Seismic anisotropy, the dependence of seismic velocity on the direction of propagation, is a common feature of modern global seismic velocity models [Dziewonski and Anderson, 1981] and a well documented property of olivine, the mineral that composes the bulk of the upper mantle [Crampin et al. 1984]. A prominent effect of seismic anisotropy is the so-called splitting of shear waves. Unlike the two shear modes in isotropic materials, which have equal phase velocities, the two corresponding modes in anisotropic materials have different velocities. As it enters an anisotropic region, a single linearly polarized shear-wave pulse will split into two pulses of different velocity. The polarizations of these pulses are related to the projection of their propagation direction onto the axes of the anisotropic elastic tensor [Aki and Richards, 1981].
Shear-wave splitting studies measure and describe the anisotropy of the Earth. One selects shear-wave phases that are known to be linearly polarized prior to entering the anisotropic study region, and measures their particle motion after traversing the region. One seeks a coordinate rotation that separates the particle motion into distinct "fast" and "slow" pulses, each of identical shape and linearly polarized in mutually-perpendicular "fast" and "slow" directions. The delay between the two pulses is proportional to the strength of the anisotropic effect, which depends both on the intensity of seismic anisotropy and the length of the path within the anisotropic material. The axes of the rotated coordinate system provide information on the symmetry and orientation of the anisotropic elastic tensor.
In some simple cases, such as when the tensor has hexagonal symmetry with a horizontal symmetry axis, the orientation of the fast splitting axis is approximately parallel to the symmetry axis, regardless of propagation direction. This simplification is often assumed in studies of the upper mantle. Estimates of splitting time and fast axis direction from many shear waves at a given station have been averaged to estimate anisotropic strength (i.e. delay time) and symmetry-axis azimuth for the mantle beneath that station [e.g. Vinnik et al., 1995; Barruol et al, 1997; Wolfe and Silver, 1998; Fouch et al., 1999]. Such "station means" are useful in tectonic settings where uniformity of the fabric in the lithosphere is likely, e.g. on the ocean floor [Wolfe and Solomon, 1998] or in the wake of a hot spot [ Schutt et al., 1998].
Station means may be quite misleading, however, in cases where both the
delay time and the fast direction vary significantly with the
propagation direction.
Such variation can occur when an anisotropy tensor is inclined from the vertical, or has a more complicated symmetry, or both (Figure 1). Babuska et al., [1993] discusses possible scenarios that involve an inclined orientation of hexagonal and orthorhombic anisotropic tensors.
This possibility has also been considered by Plomerova et al,
[1996], Levin et al. [1996], Hirn et al., [1998] and
others.
Also, a combination of two or more layers of anisotropy with hexagonal
symmetry and horizontal symmetry axes leads to a systematic variation
of the splitting parameters with the polarization of the incoming shear
wave [Silver and Savage, 1994; Vinnik et al, 1995].
For vertically incident shear waves a simple analytic expression
describes the variation of splitting parameters in a simple 2-layer
model, predicting a 
periodicity.
2-layer models with horizontal-axis anisotropy has been invoked to explain
back-azimuth variations in splitting parameters [e.g., Ozalaybey and Savage, 1994; Russo and Silver, 1994; Granet et al., 1998].
In this paper, we demonstrate that the fast direction and delay associated with shear waves that sample the Earth's upper mantle beneath two long-lived mountain belts vary strongly with shear-wave propagation direction. We interpret these variations in terms of multilayered anisotropy, implying either complex deformation in a past collisional event or, more plausibly, a mix of active and fossil deformations.
The parameters of a split shear wave can be estimated by a grid search over possible time-delays and fast-axis directions, and using some kind of goodness-of-fit criteria to select the "best" set of values. Two criteria have been used: 1) maximal similarity in the pulse shapes of the two rotated seismogram components, as quantified by cross-correlation [e.g., Bowman and Ando, 1984; Iidaka and Niu, 1998]; and 2) that the re-assembled "original" pulse has maximal rectilinearity, as quantified by the ratio of the rectilinear and elliptical motion [Kosarev et al., 1984; Silver and Chan, 1991]. These two methods give the same results when tested on nearly noise-free data. Owing to their different treatment of noise, and to complications induced by multilayered anisotropy, "cross-correlation" and "rectilinearity" measures can give substantially different results when applied to noisy data.
If the anisotropic material is homogeneous, an observed split shear wave is exactly the sum of two pulses of different polarization, one delayed with respect to the other. If, in contrast, the anisotropic material consists of several layers (or, more generally, 3-D domains) of different anisotropy, then the observed seismogram has a more complicated form, with a sequence of pulses corresponding to mode-conversions from the various layer interfaces. In general, no rotation exists in which one component of the seismogram is exactly a delayed version of the other (Figure 2). Given a high-quality broadband waveform with no interfering seismic signals, these conversions could perhaps be individually identified and modeled. Unfortunately, most SKS data in studies of the upper mantle is low-passed, and resolving closely-spaced sequence of pulses is problematic.
Our approach is to retain the two parameter (fast direction and delay) description for shear-wave propagation in anisotropic media, but to recognize that this is an "apparent" measurement, without exact correspondence to an underlying physical process. This approach was introduced by Silver and Savage, [1994] for the case of two anisotropic layers with horizontal symmetry axes, and more recently expanded to the case of a smoothly varying medium by [Rumpker and Silver, 1998]. The apparent splitting parameters (Figure 3) contain significant information about the anisotropic medium. Most importantly, the apparent splitting parameters are different from what one would expect for a homogeneous medium with the same "mean" anisotropy, so that some information on the depth-dependence of the anisotropy is preserved. Unfortunately, owing to interference between the mode-conversions, the measured values of the apparent parameters are somewhat sensitive to the frequency band of the seismic data (Figure 4). Rumpker and Silver, [1998] show that even for a case of vertical incidence in a flat-layered model with horizontal axes of hexagonal symmetry apparent splitting parameters exhibit strong dependence on the ratio between the cumulative splitting effect of the medium and the frequency content of the shear wave. This sensitivity does not present any fundamental problem when modeling the apparent splitting. One simply compares observed apparent values with predicted ones that have been computed from synthetic seismograms with the same frequency content. However, it makes difficult the comparison of data collected by different authors using different processing schemes.
In this paper we measure apparent splitting parameters at a few locations within two Paleozoic mountain belts - the Appalachians and the Urals. We focus on the anisotropic structure of the upper mantle, and so use mostly SKS and SKKS phases. We mainly use stations with long duration of operation, and thus we are able to obtain measurements from a wide variety of azimuths and angles of incidence. As we show below, the strong directional behavior of these data indicate that the upper mantle beneath these two mountain belts possesses multiple layers of different anisotropy.