Noise interferometry in an inhomogeneous environment in the geometric limit

An approximation to the transient Green's function G(x(a)∣x(b),t) between points x(a) and x(b) can be estimated by taking the time derivative of the correlation function C(ab)(t) of records of ambient noise measured at locations x(a) and x(b). From the general relationship between C(ab)(t) and G(x(a)∣x(b),t) it is shown, using a stationary-phase-like argument, that in an inhomogeneous environment in the geometric limit C(ab)(t) consists of a superposition of signed step functions and two-sided logarithmic singularities that are delayed in time by the travel times of the rays connecting x(a) and x(b).