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 | Random noise attenuation using - regularized nonstationary autoregression |  |
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We first consider a seismic section
that consists of a single
linear event with the slope
and constant amplitude.
The frequency domain representation of
is given by
 |
(1) |
where
is the wavelet spectrum,
is the temporal frequency,
and
is the spatial variable. We assume
, where
,
is the number of traces in the whole section.
The relationship between the n-th trace and (n-1)-th trace can be easily
shown as
 |
(2) |
where
. This recursion is a first-order
differential equation also known as an AR model of order 1 and represents a
single complex-valued harmonic (Bekara and van der Baan, 2009). If there are
linear
events in x-t domain, we can have a difference equation of order
(Sacchi and Kuehl, 2001)
 |
(3) |
The recursive filter
can be found for predicting a
noise-free superposition of complex harmonics. Considering seismic data with additive
random noise and non-causal prediction with order
which includes both forward
and backward prediction equations (Spitz, 1991; Naghizadeh and Sacchi, 2009), we can obtain
 |
(4) |
where
is a complex noise sequence. Canales (1984) argues
a causal estimate of signal
is the predictable part of data obtained by an AR model. This operation is usually
called
-
deconvolution (Gulunay, 1986). Noise-free events that are linear in
the
-
domain manifest as a superposition of harmonics in the
-
domain and these
harmonics can be perfectly predicted using AR filter. If seismic events are not linear,
or the amplitudes of wavelet are varying from trace to trace, they no longer follow
Canales’s assumptions (Canales, 1984). One needs to perform
-
deconvolution over a
short sliding window in time and space. This leaves the choice of window parameters
(window size and length of overlapping between adjacent windows). Bekara and van der Baan (2009)
discuss some limitations of conventional
-
deconvolution in detail.
 |
 |
 |
 | Random noise attenuation using - regularized nonstationary autoregression |  |
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Next: - domain regularized nonstationary
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2013-11-13