On: “The reflection, refraction, and diffraction of waves in media with an elliptical velocity dependence”, by Franklyn K. Levin (GEOPHYSICS, April 1978, p. 528–537.)
Authors: Helbig K. 
Published: October 2002

Levin treats the subject concisely and exhaustively. Nevertheless, I feel a few comments to be indicated. My first point is rather general: of the three surfaces mentioned in the Appendix, the phase velocity surface (or normal surface) is easiest to calculate, since it is nothing but the graphical representation of the plane‐wave solutions for each direction. The wave surface has the greatest intuitive appeal, since it has the shape of the far‐field wavefront generated by an impulsive point source. The slowness surface, though apparently an insignificant transformation of the phase‐velocity surface, has the greatest significance for two reasons: (1) The projection of the slowness vector on a plane (the “component” of the slowness vector) is the apparent slowness, a quantity directly observed in seismic measurement. Continuity of wave‐fronts across an interface—the idea on which Snell’s law is based—is synonymous with continuity of apparent (or trace) slownesses; and (2) the slowness surface is the polar reciprocal of the wave surface; that is to say, not only has the radius vector of the slowness surface the direction of the normal to the wave surface (which follows from the definition of the two surfaces), but the inverse is also true. That is, the normal to the slowness surface has the direction of the corresponding ray (the radius vector of the wave surface). The fact that this surface so conveniently embodies all relevant information—direction of wave normal and ray, inverse phase velocity, inverse ray velocity (projection of the slowness vector on the ray direction), and the trace slowness along an interface—was the main reason for its introduction by Hamilton (1837) and McCullagh (1837). It is true that this information also can be obtained from the other surfaces, but only in a somewhat roundabout way, which can lead to serious complications. That only few of these complications are apparent in Levin’s article is a consequence of the fact that the polar reciprocal of a surface of second degree is another surface of second degree, in this case an ellipsoid. For more complicated and realistic types of anisotropy, one has to expect much more complicated surfaces. For transverse anisotropy, the slowness surface consists of one ellipsoid (SH‐waves) and a two‐leaved surface of fourth degree, the wave surface of an ellipsoid and a two‐leaved surface of degree 36. More general types of elastic anisotropy can lead to wave surfaces of up to degree 150, while the slowness surface is at most of degree six. It is, therefore, in the interest of a unified theory of wave propagation in anisotropic media to use, wherever possible, the slowness surface. The advantages of this are exemplified by Snell’s law in its general form. While it is impossible to base a concise formulation on the wave surface (reflected and refracted rays do not always lie in the plane containing the incident ray and the normal to the interface), the use of the slowness surface allows the following simple statement (Helbig 1965): “The slowness vectors of all waves in a reflection/refraction process have their end points on a common normal to the interface; the direction of the rays is parallel to the corresponding normals to the slowness surfaces”. A method to interpret refraction seismic data with an anisotropic overburden based on this form of Snell’s law has been described in Helbig (1964).