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and the inverse discrete wavelet transform (IDWT)
is:
,
,
(5)
The discrete wavelet transform (DWT), using the
property of localization of wavelet bases has been
used as a powerful tool in filtering and separation
problems. The continuous wavelet transform (CWT)
exploits the upward continuation properties of the
field horizontal derivative and allows the location of
potential field singularities in a simple geometrical
manner (Fedi et al., 2004).
2.3 Thresholding
Separation is how to manipulate the wavelet
correlation coefficients produced by the DWT in
order to obtain the best residual-free data set, known
as smoothed out regional. Residuals in real data are
often seen as high-frequency or spike-like
components and predefined feature corresponds to
i.e. shallow micro-anomalies. With real data, there
are only two practical choices of thresholding: hard
or soft. With hard thresholding, all values of the
wavelet correlation coefficients below (or above,
depending on the application) the threshold value λ
are set to zero. In soft thresholding, the values
approach zero at a linear rate (Fedi et al., 2004).
The explicit difference between hard and soft
thresholding is when |x(t)| > λ. In the case |x(t)| ≤ λ,
λ for both hard and soft thresholding is zero. For
hard thresholding, λ is equal to x(t) but for soft
thresholding is determined by this equation:
sign(x(t))(|x(t)| − λ). Where x(t) is the value of the
wavelet correlation coefficient at some level
dependent observation points (Strang and Nguyen,
1996). Soft thresholding of these same data was
found to reconstruct the signal in a more continuous
form that did not induce obvious artefacts. This
same conclusion has been reached by other studies
(Donoho and Johnstone, 1994); (Moreau et al.,
1999) therefore, soft thresholding has been applied
to all the data of this study
3 GRAVITY DATA SEPARATION
TECHNIQUE
High frequency events are a drastic deviation from
the general trend of the local data in either frequency
content or amplitude or both (Fedi et al., 2000). In
the other words high frequency components is a
subjective feature of all real data. The perception of
what residual is and what it is not varies with the
intent of the end use of the data what may be
considered residual to one observer may be regional
to another. This leads to the realization that no
matter what the application is, a measured value will
always have some amount of unwanted signal. As a
result, the need for separating the unwanted portion
from the portion of interest is essential to all users
and is the motivational concept behind separation
(Leblanc and Morris, 2001).
Although separation methods have sound basis
for specific applications under specific conditions,
each has variable degrees of success when applied to
high frequency features such as aeromagnetic spike
anomalies. A data spike is a single point anomaly
whose magnitude is usually, but not necessarily, of
significant deviation from the trend of the data. It is
generally smaller in spatial extent and larger in
amplitude than the local trend of the geologically
sourced data. The ambiguity of this definition is a
result of the signal associated with non geologic
sources that cause the spike-like anomalies. These
sources include acquisition errors, levelling, latitude,
terrain, tides, drift etc. and shallow small anomalies.
Surface micro-anomalies create high-frequency
portion at signal. Sometimes the purpose of the
analysis is diagnosis of these shallow anomalies. In
such a case low pass filter damage useful
information of the data. Remember that random high
frequency signals can not always describe the
behaviour of gravity residuals; so the arithmetic is
used to remove such high frequency features have
limited application in practice.
Maximum (abs( main signal-long wavelength
signal produced by SWT))= Maximum Residual
(MR)
By applying the discrete stationary wavelet filter and
soft thresholding all high frequency effects are
removed. So, regression of the effects of surface
anomalies should be maximal. Continuity of soft
thresholding reduces the high frequency content of
signal that is occurred by growing scales until the
overall form of the signal has not been deformed.
An optimal separation also let the effect of
deeper anomalies that were not seen because of
micro anomaly is now evident. Minimum deviation
of processed signal under wavelet thresholding
occurs at lower scales. Going to larger scales causes
separation of larger residuals. The signal to noise
ratio or Regional to Residual Ratio (RRR) also
decreases with increasing scale. Since the residual
amplitude is in the denominator of the ratio, small
EffectiveResidualandRegionalGravityAnomalySeparation-Using1-D&2-DStationaryWaveletTransform
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