Anisotropic velocity (VTI) and the η parameter

Part 3, Velocity Analysis & NMO

Learning objectives

State why a VTI medium requires a non-hyperbolic NMO correction
Identify the η (eta) parameter and describe what it measures physically
Recognize the far-offset signature of an un-corrected η on a NMO-flattened gather
Pair (V_NMO, η) as the two parameters that must BOTH be picked in an anisotropic layer

So far the NMO equation has been hyperbolic: t² = t₀² + x²/V². Real earth often is not. Especially in shales and fractured carbonates, velocity depends on the direction the wave is travelling. Horizontal-layer P-waves travel faster than vertical ones, and the move-out curve on a CMP gather is NOT a pure hyperbola. Ignore the anisotropy and the deep reflectors stay curved after NMO, no matter how well you pick V.

1. What VTI means

VTI = Vertically Transversely Isotropic. The rock’s elastic properties have one special direction (vertical, set by gravity during deposition), and isotropic in any 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 derived the exact fourth-order move-out equation:

t^{2}(x) = t_{0}^{2} + \frac{x^{2}}{V_{\text{NMO}}^{2}} - \frac{2\eta\, x^{4}}{V_{\text{NMO}}^{2}\bigl(t_{0}^{2}V_{\text{NMO}}^{2} + (1 + 2\eta)\,x^{2}\bigr)}

Two parameters to pick: V_NMO (the short-offset velocity, same as before) and η (eta), the anisotropy parameter. When η = 0 the equation reduces to the hyperbolic one. When η > 0, the 4th-order term subtracts from the hyperbolic prediction at long offsets, making the true travel-time curve flatter than a hyperbola.

3. What η physically measures

η is a compact combination of Thomsen’s anisotropy parameters ε (velocity asymmetry) and δ (near-vertical move-out coefficient):

\eta = \frac{\varepsilon - \delta}{1 + 2\delta}

For typical shales, η is 0.05-0.20. Values above 0.10 produce a clearly non-hyperbolic signature on any offset-to-depth ratio greater than 1.

4. The signature you look for

Apply a hyperbolic NMO (pretend η = 0) with the best V you can pick. The near offsets will flatten. The far offsets will not, they’ll bend upward (a “smile”) because the hyperbolic prediction overcorrected them. The widget below lets you see this directly.

True V = 2500, true η = 0.15. The offset range (up to 3100 m) is roughly 2.5× the reflector depth at t₀ = 1.0 s, well into the VTI regime. Try:

η = 0, V = 2500: near offsets flat, far offsets still bent.
η = 0, V = 2700: some trade-off; far offsets flatter but near offsets now under-corrected (they bend down).
η = 0.15, V = 2500: the event flattens across the full offset range, the TUNED status confirms.

5. Picking (V, η) in practice

Like velocity, η is picked via semblance, but over a 2D (V, η) panel rather than a 1D V scan. Or you can do two sweeps:

Pick V on near-offset data with η = 0.
Fix V, pick η on far-offset data to flatten the residual smile (the far-offset over-correction).
Iterate.

6. Why you cannot afford to ignore η

AVO. The amplitude at angle θ depends on how you measure θ, which depends on V_NMO. A mis-estimated V (because you ignored η) biases every AVO attribute.
Migration. Anisotropic migration needs (V, η) to properly image steep events. Ignoring η places deep reflectors at the wrong depth.
Depth conversion. Using a Dix-derived V_int from VTI V_NMO gives a biased interval velocity, depths come out ~5-10 % deep.
4D. Time-lapse amplitude changes can be 10-50 %, anisotropy effects can leak into the 4D signal if not handled consistently.

7. Orthorhombic, TTI, and more

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

TTI (Tilted TI), the symmetry axis is not vertical (e.g., dipping shales near a salt flank). Adds a tilt angle to the parameter set.
Orthorhombic, three distinct orthogonal symmetry planes (e.g., a fractured VTI shale). 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.

**The one sentence to remember**

VTI earth requires (V_NMO, η) as a pair; hyperbolic NMO can flatten near offsets OR far offsets but not both; ignoring η systematically biases every downstream product.

Where this goes next

§3.5 adds the final residual NMO refinement: after (V, η) picking, small time shifts and fourth-order residuals still remain because of velocity heterogeneity, dip, and HOMO (higher-order move-out). Fixing them is iterative, residual velocity picking plus residual statics plus a HOMO correction.

References

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.