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and the mathematical representation of the
filter, and its
parameters, is the operator. The dataset is the input to the
filter which alters or transforms the data, and the output is
input and the output depends on characteristics of the
lter
and is known as the
lter
and indeed, there exists a vast literature describing digital
filters for all types of numerical data. In particular, a large
number of
filters have been developed by the petroleum
industry for enhancing seismic data (see
Section 6.5.2
).
Fortunately, in mineral geophysics there are a small
number of commonly used
filters, some method speci
c,
that are highly effective for most requirements. They are
described in
Section 2.7.4.4
.
There are two basic ways of mathematically enacting a
filter on a dataset, using transforms and using convolution
(
Fig. 2.21
). Most of the common filtering operations can be
enacted in either way, although for some types of filtering
it is easier to understand the process in one form rather
than the other. A third class of filter is based on the
statistical properties of the data. A common example is
the median
'
is response. Some examples of
elementary
filters are: change a dataset
s polarity by simply
multiplying every data point in the dataset by
'
-
1; slightly
more complex, convert the data to an equivalent series
having a mean value of zero by subtracting the overall
mean of the dataset from each data point.
Filters can be applied to 1D and 2D data. In principle,
one can design a filter to do almost anything to the data,
a)
Input
Output
filter (see Smoothing in
Section 2.7.4.4
), which
simply determines the median value of a moving window
of data points and uses this as the
filtered value.
Filter
b)
Input
Transforms
A data series exists in a particular domain (see
Section 2.2
)
and it is often convenient to transform it into a different
domain to facilitate its analysis and
filtering. There are
several commonly used transforms, but the most import-
ant one is the Fourier transform (see
Appendix 2
). Filtering
Output
Inverse
filter
Figure 2.20
Schematic illustration of a
filter with a
blocking function, i.e. it converts the data to a step-like form. (b) The
inverse filter which reverses the action of the filter in (a).
filter. (a) A
Time/space domain
Frequency domain
Input
Fourier transform
0
Time/location
Inverse Fourier transform
Frequency
0
Multiply
spectra
Convolution
Filter
operator
Fourier transform
0
Time/location
Inverse Fourier transform
Frequency
0
Output
Fourier transform
0
Time/location
Inverse Fourier transform
Filter
cutoff
frequency
Frequency
0
Figure 2.21
Filtering of a 1D dataset in the space/time domain and the frequency domain. The filter is a low-pass frequency filter. Redrawn,
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