Slowness domain offset and traveltime approximations in layered vertical transversely isotropic media

Abstract

Considering horizontally layered transversely isotropic media with vertical symmetry axis and all types of pure‐mode and converted waves we present a new wide‐angle series approximation for the kinematical characteristics of reflected waves: horizontal offset, intercept time, and total reflection traveltime as functions of horizontal slowness. The method is based on combining (gluing) both zero‐offset and (large) finite‐offset series coefficients. The horizontal slowness is bounded by the critical value, characterised by nearly horizontal propagation within the layer with the highest horizontal velocity. The suggested approximation uses five parameters to approximate the offset, six parameters to approximate the intercept time or the traveltime, and seven parameters to approximate any two or all three kinematical characteristics. Overall, the method is very accurate for pure‐mode compressional waves and shear waves polarised in the horizontal plane and for converted waves. The application of the method to pure‐mode shear waves polarised in the vertical plane is limited due to cusps and triplications. To demonstrate the high accuracy of the method, we consider a synthetic, multi‐layer model, and we plot the normalised errors with respect to numerical ray tracing.

Slowness‐domain kinematical characteristics for horizontally layered orthorhombic media. Part I: Critical slowness match

“…These results coincide completely with the relationships obtained for VTI in the slowness domain by Koren and Ravve (2018). For an elliptic VTI medium, ε = δ, the second-order coefficients simplify to M 2,P = 1 + 2δ , M 2,SV = 1 , M 2,S H = 1 + 2γ, (C21)…”

“…Starting with an original approximation function for the radial offset, we suggest an approximation function similar to the one applied in the VTI case (Koren and Ravve 2018), where the coefficients A, B, C, D, E are now azimuthally dependent. These coefficients have units of distance, and the radial offset is given by where, again,p h is the normalized slowness andq is the normalized complementary slowness.…”

“…This approach is called 'two-interval approximation' because it assumes two different moveout formulae for two offset intervals. Alternative formulae, which are valid for an extended range, up to the case of nearly horizontal rays in multi-layer media, were presented in the offset domain by Ravve and Koren (2017b) and in the slowness domain by Koren and Ravve (2018).…”

“…For a given reflection point with a normal-incidence time t o , and for a given slowness azimuth ψ slw , the formulae include five coefficients and an additional parameter, which is the critical slowness p c (ψ slw ). This set of five coefficients plus the critical slowness is similar to the set used in the formulae proposed for VTI layered models by Koren and Ravve (2018), but in this case the parameters are azimuthally dependent. Unlike the VTI model case where only a radial offset component exists, in the case of azimuthally anisotropic models, the transverse offset component can have significant value due to the difference between the slowness and the offset azimuths, tan(ψ off − ψ slw ) = h T /h R .…”

S‐wave in 2D acoustic transversely isotropic media with a tilted symmetry axis

“…The P-and SV-waves polarized in the vertical symmetry planes have kinematics defined by four parameters: the vertical P-and S-waves phase velocities and two anisotropy parameters ε and δ (Thomsen 1986). Various moveout approximations have been proposed for VTI media (Stovas 2010;Ravve and Koren 2017;Koren and Ravve 2018). However, it was found that the P-wave moveout is weakly dependent on vertical S-wave phase velocity for practical weakly anelliptic VTI media (Tsvankin and Thomsen 1994;Alkhalifah and Tsvankin 1995).…”