What Anisotropy Is
Learning objectives
- Define isotropy and anisotropy in terms of velocity and direction
- See a circular wavefront become an ellipse
- Read the vertical and horizontal velocity difference
- Meet the Thomsen epsilon and delta parameters
Velocity That Depends on Direction
Every engine in this course so far, acoustic and elastic alike, made a silent assumption: that velocity is the same in every direction. Fire a wave from a point in such a medium and the wavefront expands as a perfect circle, a sphere in three dimensions, because the rock has no preferred direction. That is isotropy.
Real sedimentary rock is not like that. It is layered, deposited bed on bed, and often threaded with aligned fractures. A wave travelling along the beds moves faster than one crossing through them. Velocity now depends on the direction of travel, and the wavefront from a point source is no longer a circle but an ellipse, stretched along the fast direction. That is anisotropy.
Naming the Shape: Thomsen Parameters
To model anisotropy we need to describe the shape of that velocity surface with a few numbers. Thomsen's parameters do exactly this for the common weakly anisotropic case. (epsilon) measures how much faster the horizontal velocity is than the vertical: at ninety degrees the P velocity is . (delta) controls the curvature of the surface near the symmetry axis, the part that governs normal moveout and short-offset amplitudes even though it barely changes the horizontal velocity. A third parameter (gamma) describes the shear anisotropy.
Set both to zero and the surface snaps back to a circle: isotropy is just the special case where the parameters vanish. That is the fit-for-purpose hook for this whole part. If your rock is nearly isotropic, the earlier engines are fine. If it is not, ignoring in particular puts reflectors at the wrong depth. The next section explains WHY rocks become anisotropic in the first place: layering and fractures.