculated, so changes in geology oriented perpendicular
to the gradient direction produce the strongest gradient
response. They can be combined using Pythagoras
the random noise, spikes and step, are demonstrated
clearly. Finally, note the very weak responses associated
with gradual changes, i.e. the linear gradient and broad
smooth sections
Derivatives of any order can be computed, with the
advantage that the responses become narrower as the order
increases. A major disadvantage of all, but especially higher
order, derivatives is that they are more sensitive to noise;
but this property can be used to investigate the high-
frequency noise characteristics of a data series. For
example, residual errors not apparent in levelled data
may be very obvious in its derivatives.
Although somewhat counter-intuitive, it is possible to
compute fractional derivatives, e.g. derivatives with order
ing the order of the derivative to the point where noise
levels become unacceptable allows spatial resolution to be
increased to the limits imposed by the quality of the data.
'
r) at location (x
n
, y
n
) in the resultant
direction (r) of its horizontal component as follows:
∂
P/
∂
s
∂
2
+
2
∂
P
P
P
∂
r
ð
x
n
y
n
Þ¼
ð
2
:
4
Þ
,
∂
∂
x
y
∂
The total horizontal gradient represents the maximum
gradient in the vicinity of the observation point and, there-
fore, is perpendicular to contours of the measured param-
eter (
Fig. 2.24a
). In the same way, the total gradient
(derivative) of P (
r) can be obtained from the deriva-
tives (gradients measured)
∂
P/
∂
in all
three perpendicular
directions:
s
∂
2
+
2
+
2
Frequency/wavelength
Enhancement of particular wavelength or frequency vari-
ations in a data series is applied to virtually all types of
geophysical data. As with gradient
filtering, this is possible
in either the time/spatial or the frequency domains. A
lter
designed to remove (attenuate) all frequencies above a
certain cut-off frequency, and allow only those frequencies
lower than the cut-off frequency to pass through to its
output, is known as a low-pass filter. Conversely, a filter
that retains only the shorter wavelengths (higher frequen-
cies) in its output is a high-pass filter. Filters that either
remove (attenuate) or pass frequencies within a defined
frequency interval are called band-stop and band-pass
filters, respectively. As demonstrated with a low-pass filter
in
Fig. 2.21
, wavelength/frequency
¼
∂
P
∂
P
P
P
∂
∂
∂
x
n
y
n
ð
2
:
5
Þ
,
r
∂
x
y
z
∂
Its most common application is found in gravity and
magnetics where it is known as the analytic signal (see
The gradient of the gradient data, i.e. the derivative of
the
first derivative, may also be computed and is known as
the second derivative. This is the curvature of the field of
the parameter being measured. Both first and second hori-
zontal derivatives in the along-line direction are shown in
wavelength) changes and are very useful for mapping
geological contacts and shallow features (see
Section
more localised than the broader response of the original
data (compare the relevant curves in
Fig. 2.25
and also see
of interference from adjacent contacts etc. so the resolution
of the data is increased. The smoothly varying intervals in
Fig. 2.25
show that the gradient response can have a
complicated relationship with the form of the input. The
gradient is often asymmetrical and has positive and nega-
tive parts making quantitative analysis less intuitive than
when working with the original data. Also, calculated
derivatives are very sensitive to noise, especially the
Y-derivatives as they are measured across-line where
the samples are more widely spaced. Amplification of the
abruptly changing sections of the test dataset, such as
filters can be enacted in
either the frequency domain or as a convolution in the
space/time domains.
The actions of 1D band-pass, high-pass and low-pass
filters are demonstrated in
Fig. 2.25
. Note how the sections
comprising random variations and spikes are not com-
pletely removed by any of the
filters. This is because
random variations and spikes contain all frequencies (see
Appendix 2
)
and so some components always remain after
frequency/wavelength
filtering. The low-pass
filter has
been designed to pass frequencies lower than those com-
prising the central sinusoidal section of the dataset, and the
cut-off frequency of the high-pass filter is greater than the
highest frequency in this section. The band-pass filter
passes frequencies between the cut-off frequencies of the
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