Marine Magnetic Anomalies - Quantitative Plate Tectonics - page 169

Fig. 5.22 Fourier filtering of magnetic anomaly data in
the phase preceding forward modelling. In this example,
wavenumbers below 0.01 km 1

The original signal ( dashed line ) had an almost constant
offset T 0 Š 58.3 nT. The filtered magnetic anomalies

have zero average and are more appropriate for modelling have been filtered away.

7. Parseval Identity . The “energy” of a real func-
tion is invariant under Fourier transformation:
C Z

from the magnetic sources. This transformation
is called upward continuation and is useful when
aeromagnetic data observed at different altitudes
must be merged, or when an investigator wants
to attenuate the short-wavelength components
of the signal and enhance the complementary
range, which is a form of data filtering.
For example, one could wish to enhance
the anomalies associated with deep sources,
which have longer wavelengths, with respect
to the short-wavelength anomalies generated by
near-surface sources. The theoretical basis for
upward continuation is the Green's third identity
( 4.70 ) . Let us assume to know the potential V
everywhere on the plane z D z 0 .Wewantto
calculate the potential at some point with greater
elevation ( x , y , z 0 - z ). To this purpose, let us
choose a harmonicity region R as in Fig. 5.23 ,
and assume that all magnetic sources are located
at altitude z > z 0 .
In this case, we can separate the integral ( 4.70 )
in two parts, one that performs integration over
the hemisphere surface, and one that operates on
acircleintheplane z D z 0 (Fig. 5.23 ). Therefore,

C Z

1
2

2 dx D 2 dk (5.84)

j f.x/ j j F.k/ j

1 1

One of the simplest applications of Fourier's
transform is the filtering of magnetic anomaly
data before initiating the forward modelling step.
Figure 5.22 illustrates an example where the
original data had a common offset of a few
tens nT. The signal was first transformed to the
Fourier domain, then the low wavenumbers (long
wavelengths) that were responsible for the signal
displacement were removed from the spectrum.
Finally, the filtered Fourier domain representa-
tion was converted back to the space domain
through an inverse transform. The resulting signal
is clearly more suitable for the subsequent inter-
pretation phase. Soon we will introduce a more
complex application of the Fourier transform.
Let us consider now the problem of
transforming the potential field data observed at
some altitude to a different surface, more distant