What does equation (1) represent? Figure 1 shows a tiny portion of a wavefront at two closely-spaced instants in time. On this scale, the wavefront can be approximated as a plane wave. The plane wave is moving along in some arbitrary direction with a velocity V, so it travels a distance between the two times. At the same time, the plane wave also moves a distance along the X axis and along the Z axis.
The triangle to the lower right of Figure 1 has sides of length and meeting at a right angle, and a height of (if the hypotenuse is considered the base). Three applications of Pythagoras' theorem (one for each of the three right triangles involved) and a little algebra show that
How do we make equation (1) anisotropic? From the previous two paragraphs, we can see that in equation (1) (or in equation (3)) is just the horizontal phase slowness. For a monochromatic plane wave this is .Similarly, .This lets us rewrite equation (1) as a dispersion relation:
Recast as a dispersion relation, F(u) in equation (4) of Van Trier and Symes has a simple interpretation: u is a horizontal phase slowness , and F(u) is a vertical phase slowness .Replacing the isotropic dispersion relation in their paper with an appropriate anisotropic one gives an anisotropic traveltime extrapolation equation. Unfortunately, solving for kz as a function of kx can be difficult for some anisotropic media. In the general case of arbitrary three-dimensional anisotropy it can require solving a sixth order polynomial.