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![]() | Angle gathers in wave-equation imaging for transversely isotropic media | ![]() |
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Using equation (5) we evaluated angle gathers as a function
of offset and midpoint wavenumbers for a given frequency. We tested
such mapping for various models using different strengths of
anisotropy as we varied
,
, and the NMO velocity
.
Figures 2-3 show contour
density plots of angle as a function of offset and
midpoint wavenumbers, for a 60-Hz frequency slice. In Figure 2 the
medium is isotropic, with a velocity of 2 km/s. Clearly, for
, the angle is zero regardless of the midpoint
wavenumber, which is expected, because for zero-offset the scattering or
opening angle is equal to zero. Also, we observe that angles
decrease with dip (or
) for a given offset wavenumber, which is also
expected, because for any offset a scattering angle becomes zero in the case of a
vertical reflector. The areas given in white in the Figures 2-5
and throughout correspond to regions where the
or
become complex, and thus represent evanescent waves.
AnglesEta0
Figure 2. Constant-depth constant-frequency (60 Hz) slice mapped to reflection 2opening angles for an isotropic medium with velocity equal to 2 km/s. Zero-offset wavenumber maps to zero (normal incidence) angle. The four blank corners represent evanescent regions. 2Negative angles correspond to a switch in the source-receiver direction, and thus, the result is symmetric based on the principal of reciprocity |
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In anisotropic media, as illustrated in Figure 3,
for
equal to 0.1 and 0.3,
the angles decrease with dip for a constant offset wavenumber faster
than in the isotropic case. In the example, considering that
is lower in the
anisotropic models, the higher horizontal velocities given by the
larger
resulted in smaller scattering angles because reflection
occurs more updip for larger
.
AnglesEta
Figure 3. Constant-depth constant-frequency (60 Hz) slice mapped to reflection 2opening angles as in Figure 2, but for a VTI model with ![]() ![]() ![]() ![]() |
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Whereas the influence of
is clearly large, the
change in vertical velocity has a minor influence on the angles as a
function of the midpoint wavenumber (or dip), as demonstrated by the
difference plot in Figure 4. A 0.6 km/s
difference in vertical velocity of an elliptical isotropic model with
=0 (left) and a VTI model with
=0.3 resulted in differences mainly in the offset wavenumber
direction, because depth change caused by the different vertical
velocity provides variations in angles with offset.
Anglesdiffvz
Figure 4. Left: The difference between reflection 2opening angles for an elliptical anisotropic model with ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Anglesdiffv
Figure 5. Left: The difference between reflection angles for an elliptical anisotropic model with ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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![]() | Angle gathers in wave-equation imaging for transversely isotropic media | ![]() |
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