On seismic gyrotropy

Geophys. J. Int. (1996) 124,415-426 On seismic gyrotropy I. R. Obolentseva Institute of Geophysics of the Russian Academy of Sciences (Siberian Division), University p i . , 3, Novosibirsk 630090, Russia Accepted 1995 August 17. Received 1995 August 14; in original form 1995 March 21 Key words: seismic-wave propagation, S waves. 1 INTRODUCTION Multiwave seismic investigations, based on recordings not only of compressional (P) waves, but also of shear (S) and converted ( P S , S P ) waves, have shown that, as a rule, waves with S polarizations exhibit ‘anomalous’ features, There have been many such investigations carried out in the former USSR and worldwide (see e.g. Puzyrev et al. 1985; Obolentseva & Gorshkalev 1986; Alford 1986; Winterstein & Meadows 1991). ‘Anomalous’ polarization refers to the presence of large accessory components of S waves: y components of PS waves and also SS waves, generated by X , Z sources, and correspondingly x,z components of SS waves from Y sources. (The main components are x and z in the first case and y in the second case. The x-axis is the direction of the source-receiver, usually coinciding with the profile orientation. The letters x, y and z refer to receivers, and Y, Y and Z to sources.) The terms ‘main’ and ‘accessory’ components are used in Russian literature, whereas descriptions using such notions as ‘cross-line’ and ‘in-line’ components and sources (Alford 1986) are more common in other literature. Possible explanations of the phenomenon of ‘anomalous polarization’ include ( 1) dipping boundaries, if dip angles are more than 15-20”, (2) large lateral heterogeneities, (3) diffraction objects that are lateral to the profile, (4)azimuthal anisotropy, and (5) gyrotropy. The first three reasons are obvious and may easily be recognized. The fourth and the fifth factors are not so obvious, because they are due to specific properties of the microstructure of the medium. Azimuthal anisotropy is considered by many authors to be the main factor involved (e.g. Crampin 1981; Obolentseva & Grechka 0 1996 RAS 1987; Obolentseva, Grechka & Nicolsky 1987; Obolentseva & Gorshkalev 1986; Alford 1986); however, the present author believes that it is not the only factor. The concept of seismic gyrotropy is a recent introduction (Obolentseva 1988, 1992, 1993). The so-called ‘polarization anomalies’ can be completely explained using gyrotropic models of geological media. In general, the most useful model of a geological medium (for describing polarization peculiarities observed experimentally) is an anisotropic gyrotropic one. Anisotropic gyrotropic models are effective models of microheterogeneous media containing elementary objects such as grains, thin layers, microcracks and microbreaks (their linear dimensions are much less than the wave length) that are situated in a particular way in space. If they are parallel to certain lines or planes, an anisotropy arises; if these microobjects are arranged so that the medium has no centre of symmetry, a gyrotropy may appear: it is a necessary condition. The main feature of waves propagating in gyrotropic media is their elliptical polarization. If, moreover, right or left orientations of the elementary objects dominate in the medium, i.e. it is enantiomorphous, then rotation of the polarization plane, as it is termed in optics, occurs. Optical gyrotropy is well known (Landau & Lifshits 1992); seismic gyrotropy is introduced by analogy. 2 HOOKE’S LAW FOR ANISOTROPIC GYROTROPIC M E D I A The essence of gyrotropy is the phenomenon caused by the spatial dispersion (of the first order) of elastic properties of a 415 SUMMARY A new concept of seismic gyrotropy is introduced. Mathematically, it means that in Hooke’s law new terms containing strain derivatives with respect to spatial coordinates are added. Physically, the concept of gyrotropy makes it possible to take into account the microheterogeneity of rocks. The symmetry characteristics of the gyration tensor (antisymmetric tensor of the fifth rank) have been established and its form for different symmetry groups has been found. It is shown that the gyration tensor describes small rotations. The wave equation with gyrotropic terms has four solutions: one P wave and three S waves, all with elliptical polarizations. One example of a theoretical seismogram and some experimental data are given. The main conclusion is that the anisotropic gyrotropic model enables us to explain ’anomalous’ polarization of shear waves observed in experiments. I. R . Obolentseva 416 medium. In a medium with spatial dispersion, Hooke’s law, g ij. . = arjkl . . &k l , should be written in the form a(r) = - a(w, k) = a@, k)E(W, k) , where lw a(r, r’)E(r’)dr’ a(w,k)= to account for the contribution of neighbouring points r‘ to the stressed state at the point r. Eq. (1) was written by analogy with optics (Landau & Lifshits 1992; Kisel & Burkov 1980). In optics, the equation of non-local coupling between the electric induction, D, and the intensity of the electrical field, E, is T=t-t‘. The relationship between the stresses and the strains can be represented in the form of a series The second term of this expansion describes the first-order spatial dispersion, called the gyrotropy, the third term describes the second-order spatial dispersion etc. It can be shown (e.g. Chichinin 1993) that only the second term in eq. (2) is responsible for a rotation of the polarization plane in the propagation of elastic shear waves in such a medium; if the second term equals zero and the third one differs from zero, the rotation of the polarization plane might occur if the wave vector k = wn/V were imaginary. So, in our further considerations we shall use eq. (2) with two terms. Thus in gyrotropic media, Hooke’s law o..= 1.1 c.. lJkl &kl j; I:., a(t, t’, r, r’)E(r‘,t’) dr‘ dt’. If the medium is homogeneous and its properties are stationary, then a(t - t’, r - r’)E(r’,t’) dr’ dt’. + bijkln a e k l / a x n (3) contains additional terms which depend on strain derivatives with respect to the spatial coordinates. Hooke’s law in the form of eq. (3) was given in Sirotin & Shascolskaya (1979), where it was applied to crystals. The gyration tensor b is a tensor of the fifth (odd) rank, therefore it is equal to zero if a medium has a centre of symmetry. Hence gyrotropy may be inherent only to media belonging to acentric groups: 21 point groups (out of 32) and four limit groups (out of seven). 2.1 Inner symmetry of the tensor b What properties of inner symmetry must the gyration tensor b possess? In choosing them we must take into account the symmetry properties of the elasticity moduli tensor c. The tensor c is symmetric in the permutation of the indices in the first and second pairs: Cijkl a(t , t’)E( t’) dt‘. Frequency dispersion means that the stress at a given moment depends on the stresses at the preceding times. In a medium with spatial and frequency dispersions we have t)= R=r- (...truncated)