Geometrical spreading in a layered transversely isotropic medium with vertical symmetry axis

Abstract

Compensation for geometrical spreading is important in prestack Kirchhoff migration and in amplitude versus offset/amplitude versus angle (AVO/AVA) analysis of seismic data. We present equations for the relative geometrical spreading of reflected and transmitted P-and S-wave in horizontally layered transversely isotropic media with vertical symmetry axis (VTI). We show that relatively simple expressions are obtained when the geometrical spreading is expressed in terms of group velocities. In weakly anisotropic media, we obtain simple expressions also in terms of phase velocities. Also, we derive analytical equations for geometrical spreading based on the nonhyperbolic traveltime formula of Tsvankin and Thomsen, such that the geometrical spreading can be expressed in terms of the parameters used in time processing of seismic data.Comparison with numerical ray tracing demonstrates that the weak anisotropy approximation to geometrical spreading is accurate for P-waves. It is less accurate for SV-waves, but has qualitatively the correct form. For P waves, the nonhyperbolic equation for geometrical spreading compares favorably with ray-tracing results for offset-depth ratios less than five. For SV-waves, the analytical approximation is accurate only at small offsets, and breaks down at offset-depth ratios less than unity. The numerical results are in agreement with the range of validity for the nonhyperbolic traveltime equations.

Seismology of Azimuthally Anisotropic Media and Seismic Fracture Characterization

“…Note that if the medium is azimuthally anisotropic, even the traveltime derivatives with respect to offset depend on the orientation of the source-receiver line. For isotropic and VTI media, where ∂ 2 t/∂α 2 = 0 and ψ s = ψ r = ψ, equation 8.14 further simplifies to (Ursin and Hokstad, 2003) L(x) = cos ψ � 1 x The main practical importance of equation 8.13 is in providing the analytic foundation for geometrical-spreading correction of wide-azimuth reflection data. As shown in Chapter 3, long-spread P-wave moveout in layered orthorhombic media is accurately described by the generalized Alkhalifah-Tsvankin equation 3.37.…”

“…For example, an algorithm to compute geometrical spreading from coarsely-gridded traveltime tables was introduced by Vanelle and Gajewski (2003). Ursin and Hokstad (2003) represented the geometrical-spreading factor of P-waves in horizontally layered VTI media in terms of reflection traveltime described by the Tsvankin-Thomsen nonhyperbolic moveout equation 3.3. A simpler geometrical-spreading approximation for VTI media can be obtained using the Alkhalifah-Tsvankin moveout equation 3.36 parameterized by the NMO velocity and the anellipticity parameter η.…”

Improved geometric-spreading approximation in layered transversely isotropic media

“…Assuming that a source and receiver are placed in the same homogeneous layer at the same depth, and the emitted and received wave type is qP-wave, we obtain the expression for the radiation pattern ͑for more details, see Ursin and Hokstad, 2003;Xu et al, 2005͒ …”

Ray tracing and amplitude calculation in anisotropic layered media

“…The geometrical spreading parameter can be expressed physically as a measure of energy per unit area of wave front, assuming a fixed amount of energy within a ray tube. Ursin (2002) presented simplified expressions for the geometrical spreading in weakly anisotropic media. The geometrical spreading factor L which denotes the energy attenuation inside each layer is expressed as…”