Slant Stacking and Its Significance for Anisotropy*

Abstract

Slant stacking transforms seismic data, recorded as a function of source-receiver offset and traveltime, into the domain of intercept time r and ray parameter p . The shape of the 7-p-curves thus obtained is closely related to the slowness surfaces of the layers. A layerstripping operation in the z-p-domain removes all effects of the layers above the target layer. The resulting curve is equal to the slowness surface of the layer except for a scaling factor containing the thickness and dip of the layer. The slowness surface is a characteristic surface for anisotropic media. This makes the r-p-domain very suitable for detecting and describing anisotropic layers. The relationship between the shape of r-p-curves, the slowness surfaces, and the geometry of the layers is derived. Synthetic 7-p-curves calculated with the reflectivity method show some difficulties that can arise in determining the shape of the curves and in applying the stripping operation. It is shown that the effects of vertical inhomogeneity on the interpretation of r-p-curves in terms of anisotropy are small.

A Student's Garden of Anisotropy

“…Theoretically, this problem can arise even if the line of sources lies in a vertical plane of symmetry (con s ider the outer slowness surface in Figure 5), although this is unlikely in naturally occurring anisotropy. We see that we cannot exploit the invertability of the Radon transform (Chapman 1978, Hake 1986) to determine the delay-time surface reliably, even only in cross section, from single lines of sources. The only general solution to this problem is to acquire data using a two-dimensional array of sources shooting to receivers whose locations are independent of the sources.…”

“…In connection with the inverse problem of deter mining anisotropic layered structure, Hake (1986) has also noted the greater simplicity of working in the slowness domain, even in the presence of dipping interfaces. We now turn to this viewpoint.…”

“…Hake 1986, Garmany 1988a Figure 7 Huygens wave-front construction using several line sources to approximate a plane-wave source. The resulting plane wave has the same normal as the plane containing the sources.…”

“…Stephen also measured P-wave anisotropy, fi nding the total excursion in P velocity to be about 20%, and he determined that the anisotropy was confi ned to the upper 500 m of the crust. Shearer & Orcutt (1985, 1986) utilized a large set of travel times and isotropic waveform modeling to infer that cracking in the upper 1-1.5 km of basaltic rock was responsible for P-and S-wavc anisotropy at a site in old (140 Myr) ocean crust. The observed orientation of the anisotropy in these studies is consistent with cracks parallel to the midocean ridges, so the anisotropy is presumably tectonically induced in the upper crust as well as in the upper mantle.…”

Multicomponent Seismic Complete Session

“…In a layered medium, the intercept time τ represents the aggregate vertical slowness-thickness product shown in equation 1 (Bessonova et al, 1974;Diebold and Stoffa, 1981;Hake, 1986).…”