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 timeseries, x(t) and y(t), that contain information about 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 timeseries are defined by the rules, x(t)=hpre(m,t)*Vobs(t) and y(t)=vpre(m,t)*Hobs(t), where Vobs(t) and Hobs(t) are the observed radial−horizontal and tangential−horizontal component seismograms, respectively, vpre(m,t) and hpre(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. multi−layer anisotropy) without recourse to approximate “apparent splitting” parameters. We apply the method to synthetic SKS pulses generated for two−layer