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| Nonhyperbolic reflection moveout of -waves:
An overview and comparison of reasons | |
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Next: ANISOTROPY VERSUS LATERAL HETEROGENEITY
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Previous: Curved reflector beneath isotropic
For a dipping curved reflector in a homogeneous VTI medium,
the ray trajectories of the incident and reflected waves are straight,
but the location of the reflection point is no longer controlled by
the isotropic laws. To obtain analytic expressions in this
model, we use the theoremthat connects the derivatives of the
common-midpoint traveltime with the derivatives of the one-way
traveltimes for an imaginary wave originating at the reflection point
of the zero-offset ray. This theorem, introduced for the
second-order derivatives by Chernjak and Gritsenko (1979),
is usually called the normal incidence point (NIP) theorem
(Hubral, 1983; Hubral and Krey, 1980). Although the original proof did not
address anisotropy, it is applicable to anisotropic media because it is based
on the fundamental Fermat's principle. The ``normal incidence'' point
in anisotropic media is the point of incidence for the zero-offset ray
(which is, in general, not normal to the reflector). In Appendix A, we
review the NIP theorem, as well as its extension to the high-order
traveltime derivatives (Fomel, 1994).
Two important equations derived in Appendix A are:
where is the one-way traveltime of the direct wave propagating from
the reflection point to the point at the surface . All
derivatives in equations (66) and (67) are evaluated
at the zero-offset ray. Both equations are based
solely on Fermat's principle and, therefore, remain valid in any type of
media for reflectors of an arbitrary shape, assuming that the traveltimes
possess the required order of smoothness. It is especially convenient
to use equations (66) and (67) in
homogeneous media, where the direct traveltime can be expressed explicitly.
To apply equations (66) and (67) in VTI
media, we need to start with tracing the zero-offset ray. According to
Fermat's principle, the ray trajectory must correspond to an extremum of
the traveltime. For the zero-offset ray, this simply means that the
one-way traveltime satisfies the equation
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(68) |
where
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(69) |
Here, the function describes the reflector shape, and is the ray angle given by the trigonometric relationship
(Figure 5)
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(70) |
Substituting approximate equation (5) for the group velocity
into equation (69) and linearizing it with respect to the
anisotropic parameters and , we can solve equation
(68) for , obtaining
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(71) |
or, in terms of ,
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(72) |
where is the local dip of the reflector at the reflection
point . Equation (72) shows that, in VTI media, the
angle of the zero-offset ray differs from the reflector dip
(Figure 5). As one might expect, the relative
difference is approximately linear in Thomsen anisotropic parameters.
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nmoray
Figure 5. Zero-offset reflection from a
curved reflector beneath a VTI medium (a scheme). Note that the ray angle
is not equal to the local reflector dip .
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Now we can apply equation (66) to evaluate the second term of the
Taylor series expansion (22) for a curved
reflector. The linearization in anisotropic parameters leads to the expression
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(73) |
which is equivalent to that derived by Tsvankin (1995). As in
isotropic media, the normal-moveout velocity does not depend on the
reflector curvature. Its dip dependence, however, is an important
indicator of anisotropy, especially in areas of conflicting dips
(Alkhalifah and Tsvankin, 1995).
Finally, using equation (67), we
determine the third coefficient of the Taylor series. After
linearization in anisotropic parameters and lengthy algebra, the
result takes the form
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(74) |
where
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(75) |
and the coefficient is defined by equation (58). For zero curvature (a plane reflector) , and
the only term remaining in equation (75) is
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(76) |
If the reflector is curved, we can rewrite the
isotropic equation (63) in the form
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(77) |
where the normal-moveout velocity and the quantity are given by
equations (73) and (75), respectively.
Equation (77) approximates the nonhyperbolic moveout
in homogeneous VTI media above a curved reflector. For small
curvature, the accuracy of this equation at finite offsets
can be increased by modifying the denominator in the quartic term similarly
to that done by Grechka and Tsvankin (1998) for VTI media.
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| Nonhyperbolic reflection moveout of -waves:
An overview and comparison of reasons | |
|
Next: ANISOTROPY VERSUS LATERAL HETEROGENEITY
Up: CURVILINEAR REFLECTOR
Previous: Curved reflector beneath isotropic
2014-01-27