Anisotropic Moveout

Part 7, Part 7: Anisotropy, the Physics

Learning objectives

Recall hyperbolic NMO in an isotropic earth
See that the short-offset NMO velocity is set by delta
Watch far-offset moveout go non-hyperbolic with eta
Read the residual hockey stick as the anisotropy signature

The Gather That Will Not Flatten

On a common-midpoint gather, a reflection arrives later at longer offset. In an isotropic earth that delay is a clean hyperbola, $t(x)^2 = t_0^2 + x^2/V^2$ 02+x2/V2, and choosing the correct NMO velocity $V$ subtracts the moveout and lays the event flat at $t_0$ . One velocity, one flat gather. This is the engine of stacking.

VTI anisotropy breaks it in two places. First, the velocity that flattens the near offsets is not the true vertical velocity but $V_{nmo} = V_{p0}\sqrt{1+2\delta}$ sqrt1+2delta. The near-offset moveout is governed by $\delta$ , so a gather can look perfectly flat at short offset and still be built on the wrong velocity. Second, at long offset the moveout stops being a hyperbola at all.

Eta, and the Hockey Stick

The departure from a hyperbola is controlled by the anellipticity $\eta = (\epsilon-\delta)/(1+2\delta)$ , and the industry-standard description is the Alkhalifah-Tsvankin non-hyperbolic moveout equation. When $\eta$ is zero (the elliptical case, $\epsilon=\delta$ ) the moveout is exactly hyperbolic and one velocity flattens everything. When $\eta$ is not zero, correct the gather with the best single hyperbola and the near offsets snap flat while the far offsets curl up into a residual hockey stick. The size of that curl is a direct measurement of $\eta$ .

This is why long-offset velocity analysis needs two numbers, $V_{nmo}$ nmo and $\eta$ , not one. It is also the fit-for-purpose boundary for modelling: if you only care about a short-offset stack, an isotropic velocity that absorbs $\delta$ is enough, but the moment you use far offsets, for AVO or for imaging, ignoring $\eta$ leaves that hockey stick in your data and mis-positions your reflectors. The next section takes the same idea sideways, into the azimuth, where fractures live.