Angle gathers in wave-equation imaging for transversely isotropic media

Numerical tests: The anisotropy influence

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 $\eta$ , $v_z$ , and the NMO velocity $v$ .

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 $k_{\text{hx}}=0$ , 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 $k_{mx}$ ) 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 $k_{sz}$ or $k_{rz}$ 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

In anisotropic media, as illustrated in Figure 3, for $\eta$ 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 $v_z$ is lower in the anisotropic models, the higher horizontal velocities given by the larger $\eta$ resulted in smaller scattering angles because reflection occurs more updip for larger $\eta$ .

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 $v_z$ =1.8 km/s, $v$ =2 km/s, and $\eta =0.1$ (left) and $\eta =0.3$ (right).

Whereas the influence of $\eta$ 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 $\eta$ =0 (left) and a VTI model with $\eta$ =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 $v_z$ =1.8 km/s, $v$ =2 km/s and that of a similar model, with $v_z$ =1.2 km/s. Right: The difference betweenreflection 2opening angles for a VTI model of Figure 3 (right) for $v_z$ =1.8 km/s, $v$ =2 km/s, and $\eta =0.3$ and that of a similar model, with $v_z$ =1.2 km/s.

In comparison, if we change the NMO velocity, $v$ , the angles hardly change at all, especially around small dips and small offsets. This fact is evident in Figure 5, where we change NMO velocity 0.6 km/s, and the general difference is small until we get to large offset and midpoint wavenumbers. This difference implies that the mapping is practically NMO-velocity independent. This is the case for $\eta =0$ (left) and $\eta =0.3$ (right) in Figure 5, which implies, that for a given elliptical anisotropic model one can find an isotropic model that produces similar mapping granted that the velocity of the isotropic model is equal to the vertical velocity for the elliptical anisotropy.

Anglesdiffv
Figure 5. Left: The difference between reflection angles for an elliptical anisotropic model with $v_z$ =1.8 km/s, $v$ =2 km/s and that of a similar model, with $v$ =1.4 km/s. Right: The difference between reflection angles for a VTI model of Figure 3 (right) for $v_z$ =1.8 km/s, $v$ =2 km/s, and $\eta =0.3$ and that of a similar model, with $v$ =1.4 km/s.

Angle gathers in wave-equation imaging for transversely isotropic media