Page 478 - Acquisition and Processing of Marine Seismic Data

10.2 NMO STRETCHING 469

the rocks using Lame constants λ and μ as well as ð ε δÞ
η ¼ (10.15)
bulk modulus k, and he denoted a “weak anisot- ð 1+2δÞ
ropy” if these parameters are small enough (e.g.,
≪1). For specific cases, ε ¼ δ is known as ellipti- In general, η is positive, and larger values result
cal anisotropy, and δ < ε indicates isotropic in a decrease in NMO correction time.
layers. Thomsen (1986) also proposed that most
elastic media can be considered weakly aniso-
tropic, which can be mathematically expressed 10.2 NMO STRETCHING
by a certain anisotropy parameter δ. Tsvankin
(1995) suggested an NMO velocity equation As a result of the dynamic nature of NMO
for a small-spread approximation depending correction, a frequency distortion occurs in far
on the phase velocity in a homogenous aniso- offsets along the shallow parts of NMO cor-
tropic VTI medium. Following this approxima- rected CDP gathers, known as NMO stretching.
tion, Alkhalifah and Tsvankin (1995) replaced As schematically indicated in Fig. 10.5A and B,
dip angle with ray parameter p (horizontal slow- the waveform with a dominant period of t A will
ness) corresponding to the zero-offset reflection be stretched into a larger period of t B after NMO
to get correction. Therefore, the amplitude spectrum
of an NMO corrected wavelet has lower fre-
V NMO 0ðÞ quency content since the time length of the
(10.13)
V NMO pðÞ ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
1 p V 2 0 ðÞ wavelet is longer than that in the zero-offset
NMO
trace. This is clear in the synthetic data example
where V NMO (0) is the NMO velocity in case of a in Fig. 10.9. The close-ups of NMO corrected
horizontal (zero dip) reflector. Using anisotropy gather in Fig. 10.9C and D compare the differ-
parameters defined by Thomsen (1986), ence of dominant periods of the wavelets for
Tsvankin and Thomsen (1994) suggested a long near and far offset traces after correction.
offset travel time equation which depends only According to their corresponding mean ampli-
on two parameters, V NMO (0) and η, for any ori- tude spectra in Fig. 10.9E and F, the dominant
entation of the CMP line with respect to the frequency of the seismic wavelet is reduced
dip plane of the reflector, from 60 to 30 Hz for far offset traces. This situa-
tion also lowers the resolution of the final stack
x 2 sections. Shatilo and Aminzadeh (2000) propose
2
2
t xðÞ ¼ t 0ðÞ +
V 2 0 ðÞ that this low-pass filtering effect of the NMO
NMO
2ηx 4 correction is analogous to the effect of frequency
2 2 2 2
V 0 ðÞ t 0ðÞV 0 ðÞ +1 + 2ηÞx dependent attenuation, which therefore adver-
ð
NMO NMO
sely affects the attenuation calculations from
(10.14)
the surface seismic data.
In Eq. (10.14), the anisotropic coefficient η When two reflection hyperbolas intersect due
defines the degree of deviation from hyperbolic to the increase in velocity with depth, the
moveout for a given V NMO (0) and t(0). stretching leads to much more drastic results:
η ¼ 0 represents an elliptical medium and the for the intersecting events on the CDP gathers,
moveout is hyperbolic. In practice, anisotropic conventional NMO correction completely fails
NMO correction can be applied to long offset since it assigns a unique time position to every
seismic data, where only η is needed to describe time sample in the input traces (Shatilo and
the anisotropy, which can be defined in terms of Aminzadeh, 2000). However, for the area of
Thomsen (1986)’s anisotropic parameters as intersection, an uncorrected time sample may