On establishing the accuracy of noise tomography travel-time measurements in a realistic medium

P>It has previously been shown that the Green's function between two receivers can be retrieved by cross-correlating time series of noise recorded at the two receivers. This property has been derived assuming that the energy in normal modes is uncorrelated and perfectly equipartitioned, or that the distribution of noise sources is uniform in space and the waves measured satisfy a high frequency approximation. Although a number of authors have successfully extracted travel-time information from seismic surface-wave noise, the reason for this success of noise tomography remains unclear since the assumptions inherent in previous derivations do not hold for dispersive surface waves on the Earth. Here, we present a simple ray-theory derivation that facilitates an understanding of how cross correlations of seismic noise can be used to make direct travel-time measurements, even if the conditions assumed by previous derivations do not hold. Our new framework allows us to verify that cross-correlation measurements of isotropic surface-wave noise give results in accord with ray-theory expectations, but that if noise sources have an anisotropic distribution or if the velocity structure is non-uniform then significant differences can sometimes exist. We quantify the degree to which the sensitivity kernel is different from the geometric ray and find, for example, that the kernel width is period-dependent and that the kernel generally has non-zero sensitivity away from the geometric ray, even within our ray theoretical framework. These differences lead to usually small (but sometimes large) biases in models of seismic-wave speed and we show how our theoretical framework can be used to calculate the appropriate corrections. Even when these corrections are small, calculating the errors within a theoretical framework would alleviate fears traditional seismologists may have regarding the robustness of seismic noise tomography.