Geophysical data acquisition, processing and interpretation - Geophysics for the Mineral Exploration Geoscientist - page 41

using transforms involves applying a forward transform to
the dataset, transforming it into the new domain. Next, the
filtering operation is enacted by modifying the transformed
data in some desired way. Applying the inverse transform
returns the modi ed transformed data series to its original
domain as the filtered data series.
In the case of the Fourier transform, the data are trans-
formed to a series of component sine waves of different
frequencies, each represented in terms of its amplitude and
phase, i.e. the amplitude and phase spectra; these depend
upon the nature of the original data series (see Appendix 2 ) .
This is known as the frequency domain (also referred to as
the Fourier domain), and represents an alternative to the
time and spatial domains in which most data are recorded.
The filter operator is defined by its frequency spectrum, in
the same way as the transformed dataset, and the two
amplitude spectra are multiplied and their phase spectra
added to obtain the spectra of the

Input (unfiltered data)

1 2 4 7 9 9 7 4 2 1

1/3 1/3 1/3 Filter operator

Filter
operator
'slides'
past input
data series

1/3 1/3 1/3

1/3 1/3 1/3

1/3 1/3 1/3

1/3 1/3 1/3

1/3 1/3 1/3

1/3 1/3 1/3

1/3 1/3 1/3

filtered output. Filtering
attenuates or ampli es particular frequencies, and/or
changes their phases. Recombining the component sine
waves transforms the filtered spectra back to the original
domain to form the filtered data series. Frequency domain
filtering of both 1D and 2D data is possible. Filtering of a
1D dataset is illustrated schematically in Fig. 2.21 . In this
case the filter removes all frequencies above a de ned cut-
off frequency but does not affect phase, i.e. it is a form of
wavelength/frequency filter (see Frequency/wavelength in
Section 2.7.4.4 ).

(1/3 x 1) (1/3 x 2) (1/3 x 4) (1/3 x 7) (1/3 x 9) (1/3 x 9) (1/3 x 7) (1/3 x 4)

(1/3 x 2)

+ + + + + + + +

(1/3 x 2) (1/3 x 4) (1/3 x 7) (1/3 x 9) (1/3 x 9) (1/3 x 7) (1/3 x 4)

+ + + + + + + +

(1/3 x 4) (1/3 x 7) (1/3 x 9) (1/3 x 9) (1/3 x 7) (1/3 x 4) (1/3 x 2) (1/3 x 1)

= = = = = = = =

2.3 4.3 6.7 8.3 8.3 6.7 4.3 2.3

Output (filtered data)

Figure 2.22 Convolution in 1D with a three-point filter kernel.

See text for details.

Convolution
A common form of filtering, familiar to most readers,
involves computing the running (moving) average of a
groupofdatapoin .Con idera1Ddatase ie .
Obtaining the three-point running average of the data
involves summing three consecutive data points and
dividing by the number of points (3). This is a

created. The process is known as convolution and is
illustrated for 1D data in Fig. 2.22 . Itisalsoreferredto
as filtering in the time/spatial domain.
The example shown above has coefficients of equal
value, but an unlimited variety of

filters can be produced

by varying the number of cients and their indi-

vidual values. For example, to change the polarity of every
data point (mentioned earlier) involves the very simple,
one coef cient operator {

filter coef

filter

operator de cients of equal

value, in this case 1/3, and denoted as the series {1/3,
1/3, 1/3}. The set of filter coef cients is also known as
the filter kernel. It is applied to a subset or window of
three data points in the data series by multiplying each
filter coef cient with its respective data point, and sum-
ming the three multiplied points to obtain the new
filtered (averaged) output value. The output is assigned
to the point/location at the centre of the window, the
operator moved to an adjacent position to window the
next consecutive group of data points, and the process
repeated until a new, filtered version of the dataset is

ned by a series of coef

- 1}; the gradient ( (first derivative)

filter has the coef cients {

1, 1}; and the curvature (second
derivative) filter has the coef cients {1,

-

2, 1}.
Convolution can also be implemented on 2D data
( Fig. 2.23 ), in which case the filter operator has the form
of a matrix which is progressively moved through the
dataset and the new filtered value (P Output ) is assigned to
the centre point of the window. The simple three-point
running average filter above would then have the form of a
3

-

3 matrix of nine coefficients each equal to the inverse of
their average value, i.e. 1/9.