RAYS AND WAVEFRONT CHARTS IN GRADIENT MEDIA 1

A bstract Wavefront charts in anisotropic gradient media are a useful tool in ray geometric constructions, particular in shear‐wave exploration. They can be constructed by: (i) a family of wavefronts that contains a vertical plane as member ‐ it is convenient to choose constant time increments; (ii) tracing one ray that makes everywhere the angle with the normal to the wavefront that is required by the anisotropy of the medium; (iii) scaling this ray to obtain a set of rays with different ray parameters; (iv) shifting these rays (with wavefront elements attached) so that they pass through a common source point; (v) interpolating the wavefronts between the elements. The construction is particularly simple in linear ‐gradient media, since here all members of the family of wavefronts are planes. Since the ray makes everywhere the angle prescribed by the anisotropy with the normal of the (plane) wavefronts, the ray has the shape of the slowness curve rotated by −π/2. For isotropic media the slowness curve is a circle, and thus rays are circular arcs. The circles themselves intersect in the source point and in a second point above the surface of the earth. This provides a simple proof that wavefronts emanating from a point source in an isotropic linear‐gradient medium are spheres: inversion of the set of circular rays with the source as centre maps the pencil of circular rays into a pencil of straight lines passing through a point. A pencil of concentric spheres around this point is perpendicular to the pencil of straight lines. On inverting back the pencil of spheres is mapped into another pencil of spheres that is perpendicular to the circular rays.