The Thomsen Parameters

Part 7, Part 7: Anisotropy, the Physics

Learning objectives

  • Connect epsilon, delta, and gamma to the velocity surfaces
  • See epsilon stretch P horizontally and gamma stretch SH
  • Understand delta as the near-axis, moveout-controlling term
  • Watch the SV surface bulge from sigma proportional to epsilon minus delta

Three Numbers for a Whole Tensor

A general anisotropic rock is described by a stiffness tensor with twenty-one numbers. Thomsen's insight was that for the weak, transversely isotropic case that dominates in the subsurface, three combinations carry almost all the observable behaviour, and each has a clean physical meaning you can watch on the velocity surface.

epsilon\epsilon is the P-wave anisotropy: at ninety degrees the P velocity is Vp0(1+epsilon)V_{p0}(1+\epsilon)p0(1+epsilon), so raising it stretches the P surface horizontally. gamma\gamma is the SH anisotropy: the SH surface is a simple ellipse reaching Vs0(1+gamma)V_{s0}(1+\gamma)s0(1+gamma) at ninety degrees. delta\delta is the subtle one. It barely changes the horizontal velocity, but it sets the curvature of the P surface close to the axis, and that near-axis shape is what governs normal moveout and short-offset amplitudes. Ignore delta\delta and your depths are wrong even when your horizontal velocity looks right.

The Thomsen parameters, made visiblePSVSHEpsilon stretches P horizontally; gamma stretches SH; the SV surface bulges at 45 degrees from sigma, proportional to epsilon minus delta.

The SV Surface Tells the Truth

The clearest reason delta\delta is not redundant with epsilon\epsilon is the SV wave. Its anisotropy is governed by the combination sigma=(Vp0/Vs0)2(epsilon−delta)\sigma = (V_{p0}/V_{s0})^2(\epsilon-\delta)epsilon−delta), and it peaks not at the horizontal but at forty-five degrees. Set epsilon\epsilon and delta\delta equal and sigma\sigma vanishes: the SV surface stays round, the elliptical-anisotropy special case. Pull them apart and the SV surface grows a pronounced bulge at forty-five degrees that neither epsilon\epsilon nor delta\delta alone would predict. Because the multiplier (Vp0/Vs0)2(V_{p0}/V_{s0})^2 is large, even a small epsilon−delta\epsilon-\delta produces a big SV effect.

That is the whole reason anisotropic modelling needs more than one parameter. A single velocity, or even a single anisotropy number, cannot reproduce a surface that behaves differently at zero, forty-five, and ninety degrees. Switch to the shape-normalized view to compare the three surfaces directly, and slide epsilon\epsilon and delta\delta apart to see sigma\sigma do its work. The next section takes this surface to the gather and asks what it does to moveout.

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