Thomsen Parameters Calculator: Epsilon, Gamma, and Delta from Stiffnesses

Most seismic processing quietly assumes rock is the same in every direction, and most rock quietly disagrees. Shales, which make up the bulk of any sedimentary column, are built from plate-shaped clay minerals that settle flat, so waves travel measurably faster along the bedding than across it. Before 1986 this was described with an intimidating stiffness tensor; then Leon Thomsen showed that for the transversely isotropic case (VTI, symmetry axis vertical) the whole story compresses into a vertical Vp and Vs plus three dimensionless numbers small enough to reason about: epsilon, gamma, and delta.

What each parameter tells you

• Epsilon is the fractional difference between horizontal and vertical P velocity: epsilon of 0.1 means the rock is about 10 percent faster along the layers
• Gamma plays the same role for horizontally polarized shear waves, and drives shear-wave splitting
• Delta has no such tidy picture; it is a combination involving C13 that governs how velocity varies for near-vertical ray paths, which happens to be exactly where surface seismic lives

The formulas, computed by this calculator from the five independent stiffnesses of a VTI medium: epsilon = (C11 – C33)/(2 C33), gamma = (C66 – C44)/(2 C44), and delta = [(C13 + C44)^2 – (C33 – C44)^2] / [2 C33 (C33 – C44)]. Vertical velocities come along free: Vp0 = sqrt(C33/rho) and Vs0 = sqrt(C44/rho).

Why delta is the one that costs money

The near-offset moveout that velocity analysis picks is not the vertical velocity; it is the NMO velocity, and in a VTI medium Vnmo = Vp0 x sqrt(1 + 2 delta). With a delta of 0.1, moveout-derived velocities run about 10 percent fast relative to the vertical velocity that converts time to depth. Stack with them and the image looks fine; depth-convert with them and every horizon comes in deep, which is the classic well-tie surprise in shale-heavy basins. Epsilon, meanwhile, matters mostly at long offsets and for the horizontal velocity that migration needs; the difference between epsilon and delta (the anellipticity) is what long-offset moveout inversion actually senses.

Worked example

• Stiffnesses C11 = 24.84, C13 = 11.35, C33 = 20.70, C44 = 5.18, C66 = 7.25 GPa at 2.30 g/cc
• Epsilon = 0.100, gamma = 0.200, delta = 0.050
• Vertical Vp0 = 3.00 km/s but Vnmo = 3.15 km/s: a 5 percent depth bias if ignored
• Horizontal Vp = 3.29 km/s, the velocity a migration algorithm needs at 90 degrees

Typical magnitudes from published core studies: clean sandstones sit near zero on all three, most shales run epsilon of 0.1 to 0.3 with delta anywhere from slightly negative to 0.2, and organic-rich shales can push epsilon past 0.3. Negative delta is real and documented, so resist the urge to treat it as a data error. And a scope reminder: these definitions assume VTI symmetry; steeply dipping fabric or aligned vertical fractures need HTI or orthorhombic descriptions instead.

Related Calculators and Reference

The vertical velocities here connect to the elastic moduli interconversion calculator for the isotropic limit, the NMO velocity bias ties into the NMO correction calculator, and time-to-depth consequences show up in the time-depth converter. The SEG Wiki entry on Thomsen parameters covers the 1986 paper and typical values.