# Why can Thomsen's parameter $\epsilon$ be negative in VTI media?

I just got several well log curves, some of which are Thomsen's parameters. They are a result of interpretation of Sonic Scanner tool measurements (Schlumberger). Negative epsilon values appear quite frequently, which is not physical for VTI media (as far as I can conclude from books). Am I right?

The question is not about measurements with a specific tool, but rather about the physical and geological interpretation of negative epsilon values. Gamma is also negative, by the way.

Negative values are physical and are expected.

Indeed, Thomsen's 1986 paper has several negative values for $\delta$, $\epsilon$, and $\gamma$ — look at the data in Table 1. He also discusses negative values, at least for $\delta$ (e.g. on page 1961) . You can quickly get a sense of how common negative values are from his Figure 4: As for physical meaning, it depends on the parameter. I recently wrote this about the physical meaning of the Thomsen parameters (edited for brevity):

• $\delta$ or delta — the short offset effect — captures the relationship between the velocity required to flatten gathers (the NMO velocity) and the zero-offset average velocity as recorded by checkshots.
• $\epsilon$ or epsilon — the long offset effect — is, according to Thomsen himself: "the fractional difference between vertical and horizontal P velocities; i.e., it is the parameter usually referred to as 'the' anisotropy of a rock".
• $\gamma$ or gamma — the shear wave effect — relates, as rock physics meister Colin Sayers put it on Twitter, a horizontal shear wave with horizontal polarization to a vertical shear wave.

Usually these parameters are, at a gross scale anyway, positive, because usually velocity is faster along bedding (roughly horizontally) than across it (roughly vertically). But it's easy to imagine scenarios where the relationship is reversed, especially if we're talking about small scales — as in log measurements.

Intuitively, we can draw these conclusions about negative parameters:

• A rock with $\delta < 0$ has the zero-offset reflections 'seeing' faster velocities than the long-offset reflections.
• A rock with $\epsilon < 0$ has a faster vertical velocity than horizontal velocity.
• A rock with $\gamma < 0$ has a faster vertical shear velocity than horizontal shear velocity.

Reference

Thomsen, L (1986). Weak elastic anisotropy. Geophysics 51 (10), 1954–1966. DOI 10.1190/1.1442051.

• Berryman, Grechka and Berge's article from 1999 also does a rather good job to explain the formulas and the thin layer setting. They also ran a Monte Carlo on synthetic data resulting in negative delta and epsilon. Available here: researchgate.net/profile/James_Berryman/publication/… – Tactopoda Mar 16 '15 at 21:39
• Thank you Matt and Mr. tobbe for such great answers, and I'm especially grateful for the literature references you have provided. Just an interesting fact (from that Berryman's article): if we have a Backus'-style stack of horizontal isotropic layers, $\epsilon$ can be negative in some cases, whereas $\gamma$ is always non-negative (as well as anellipticity parameter $\eta$). – antongrin Mar 18 '15 at 14:54
• antongrin: be aware that Thomsen is dealing with intrinsic anisotropy. It's a slippery concept, but extrinsic anisotropy (from fine layering) will have other characteristics — this is what Berryman is writing about. See also Chris Liner's recent writings about negative $Q$. Interesting stuff! – kwinkunks Mar 18 '15 at 15:46
• Thank you very much for your advice. However, I thought that Thomsen's parameters can be applied to any weakly anisotropic medium with vertical axis of symmetry (we don't care whether anisotropy is an intrinsic property or it is caused by fine layering). By the way, that is close to the problem I am focused on right now. To briefly describe, the objective is to explain the difference between the Thomsen's parameters derived from surface seismic data and from well logs. My point of view is that fine layering should be taken into account. But that's definitely a different discussion. – antongrin Mar 19 '15 at 12:20

VTI anisotropy can also result from stress induced anisotropy when the two horizontal stresses are equal and the overburden is the largest stress acting on a stress sensitive formation such as a weakly consolidated sandstone. In such situations grain contacts are aligned in the vertical direction resulting in a stiffer (faster compressional velocity) response. See e.g. Detection of stress-induced velocity anisotropy in unconsolidated sands, Vega. The Leading Edge 2006. Note that the anisotropy values provided in Thomsen 1986 are not necessarily reliable and predate modern laboratory measurements. More recent measurements of anisotropy are available from a number of studies such as Wang, 2002 (Part II Geophysics). A more recent compilation of published mudrock anisotropy parameters is available which shows that very few of the mudrock Thomsen epsilon anisotropy values are negative, see http://www.rockphysicists.org/data The other issue is that you do not measure Thomsen's epsilon with a sonic log in a single well. The Thomsen epsilons in your log suite will have been estimated from some other supplementary information, most likely an empirical relation based on observed core measurements. There is a model developed by Schlumberger referred to as the Modified Annie model that may have been used.

• Thank you for an explanation and links provided. And yes, epsilon log is a result of: 1) estimating Stoneley wave velocity versus frequency; 2) inverting this dispersion function to obtain Vshear in horizontal plane and, therefore, C66; 3) obtaining Thomsen's gamma from c44 and c66; 4) calculating Epsilon and Delta using an empirical MAnnie model. Following this technique, negative gamma (and epsilon) values occur when Stoneley wave velocity is too small. This happens if the mobility of pore fluid is high. So negative epsilons correspond to permeable zones rather than anisotropic ones. – antongrin Jul 28 '15 at 8:17