A short tour through the Stanford Exploration Project contributions to anisotropy

**Tariq Alkhalifah
tariq@sep.stanford.edu**

Browsing through the countless Stanford exploration Project (SEP) reports, I was amazed to find a large number of quality papers that tackle the sensitive issue of anisotropy. The amazement stems from the fact that a hidden principle, which states ``anisotropy is not an important'', existed in SEP. This principle evolved from statements like ``If you believe in anisotropy!'' and ``avoid the two A's (anisotropy and AVO)'' uttered by SEP's leader and chief executive, Jon Claerbout, through the years. Despite such statements Jon has maintained an environment in SEP that allows its students to work on any subject they desire. The fact that some students choose anisotropy is a testament to the importance of this subject, and a testament to Francies Muir (past SEP professor and a devote anisotropist) positive influence on the group.

SEP exploits into this subject ranged for modeling seismic waves in anisotropic media
to imaging data
from such media. Jacobs (1979) was the first to write an anisotropic
paper, titled ``Velocity anisotropy'', in an SEP report. This paper, among other things,
suggested stretching the z-axis to accommodate the ambiguity of resolving depth from
surface seismic data in anisotropic media. This was one of the earliest
papers to acknowledge the depth problem faced in practice due to anisotropy-
the difference between the vertical and imaging velocities. Jacobs (1982)
later included anisotropy in his thesis in a chapter
titled ``Pseudo-*P* and Pseudo-*S* Waves in a Hexagonally
Anisotropic Earth'', where he discussed the downward continuation of the Pseudo-*S*
wave despite that shear waves was another *taboo* at SEP.

Consistent with the low popularity of anisotropy in the early 80's, the next paper to treat anisotropy at SEP appeared years later Dellinger and Muir (1985) and was titled ``Axisymmetric Anisotropy I : Kinematics''. This paper seemingly had opened a flood gate on this subject because the very next report included four papers Dellinger (1985a,b); Muir and Dellinger (1985); Woodward and Muir (1985) tackling issues ranging from modeling to moveout. The following year, Muir (1987) looked at Dix and Backus averaging for obtaining equivalent parameters and Etgen (1987) applied finite difference to the anisotropic elastic wave equation. One report later, Muir and Dellinger (1988) examined the relation between layering and anisotropy; they sought to obtain an equivalent homogeneous anisotropic media that approximated the layered model. In the same report, Nichols (1988) generated synthetic seismograms for anisotropic models and Guiziou (1988) did some 3-D raytracing in anisotropic media.

Dellinger (1988) built a filter capable of separating the *P*
and *S*-wave components of the elastic anisotropic wavefield. Inversion of anisotropic
parameters was discussed first in SEP by Dellinger (1989), where he examined
the feasibility of inverting for the elastic coefficients in a Vertical-seismic-profile (VSP)
type of setting. Nichols (1989) showed the equivalence between the
Hudson model of the elastic properties of fractured rocks and the Schoenberg model.
Karrenbach (1989) uses a paraxial approximation to separate the effect
of anisotropy versus *v*(*z*) variation of velocity. Popovici and Muir (1989) use an algorithm
based on a Markov random field to model anisotropic porous rocks. Muir and Nichols (1989) use the
compliance rather than the stiffness tensor to show that rocks can be fractured in any order.

A year later, Dellinger and van Trier (1990) devised one of the earliest implementations of finite difference scheme to the eikonal equation in anisotropic media. Though they had some stability problems at the time, that was treated later Dellinger (1991), they managed to produce interesting traveltime curves in inhomogeneous media. Ji (1990) compares the nonhyperbolic moveout equation of Muir with that of Byun in inverting for an anellipticity parameter. Dellinger et al. (1990) show the success of the Schoenberg-Muir technique of equivalent media on synthetic data. More study on equivalent media followed Karrenbach (1990); Muir (1993); Nichols and Karrenbach (1990). Cunha (1990) inverted for an elliptical anisotropic parameters using data from a cross well configuration.

In 1991, the double elliptic approximation of Muir (1990) was used in moveout and imaging applications Dellinger and Muir (1991); Karrenbach (1991a); an approximation built to utilize the simplicity of the elliptical anisotropic model to represent a transversely isotropic model. Also, Michelena and Muir (1991) looked at anisotropic tomography. Many other papers that range from azimuthal anisotropy to fine layering to imaging exists in the newer SEP reports Alkhalifah et al. (1997); Alkhalifah and Fomel (1997); Alkhalifah and Rampton (1997a,b); Alkhalifah (1997a,b,c, 1998); Berryman (1996); Fomel and Grechka (1996); Karrenbach et al. (1992); Karrenbach and Muir (1994); Karrenbach (1991b); Michelena (1992).

The above is just a highlight of the many papers at SEP that promote and discuss the important subject of anisotropy. The above is in no way inclusive of all the SEP papers published on this subject. The only way to benefit from and appreciate the level of anisotropy contribution in SEP is to directly read and study the papers involved.

4/20/1999