Page 240 - Acquisition and Processing of Marine Seismic Data

4.8 HILBERT TRANSFORM 231

etc. For example, let’s find out the inverse of the applying a π/2 rad phase difference to each fre-
wavelet w(t) ¼ (2, 1). The z transform of the quency component of the input function. The
wavelet is w(z) ¼ (2 + z) and its inverse is theoretical basis of the transform depends on
wzðÞ ¼ 1= 2+ zÞ. In order to determine the the definition of analytical function. The time-
ð
inverse wzðÞ, derivatives of 1/(2 + z) are calcu- dependent analytical function u(t) is a complex
lated. The first four derivatives of function and can be defined as
wzðÞ ¼ 1= 2+ zÞ are
ð
utðÞ ¼ xtðÞ + iy tðÞ (4.29)
1 2
0 00 000
z
w zðÞ ¼ w zðÞ ¼ w ðÞ where x(t) and y(t) are real and imaginary com-
2 3
ð 2+ zÞ ð 2+ zÞ ponents, respectively, and are known as the
6 24
000 0
z
¼ w ðÞ ¼ Hilbert transform pair (Fig. 4.17A). Bracewell
4 5
ð 2+ zÞ ð 2+ zÞ (1965) suggested that the imaginary component
y(t) is obtained by the Hilbert transform of the
Substituting z ¼ 0 into the derivatives of wzðÞ,
we can obtain the coefficients in Eq. (4.28). Using real component x(t)by
∗
a MacLaurin series expansion, the z transform of ytðÞ ¼ htðÞ xtðÞ (4.30)
the inverse of wavelet w(t) ¼ (2, 1) can be
obtained as where * denotes convolution, h(t) is the Hilbert
transform operator (Fig. 4.17B) defined by
1 1 1 2 1 3 1 4
wzðÞ ¼ z + z z + z ⋯ 1
2 4 8 16 32 htðÞ ¼ (4.31)
πt
Getting the coefficients of z, the inverse of the
wavelet w(t) ¼ (2, 1) can be found as and its Fourier transform H(ω) is described as
wtðÞ ¼ 1=2, 1=4, 1=8, 1=16, 1=32, …Þ. (Fig. 4.17C):
ð
i ω > 0
8
>
>
<
4.8 HILBERT TRANSFORM H ωðÞ ¼ isgn ωðÞ ¼ +i ω < 0 (4.32)
>
>
:
0 ω ¼ 0
The Hilbert transform is a useful tool to
obtain extra information from the seismic data, where sgn(ω) is the signum function. Substitut-
known as complex trace attributes, explained ing Eq. (4.31) into Eq. (4.29), we get (Barnes,
in Section 12.5. It is a mathematical process 2007):

FIG. 4.17 (A) Analytical signal, (B) Hilbert transform operator, (C) Fourier transform of (B).