Approximate polarization of plane waves in a medium having weak transverse isotropy

Abstract

Many real rocks and sediments relevant to seismic exploration can be described by elastic, transversely isotropic media. The properties of plane waves propagating in a transversely isotropic medium can be given in an analytically exact form. Here the polarization is recast into a comprehensive form that includes Daley and Hron’s normalization and Helbig’s full range of elastic constants. But these formulas are rather lengthy and do not easily reveal the features caused by anisotropy. Hence Thomsen suggested an approximation scheme for weak transverse isotropy. His derivation of the approximate polarization, however, is based on a property that is not suitable to measure small differences between an isotropic and a weakly transversely isotropic medium. Therefore the approximation of the polarization is recast. The corrected approximation does show a dependence on weak transverse isotropy. This feature can be viewed as an additional rotation of the polarization with respect to the wavenormal. It depends on the anisotropy as well as the inverse velocity ratio. An approximate condition of pure polarization, which occurs in certain directions, is also obtained. The corrected approximation results in a better match of the approximate polarization with the exact one, providing the assumption of weak transverse isotropy is met. When comparing the additional rotation with the deviation of the (observable) ray direction from the wavenormal, the qSV‐wave indicates transverse isotropy most clearly.

Anisotropic parameters and P‐wave velocity for orthorhombic media

“…Adapting the results of Rommel (1994) and Tsvankin (1996) obtained for TI media, we find the following weak-anisotropy approximation for equation (37) tan v = tan 9 { 1 + B {2S 121 + 4(F 12l -&l 2) ) sinz B] }.…”

Exact elastic impedance matrices for transversely isotropic medium

“…Substituting equations A-4 and A-7 into equation A-9, and using Thomsen's parameters to represent elastic modulus, we have Rommel (1994), but it has high accuracy for strong anisotropy .…”

“…The components of the polarization vectors are discussed by Daley and Hron (1997), Fryer and Frazer (1987), and Rommel (1994). They correspond to the polarization directions of the qP-wave θ Ã and qS-wave φ Ã as follows:…”

Constraining the anisotropy structure of the crust by joint inversion of seismic reflection travel times and wave polarizations

“…Abundant previous work on seismic anisotropy has been made up to date, from the pioneer studies (Postma 1955;Backus 1962;Berryman 1979;Levin 1979;Helbig 1983Helbig , 1984Crampin 1986;Vidale 1986;Park et al 1987;Jurkevics 1988;Rommel 1994;Eduardo and Paul 1994;Alkhalifah and Tsvankin 1995;Tsvankin 1996) to more recent papers (Grechka et al 2001;Yang et al 2002;Qin et al 2003a, b). Starting from the fact that the common (weak) anisotropy that is present in the nature is however for transversely isotropic media with a vertical symmetry axis (VTI media), Thomsen (1986) quantified this sort of anisotropy through three basic parameters, namely, ε, δ, and γ , which are more intuitive and easier to measure and to be implemented than earlier measures of anisotropy; the parameters ε and δ define the Pwave anisotropy while the parameter γ controls the S-wave anisotropy.…”

“…Several papers discuss the inversion for crustal velocity structure using seismic reflection travel times (Lutter and Nowack 1990;Zelt and Smith 1992;Pullammanappallil and Louie 1993, 1994, 1997Bai and Wang 2004) and also the formal inversion for laterally heterogeneous velocity structure from P-wave polarization data (Hu and William 1992). These papers have two features in common, namely, the assumption of an isotropic seismic velocity model, neglecting any effect coming from seismic anisotropy, and a fixed seismic reflector geometry defining the structure of the layered elastic physical medium.…”