Page 336 - Acquisition and Processing of Marine Seismic Data

6.3 SPIKING DECONVOLUTION 327

TABLE 6.3 Deconvolution Output Obtained by Convolving Input Minimum Phase Wavelet w(t) ¼ (2, 1) With the
First Four Terms of the Inverse Filter Operator h(t) ¼ (1/2, 1/4, 1/8, 1/16)

2 21 Output
1/16 1/8 1/4 1/2 1
1/16 1/8 1/4 1/2 0
1/16 1/8 1/4 1/2 0
1/16 1/8 1/4 1/2 0
1/16 1/8 1/4 1/2 0.0625

better result, we obtain the output in Table 6.3, TABLE 6.4 Deconvolution Output Obtained by
which is much closer to a spike. As the num- Convolving Maximum Phase Input Wavelet w(t) ¼ ( 1, 2)
ber of involved terms increases, the output With First Three Terms h(t) ¼ ( 1, 2, 4) of the
Inverse Filter Operator
approaches to an ideal spike. However, the
resultant wavelet will never be an exact spike, 21 2 Output
because the operator must always be truncated
4 2 1 1
to make a finite length time series.
The energy of the wavelet in these examples is 4 2 1 0
front-loaded, that is, the wavelet is minimum 4 2 1 0
phase, and therefore it has a stable and conver- 4 2 1 8
gent inverse and the inverse filtering is success-
ful in converting this wavelet into a spike. What
if the wavelet is not minimum phase? If we con- TABLE 6.5 Deconvolution Output Obtained by
sider a maximum phase wavelet w(t)¼( 1, 2),
Convolving Maximum Phase Input Wavelet w(t) ¼
its z transform is ( 1, 2) With First Four Terms h(t) ¼ ( 1, 2, 4, 8)
of the Inverse Filter Operator
wzðÞ ¼ 1+2z
21 2 Output
The inverse of this wavelet e wzðÞ obtained again
by a MacLaurin series expansion is 8 4 2 1 1
1 8 4 2 1 0
e wzðÞ ¼
1+2z 8 4 2 1 0
4
3
2
¼ 1 2z 4z 8z 16z + ⋯⋯:
8 4 2 1 0
The coefficients of variable z ( 1, 2, 4, 8, 8 4 2 1 16
16,…) provide the deconvolution operator
h(t). However, this time we get a divergent series
of an infinite number of samples. When we con- higher filter coefficients of the operator into
volve them with the input wavelet w(t)¼( 1, 2) the deconvolution process, such as the first four
after truncating to use the first three terms h(t)¼ terms, makes the output much more unfavor-
( 1, 2, 4), we get the deconvolved output able (Table 6.5). The reason for this is that the fil-
shown in Table 6.4. The result is quite different ter coefficient values increase with time, and if
from an ideal spike, even though it is less spiky we apply them to the input after truncation,
than the input. Furthermore, incorporating the larger terms are excluded since it is a divergent