Apparent shear-wave splitting parameters in the presence of vertically varying anisotropy

In this paper we study teleseismic shear-wave splitting due to multiple ani sotropic layers. The effects can be described in terms of a multilayered sp litting operator that may be calculated using a matrix formulation. As a sp ecial case, we consider smooth variations of anisotropy with depth. The mod el is characterized by the (horizontal) fast-axis directions at the top (ph i(t)) and bottom (phi(b)) of the anisotropic region, and the total delay ti me (Delta t). Numerical results show that the particle motion at relatively long periods (T/Delta t greater than or similar to 5) is clearly elliptica l and is a function of the polarization direction of the incident wave. At shorter periods (T/Delta t less than or similar to 1); however, the seismog ram is characterized by two distinctly separated arrivals corresponding to the fast- and slow-axis directions at the top of the anisotropic region. Re lative amplitudes of the fast and slow arrivals depend on the orientation o f the initial polarization with respect to the axis directions at the botto m. Only the high-frequency results agree with predictions based on ray theo ry. At fixed initial polarization, the longer-period results can be describ ed in terms of an apparent fast-polarization direction (phi(a)) and an appa rent delay time (delta t(a)). The apparent splitting parameters exhibit a n /2-periodicity as functions of initial polarization (or backazimuth), which is similar to the two-layer case considered by Silver & Savage (1994). Ass uming that phi(t) - phi(b) < 45 degrees, we find variations of phi(a) and d elta t(a) to be smaller than 10 per cent over most (approximate to 2/3) of the backazimuth range. In this case, phi(a) similar or equal to (phi(t) + p hi(b))/2 and delta t(a) similar or equal to Delta t. The calculated apparen t splitting parameters agree well with direct measurements using synthetic waveforms and may thus be applied to the inversion of observations. Long-pe riod apparent splitting parameters for different types of depth variations cannot be resolved uniquely without further constraints from the deformatio n history at a given station. For example, we find that recently observed a zimuthal variations of splitting parameters (Ozalaybey & Savage 1994; Brech ner et al. 1998) can be modelled equally well in terms of two distinct laye rs (a four-parameter model) or smooth variations of fast-axis directions (a three-parameter model). Depth-dependent models with randomly chosen Cast-a xis directions and delay times in each layer display a characteristic decre ase of the most probable apparent delay time, delta t(a)(P-max), with incre asing number of layers N. We find that delta t(a)(P-max) similar to 1 / roo t N. The fact that most measurements yield values of delta t(a) similar or equal to 1 s (Silver 1996) suggests that the majority of observations can b e explained in terms of one or two anisotropic layers, provided the fast-ax is directions in different layers are independent.