Formulation of the spatial autocorrelation (SPAC) method in dissipative media

Spatial autocorrelation (SPAC) method is formulated in dissipative media for one-, two- and three-dimensional (1-D, 2-D and 3-D) scalar wavefields based on the generalized wave equation or the generalized telegraph equation. A rather straightforward derivation is possible by using a close mathematical relation between the SPAC method and seismic interferometry, though a model set-up should be modified by including attenuation. For 3-D cases, the normalized cross spectrum of a scalar wavefield in a dissipative medium is found to be different from that in a non-dissipative medium merely by an exponentially damping term. However, expressions for 1-D and 2-D cases are not as simple as 3-D cases meaning that not only amplitude but also phase in the normalized cross-spectrum are modified by attenuation. The SPAC expressions derived are considered to be applied to rather homogeneous distribution of attenuation. For the estimation of attenuation, the SPAC method needs to be used for larger station separations than the traditional SPAC method does. Analysis of the SPAC expressions for 2-D cases shows that the conjecture of Prieto (2009) is not strict but approximately good for small attenuation. This study will provide a theoretical basis to estimate not only phase velocity but also attenuation from analysis of ambient noises.