Theoretical and numerical results on effects of attenuation on correlation functions of ambient seismic noise

We study analytically and numerically effects of attenuation on cross-correlation functions of ambient noise in a 2-D model with different attenuation constants between and outside a pair of stations. The attenuation is accounted for by quality factor Q(omega) and complex phase velocity. The analytical results are derived for isotropic far-field source distribution assuming the Fresnel approximation and mild attenuation. More general situations including cases with non-isotropic source distributions are examined with numerical simulations. The results show that homogeneous attenuation in the interstation regions produces symmetric amplitude decay of the causal and anticausal parts of the noise cross-correlation function. The attenuation between the receivers and far-field sources generates symmetric exponential amplitude decay and may also cause asymmetric reduction of the causal/anticausal parts that increases with frequency. This frequency dependence can be used to distinguish asymmetric amplitudes due to attenuation from frequency-independent asymmetry in noise correlations generated by non-isotropic source distribution. The attenuations both between and outside station pairs also produce phase shifts that could affect measurements of group and phase velocities. In terms of noise cross-spectra, the interstation attenuation is governed by Struve functions while the attenuation between the far-field sources and receivers is associated with exponential decay and the imaginary part of complex Bessel function. These results are fundamentally different from previous studies of attenuated coherency that append the Bessel function with an exponential decay that depends on the interstation distance.