Page 239 - Acquisition and Processing of Marine Seismic Data

230 4. FUNDAMENTALS OF DATA PROCESSING

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(ii) cutting out the high-frequency components z , we shift the time series one sampling rate
with a low-pass filter along the time axis.
(iii) decreasing the group interval The z transform has several applications in
signal processing. The Fourier transform can
The first two approaches are practically infea- be done using the z transform. The convolution
sible, since the first one will change the inclina-
process can be performed with a multiplication
tion of the events in the data, and the second will
process by the z transform. The multiplication of
remove the high-frequency information. The
the z transforms of two time series equals the z
third approach requires increasing the total
transform of their convolution. Let’s consider
number of recording channels for a standard
two series as x(t) ¼ (2, 1) and y(t) ¼ (3, 2, 1).
streamer length during the acquisition. If this
Their convolution is x(t)*y(t) ¼ (6, 1, 4, 1).
is not possible, the group interval can be
The z transforms of both series are x(z) ¼
reduced by using a data-dependent interpola- 2
(2 z) and y(z) ¼ (3 + 2z z ). The multiplica-
tion method, which provides extra synthetic
tion of both z transforms gives x(z) y(z) ¼
traces between the successive channels. 2 3
(6 + z 4z + z ), and the coefficients of the resul-
tant polynomial define the inverse z transform,
z{x(z) y(z)} 1 ¼ (6, 1, 4, 1), which corresponds
4.7 Z TRANSFORM
to the convolution of the x(t) and y(t) series.
The z transform is also used in wavelet pro-
The z transform converts discrete time series
cessing. The inverse of the seismic wavelet,
into a polynomial expressed by the powers of which is used in deterministic deconvolution
the variable z. That is, the z transform of a time applications (Section 6.3), can be obtained by
series is a polynomial of z, and the coefficients of the z transform. The relation between the wave-
z are the discrete samples of the input series. The let w(t) and its inverse wtðÞ is given by
discrete Fourier spectrum of x(t) is given by
∗ (4.25)
wtðÞ wtðÞ ¼ δ tðÞ
∞
X iωnΔt
x ωðÞ ¼ x n e (4.20) where δ(t) is the Dirac delta function and its z
∞ transform is z{δ(t)} ¼ 1. Hence, we get the
and if we expand the summation, we get inverse
x ωðÞ ¼ x 0 + x 1 e iωΔt + x 2 e iω2Δt + ⋯ (4.21) wzðÞ wzðÞ ¼ 1 (4.26)
Substituting z¼e iωΔt into Eq. (4.20), we get wzðÞ ¼ 1=wzðÞ (4.27)
∞
X n The inverse of the z transform of any function
xzðÞ ¼ x n z (4.22) can be obtained by Taylor’s series expansion.
∞
Taylor’s series approximation of an f(x) function
If the elements of the discrete x n series are in the neighborhood of zero is known as the
MacLaurin series and can be found in the fol-
(4.23)
x n ¼ x 0 ,x 1 ,x 2 ,x 3 ,x 4 ,…
lowing form of infinite series expansion:
then its z transform is expressed as
0
00
0
000
f 0ðÞ f 0ðÞ 2 f ðÞ 3
fxðÞ ¼ f 0ðÞ + x + x + x + ⋯
2
3
xzðÞ ¼ x 0 + x 1 z + x 2 z + x 3 z + …: (4.24) 1! 2! 3!
(4.28)
Thepowerofthevariablezcorrespondstothetime
lag of the discrete time samples of the x n series. where f (0) denotes the first derivative of f(x)
0
Therefore, if we multiply the transformation by around zero, f (0) denotes the second derivative,
00