The cross-convolution method for interpreting SKS splitting observations, with application to one and two-layer anisotropic earth models

We present a new method for determining anisotropic earth models using observations of split shear waves (such as SKS). The method consists of first constructing two time-series, x(t) and y(t), that contain information concerning both the observed seismograms and a hypothetical earth model, and then varying the earth model so as to minimize the misfit, e(t) = x(t) - y(t). The time-series are defined by the rules, x(t) = h^pre (m, t)* V^obs(t) and y(t) = v^pre(m, t)* H^obs(t), where V^obs(t) and H^obs(t) are the observed radial-horizontal and tangential-horizontal component seismograms, respectively, v^pre(m, t) and h^pre(m, t) are the radial-horizontal and tangential-horizontal impulse responses, respectively, predicted by some earth model, m, and ^* denotes convolution. The best-fitting earth model is the one that minimizes ∥e(t)∥² with respect to m, where ∥ · ∥ is an amplitude-normalized L₂ norm. This definition of misfit is insensitive to the source wavelet, and thus eliminates that unknown quantity from the problem. We show that this method yields estimates of one-layer splitting parameters that are very similar to those achieved through traditional means, but that unlike those traditional methods it can be applied to more complicated models (e.g. multilayer anisotropy) without recourse to approximate 'apparent splitting' parameters. We apply the method to synthetic SKS pulses generated for two-layer anisotropic earth models, and show that it can successfully recover information concerning both layers. We then apply the method to SKS and PKS data from Piñon Flat, California (PFO) on the San Andreas fault. This station has been studied previously by Özalaybey and Savage, Liu et al. and Polet and Kanamori, who provide two-layer solutions based on modelling of the azimuthal variation of apparent splitting parameters. Our results are broadly consistent with theirs, but provide additional information concerning the statistical significance of the two-layer solution. By this measure, none of the published two-layer solutions is significantly better than a one-layer solution, although all fit the waveform data better than the best-fitting one-layer solution. This result is consistent with the interpretations of Özalaybey and Savage and Liu et al., who provide similar assessments of their two-layer solutions. The fact that neither our one-layer nor two-layer solutions does very well at reducing the overall misfit strongly suggests that some process other than two-layer anisotropy is affecting SKS and PKS waveforms at PFO.