Anisotropic velocity (VTI) and the η parameter

So far the NMO equation has been hyperbolic: $t^2 = t\_0^2 + x^2/V^2$. Real rock often is not. In shales and fractured carbonates the velocity depends on the direction the wave travels: P-waves travelling horizontally through layered shale are faster than P-waves travelling vertically, so the moveout on a CMP gather is not a pure hyperbola. Ignore the anisotropy and shallow-to-intermediate reflectors, where the spread is long compared with the depth, stay bent at far offsets after NMO, or flatten only if you pick a $V\_{\\mathrm{NMO}}$ that is too high.

1. What VTI means

VTI = vertically transversely isotropic. The rock’s elastic properties have one special direction (vertical, set by gravity during deposition) and are isotropic in every horizontal plane. Shale is the classic VTI medium: platy clay minerals align horizontally during deposition, so horizontal P-wave velocity can be 5-20 % higher than vertical.

2. The non-hyperbolic NMO equation

For a VTI layer, Alkhalifah and Tsvankin (1995) derived a non-hyperbolic moveout approximation that stays accurate to large offsets:

Two parameters to pick: $V\_{\\mathrm{NMO}}$ (the short-offset velocity, same as before; in VTI it equals $V\_{P0}\\sqrt{1+2\\delta}$, not the vertical velocity) and $\\eta$ (eta), the anisotropy parameter. When $\\eta = 0$ the equation reduces to the hyperbolic one. When $\\eta > 0$, the fourth-order term subtracts from the hyperbolic prediction at long offsets, making the true traveltime curve flatter than a hyperbola.

3. What η physically measures

$\\eta$ is a compact combination of Thomsen’s anisotropy parameters $\\varepsilon$ (the fractional difference between horizontal and vertical P velocity) and $\\delta$ (which controls near-vertical moveout):

For typical shales, $\\eta$ is 0.05-0.20. Values above 0.10 give a clear non-hyperbolic signature once the offset exceeds about 1.5 to 2 times the reflector depth.

4. The signature you look for

Apply a hyperbolic NMO (pretend $\\eta = 0$) with the short-spread velocity. The near offsets flatten. The far offsets do not: they curl upward to earlier times (the “hockey stick”) because the hyperbolic prediction over-corrects them. In Figure 3.4 you correct three reflections from a shale with $\\varepsilon = 0.20$ and $\\delta = 0.04$ (so $\\eta = 0.148$ and $V\_{\\mathrm{NMO}}$ = 2494 m/s) with your own $V\_{\\mathrm{NMO}}$ and $\\eta$, and read the residual moveout and the semblance your pair leaves.

The recorded traveltimes in the figure are exact for the shale, while the correction uses the Alkhalifah-Tsvankin equation, so even the right pair leaves a few milliseconds. The offsets reach 3100 m, 2.6 times the 1200 m depth of reflector 2, well into the range where $\\eta$ shows. Try:

• $\\eta = 0$, $V\_{\\mathrm{NMO}}$ = 2490 m/s: reflector 2 is flat out to about its depth, but its far offsets curl up, 118 ms early at 3100 m: the hockey stick.
• Best hyperbola: at $\\eta = 0$ the semblance peaks at 2680 m/s. The event looks nearly flat (its largest residual is +13 ms, at 1800 m), but that velocity is 7.5 % above $V\_{\\mathrm{NMO}}$. The hyperbola has absorbed $\\eta$ into a biased velocity, and that error goes on into Dix, depth and AVO.
• $\\eta = 0.15$, $V\_{\\mathrm{NMO}}$ = 2490 m/s: the event flattens across the whole spread, to within 8 ms, and its semblance on the recorded gather rises from 0.29 to 0.97. What is left is the error of the approximation itself.
• Shallow against deep: at $\\eta = 0$, reflector 1, with offsets to 4.3 times its depth, lands 331 ms early at 3100 m; reflector 3, at 1.6 times its depth, only 47 ms.
• A short spread: cut the maximum offset to 1200 m, the depth of reflector 2, and every $\\eta$ from 0 to 0.30 leaves it flat to within 6 ms: without long offsets $\\eta$ cannot be picked.

5. Picking (V, η) in practice

Like velocity, $\\eta$ is picked with semblance, but over a 2D $(V\_{\\mathrm{NMO}}, \\eta)$ panel rather than a 1D $V$ scan; plate (d) of the figure is one. Its bright ridge runs from high velocity at $\\eta = 0$ to lower velocity at larger $\\eta$: the two parameters trade off, and only long offsets pin the pair down. Or you can do two sweeps:

1. Pick $V\_{\\mathrm{NMO}}$ with $\\eta = 0$ on offsets up to about the reflector depth.
2. Fix $V\_{\\mathrm{NMO}}$ and pick $\\eta$ on the far offsets to flatten the hockey stick (the far-offset over-correction).
3. Iterate.

6. Why you cannot afford to ignore η

• AVO. The amplitude at angle $\\theta$ depends on how you map offset to $\\theta$, which depends on $V\_{\\mathrm{NMO}}$. A $V$ biased high because $\\eta$ was ignored biases every AVO attribute.
• Migration. Anisotropic migration needs $(V, \\eta)$ to image steep events properly. Ignoring $\\eta$ mispositions and defocuses steeply dipping reflectors.
• Depth conversion. Dix applied to VTI $V\_{\\mathrm{NMO}}$ gives $V\_{P0}\\sqrt{1+2\\delta}$, not the vertical velocity, so depths come out too deep by roughly $\\delta$ (often 5-10 %) unless $\\delta$ is calibrated at wells.
• 4D. Time-lapse amplitude changes are often 10-50 %, and anisotropy handled differently between vintages can leak into that signal.

7. Orthorhombic, TTI, and more

VTI is the simplest useful anisotropy. Two more common cases:

• TTI (tilted TI): the symmetry axis is not vertical (for example, dipping shales near a salt flank), which adds a tilt angle to the parameter set.
• Orthorhombic: three mutually orthogonal symmetry planes (for example, a VTI shale cut by vertical fractures), which adds azimuth dependence.

Production flows handle each at increasing cost. This section stops at VTI; the advanced cases are covered in Part 6 FWI and Part 7 QI.

Where this goes next

Section 3.5 adds the final residual NMO refinement: after $(V, \\eta)$ picking, small time shifts and fourth-order residuals still remain because of velocity heterogeneity, dip, and HOMO (higher-order moveout). Fixing them is iterative: residual velocity picking, residual statics and a HOMO correction.

References

• Alkhalifah, T., Tsvankin, I. (1995). Velocity analysis for transversely isotropic media. Geophysics, 60, 1550.
• Thomsen, L. (1986). Weak elastic anisotropy. Geophysics, 51, 1954.
• Tsvankin, I. (1996). P-wave signatures and notation for transversely isotropic media: an overview. Geophysics, 61, 467.
• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Taner, M. T., Koehler, F. (1969). Velocity spectra, digital computer derivation and applications of velocity functions. Geophysics, 34, 859.
• Etgen, J., Gray, S. H., Zhang, Y. (2009). An overview of depth imaging in exploration geophysics. Geophysics, 74, WCA5.
• Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge UP.