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.
Analytical inversion for Thomsen's γ parameter in weakly anisotropic layered media
An analytic scheme for traveltime inversion in horizontally layered media exhibiting transverse isotropy with a vertical symmetry axis (TIV) is presented. It pertains to SH waves, i.e., characterized by a single ‘Ihomsen parameter, y, or to other waves exhibiting approximately elliptical velocity dependence. The proposed method incorporates ray bending at interfaces. The suggested approach is particularly suitable for vertical seismic profiling as best providing required information. The input information consists of traveltime for obliquely travelling downgoing waves which can be measured by an offset survey, vertical wave speed from the zero-offset survey, and layer thicknesses directly from the wellbore. Correct results necessitate high accuracy of all input parameters. The proposed inversion method is particularly sensitive to errors in measured traveltime. ( 4 )
The crystal structure of stilpnomelane. Part II. The full cell
Summary The structural formula of stilpnomelane averaged from 37 representative literature analyses and based on the determined structure is (Ca,Na,K)4(Ti0·1Al2·3Fe35·5Mn0·8Mg9·3)[Si63Al9](O,OH)216·nH2O. Ferrostilpnomelanes have up to 12 R 3+ cations in the octahedral sheet, ferristilpnomelanes commonly have between 24 and 30 R 3+ cations, rarely up to 37. The two groups can be distinguished by accurate X-ray powder diffraction, ferrostilpnomelanes having d 001 less and a greater than have ferristilpnomelanes. The unit cell is triclinic, a = b∼21·8 Å, γ = 120°, d 001∼12·2 Å, P . Hexagonal groups or ‘islands’ of 24 silicon-oxygen tetrahedra co-ordinate to the octahedral sheet, the centres of the ‘islands’ are displaced by 5a/12, 8b/12 on either side of the octahedral sheet as a result of warping of this sheet to permit articulation with the smaller tetrahedral sheet. The 24-tetrahedra ‘islands’ are linked laterally by 6-member rings of inverted tetrahedra, which in turn link at their apices to like rings connecting the next sheet of ‘islands’. Thus a single tetrahedral layer is 4 tetrahedra thick and has symmetry 6/m. The 6-member rings of tetrahedra act as hinges permitting up to 0·5 Å extension or contraction of the tetrahedral sheet to accommodate variations in the octahedral sheet dimensions with varying R 2+/R 3+ ratio.