Page 341 - Acquisition and Processing of Marine Seismic Data

332 6. DECONVOLUTION

FIG. 6.19 Schematic illustration of calculation and
application of deconvolution with optimum Wiener
filters. (*) denotes convolution.

2 3 2 3 2 3
a 0 a 1 a 2 … a n 1 1 a mixed, maximum or zero phase wavelet. The
h 0
method to calculate the minimum phase equiv-
6 7 6 7 6 7
a 1 a 0 a 1
alent of a wavelet is to compute the inverse of its
6 … a n 2 7 6 h 1 7 6 0 7
6 7 6 7 6 7
6 7 6 7 6 7
0
6 a 2 a 1 a 0 … a n 3 7 6 h 2 7 6 7 spiking deconvolution operator (Yılmaz, 2001).
Fig.6.20showsanapplicationofstatisticaland
6 7 6 7 6 7
6 7 6 7 6 7
: : : : : :
6 … 7 6 7 ¼ 6 7
deterministic Wiener spiking deconvolution to a
6 7 6 7 6 7
6 7 6 7 6 7
: : : : : : synthetic seismic trace consisting of a minimum
6 … 7 6 7 6 7
6 7 6 7 6 7
phase wavelet. Although deterministic deconvo-
6 7 6 7 6 7
:
: : :
6 7 6 : 7 6 7
4 … : 5 4 5 4 5 lution produces almost perfect results, as
h n 1 0 expected (Fig. 6.20C), it has very limited applica-
a n 1 a n 2 a n 3 … a 0
tion in seismic exploration because it requires the
(6.19)
source waveform to be known. Statistical decon-
In fact, Wiener spiking deconvolution mathe- volution today is a common technique in seismic
matically equals to least-squares inverse filter- processing and it produces satisfactory results,
ing. The difference between them is that a even if the source wavelet is not known, provid-
known source wavelet is used in the calculation ing that the seismogram is composed of a mini-
of the autocorrelation matrix of normal equa- mum phase wavelet (Fig. 6.20D). A similar
tions in the case of least-squares inverse filtering analysis is shown in Fig. 6.21, this time for a seis-
(implying a deterministic deconvolution), mogram composed of a mixed phase wavelet.
whereas this matrix is obtained from the seismic Forbothstatisticalanddeterministicapproaches,
trace itself in the case of Wiener spiking decon- Wiener spiking deconvolution yields unaccept-
volution (implying a statistical deconvolution). able results for mixed phase wavelets.
The amplitude spectrum of the Wiener spiking As a result, if the source waveform is not
deconvolution operator is approximately the minimum phase, spiking deconvolution can-
inverse of the amplitude spectrum of the source not convert it into a zero-lag spike. For
wavelet. This type of deconvolution can also be instance, wavelet w(t)¼( 1, 2) cannot be con-
used to derive the minimum phase equivalent of verted into a zero-lag spike (1, 0, 0) by spiking