(1) |

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.

Figure 1

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

(2) |

(3) |

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:

(4) |

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 *k*_{z} as a function of *k*_{x} can be difficult
for some anisotropic media. In the general case of arbitrary three-dimensional
anisotropy it can require solving a sixth order polynomial.

1/13/1998