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Rays not perpendicular to fronts

Anisotropy means that waves propagating in different directions propagate at different speeds. Anisotropy does not mean that velocity is a function of spatial location, and thus anisotropy does not cause rays to bend. The peculiar thing about anisotropy is that rays are not perpendicular to wavefronts. Figure 7 illustrates this idea.

 
aniso
aniso
Figure 7
Wavefronts in an isotropic medium (left) and an anisotropic medium (right). Note that on the right, the rays are not perpendicular to the wavefronts. (Rothman)


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The diagram on the left shows spherical wavefronts emanating from a point source at the origin. This is the usual, isotropic case. The diagram on the right shows the nonspherical wavefronts of the 15$^\circ$ migration equation. Note that near the z-axis they are nearly spherical, but further away they do a poor job of matching a sphere with its center at the origin.

The ideal wavefront from a Huygens secondary source is a semicircle. The secondary source that results from the 15$^\circ$extrapolation equation is an ellipse. The secondary source that results from the 45$^\circ$ extrapolation equation is an interesting, heartlike shape. These are drawn in Figure 8.

 
anisoxz
anisoxz
Figure 8
Wavefronts of 15$^\circ$ (left) and 45$^\circ$ (right) extrapolation equations, inscribed within the exact semicircle. Waves with $\sin\,\theta = v k_x /\omega = \pm 1$ are marked with small dots. Evanescent energy lies beyond the dots. (Rothman)


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In practice, the top parts of the ellipse and the heart are rarely observed because they are in the evanescent zone, and the x-axis is seldom refined enough for them to be below the aliasing frequency. The center of the heart is sometimes seen in the (x,t)-plane when the 45$^\circ$ program is used. It is shown by a line drawing in Figure 9 and shown using a 45$^\circ$diffraction program in Figure 16.

 
anisoxt
Figure 9
45$^\circ$ heart theory. The cusp arises in the evanescent region. (Rothman)

anisoxt
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previous up next print clean
Next: Wavefront direction and energy Up: ANISOTROPY DISPERSION IN MIGRATION Previous: ANISOTROPY DISPERSION IN MIGRATION
Stanford Exploration Project
10/31/1997