Direct traveltime inversion of VSP data for elliptical anisotropy in layered media
This paper presents an exact method for inversion of VSP data for anisotropic parameters that uses traveltimes directly, rather than phase slownesses computed from traveltime differences. There is a tradeoff in the present traveltime-based method compared with the slowness-based method. In our method we have to assume some function representing the vertical velocity variation, something that is not necessary in the phase-slowness method. There is, however, a higher numerical accuracy in the direct traveltime method. This is due partly to our using observed traveltimes directly in the computations rather than taking differences between values of comparable size, which greatly magnifies the relative errors. Error analysis shows that there also are other intrinsic reasons why our technique has less error. A numerical example representing a typical VSP yielded errors from the phase-slowness method (requiring two sources and two receivers) that were about 18 times larger than for a single determination by the direct traveltime method. Furthermore, in cases where the VSP has been acquired with a single source offset, e.g., for some offshore wells, the traveltime method will yield results, whereas the slowness method will not. INTRODUCTION In the last decade or two there has been a greatly increased recognition that significant seismic anisotropy is rather common in sedimentary rocks. It has also been shown that neglecting even a seemingly modest degree of anisotropy can lead to significant degradation in processed seismic images (e.g., Winterstein, 1986; Wright, 1987; Larner, 1993; Tsvankin, 1996; Chen and Castagna, 2000) or to significant errors in vertical or lateral positioning of subsurface features (e.g. Banik, 1984, 1987; Isaac and Lawton, 1999; Vestrum et al., 1999). Gaiser (1990) determined how to estimate the anisotropic parameters of a transversely isotropic medium from VSP data using vertical and horizontal phaseslowness measurements. Miller and Spencer (1994) and Miller et al. (1994) presented methods for inverting phase slowness estimates from walkaway VSPs for anisotropy parameters. Using a similar method, Leaney et al. (1999) have inverted multiazimuthal multioffset VSP P-wave data over a marine carbonate reservoir to determine azimuthal anisotropy in the carbonate and transverse isotropy in the overlying shale. Leslie and Lawton (1999) present a method for inverting refraction seismic data for the anisotropic parameters of steeply dipping strata. Tsvankin and Thomsen (1994) and Alkhalifah and Tsvankin (1995) deal with the problem of inverting for longer spreads, beyond the near-vertical elliptic-anisotropic approximation, in order to recover velocity fields for processing. * Authors listed in alphabetical order.
Time-depth and velocity-depth relations in sedimentary basins- A study based on current investigation in the Arctic Islands and an interpretation of experience elsewhere
Velocities of P‐waves in the Arctic Islands obey the same power‐law equations (1) t=azn+b, (2) V=cz1-n that apply to Western Canada (Acheson, 1963). They appear to be closely related to the theoretical one‐sixth power law (n = 5/6) derived by Gassmann (1950), Brandt (1955), and others. Equations (1) and (2) are piece‐wise continuous at any given location and consist of a number of discrete segments corresponding to geologic units, each characterized by a distinct set of values for a, b, c, and n. Except for a limited number of velocity anomalies such as permafrost, these units relate to formations or groups of formations. Velocity changes in a, c, and n correspond to major changes in lithology, pressure, or compaction. Velocity fluctuations within the formations caused by variation in grain size, porosity, fluid‐to‐matrix coupling, and so on are not indicated in this analysis, and the velocity‐depth relation refers to secular velocity variation only. Correlation of reported velocity findings from many par...
“Zoeppritz” rationalized, and generalized to anisotropic media
This paper is meant to be a guide through a labyrinth of reflectivity and transmissivity at interfaces between anisotropic elastic media. The Zoeppritz equations for isotropic plane‐wave reflection and transmission coefficients are cast in a form from which explicit elementary expressions for the reflectivity and transmissivity matrices are derived in terms of four submatrices of the Zoeppritz coefficient matrix, two submatrices associated with each of the two media. The two submatrices associated with a medium are also the building blocks for a simple construction of the propagator matrix for a layer of that medium. All expressions are applicable to anisotropic media, subject to the requirement that the media involved must be at least monoclinic with a mirror plane of symmetry parallel to the reflecting plane. The submatrices associated with a medium depend only on the elastic properties of that medium and are functions of the eigenvalues and eigenvectors of the Christoffel equations governing plane‐wave...